The Formation of Shocks in 3-Dimensional Fluids (EMS Monographs in Mathematics)

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Author: Demetrios Christodoulou

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EMM_christodoulou_titelei

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EMS Monographs in Mathematics Edited by Ivar Ekeland (Pacific Institute, Vancouver, Canada) Gerard van der Geer (University of Amsterdam, The Netherlands) Helmut Hofer (Courant Institute, New York, USA) Thomas Kappeler (University of Zürich, Switzerland) EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and self-contained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature.

Previously published in this series: Richard Arratia, A.D. Barbour, Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach


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Demetrios Christodoulou

The Formation of Shocks in 3-Dimensional Fluids M

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European Mathematical Society


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Author: Prof. Demetrios Christodoulou Department of Mathematics ETH-Zentrum 8092 Zürich Switzerland

2000 Mathematical Subject Classification (primary; secondary): 35L67; 35L65, 35L70, 58J45, 76L05, 76N15, 76Y05

ISBN 978-3-03719-031-9 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Printed in Germany 987654321

This work is dedicated to the memory of my father L AMBROS C HRISTODOULOU born Alexandria 1913 deceased Athens 1999 whose kindness is the fondest memory I have

Contents

Prologue and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1

Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variation . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2

The Basic Geometric Construction . . . . . . . . . . . . . . . . . . . . . .

39

3

The Acoustical Structure Equations . . . . . . . . . . . . . . . . . . . . .

53

4

The Acoustical Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5

The Fundamental Energy Estimate . . . . . . . . . . . . . . . . . . . . . .

99

6

Construction of the Commutation Vectorfields . . . . . . . . . . . . . . . .

139

7

Outline of the Derived Estimates of Each Order . . . . . . . . . . . . . . .

169

8

Regularization of the Propagation Equation for d/trχ. Estimates for the Top Order Angular Derivatives of χ . . . . . . . . . . . .

203

Regularization of the Propagation Equation for / µ. Estimates for the Top Order Spatial Derivatives of µ . . . . . . . . . . . . .

275

9 10

11

12

Control of the Angular Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to χ . . . . . . . . . . . . . . . . . . Part 1: Control of the angular derivatives of the first derivatives of the x i . . . . . . . . . . . . . . . . . . . Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl . . . . . . . . . .

329 329 403

Control of the Spatial Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to µ . . . . . . . . . . . . . . . . . . Part 1: Control of the spatial derivatives of the first derivatives of the x i . . . . . . . . . . . . . . . . . . . . . . . . Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l . . . . . . .

473 589

Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of χ and Spatial Derivatives of µ . . .

665

473

viii

13

Contents

The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities. The Energy Estimates. Recovery of the Bootstrap Assumptions. Statement and Proof of the Main Theorem: Existence up to Shock Formation . . . . . . . . . . . . . . . . . . . . Part 1: Derivation of the properties C1, C2, C3 . . . . . . . . . . . . Part 2: The error estimates of the acoustical entities . . . . . . . . . Part 3: The energy estimates . . . . . . . . . . . . . . . . . . . . . . Part 4: Recovery of assumption J. Recovery of the bootstrap assumption. Proof of the main theorem . . . . . . . . . . . .

. . . .

741 741 757 831

. . .

874

Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution . . . . . . . . . . . . . . . . . . . . . . . . . .

893

The Nature of the Singular Hypersurface. The Invariant Curves. The Trichotomy Theorem. The Structure of the Boundary of the Domain of the Maximal Solution . . . . . . . . . . . . . . . . . . . . .

927

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

977

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

987

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

989

14 15

. . . .

. . . .

Prologue and Summary

The equations describing the motion of a perfect fluid were first formulated by Euler in 1752 (see [Eu1], [Eu2]), based, in part, on the earlier work of D. Bernoulli [Be]. These equations were among the first partial differential equations to be written down, preceded, it seems, only by D’Alembert’s 1749 formulation [DA] of the one-dimensional wave equation describing the motion of a vibrating string in the linear approximation. In contrast to D’Alembert’s equation however, we are still, after the lapse of two and a half centuries, far from having achieved an adequate understanding of the observed phenomena which are supposed to lie within the domain of validity of the Euler equations. The phenomena displayed in the interior of a fluid fall into two broad classes, the phenomena of sound, the linear theory of which is acoustics, and the phenomena of vortex motion. The sound phenomena depend on the compressibility of a fluid, while the vortex phenomena occur even in a regime where fluid may be considered to be incompressible. The formation of shocks, the subject of the present monograph, belongs to the class of sound phenomena, but lies in the nonlinear regime, beyond the range covered by linear acoustics. The phenomena of vortex motion include the chaotic form called turbulence, the understanding of which is one of the great challenges of science. Let us make a short review of the history of the study of the phenomena of sound in fluids, in particular the phenomena of the formation and evolution of shocks in the nonlinear regime. At the time when the equations of fluid mechanics were first formulated, thermodynamics was in its infancy, however it was already clear that the local state of a fluid as seen by a comoving observer is determined by two thermodynamic variables, say pressure and temperature. Of these, only pressure entered the equations of motion, while the equations involve also the density of the fluid. Density was already known to be a function of pressure and temperature for a given type of fluid. However in the absence of an additional equation, the system of equations at the time of Euler, which consisted of the momentum equations together with the equation of continuity, was underdetermined, except in the incompressible limit. The additional equation was supplied by Laplace in 1816 [La] in the form of what was later to be called the adiabatic condition, and allowed him to make the first correct calculation of the speed of sound. The first work on the formation of shocks was done by Riemann in 1858 [Ri]. Riemann considered the case of isentropic flow with plane symmetry, where the equations of fluid mechanics reduce to a system of conservation laws for two unknowns and with two independent variables, a single space coordinate and time. He introduced for such

2


systems the so-called Riemann invariants, and with the help of these showed that solutions which arise from smooth initial conditions develop infinite gradients in finite time. Riemann also realized that the solutions can be continued further as discontinuous solutions, but here there was a problem. Up to this time the energy equation was considered to be simply a consequence of the laws of motion, not a fundamental law in its own right. On the other hand, the adiabatic condition was considered by Riemann to be part of the main framework. Now as long as the solutions remain smooth it does not matter which of the two equations we take to be the fundamental law, for each is a consequence of the other, modulo the remaining laws. However this is no longer the case once discontinuities develop, so one must make a choice as to which of the two equations to regard as fundamental and therefore remains valid thereafter. Here Riemann made the wrong choice. For, only during the previous decade, in 1847, had the first law of thermodynamics been formulated by Helmholtz [He], based in part on the experimental work of Joule on the mechanical equivalence of heat, and the general validity of the energy principle had thereby been shown. In 1865 the concept of entropy was introduced into theoretical physics by Clausius [Cl2], and the adiabatic condition was understood to be the requirement that the entropy of each fluid element remains constant during its evolution. The second law of thermodynamics, involving the increase of entropy in irreversible processes, had first been formulated in 1850 by Clausius [Cl1] without explicit reference to the entropy concept. After these developments the right choice in Riemann’s dilemma became clear. The energy equation must remain at all times a fundamental law, but the entropy of a fluid element must jump upward when the element crosses a hypersurface of discontinuity. The formulation of the correct jump conditions that must be satisfied by the thermodynamic variables and the fluid velocity across a hypersurface of discontinuity was begun by Rankine in 1870 [Ra] and completed by Hugoniot in 1889 [Hu]. With Einstein’s discovery of the special theory of relativity in 1905 [Ei], and its final formulation by Minkowski in 1908 [Mi] through the introduction of the concept of spacetime with its geometry, the domain of geometry being thereby extended to include time, a unity was revealed in physical concepts which had been hidden up to this point. In particular, the concepts of energy density, momentum density or energy flux, and stress, were unified into the concept of the energy-momentum-stress tensor and energy and momentum were likewise unified into a single concept, the energy-momentum vector. Thus, when the Euler equations were extended to become compatible with special relativity, it was obvious from the start that it made no sense to consider the momentum equations without considering also the energy equation, for these two were parts of a single tensorial law, the energy-momentum conservation law. This law together with the particle conservation law (the equation of continuity of the non-relativistic theory), constitute the laws of motion of a perfect fluid in the relativistic theory. The adiabatic condition is then a consequence for smooth solutions. A new basic physical insight on the shock development problem was reached first, it seems, by Landau in 1944 [Ln]. This was the discovery that the condition that the entropy jump be positive as a hypersurface of discontinuity is traversed from the past to the future, should be equivalent to the condition that the flow is evolutionary, that is, that conditions


3

in the past determine the fluid state in the future. More precisely, what was shown by Landau was that the condition of determinism is equivalent, at the linearized level, to the condition that the tangent hyperplane at a point on the hypersurface of discontinuity, is on one hand contained in the exterior of the sound cone at this point corresponding to the state before the discontinuity, while on the other hand intersects the sound cone at the same point corresponding to the state after the discontinuity, and that this latter condition is equivalent to the positivity of the entropy jump. This is interesting from a general philosophical point of view, because it shows that irreversibility can arise, even though the laws are all time-reversible, once the solution ceases to be regular. To a given state at a given time there always corresponds a unique state at any given later time. If the evolution is regular in the associated time interval, then the reverse is also true: to a given state at a later time there corresponds a unique state at any given earlier time, the laws being time-reversible. This reverse statement is however false if there is a shock during the time interval in question. Thus determinism in the presence of hypersurfaces of discontinuity selects a direction of time and the requirement of determinism coincides, modulo the other laws, with what is dictated by the second law of thermodynamics which is in its nature irreversible. This recalls the interpretation of entropy, first discovered by Boltzmann in 1877 [Bo], as a measure of disorder at the microscopic level. An increase of entropy was thus understood to be associated to an increase in disorder or to loss of information, and determinism can only be expected in the time direction in which information is lost, not gained. An important mathematical development with direct application to the equations of fluid mechanics in the physical case of three space dimensions, was the introduction by Friedrichs of the concept of a symmetric hyperbolic system in 1954 [F] and his development of the theory of such systems. It is through this theory that the local existence and domain of dependence property of solutions of the initial value problem associated to the equations of fluid mechanics are established. Another development in connection to this was the general investigation by Friedrichs and Lax in 1971 [F-L] (see also [Lx1]) of nonlinear first order systems of conservation laws which for smooth solutions have as a consequence an additional conservation law. This is the case for the system of conservation laws of fluid mechanics, which consists of the particle and energy-momentum conservation laws, which for smooth solutions imply the conservation law associated to the entropy current. It was then shown that if the additional conserved quantity is a convex function of the original quantities, the original system can be put into symmetric hyperbolic form. Moreover, for discontinuous solutions satisfying the jump conditions implied by the integral form of the original conservation laws, an inequality for the generalized entropy was derived. The problem of shock formation for the equations of fluid mechanics in one space dimension, and more generally for systems of conservation laws in one space dimension, was studied by Lax in 1964 [Lx2], and 1973 [Lx3], and John [J1] in 1974. The approach of these works was analytic, the strategy being to deduce an ordinary differential inequality for a quantity constructed from the first derivatives of the solution, which showed that this quantity must blow up in finite time, under a certain structural assumption on the system called genuine nonlinearity and suitable conditions on the initial data. The gen-

4


uine nonlinearity assumption is in particular satisfied by the non-relativistic compressible Euler equations in one space dimension provided that the pressure is a strictly convex function of the specific volume. A more geometric approach in the case of systems with two unknowns was developed by Majda in 1984 [Ma1] based in part on ideas introduced by Keller and Ting in 1966 [K-T]. In this approach, which is closer in spirit to the present monograph, one considers the evolution of the inverse density of the characteristic curves of each family and shows that under appropriate conditions this inverse density must somewhere vanish within finite time. In this way, not only were the earlier blow-up results reproduced, but, more importantly, insight was gained into the nature of the breakdown. Moreover Majda’s approach also covered the case where the genuine nonlinearity assumption does not hold, but we have linear degeneracy instead. He showed that in this case, global-in-time smooth solutions exist for any smooth initial data. The problem of the global-in-time existence of solutions of the equations of fluid mechanics in one space dimension was treated by Glimm in 1965 [Gl] through an approximation scheme involving at each step the local solution of an initial value problem with piecewise constant initial data. The convergence of the approximation scheme then produced a solution in the class of functions of bounded variation. Now, by the previously established results on shock formation, a class of functions in which global existence holds must necessarily include functions with discontinuities, and the class of functions of bounded variation is the simplest class having this property. Thus, the treatment based on the total variation, the norm in this function space, in itself an admirable investigation, would be insuperable if the development of the one-dimensional theory was the goal of scientific effort in the field of fluid mechanics. However that goal can only be the mathematical description of phenomena in real three-dimensional space and one must ultimately face the fact that methods based on the total variation do not generalize to more than one space dimension. In fact it is clear from the study of the linearized theory, acoustics, which involves the wave equation, that in more than one space dimension only methods based on the energy concept are appropriate. The first and thus far the only general result on the formation of shocks in threedimensional fluids was obtained by Sideris in 1985 [S]. Sideris considered the compressible Euler equations in the case of a classical ideal gas with adiabatic index γ > 1 and with initial data which coincide with those of a constant state outside a ball. The assumptions of his theorem on the initial data were that there is an annular region bounded by the sphere outside which the constant state holds, and a concentric sphere in its interior, such that a certain integral in this annular region of ρ − ρ0 , the departure of the density ρ from its value ρ0 in the constant state, is positive, while another integral in the same region of ρvr , the radial momentum density, is non-negative. These integrals involve kernels which are functions of the distance from the center. It is also assumed that everywhere in the annular region the specific entropy s is not less than its value s0 in the constant state. The conclusion of the theorem is that the maximal time interval of existence of a smooth solution is finite. The chief drawback of this theorem is that it tells us nothing about the nature of the breakdown. Also the method relies on the strict convexity of the pressure as


5

a function of the density displayed by the equation of state of an ideal gas, and does not extend to more general equations of state. The other important work on shocks in three space dimensions was the 1983 work of Majda [Ma2], [Ma3], on what he calls the shock front problem. In this problem we are given initial data in 3 which is smooth in the closure of each component of 3 \ S, where S is a smooth surface in 3 . The data is to satisfy the condition that there exists a function σ on S such that the jumps of the data across S satisfy the Rankine–Hugoniot jump conditions as well as the entropy condition with σ in the role of the shock speed. The higher order compatibility conditions associated to an initial boundary value problem are also required to be satisfied. We are then required to find a time interval [0, τ ], a smooth hypersurface K in the spacetime slab [0, τ ] × 3 and a solution of the compressible Euler equations which is smooth in the closure of each component of [0, τ ] × 3 \ K , and satisfies across K the Rankine–Hugoniot jump conditions as well as the entropy condition. We may think of this problem as the local-in-time shock continuation problem. Majda solved this problem under an additional condition on the initial data which seems to be necessary for the stability of the linearized problem. The additional condition follows from the other conditions in the case of a classical ideal gas, but it does not follow for a general equation of state. The present monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider regular initial data on a space-like hyperplane 0 in Minkowski spacetime which outside a sphere coincide with the data corresponding to a constant state. We consider the restriction of the initial data to the exterior of a concentric sphere in 0 and we consider the maximal classical development of this data. Then, under a suitable restriction on the size of the departure of the initial data from those of the constant state, we prove certain theorems which give a complete description of the maximal classical development, which we call maximal solution. In particular, the theorems give a detailed description of the geometry of the boundary of the domain of the maximal solution and a detailed analysis of the behavior of the solution at this boundary. A complete picture of shock formation in threedimensional fluids is thereby obtained. Also, sharp sufficient conditions on the initial data for the formation of a shock in the evolution are established and sharp lower and upper bounds for the temporal extent of the domain of the maximal solution are derived. The reason why we consider only the maximal development of the restriction of the initial data to the exterior of a sphere is in order to avoid having to treat the long time evolution of the portion of the fluid which is initially contained in the interior of this sphere. For, we have no method at present to control the long time behavior of the pointwise magnitude of the vorticity of a fluid portion, the vorticity satisfying a transport equation along the fluid flow lines. Our approach to the general problem is the following. We show that given arbitrary regular initial data which coincide with the data of a constant state outside a sphere, if the size of the initial departure from the constant state is suitably small, we can control the solution for a time interval of order 1/η0 , where η0 is the sound speed in the surrounding constant state. We then show that at the end of this interval a thick annular region has formed, bounded by concentric spheres, where the flow is irrotational and isentropic, the constant state holding outside the outer sphere. We then

6


study the maximal classical development of the restriction of the data at this time to the exterior of the inner sphere. In the irrotational isentropic case there is a function φ which we call a wave function, the differential of which at a point determines the state of the fluid at that point, and the fluid equations reduce to a nonlinear wave equation for φ, as is shown in Chapter 1. The order of presentation in this monograph is however the reverse of that just outlined. After the first four chapters which set up the general framework, we confine attention to the irrotational isentropic problem up to Chapter 13, where the main theorem, Theorem 13.1, is proved. We return to the general problem in Chapter 14, after establishing a theorem, Theorem 14.1, which, in the irrotational isentropic context, gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution. It is at this point where our treatment of the general problem resumes, and we analyze the solution of the general problem during the initial time interval. In fact, our analysis allows us to find which conditions on the data at the beginning of the time interval result in data at the end of the time interval verifying the assumptions of Theorem 14.1. In this way we are able to establish a theorem, Theorem 14.2, which, in the general context of fluid mechanics, gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution. We should emphasize at this point that if we were to restrict ourselves from the beginning to the irrotational isentropic case, we would have no problem extending the treatment to the interior region, thereby treating the maximal solution corresponding to the data on the complete initial hyperplane 0 . In fact, it is well known that sound waves decay in time faster in the interior region and our constructions can readily be extended to cover this region. It is only our present inability to achieve long time control of the magnitude of the vorticity along the flow lines of the fluid, that prevents us from treating the interior region in the general case. The geometry of the boundary of the domain of the maximal solution is studied in Chapter 15, the main results being expressed by Theorem 15.1 and Propositions 15.1, 15.2, and 15.3. The boundary consists of a regular part and a singular part. Each component of the regular part C is an incoming characteristic hypersurface with a singular past boundary. The singular part of the boundary of the domain of the maximal solution is the locus of points where the inverse density of the wave fronts vanishes. It is the union ∂− H H , where each component of ∂− H is a smooth embedded surface in Minkowski spacetime, the tangent plane to which at each point is contained in the exterior of the sound cone at that point. On the other hand each component of H is a smooth embedded hypersurface in Minkowski spacetime, the tangent hyperplane to which at each point is contained in the exterior of the sound cone at that point, with the exception of a single generator of the sound cone, which lies on the hyperplane itself. The past boundary of a component of H is the corresponding component of ∂− H . The latter is at the same time the past boundary of a component of C. As is explained in the Epilogue, the maximal classical solution is the physical solution of the problem up to C ∂− H , but not up to H . The problem of the physical continuation of the solution is set up in the Epilogue as the shock development problem. This problem is associated to each component of ∂− H and its solution requires the construction of a hypersurface of discontinuity K , lying in the past of the cor-


7

responding component of H , but having the same past boundary as the latter, namely the given component of ∂− H . Thus, although the notion of maximal classical solution is not physically appropriate up to H , it does provide the basis for constructing the physical continuation, the solution of the shock development problem, by providing not only the right boundary conditions at C ∂− H , but also a barrier at H which is indispensable for controlling the physical continuation. The actual treatment of the shock development problem and the subsequent shock interactions shall be the subject of a subsequent monograph. The present monograph concludes with a derivation of a formula for the jump in vorticity across K , which shows that while the flow is irrotational ahead of the shock, it acquires vorticity immediately behind, which is tangential to the shock front and is associated to the gradient along the shock front of the entropy jump. We have chosen to work in this monograph with the relativistic Euler equations rather than confining ourselves to their non-relativistic limit, for three reasons. The first is the obvious reason that there is a class of natural phenomena, those of relativistic astrophysics, which lie beyond the domain of the non-relativistic equations. The second reason is that there is a substantial gain in geometric insight in considering the relativistic equations. At a fundamental level, the picture looks simpler from the relativistic perspective, because of the aforementioned unity of physical concepts brought about by the spacetime geometry viewpoint of special relativity. As an example we give the equation (1.51) of Chapter 1: (1) i u ω = −θ ds. Here ω is the vorticity 2-form. According to the definitions of Chapter 1, ω = dβ where β is the 1-form defined, relative to an arbitrary system of coordinates, by: √ βµ = − σ u µ , u µ = gµν u ν ,

(2)

(3)

√ σ being the relativistic enthalpy per particle, u µ the fluid velocity and gµν the Minkowski metric. In (1), θ is the temperature and s the entropy per particle, while i u denotes contraction on the left by the vectorfield u. Equation (1) is arguably the simplest explicit form of the energy-momentum equations. Our derivation in the Epilogue of the jump in vorticity behind a shock relies on this equation. The 1-form β plays a fundamental role in this monograph. In the irrotational isentropic case it is given by β = dφ, where φ is the wave function. The third reason why we have chosen to work with the relativistic equations is that no special care is needed to extract information on the non-relativistic limit. This is due to the fact that the non-relativistic limit is a regular limit, obtained by letting the speed of light in conventional units tend to infinity, while keeping the sound speed fixed. To allow the results in the non-relativistic limit to be extracted from our treatment in a straightforward manner, we have chosen to avoid summing quantities having different physical dimensions when such sums would make sense only when a unit of velocity has been chosen, even though we follow the natural choice within the framework of special

8


relativity of setting speed of light equal to unity in writing down the relativistic equations of motion. We shall presently give an example to illustrate what we mean. Consider the vectorfield K 0 defined by equation (5.15) of Chapter 5: K 0 = (η0−1 + α −1 κ)L + L,

L = α −1 κ L + 2T.

(4)

The terms here have not yet been defined, but the reader may return to this example after assimilating the appropriate definitions. In any case, η0 is, as mentioned above, the sound speed in the surrounding constant state. The function α is the inverse density of the hyperplanes t corresponding to the constant values of the time coordinate t, with respect to the acoustical metric h µν : h µν = gµν + (1 − η2 )u µ u ν ,

u µ = gµν u ν ,

(5)

gµν being again the Minkowski metric, η the sound speed, and u µ the fluid velocity. This is a Lorentzian metric on spacetime, the null cones of which are the sound cones. The function α has the physical dimension of velocity. The function κ is the inverse spatial density of the wave fronts with respect to the acoustical metric, a dimensionless quantity. Thus in the sum η0−1 +α −1 κ, which is the coefficient of L in the first term of (4), each term has the physical dimension of inverse velocity. The vectorfield L is the tangent vectorfield of the bicharacteristic generators, parametrized by t, of a family of outgoing characteristic sets of an acoustical function u. The wave fronts St,u are the hypersurfaces Cu , the level surfaces of intersection Cu t . The physical dimension of L is inverse time. Thus the first term in (4) has the physical dimension of inverse length. The vectorfield T defines a flow on each of the t , taking each wave front onto another wave front, the normal, relative to the induced acoustical metric h, the flow of the foliation of t by the surfaces St,u . It has the physical dimension of an inverse length. The first term in the second part of (4) also has the same physical dimension, hence the physical dimension of the vectorfield L is inverse length as well. We conclude that each term in (4) has the physical dimension of inverse length, thus the physical dimension of K 0 is inverse length. Denoting, as above, by σ the square of the relativistic enthalpy per particle, we have: √ σ = e + pv (6) where e is the relativistic energy per particle, p the pressure and v is the volume per particle. Let H be the function defined by: 1 − η2 = σ H.

(7)

The derivative of H with respect to σ at constant s plays a central role in shock theory . This quantity is expressed by (see equation (E.47) in the Epilogue): dH d 2v 3v dv (8) = −a +√ dσ s d p2 σ dp s s

where a is the positive function: a=

η4 . 2σ v 3


9

The sign of (d H /dσ )s in the state ahead of a shock determines, as is shown in the Epilogue, the sign of the jump in pressure in crossing the shock to the state behind. The jump in pressure is positive if this quantity is negative, the reverse otherwise. The value of (d H /dσ )s in the surrounding constant state is denoted by in this monograph. This constant determines the character of the shocks for small initial departures from the constant state. In particular when = 0, no shocks form and the domain of the maximal classical solution is complete. Consider the function (d H /dσ )s as a function of the thermodynamic variables p and s. Suppose that we have an equation of state such that at some value s0 of s we have (d H /dσ )s=s0 = 0, that is, the function (d H /dσ )s vanishes everywhere along the adiabat s = s0 . We show in Chapter 1 that in this case the irrotational isentropic fluid equations corresponding to the value s0 of the entropy are equivalent to the minimal surface equation, the wave function φ defining a minimal graph in a Minkowski spacetime of one more spatial dimension. Thus the minimal surface equation defines a dividing line between two different types of shock behavior. Now, the relativistic enthalpy is dominated by the term mc2 , the contribution of the particle rest mass m, to the energy per particle, c being the speed of light. We note here that the particle rest mass may be taken to be unity, so that all quantities per particle are quantities per unit rest mass. Thus in the non-relativistic limit the second term in parenthesis in (8) vanishes and the expression in parenthesis reduces simply to: (d 2 v/d p2 )s . Now, the case where (d 2 v/d p2 )s > 0, the adiabats being convex curves in the p, v plane, so that (d H /dσ )s < 0, is the more commonly found in nature, however the reverse case does occur in the gaseous region near the critical point in the liquid-to-vapor phase transition and in similar transitions at higher temperatures associated to molecular dissociation and to ionization (see [Z-R]). One of the basic concepts on which our approach relies is the general concept of variation, or variation through solutions, on which our treatment not only of the irrotational isentropic case but also of the general equations of motion is based. This concept has been discussed in the general context of Euler–Lagrange equations, that is, systems of partial differential equations arising from an action principle, in the monograph [Ch]. It was shown there that to a variation is associated a linearized Lagrangian, and it was also shown how energy currents are in general constructed on the basis of this linearized Lagrangian. It is through energy currents and their associated integral identities that the estimates essential to our approach are derived. Here the first order variations correspond to the one-parameter subgroups of the Poincaré group, the isometry group of Minkowski spacetime, extended by the one-parameter scaling or dilation group, which leave the surrounding constant state invariant. The higher order variations correspond to the oneparameter groups of diffeomorphisms generated by a set of vectorfields, the commutation fields, to be discussed below. The construction in [Ch] of an energy current requires a multiplier vectorfield which at each point belongs to the closure of the positive component of the inner characteristic core in the tangent space at that point. In the irrotational isentropic case the characteristic in the tangent space at a point consists only of the sound cone at that point and this requirement becomes the requirement that the multiplier vectorfield be non-space-like and future-directed with respect to the acoustical metric (5). We use two multiplier vectorfields in our analysis of the isentropic irrotational problem.

10


The first is the vectorfield K 0 defined by (4) and the second is the vectorfield K 1 defined by equation (5.16) of Chapter 5: K 1 = (ω/ν)L.

(9)

Here ν is the mean curvature of the wave fronts St,u , sections of the outgoing characteristic hypersurfaces Cu , relative to their characteristic normal L, the tangent vectorfield to the bi-characteristic generators of Cu , parametrized by t. However ν is defined not relative to the acoustical metric h µν but rather relative to a conformally related metric h˜ µν : h˜ µν = h µν .

(10)

It turns out that there is a choice of conformal factor such that in the isentropic irrotational case a first order variation φ˙ of the wave function φ satisfies the wave equation relative to the metric h˜ µν . This is shown in Chapter 1 and this choice defines in the remainder of the monograph. The definition makes the ratio of a function of σ to the value of this function in the surrounding constant state, thus is equal to unity in the constant state. It turns out moreover that is bounded above and below by positive constants. The function ω appearing in (9) is required to satisfy certain conditions (conditions D1–D5 of Chapter 5) and it is shown in Proposition 13.4 that the function ω = 2η0 (1 + t) does satisfy these requirements. A similar analysis to the one done above in the case of the multiplier field K 0 shows, taking into account the fact that the physical dimension of ν is inverse time, that the multiplier field K 1 has the physical dimension of length. The vectorfield K 1 corresponds to the generator of inverted time translations, which are proper conformal transformations of the Minkowski spacetime with its Minkowskian metric gµν . The latter was first used by Morawetz [Mo] to study the decay of solutions of the initial boundary value problem for the classical wave equation outside an obstacle. The vectorfield K 1 is an analogue of the multiplier field of Morawetz for the acoustical spacetime which is the same underlying manifold but equipped with the acoustical metric h µν . The energy currents associated to K 0 and K 1 are defined by equations (5.18) and (5.19) of Chapter 5, respectively. The energy current associated to K 1 contains certain additional lower order terms, defined through the function ω. Analogous terms were present in the work of Morawetz. The general structure of these terms has been investigated, in the general context of Euler–Lagrange equations, in [Ch]. To each variation ψ, of any order, there are energy currents associated to ψ and to K 0 and K 1 respectively. These currents define the energies E u0 [ψ](t), E1u [ψ](t), and fluxes F0t [ψ](u), F1t [ψ](u). For given t and u the energies are integrals over the exterior of the surface St,u in the hyperplane t , while the fluxes are integrals over the part of the outgoing characteristic hypersurface Cu between the hyperplanes 0 and t . To obtain the energy and flux associated to K 1 , certain integrations by parts are performed. This construction is presented in Chapter 5. The precise choice of the factor ω/ν in (9) is dictated by the need to eliminate certain error integrals which would otherwise be present. It is these energy and flux integrals, together with a spacetime integral K [ψ](t, u) associated to K 1 , to be discussed below, which are used to control the solution. It is evident from the above that the means by which the solution is controlled depend on the choice of the acoustical function u, the level sets of which are the outgoing


11

characteristic hypersurfaces Cu . The function u is determined by its restriction to the initial hyperplane 0 . The divergence of the energy currents, which determines the growth of the energies and fluxes, itself depends on (K 0 ) π, ˜ in the case of the energy current associated to K 0 , and (K 1 ) π, ˜ in the case of the energy current associated to K 1 . Here for any vectorfield X in spacetime, we denote by (X ) π˜ the Lie derivative of the conformal acoustical metric h˜ with respect to X. We may call (X ) π˜ the deformation tensor corresponding to X. In the case of higher order variations, the divergences of the energy currents depend ˜ for each of the commutation fields Y to be discussed below. also on the (Y ) π, All these deformation tensors ultimately depend on the acoustical function u, or, which is the same, on the geometry of the foliation of spacetime by the outgoing characteristic hypersurfaces Cu , the level sets of u. The most important geometric property of this foliation from the point of view of the study of shock formation is the density of the packing of its leaves Cu . One measure of this density is the inverse spatial density of the wave fronts, that is, the inverse density of the foliation of each spatial hyperplane t by the surfaces St,u . This is the function κ which appears in (4) and is given in arbitrary coordinates on t by: −1 κ −2 = (h )i j ∂i u∂ j u (11) where h i j is the induced acoustical metric on t . Another measure is the inverse temporal density of the wave fronts, the function µ given in arbitrary coordinates in spacetime by: 1 = −(h −1 )µν ∂µ t∂ν u. µ

(12)

The two measures are related by: µ = ακ

(13)

where α is the inverse density, with respect to the acoustical metric, of the foliation of spacetime by the hyperplanes t . The function α also appears in (4) and is given in arbitrary coordinates in spacetime by: α −2 = −(h −1 )µν ∂µ t∂ν t.

(14)

It is expressed directly in terms of the 1-form β in the general case, or dφ in the irrotational isentropic case. It turns out moreover, that it is bounded above and below by positive constants. Consequently µ and κ are equivalent measures of the density of the packing of the leaves of the foliation of spacetime by the Cu . Shock formation is characterized by the blowup of this density or equivalently by the vanishing of κ or µ. The above and the basic geometric construction are discussed in Chapter 2. The other entity, besides κ or µ which describes the geometry of the foliation by the Cu , is the second fundamental form of the Cu . Since the Cu are null hypersurfaces with respect to the acoustical metric h, their tangent hyperplane at a point is the set of all vectors at that point which are h-orthogonal to the generator L, and L itself belongs to the tangent hyperplane, being h-orthogonal to itself. Thus the second fundamental form χ of Cu is intrinsic to Cu and in terms of the metric h/ induced by the acoustical metric on the St,u sections of Cu , it is given by: L / L h/ = 2χ

(15)

12


where L / X ϑ for a covariant St,u tensorfield ϑ denotes the restriction of L X ϑ to T St,u . The acoustical structure equations such as the propagation equation for χ along the generators / χ, the divergence of χ intrinsic to St,u , in of Cu , the Codazzi equation which expresses div terms of d/trχ, the differential on St,u of trχ, and a component of the acoustical curvature and of k, the second fundamental form of the t relative to h, are presented in Chapter 3. Also included in the acoustical structure equations is the Gauss equation which expresses /) in terms of χ and a component of the acoustical curvature the Gauss curvature of (St,u , h and of k, and an equation which expresses L /T χ in terms of the Hessian of the restriction of µ to St,u and another component of the acoustical curvature and of k. In the same chapter the components of k are analyzed. These acoustical structure equations contain terms which blow up as κ or µ tend to zero. The regular form of these equations is given in the next chapter, Chapter 4, the subject of which is the analysis of the acoustical curvature. It is there shown that the terms which blow up as κ or µ tend to zero cancel. The most important acoustical structure equation from the point of view of the formation of shocks is the propagation equation for µ along the generators of Cu , equation (3.96): Lµ = m + µe. (16) An equivalent equation, (3.99), is satisfied by κ. This equation is derived in Chapter 3 only in the irrotational isentropic case, in contrast to the other structure equations which hold in general. The function m is in this case given by: m=

dH 1 (Lφ)2 (T σ ) 2 dσ

(17)

while the function e, given by (3.98), depends only on the derivatives along L of the ψα , α = 0, 1, 2, 3, the first variations corresponding to the spacetime translations. A similar propagation equation holds in the general case with the function m given by: 1 2 dH m = (β L ) (T σ ) (18) 2 dσ s and the function e depending only on the derivatives of the βα , the rectangular components of β, tangential to the Cu . We shall presently indicate why this has to be the case without performing the calculations, which in any case parallel those of Chapter 3, with the general formulas (3.4) and (3.10) for the vectorfield V i and the metric h i j induced on the t , in place of the corresponding formulas (3.6) and (3.11) in the irrotational isentropic case. The fluid velocity being transversal to Cu , by virtue of the adiabatic condition u · ds = 0, any derivative of s can be expressed as a derivative tangential to Cu . On the other hand, according to (1) for any vector X we have: ω(u, X) = −θ X · ds. It follows that the differential of the entropy and the vorticity do not contribute to the function m, hence an equation of the form (16) results with m given by (18). It is the function m which determines shock formation, when being negative, causing µ to decrease to zero.


13

The path we have followed in attacking the problem of shock formation in 3-dimensional fluids illustrates the following approach in regard to quasilinear hyperbolic systems of partial differential equations. That is, the quantities which are used to control the solution must be defined using the causal, or characteristic, structure of spacetime determined by the solution itself, not an artificial background structure. The original system of equations must then be considered in conjunction with the system of equations which this structure obeys, and it is only through the study of the interaction of the two systems that results are obtained. The work [C-K] on the stability of the Minkowski space in the framework of general relativity was the first illustration of this approach. In the present case however, the structure, which is here the acoustical structure, degenerates as shocks begin to form, and the precise way in which this degeneracy occurs must be guessed beforehand and established in the course of the argument of the mathematical proof. The fact that the underlying structure degenerates implies that our estimates are no longer even locally equivalent to standard energy estimates, which would of necessity have to fail when shocks appear. Chapter 5 establishes the fundamental energy estimate, Theorem 5.1. This applies to a solution of the homogeneous wave equation in the acoustical spacetime, in particular to any first order variation. Now the higher order variations satisfy inhomogeneous wave equations in the acoustical spacetime, the source functions depending on the deformation tensors of the commutation fields. These source terms give rise to error integrals, that is to spacetime integrals of contributions to the divergence of the energy currents, which are written down but not estimated in Chapter 5. The remaining error integrals however, are all estimated in the proof of Theorem 5.1, and since these estimates apply to any variation of any order, Chapter 5 contributes in an essential way to the main theorem of Chapter 13. The proof of Theorem 5.1 relies on certain bootstrap assumptions on the acoustical entities. The most crucial of these assumptions concern the behavior of the function µ. These are the assumptions C1, C2, and C3, which are established in the first part of Chapter 13, by Propositions 13.1, 13.2, and 13.3, respectively, on the basis of the final set of bootstrap assumptions, which consists only of pointwise estimates for the variations up to certain order. To give an idea at this point of the nature of these assumptions, the assumption C2 required in Chapter 5 to obtain the fundamental energy estimate up to time s is (modulo assumption C1): µ−1 (T µ)+ ≤ Bs (t) : for all t ∈ [0, s] where Bs (t) is a function such that: s (1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt ≤ C

(19)

(20)

0

with C a constant independent of s. Here T is the vectorfield defined above and we denote by f+ and f − , respectively the positive and negative parts of an arbitrary function f . This assumption is then established by Proposition 13.2 with Bs (t) the following function:

(1 + τ ) Bs (t) = C δ0 √ + Cδ0 (1 + τ ) σ −τ

(21)

14


where τ = log(1 + t), σ = log(1 + s), and δ0 is a small positive constant appearing in the final set of bootstrap assumptions. Now, the spacetime integral K [ψ](t, u) mentioned above, is defined by (5.169). It is essentially the integral of 1 − (ω/ν)(Lµ)− |d/ψ|2 2 in the spacetime exterior to Cu and bounded by 0 and t . Assumption C3 on the other hand states that there is a positive constant C independent of s such that in the region below s where µ < η0 /4 we have: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1

(22)

In view of this assumption, the integral K [ψ](t, u) gives effective control of the derivatives of the variations tangential to the wave fronts in the region where shocks are to form. The same assumption, which is established by Proposition 13.3, also plays an essential role in the study of the singular boundary in Chapter 15. The final stage of the proof of Theorem 5.1 is the analysis of a system of integral inequalities in two variables t and u satisfied by the five quantities E0u [ψ](t), E1u [ψ](t), F0t [ψ](u), F1t [ψ](u), and K [ψ](t, u). This analysis is reflected at analogous stages in the course of the proof of Theorem 13.1. The commutation fields Y are defined in Chapter 6. They are five: the vectorfield T which is transversal to the Cu , the field Q = (1 + t)L along the generators of the Cu and the three rotation fields Ri : i = 1, 2, 3 which are tangential to the St,u sections. ◦

◦

The latter are defined to be Ri : i = 1, 2, 3, where the Ri i = 1, 2, 3 are the generators of spatial rotations associated to the background Minkowskian structure, while is the h-orthogonal projection to the St,u . Note that the commutation field T has the physical dimension of inverse length while the other commutation fields are dimension˜ (Q) π, ˜ and (Ri ) π˜ : i = 1, 2, 3 are less. Expressions for the deformation tensors (T ) π, then derived, which show that these depend on the acoustical entities µ and χ. The last however depend in addition on the derivatives of the restrictions to the surfaces St,u of the spatial rectangular coordinates x i : i = 1, 2, 3, as well as on the derivatives of the x i with respect to T and L, that is, on the rectangular components T i and L i of the vectorfields T and L (note that L 0 = 1, T 0 = 0). The estimates of these and their derivatives with respect to the commutation fields in terms of the acoustical entities occupies a major part of Chapters 10 and 11 as shall be discussed below. In Chapter 7 a recursion formula is obtained for the source functions associated to the higher order variations, on the basis of which an explicit formula for these source functions is obtained in Chapter 13. Then the error integrals arising from the contributions to the source functions containing the top and next to the top order derivatives of the variations are estimated. Chapters 8 and 9 are crucial for the entire work because it is here that the estimates for the top order derivatives of the acoustical entities are derived. The expressions for the source functions and the associated error integrals from Chapter 7 show that the error integrals corresponding to the energies of the n +1st order variations contain the nth order derivatives of the deformation tensors, which in turn contain the nth order derivatives of


15

χ and n + 1st order derivatives of µ. Thus to achieve closure, we must obtain estimates for the latter in terms of the energies of up to the n + 1st order variations. Now, the propagation equations of Chapter 3 for χ and µ, appearing in regular form in Chapter 4, give appropriate expressions for L / L χ and Lµ. However, if these propagation equations, which may be thought of as ordinary differential equations along the generators of the Cu , are integrated with respect to t to obtain the acoustical entities χ and µ themselves, and their spatial derivatives are then taken, a loss of one degree of differentiability would result and closure would fail. We overcome this difficulty in the case of χ in Chapter 8 by considering the propagation equation for µtrχ. We show that, by virtue of a wave equation for σ , which follows from the wave equations satisfied by the first variations corresponding to the spacetime translations, the principal part on the right-hand side of this propagation equation can be put into the form −L fˇ of a derivative of a function − fˇ with respect to L. This function is then brought to the left-hand side and we obtain a propagation equation for µtrχ + fˇ. In this equation χ, ˆ the trace-free part of χ enters, but the propagation equation in question is considered in conjunction with the Codazzi equation, which constitutes an elliptic system ˆ given trχ. We thus have an ordinary differential equation along the on each St,u for χ, generators of Cu coupled to an elliptic system on the St,u sections. More precisely, the propagation equation which is considered at the same level as the Codazzi equation is a propagation equation for the St,u 1-form µd/trχ + d/ fˇ, which is a consequence of the equation just discussed. To obtain estimates for the angular derivatives of χ of order l we similarly consider a propagation equation for the St,u 1-form: (i1 ...il )

xl = µd/(Ril . . . Ri1 trχ) + d/(Ril . . . Ri1 fˇ)

In the case of µ the aforementioned difficulty is overcome in Chapter 9 by considering the propagation equation for µ / µ, where / µ is the Laplacian of the restriction of µ to the St,u . We show that by virtue of a wave equation for T σ , which is a differential consequence of the wave equation for σ , the principal part on the right-hand side of this propagation equation can again be put into the form L fˇ of a derivative of a function fˇ with respect to L. This function is then likewise brought to the left-hand side and we /ˆ2 µ, the trace-free part of obtain a propagation equation for µ / µ − fˇ . In this equation D the Hessian of the restriction of µ to the St,u enters, but the propagation equation in question is considered in conjunction with the elliptic equation on each St,u for µ, which the specification of / µ constitutes. Again we have an ordinary differential equation along the generators of Cu coupled to an elliptic equation on the St,u sections. To obtain estimates of the spatial derivatives of µ of order m + l + 2 of which m are derivatives with respect to T , we similarly consider a propagation equation for the function: (i1 ...il ) x m,l

= µRil . . . Ri1 (T )m / µ − Ril . . . Ri1 (T )m fˇ .

This allows us to obtain estimates for the top order spatial derivatives of µ of which at least two are angular derivatives. A remarkable fact, which is shown in Chapter 13, is that the missing top order spatial derivatives do not enter the source functions, hence do not contribute to the error integrals. In fact it is shown that the only top order spatial

16


derivatives of the acoustical entities entering the source functions are those in the 1-forms (i1 ...il ) x and the functions (i1 ...il ) x . l m,l The paradigm of an ordinary differential equation along the generators of a characteristic hypersurface coupled to an elliptic system on the sections of the hypersurface as the means to control the regularity of the entities describing the geometry of the characteristic hypersurface and the stacking of such hypersurfaces in a foliation, was first encountered in [C-K]. It is interesting to note that this paradigm does not appear in space dimension less than three. In the case of the work on the stability of the Minkowski space however, in contrast to the present case, the gain of regularity achieved in this treatment is not essential for obtaining closure, because there is room for one degree of differentiability. This is because of the fact that the Einstein equations arise from a Lagrangian which is quadratic in the canonical velocities, that is, in the derivatives of the unknown functions, in contrast to the equations of fluid mechanics, or more generally of continuum mechanics, which in the Lagrangian picture are equations for a mapping of spacetime into the material manifold, the Lagrangian not depending quadratically on the differential of this mapping (see [Ch]). As a consequence, the metric determining the causal structure depends in continuum mechanics on the derivatives of the unknowns, rather than only on the unknowns themselves. In the present work, the appearance of the factor of µ, which vanishes where shocks / µ in the definitions of (i1 ...il ) xl originate, in front of d/ Ril . . . Ri1 trχ and Ril . . . Ri1 (T )m (i ...i ) l 1 and x m,l above, makes the analysis far more delicate. This is compounded with the difficulty of the slower decay in time which the addition of the terms −d/ Ril . . . Ri1 fˇ and Ril . . . Ri1 (T )m fˇ forces. The analysis requires a precise description of the behavior of µ itself, given by Proposition 8.6, and a separate treatment of the condensation regions, where shocks are to form, from the rarefaction regions, the terms referring not to the fluid density but rather to the density of the stacking of the wave fronts. To overcome the difficulties the following weight function is introduced: µm,u (t) µm,u (t) = min , 1 , µm,u (t) = min µ (23) η0 tu where tu is the exterior of St,u in t , and, in Chapter 13, the quantities E0u [ψ](t), E1u [ψ](t), F0t [ψ](u), F1t [ψ](u), and K [ψ](t, u) corresponding to the highest order variations are weighted with a power, 2a, of this weight function. Lemma 8.11, then plays a crucial role in Chapters 8 and 9 as well as in Chapter 13 where everything comes together. We present this lemma here in an imprecise manner to indicate what is involved. Let:

−1 −µ , Mu (t) = max (Lµ) − u t

Ia,u = 0

t

µ−a m,u (t )Mu (t )dt .

(24)

Then under certain bootstrap assumptions in the past of s , for any constant a ≥ 2, there is a positive constant C independent of s, u and a such that for all t ∈ [0, s] we have: Ia,u (t) ≤ Ca −1 µ−a m,u (t).

(25)


17

As mentioned in the discussion of Chapter 6 above, estimates for the derivatives of the spatial rectangular coordinates x i with respect to the commutation fields must also be obtained, the derivative of the x i with respect to the vectorfields Tˆ and L being the spatial rectangular components Tˆ i and L i of these vectorfields. Here Tˆ = κ −1 T is the vectorfield of unit magnitude with respect to h corresponding to T . Thus, although the argument depends mainly on the causal structure of the acoustical spacetime, the underlying Minkowskian structure, to which the rectangular coordinates belong, has a role to play as well, and it is the estimates in question which analyze the mutual relationship of the two structures. The major part of Chapters 10 and 11 is the derivation of estimates for the spatial derivatives of the first derivatives of the x i , in terms of the acoustical entities. In particular, Propositions 10.1 and 10.2 give estimates for the angular derivatives of the x i and of the Tˆ i = Tˆ x i , that is, their derivatives with respect to the rotation fields R j , while Propositions 11.1 and 11.2 give estimates for the spatial derivatives of the Tˆ i , that is the derivatives with respect to T and the R j , of which at least one is a T -derivative. The corollaries of these propositions provide the remaining estimates, including the required estimates for the deformation tensors of the commutation fields in terms of the acoustical entities. In particular, Corollaries 10.1.e and 10.2.e give, through Lemma 10.6, estimates for the iterated commutators of the set of rotation fields, Corollaries 10.1.i and 10.2.i give estimates for the angular derivatives of the commutators [L, Ri ] = (Ri ) Z (see Lemma 8.2), while Lemma 10.24 gives estimates for the angular derivatives of the commutators [T, Ri ] = (Ri ) (see Lemma 10.22). On the other hand, Corollaries 11.1.c and 11.2.c give estimates for the spatial derivatives of the commutators [L, T ] = . The remainder of Chapters 10 and 11 deduce the required estimates for the quantities appearing in the final estimates of Chapters 8 and 9 for the 1-forms (i1 ...il ) xl and the functions (i1 ...il−m ) x m,l−m , respectively. Chapter 12 contains the recovery of the acoustical bootstrap assumptions used in the previous chapters, in particular in Chapters 10 and 11. That is, these acoustical assumptions are established, using the method of continuity, on the basis of the final set of bootstrap assumptions, which consists only of pointwise estimates for the variations up to certain order. In the same chapter the estimates for up to the next to the top order angular derivatives of χ and spatial derivatives of µ are derived. These, when substituted in the estimates of Chapters 10 and 11, give control of all quantities involved in terms of estimates for the variations. A fundamental role in Chapter 12 is played by Propositions 12.2, 12.4, 12.5, and 12.7 which establish the coercivity hypotheses H0, H1, H2 and propositions roughly speaking show H2 on which the previous chapters depend. These / Ri ϑ|2 bounds pointwise |D /ϑ|2 . that for any covariant St,u tensorfield ϑ, the sum i |L Proposition 12.8, which shows that if X is any St,u -tangential vectorfield and ϑ any co/ X ϑ in terms of the L / Ri ϑ and the variant St,u tensorfield then we can bound pointwise L L / Ri X = [Ri , X], also plays an important role. Chapter 13 begins by establishing the basic assumptions C1, C2, and C3, on the behavior of the function µ on which the energy estimates rely. We then formulate the final bootstrap assumption, to which all other assumptions have been reduced, and which consists only of pointwise estimates for the variations up to certain order. After that we deduce an explicit formula for the source functions, using the recursion formula derived

18


in Chapter 7, and analyze the structure of the terms containing the top order spatial derivatives of the acoustical entities, showing that these can be expressed in terms of the 1-forms (i1 ...il ) x and the functions (i1 ...il−m ) x l m,l−m . These terms are shown to contribute borderline error integrals, the treatment of which is the main source of difficulties in the problem. These borderline integrals are all proportional to the constant mentioned above, the value of (8) in the surrounding constant state, hence are absent in the case = 0. We should make clear here that the only variations which are considered up to this point are the variations arising from the first order variations corresponding to the group of spacetime translations. In particular the final bootstrap assumption involves only variu (t), F t u t ations of this type, and each of the five quantities E0,[n] 0,[n] (u), E1,[n] (t), F1,[n] (u), and K [n] (t, u), which together control the solution, is defined to be the sum of the corresponding quantity E0u [ψ](t), F0t [ψ](u), E1u [ψ](t), F1t [ψ](u), and K [ψ](t, u), over all variations ψ of this type, up to order n. To estimate the borderline integrals however, we introduce an additional assumption, assumption J, which concerns the first order variations corresponding to the scaling or dilation group and to the rotation group, and the second order variations arising from these by applying the commutation field T . This assumption is later established through energy estimates of order 4 arising from these first order variations and derived on the basis of the final bootstrap assumption, just before the recovery of the final bootstrap assumption itself. It turns out that the borderline integrals all contain the factor T ψα , where ψα : α = 0, 1, 2, 3 are the first variations corresponding to spacetime translations and J is used to obtain an estimate assumption for suptu µ−1 |T ψα | in terms of suptu µ−1 |Lµ| , which involves on the right the factor ||−1 (see (13.198)). Upon substituting this estimate in the borderline integrals, the factors involving cancel, and the integrals are estimated using (25). The above is an outline of the main steps in the estimation of the borderline integrals associated to the vectorfield K 0 . The estimation of the borderline integrals associated to the vectorfield K 1 , is however still more delicate. In this case we first perform an integration by parts on the outgoing characteristic hypersurfaces Cu , obtaining hypersurface integrals over tu and 0u and another spacetime volume integral. In this integration by parts the terms, including those of lower order, must be carefully chosen to obtain appropriate estimates, because here the long time behavior, as well as the behavior as µ tends to zero, is critical. Another integration by parts, this time on the surfaces St,u , is then performed to reduce these integrals to a form which can be estimated. The estimates of the hypersurface integrals over tu are the most delicate (the hypersurface integrals over 0u only involve the initial data) and require separate treatment of the condensation and rarefaction regions, in which the properties of the function µ, in particular those established by Proposition 8.6, come into play. In proceeding to derive the energy estimates of top order, n = l + 2, the power 2a of the weight µm,u (t) is chosen suitably large to allow us to transfer the terms contributed by the borderline integrals to the left-hand side of the inequalities resulting from the integral identities associated to the multiplier fields K 0 and K 1 . The argument then proceeds along the lines of that of Chapter 5, but is more complex because account must be taken of the terms corresponding to the estimates of Chapter 7 and of all the other terms contributed by the source functions, and also because of the fact that here we are dealing with weighted


19

quantities. Once the top order energy estimates are established, we revisit the lower order energy estimates, using at each order the energy estimates of the next order in estimating the error integrals contributed by the highest spatial derivatives of the acoustical entities at that order. We then establish a descent scheme, which yields, after finitely many steps, u (t), F t u t estimates for the five quantities E0,[n] 0,[n] (u), E1,[n] (t), F1,[n] (u), and K [n] (t, u), for n = l + 1 − [a], where [a] is the integral part of a, in which weights no longer appear. It is these unweighted estimates which are used to close the bootstrap argument by recovering the final bootstrap assumption. This is accomplished by the method of continuity through the use of the isoperimetric inequality on the wave fronts St,u , and leads to the main theorem, Theorem 13.1. This theorem shows that there is another differential structure, that defined by the acoustical coordinates t, u, ϑ introduced in Chapter 2, such that relative to this structure the maximal classical solution extends smoothly to the boundary of its domain. This boundary contains however a singular part where the function µ vanishes, hence, in these coordinates, the acoustical metric h degenerates. With respect to the standard differential structure induced by the rectangular coordinates x α : α = 0, 1, 2, 3 in Minkowski spacetime, the solution is continuous but not differentiable on the singular part of the boundary, the derivative Tˆ µ Tˆ ν ∂µ βν blowing up as we approach the singular boundary. Thus, with respect to the standard differential structure, the acoustical metric h is everywhere in the closure of the domain of the maximal solution non-degenerate and continuous, but not differentiable on the singular part of the boundary of this domain, while with respect to the differential structure induced by the acoustical coordinates h is everywhere smooth, but degenerate on the singular part of the boundary. We have not sought to obtain an optimal lower bound for exponent a. This lower bound is significantly reduced by observing that only one among the first order variations actually contributes borderline integrals, namely the variation corresponding to time translations. As has already been mentioned, the first part of Chapter 14 establishes a theorem, Theorem 14.1, which gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution, in the case of irrotational isentropic initial data. The proof is through Proposition 8.6 and is based on the study of the evolution with respect to t of the mean value on the sections St,u of each outgoing characteristic hypersurface Cu of the quantity: τ = (1 − u + η0 t)Lψ0 − (ψ0 − k).

(26)

Here ψ0 is the first variation corresponding to time translations and k is its value in the surrounding constant state. The proof of Theorem 14.1 uses the estimate provided by the spacetime integral K [ψ0 ](t, u) associated to ψ0 . Theorem 14.1 is followed by the analysis of the solution of the problem with general initial data during the initial time interval of order 1/η0 , as has already been discussed above. The last part of Chapter 14 establishes a theorem, Theorem 14.2, which extends Theorem 14.1 to the general case, removing the irrotational and isentropic restrictions on the initial data. The proof of Theorem 14.2 is based on the 1-form: (27) ξµ = β˙µ + θ s˙ u µ

20


corresponding to any first order variation ( p, ˙ s˙ , u) ˙ of a general solution ( p, s, u), through solutions of the general equations of motion, and the associated functions: i = L µ ξµ ,

i = L µ ξµ .

(28)

We then study the evolution of the mean value on the St,u sections of each Cu of the quantity: τ = (1 − u + η0 t)i − v0 ( p − p0 ) (29) where v0 and p0 are respectively the volume per particle and pressure in the surrounding constant state. Here certain crucial integrations by parts on the St,u sections as well as on Cu itself are performed, in which the structure of Cu as a characteristic hypersurface comes into play. We remark that Theorems 14.1 and 14.2 also give a sharp upper bound on the time interval required for the onset of shock formation. The contents of Chapter 15 have already been briefly described above. Proposition 15.1 describes the singular part of the boundary of the domain of the maximal classical solution from the point of view of the acoustical spacetime. It shows that this singular part has the intrinsic geometry of a regular null hypersurface in a regular spacetime and, like the latter, is ruled by invariant curves of vanishing arc length. On the other hand, the extrinsic geometry of the singular boundary is that of a space-like hypersurface which becomes null at its past boundary. The invariant curves are then used to define canonical acoustical coordinates. Theorem 15.1 is the main result of the chapter. This theorem shows that at each point q of the singular boundary, the past sound cone in the cotangent space at q degenerates into two hyperplanes intersecting in a 2-dimensional plane. We thus have a trichotomy of the bi-characteristics, or null geodesics of the acoustical metric, ending at q, into the set of outgoing null geodesics ending at q, which corresponds to one of the hyperplanes, the set of incoming null geodesics ending at q, which corresponds to the other hyperplane, and the set of the remaining null geodesics ending at q, which corresponds to the 2dimensional plane. The intersection of the past characteristic cone of q with any t in the past of q similarly splits into three parts, the parts corresponding to the outgoing and to the incomings sets of null geodesics ending at q being embedded discs with a common boundary, an embedded circle, which corresponds to the set of the remaining null geodesics ending at q. All outgoing null geodesics ending at q have the same tangent vector at q. This vector is then an invariant characteristic vector associated to the singular point q. This striking result is in fact the reason why the considerable freedom in the choice of the acoustical function does not matter in the end. For as is shown in Proposition 15.2, which considers the transformation from one acoustical function to another, the foliations corresponding to different families of outgoing characteristic hypersurfaces have equivalent geometric properties and degenerate in precisely the same way on the same singular boundary. Finally, Proposition 15.3 gives a detailed description of the boundary of the domain of the maximal classical solution from the point of view of Minkowski spacetime. The contents of the Epilogue have already been adequately described above.


21

In concluding this introduction we remark that shocks develop not only in the context of fluid mechanics but also in magnetohydrodynamics, that is, the mechanics of a perfectly electrically conducting fluid in the presence of a magnetic field, in the nonlinear regime of the theory of elasticity, that is, the mechanics of deformable solids, isotropic or crystalline, as well as in the electrodynamics of continuous nonlinear media, in particular in the propagation of electromagnetic waves in electrically insulating fluids or solids with a nonlinear relationship between the electromagnetic field and the electromagnetic displacement. The pioneering work on shock formation in the theory of elasticity, in the spherically symmetric case, has been done by John [J2]. It is hoped that the present monograph will provide a springboard for those wishing to attack the general problem of shock formation in any of these fields. The present work relies more heavily on differential geometric concepts and methods than previous works on the same subject. For those prospective readers whose background is mainly in the fields of fluid mechanics or partial differential equations, but who may not have acquired an equally strong background in differential geometry, we recommend as a reference the book by Bishop and Crittenden [B-C] for the basic differential geometric concepts, and the book by Schoen and Yau [S-Y] as an excellent introduction to geometric analysis. In regard to notational conventions, Latin indices take the values 1,2,3, while Greek indices take the values 0,1,2,3. Repeated indices are meant to be summed, unless otherwise specified.

Chapter 1

Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variation The mechanics of a perfect fluid is described in the framework of the Minkowski spacetime of special relativity by a future-directed unit time-like vectorfield u, the fluid 4velocity, and two positive functions n and s, the number of particles per unit volume (in the local rest frame of the fluid) and the entropy per particle, respectively. In terms of a system of rectangular coordinates (x 0 , x 1 , x 2 , x 3 ), with x 0 a time coordinate and (x 1 , x 2 , x 3 ) space coordinates, the metric components gµν , µ, ν = 0, 1, 2, 3, are given by, g00 = −1, g11 = g22 = g33 = 1, gµν = 0 : if µ = ν.

(1.1)

The conditions on the 4-velocity components u µ , µ = 0, 1, 2, 3, are then: gµν u µ u ν = −1, u 0 > 0.

(1.2)

Here, and throughout this monograph, we follow the summation convention, according to which repeated upper and lower indices are summed over their range. The mechanical properties of a perfect fluid are specified once we give the equation of state, which expresses the mass-energy density ρ as a function of n and s: ρ = ρ(n, s).

(1.3)

According to the laws of thermodynamics, the pressure p and the temperature θ are then given by: 1 ∂ρ ∂ρ − ρ, θ = . (1.4) p=n ∂n n ∂s The functions ρ, p, θ are assumed positive. Moreover, it is assumed that p is an increasing function of n at constant s and θ is an increasing function of s at constant n. The particle current is the vectorfield I whose components are given by: I µ = nu µ .

(1.5)

24

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

The energy-momentum-stress tensor is the symmetric 2-contravariant tensorfield T whose components are: (1.6) T µν = (ρ + p)u µ u ν + p(g −1 )µν . Here (g −1 )µν , µ, ν = 0, 1, 2, 3, are the components of the reciprocal metric, (g −1 )00 = −1, (g −1 )11 = (g −1 )22 = (g −1 )33 = 1, (g −1 )µν = 0 : if µ = ν.

(1.7)

The equations of motion of a perfect fluid are the conservation laws: ∂µ I µ = 0

(1.8)

∂ν T µν = 0.

(1.9)

Here, and throughout this monograph the symbol ∂µ =

∂ ∂xµ

denotes partial derivative with respect to the rectangular coordinate x µ . Taking the component of equation (1.9) along u by contracting with u µ = gµν u ν yields the equation:

u µ ∂µ ρ + (ρ + p)∂µ u µ = 0.

(1.10)

Now, according to equation (1.8), 1 ∂µ u µ = − u µ ∂µ n. n Substituting in (1.10) reduces that equation to the form: u µ ∂µ ρ =

(ρ + p) µ u ∂µ n. n

(1.11)

On the other hand by virtue of the equation of state (1.3) and the definitions (1.4), ∂µ ρ =

(ρ + p) ∂µ n + nθ ∂µ s. n

(1.12)

Comparing (1.11) and (1.12) we conclude that u µ ∂µ s = 0.

(1.13)

That is, the entropy per particle is constant along the flow lines, that is the integral curves of the vectorfield u. This conclusion holds as long as we are dealing with a solution of the equations of motion in the classical sense, that is the variables u µ , µ = 0, 1, 2, 3; n, s are C 1 functions of the rectangular coordinates. A portion of a fluid is called isentropic if s is constant throughout this portion. It follows from the preceding that if a portion of


25

the fluid is isentropic at time x 0 = 0 then the same portion, as given by the flow of u, is isentropic for all later time x 0 > 0, as long as the solution remains C 1 . Let us denote by the 1-covariant, 1-contravariant tensorfield, which is at each point x in spacetime the operator of projection to x , the local simultaneous space of the fluid at x, namely the orthogonal complement in the tangent space at x of the linear span of u. The components of are given by: µ µ µ ν = δν + u u ν

(1.14)

µ

where δν is the Kronecker symbol. The projection of equation (1.9) to x reads, at each point, (1.15) (ρ + p)u ν ∂ν u µ + µν ∂ν p = 0 where

µ

µν = λ (g −1 )λν = (g −1 )µν + u µ u ν .

Let us introduce now the positive function σ = The function

ρ+p n

2 .

√ (ρ + p) σ = n

is called enthalpy per particle. By virtue of equations (1.3) and (1.4), ered to be a function of p and s, and its differential is given by: √ 1 d σ = d p + θ ds. n

(1.16)

√

σ can be consid-

(1.17)

We shall use in the following p and s instead of n and s as the basic thermodynamic variables. The sound speed η is defined by: ∂p 2 η = (1.18) ∂ρ s a fundamental thermodynamic assumption being that the right-hand side of (1.18) is positive. Then η is defined to be positive. Another condition on η in the present framework of special relativity is that η < 1, namely that the sound speed is less than the universal constant represented by the speed of light in vacuum. By virtue of (1.13) and the definition (1.18) we have: ∂ρ ∂ρ µ µ ∂µ p + ∂µ s u ∂µ ρ = u ∂p s ∂s p =

uµ ∂µ p. η2

(1.19)

26


The equations of motion of a perfect fluid are then seen to be equivalent to the following system in terms of the variables ( p, s, u µ : µ = 0, 1, 2, 3): u µ ∂µ s = 0 u µ ∂µ p + η2 (ρ + p)∂µ u µ = 0

(ρ + p)u ν ∂ν u µ + µν ∂ν p = 0.

(1.20)

Let ( p, s, u) be a given solution of the equations of motion (1.20) and let {( pt , st , u t ) : t ∈ I }, I an open interval of the real line containing 0, be a differentiable 1-parameter family of solutions such that: ( p0, s0 , u 0 ) = ( p, s, u). Then

( p, ˙ s˙ , u) ˙ = (d pt /dt)t =0 , (dst /dt)t =0 , (du t /dt)t =0

(1.21)

is a variation of ( p, s, u) through solutions. Note that the constraint (1.2) on u implies the following constraint on u: ˙ (1.22) u µ u˙ µ = 0. Differentiating the equations of motion for ( pt , st , u t ) with respect to t at t = 0 we obtain the equations of variation: u µ ∂µ s˙ = −u˙ µ ∂µ s µ

µ

µ

u ∂µ p˙ + η (ρ + p)∂µ u˙ = −u˙ ∂µ p − q∂ ˙ µu 2

(1.23) µ

˙ µν ∂ν p. (ρ + p)u ν ∂ν u˙ µ + µν ∂ν p˙ = −[(ρ˙ + p)u ˙ ν + (ρ + p)u˙ ν ]∂ν u µ − Here q is the function: q = η2 (ρ + p) and for any function f of the thermodynamic variables alone: ∂f ∂f ˙ f = p˙ + s˙ . ∂p s ∂s p Also:

˙ µν = u˙ µ u ν + u µ u˙ ν .

(1.24)

(1.25)

(1.26)

In equations (1.23) we have placed on the left the principal terms, which are linear in the first derivatives of the variation, and on the right the lower order terms which are linear in the variation itself, with coefficients depending linearly on the first derivatives of the background solution. To the variation ( p, ˙ s˙ , u) ˙ and the system (1.23) is associated the energy current J˙, a vectorfield with components: J˙µ = u µ s˙2 +

1 u µ p˙ 2 + 2u˙ µ p˙ + (ρ + p)u µ gνλ u˙ ν u˙ λ + p)

η2 (ρ

(1.27)


27

(where u˙ ν = gµν u˙ µ ). We have: −u µ J˙µ = s˙ 2 +

1 p˙ 2 + (ρ + p)gνλ u˙ ν u˙ λ . + p)

η2 (ρ

(1.28)

In view of the fact that u˙ is subject to the constraint (1.22), so that gνλ u˙ ν u˙ λ = νλ u˙ ν u˙ λ µ

(where νλ = gµν λ ), this is a positive definite quadratic form in the variation ( p, ˙ s˙, u). ˙ Consider for any covector ξµ in the cotangent space at a given point x the quadratic form ˙ s˙, u), ˙ ( p˙ , s˙ , u˙ ) the corresponding symmetric ξµ J˙µ (x). For any pair of variations ( p, bilinear form is: ˙ s˙ , u), ˙ ( p˙ , s˙ , u˙ )) (1.29) ξµ J˙µ (( p, 1 = ξµ u µ s˙ s˙ + 2 u µ p˙ p˙ + u˙ µ p˙ + p˙ u˙ µ + (ρ + p)u µ gνλ u˙ ν u˙ λ . η (ρ + p) Consider now the set of all non-zero covectors ξ at x such that the symmetric bilinear form (1.29) is degenerate at x, that is, there is a non-zero variation ( p, ˙ s˙ , u) ˙ such that: ˙ s˙ , u), ˙ ( p˙ , s˙ , u˙ ))(x) = 0 ξµ J˙(( p,

: for all variations ( p˙ , s˙ , u˙ ).

(1.30)

This defines the characteristic subset of the cotangent space at x. Taking into account the constraint (1.22) we see that the condition (1.30) is equivalent to the following linear system for the variation ( p, ˙ s˙ , u): ˙ (ξµ u µ )˙s = 0

It follows that either: in which case: or:

(ξµ u µ ) p˙ + ξµ u˙ µ = 0 η2 (ρ + p) ˙ λ + (ρ + p)(ξµ u µ )u˙ λ = 0. λν pξ

(1.31)

ξµ u µ = 0

(1.32)

p˙ = 0 and ξµ u˙ µ = 0

(1.33)

(h −1 )µν ξµ ξν = 0

(1.34)

in which case: s˙ = 0 and u˙ ν = −

νλ ξλ p˙ . (ρ + p)(ξµ u µ )

(1.35)

In (1.34) h −1 is the reciprocal acoustical metric (non-degenerate quadratic form in the cotangent space at each point), given by: 1 µ ν u u η2 1 −1 µν = (g ) − − 1 uµuν . η2

(h −1 )µν = µν −

(1.36)

28


It is the reciprocal of the acoustical metric h (non-degenerate quadratic form in the tangent space at each point), given by: h µν = gµν + (1 − η2 )u µ u ν

(1.37)

which plays a fundamental role in this monograph. The subset of the cotangent space at x defined by the condition (1.32) is a hyperplane Px∗ while the subset of the cotangent space at x defined by the condition (1.34) is a cone C x∗ . The dual to Px∗ characteristic subset of the tangent space at x is the linear span of u, while the dual to C x∗ characteristic subset C x is the set of all non-zero vectors X at x satisfying: h µν X µ X ν = 0.

(1.38)

Now we can construct an orthonormal basis for the tangent space at x by taking u at x to be the time-like element of the basis and complementing it with an orthonormal basis for the space-like hyperplane x , the orthogonal complement of u in the tangent space at x. The dual to such a basis is then a basis for the cotangent space at x. In other words, we may construct a new system of rectangular coordinates (x 0 , x 1 , x 2 , x 3 ) by translating the origin to the point x and choosing the positive x 0 axis in the direction of the vector u at x. In such a system the fluid at x appears to be at rest and we have: u 0 = 1, u 1 = u 2 = u 3 = 0 : at x while the components of the metric g and its reciprocal g −1 have the standard form (see (1.1) and (1.7)), hence also: u 0 = −1, u 1 = u 2 = u 3 = 0 : at x. Equations (1.34) and (1.38) reduce in such a system to the form: −η−2 (ξ0 )2 + (ξ1 )2 + (ξ2 )2 + (ξ3 )2 = 0

(1.39)

−η2 (X 0 )2 + (X 1 )2 + (X 2 )2 + (X 3 )2 = 0.

(1.40)

The requirement that η < 1 is seen to be the physical requirement that in the tangent at each point x the sound cone (1.40) lies within the light cone. This is equivalent to the requirement that in the cotangent space at each point x the light cone lies within the sound cone (1.39). The latter implies that a covector ξ at x, defining a hyperplane Hx in the tangent space at x which is space-like relative to the Minkowski metric g, and such that ξ has positive evaluation on the future-directed normal to Hx , belongs to the interior of the positive component of C x∗ . It follows from the above that the set of all covectors ξ at x such that the quadratic form ξ · J˙(x) is positive definite is the interior of the positive component of C x∗ .


29

Let us now calculate the divergence of J˙. Substituting from the equations of variation (1.23), the principal terms cancel and we find: 1 µ u ∂µ q p˙ 2 + ∂µ ((ρ + p)u µ )gνλ u˙ ν u˙ λ q2 p˙ −2˙s u˙ µ ∂µ s − 2 (u˙ µ ∂µ p + q∂ ˙ µuµ) q ˙ λν ∂λ p. −2u˙ ν [(ρ˙ + p)u ˙ λ + (ρ + p)u˙ λ ]∂λ u ν − 2u˙ ν

∂µ J˙µ = ∂µ u µ s˙2 −

(1.41)

The right-hand side is quadratic in the variation with coefficients which depend linearly on the first derivatives of the background solution. A distinguished class of solutions among all solutions of the equations of motion (1.20) are the solutions of the form, in rectangular coordinates: uµ = aµ,

p = p0 , s = s0 , hence n = n( p0 , s0 ) = n 0

(1.42)

where the a µ , µ = 0, 1, 2, 3 are constants, the components of a unit future-directed timelike vector. Moreover p0 and s0 , hence also n 0 , are constants. The solutions of the form (1.43) are the constant states. By an appropriate choice of rectangular coordinates we can set: (1.43) a 0 = 1, a i = 0 : i = 1, 2, 3. We now introduce the 1-form β whose components are given by: √ βµ = − σ u µ .

(1.44)

We calculate Lu β, the Lie derivative of β with respect to the vectorfield u. It is given by: (Lu β)µ = u ν ∂ν βµ + βν ∂µ u ν an expression valid in any system of coordinates. Taking into account condition (1.2) and then using equation (1.15) we obtain: √ (Lu β)µ = −u ν ∂ν ( σ u µ ) √ √ √ σ σ ν ∂µ p + u µ u ∂ν p − ∂ν σ . = (ρ + p) (ρ + p) Now by (1.17) the expression in the last parenthesis is equal to −θ ∂ν s, therefore by virtue of equation (1.13) the last term vanishes. Taking again account of (1.17) then yields: √ (Lu β)µ = ∂µ σ − θ ∂µ s which we may write simply as: √ Lu β = d σ − θ ds.

(1.45)

The fluid vorticity ω is the 2-form which is the exterior derivative of the 1-form β: ω = dβ.

(1.46)

30


In terms of components we have: ωµν = ∂µ βν − ∂ν βµ an expression valid in an arbitrary system of coordinates. We should note here that our vorticity 2-form does not correspond to the classical notion of vorticity. What corresponds to the classical notion of vorticity is the vectorfield: µ =

1 −1 µαβγ ( ) u α ωβγ . 2

(1.47)

Here ( −1 ) is the reciprocal volume form of the Minkowski metric g or volume form in the cotangent space at each point. We denote by the volume form of g. If (E 0 ,E 1 ,E 2 ,E 3 ) is a positive basis for the tangent space at x which is orthonormal with respect to g and (ϑ0 , ϑ1 , ϑ2 , ϑ3 ) is the dual basis for the cotangent space at x, we have: −1 (ϑ0 , ϑ1 , ϑ2 , ϑ3 ) = (E 0 , E 1 , E 2 , E 3 ) = 1. The components of and −1 are given in an arbitrary system of coordinates by: αβγ δ =

−detg[αβγ δ],

[αβγ δ] ( −1 )αβγ δ = √ −detg

(1.48)

where [αβγ δ] is the 4-dimensional fully antisymmetric symbol. In rectangular coordinates (1.48) reduces to: αβγ δ = ( −1 )αβγ δ = [αβγ δ]. The vectorfield is the obstruction to integrability of the distribution of local simultaneous spaces {x }. We call the vorticity vector . At each point x, the vector (x) belongs to x . Let us recall at this point the following general relation between the exterior derivative and the Lie derivative with respect to a vectorfield X, as applied to an exterior differential form ϑ of any rank: (1.49) L X ϑ = i X dϑ + di X ϑ. Here i X denotes contraction on the left by X. Taking X = u and ϑ to be the 1-form β, we have from (1.44): √ iu β = σ . (1.50) Comparing then (1.49) with (1.45) we conclude that: i u ω = −θ ds.

(1.51)

In the case of an isentropic fluid portion equation (1.51) reduces to: i u ω = 0.

(1.52)

Since by the definition (1.46) we have, in general: dω = 0

(1.53)


31

it follows, taking in (1.49) X = u and ϑ to be the 2-form ω, that in an isentropic fluid portion we have: (1.54) Lu ω = 0. Thus in an isentropic portion of the fluid the vorticity is Lie transported along the flow lines. A portion of the fluid is called irrotational if ω vanishes throughout this portion. Consider now a fluid portion which is isentropic as well as irrotational at time x 0 = 0. Then according to the preceding the same portion, as given by the flow of u, remains isentropic and irrotational for all later time x 0 > 0, as long as the solution remains C 1 in terms of the basic variables u, n and s, or u, p and s. If the fluid portion is simply connected, we can then introduce a function φ, determined up to an additive constant, such that β = dφ (1.55) that is, in components, βµ = ∂µ φ. It follows from the definition of β, equation (1.44), that the derivative of φ is positive along any future-directed non-space-like vector, so φ has the property that φ(y) > φ(x) whenever the point y belongs to the causal future of the point x. Equations (1.44) and (1.55) allow us to express: ∂ µφ σ = −(g −1 )µν ∂µ φ∂ν φ, u µ = − √ σ

(∂ µ = (g −1 )µν ∂ν ).

(1.56)

With (1.55) and (1.56), equation (1.45) is identically satisfied, and this is equivalent to equation (1.15). Moreover, for an isentropic fluid portion equation (1.13) is likewise identically satisfied. It follows that for an irrotational isentropic fluid portion the whole content of the equations of motion is contained in the particle current conservation law (1.8), which takes the form of a nonlinear wave equation: ∂µ (G∂ µ φ) = 0

(1.57)

G = G(σ )

(1.58)

n n2 G=√ = . ρ+p σ

(1.59)

where is given by:

Note that G can be considered to be a function of σ alone by virtue of the fact that s is constant. Summarizing, given the equation of state (1.3) expressing ρ as a function of n at some fixed value of s, equation (1.16) together with the first of equations (1.4) allows us to express n, ρ and p as functions of σ at that value of s, following which equation (1.59) gives us G as a function of σ and equation (1.57) expresses the equations of motion of an irrotational isentropic fluid, with entropy per particle equal to the given value, as a

32


nonlinear wave equation for the function φ, the fluid variables being determined in terms of this function by equations (1.56). Equation (1.57) is the Euler–Lagrange equation corresponding to a Lagrangian density function of the form L = L(σ ) (1.60) where we have, simply, L = p.

(1.61)

For, the Euler–Lagrange equation corresponding to a Lagrangian density of the form (1.60) is: dL µ ∂ φ = 0. ∂µ 2 dσ This coincides with (1.57) since, by (1.17) at constant s and (1.59), 2

dL dp 1 dp n =2 = √ √ = √ = G. dσ dσ σd σ σ

Now, not all Lagrangian densities of the form (1.60) give rise to Euler–Lagrange equations which have a fluid interpretation. First the range of σ must be restricted to the positive real line and L √ must be positive since p is to be positive. Next, L must be an increasing function of σ since dp √ =n d σ √ must be positive; thus G √ must be positive. In fact, the stronger condition that L/ σ be an increasing function of σ must hold, since (1.16) √ dp √ d σ √ (L/ σ ) = σ √ − p = ρ d σ d σ √ must be positive. Finally L must be a convex function of σ so that dn √ d σ will be positive as required by the positivity of d p/dn (at constant s). A particular Lagrangian density of the form (1.60) which does satisfy the requirements for a fluid interpretation if the range of σ is appropriately restricted is: √ L = 1 − 1 − σ. (1.62) The range of σ is a priori (−∞, 1). The requirements for a fluid interpretation are met if we restrict the range of σ to the interval (0, 1). This Lagrangian has a very simple geometric interpretation. Introducing another rectangular spatial coordinate x 4 we consider in Minkowski spacetime of four spatial dimensions the graph x 4 = φ(x),

x = (x 0 , x 1 , x 2 , x 3 )

(1.63)


33

of a function φ defined on a domain √ in the standard Minkowski spacetime of three spatial dimensions. Then the integral of 1 − σ over a subdomain U of the domain of definition of φ is simply the area of the graph over U , a time-like hypersurface, and the corresponding Euler–Lagrange equation is the minimal surface equation for such a graph. We shall see in the sequel that the minimal surface equation is of an exceptional character among all equations of the form (1.57). The nonlinear wave equation (1.57) can be written in the form: (h −1 )µν ∂µ ∂ν φ = 0

(1.64)

where h −1 is the reciprocal acoustical metric (1.36), which in the irrotational isentropic case is given by: (1.65) (h −1 )µν = (g −1 )µν − F∂ µ φ∂ ν φ. Here F = F(σ )

(1.66)

is given by: 2 dG . (1.67) G dσ √ The condition that the Lagrangian density L be a convex function of σ implies that: F=

1+σF =

1 > 0. η2

(1.68)

The acoustical metric h, given in general by (1.37), is in the irrotational isentropic case given by: h µν = gµν + H ∂µ φ∂ν φ (1.69) where H = H (σ ) is the function: H=

F 1+σF

(1.70) (1.71)

and we have: 1 − σ H = η2 .

(1.72)

In particular, if L is the Lagrangian density (1.62) then H =1 and h is the induced metric on the graph (1.63). In the √ general context of fluid mechanics we define the function H according to (1.72), with σ being, according to the above, the enthalpy. We also define in general the function F by (1.71). The physical requirement that the sound cone (1.40) lies within the light cone is then expressed by the condition F >0

(1.73)

34


In the isentropic irrotational case this is equivalent to the condition that the Lagrangian density L be a convex function of σ (1.67). Modulo the other conditions, this condition √ in fact implies the earlier condition that L be a convex function of σ . Conditions (1.68) and (1.73) on the function F are equivalent to the following condition on the function H : 0 < σ H < 1.

(1.74)

The function H or, more precisely, its differential, plays a central role in the problem of shock formation in fluid mechanics. In the isentropic case H is a function of σ and we have: 1 √ dη2 dH = σ √ + 1 − η2 . (1.75) −σ 2 dσ 2 d σ Now, from (1.16) and (1.18) we have, in the isentropic case: √ n d σ = η2 √ σ dn

(1.76)

hence, substituting in (1.75) we obtain: −σ 2

n dη dH = + 1 − η2 . dσ η dn

(1.77)

A distinguished class of solutions among all solutions of the nonlinear wave equation (1.57) are the solutions of the form, in rectangular coordinates, φ(x) = kµ x µ

(1.78)

where the kµ , µ = 0, 1, 2, 3, are constants. These solutions correspond to the constant states (1.43). The required property of φ is then equivalent to the condition that −k µ = −(g −1 )µν kν be the components of a future-directed time-like vector. For these we have: σ = k 2 where k 2 = −gµν k µ k ν > 0 and the correspondence with the constant states (1.43) is: k µ = −ka µ ,

√

σ0 =

ρ0 + p0 = k. n0

By an appropriate choice of rectangular coordinates we can set: k0 = −k 0 = k > 0, ki = k i = 0 : i = 1, 2, 3 so that φ(x) = kx 0 .

(1.79)

Let φ be a given solution of the nonlinear wave equation (1.57) and let {φt : t ∈ I }, I an open interval of the real line containing 0, be a differentiable 1-parameter family of solutions such that φ0 = φ.


Then ψ=

dφt dt

35

(1.80) t =0

is a variation of φ through solutions. Differentiating ∂µ (G(σt )∂ µ φt ) = 0, σt = −∂ µ φt ∂µ φt with respect to t at t = 0 we obtain the equation of variation: ∂µ (G(σ )(h −1 )µν ∂ν ψ) = 0.

(1.81)

Here h −1 is the reciprocal acoustical metric (1.65) corresponding to the solution φ. We can write the equation of variation in a more intelligible form in terms of the conformal ˜ given by: acoustical metric h, (1.82) h˜ µν = h µν where the conformal factor is given by: =

G/G 0 η/η0

(1.83)

where G 0 and η0 are the constant values of G and η in a reference constant state. Taking into account that by equation (1.37) deth = −1 + σ H = −η2 we obtain: deth˜ =

G/G 0 η/η0

Since also (h˜ −1 )µν = it follows that:

4 deth = −

η/η0 G/G 0

(G/G 0 )4 2 η . (η/η0 )2 0

(1.84)

(1.85)

(h −1 )µν ,

˜ h˜ −1 )µν = η0 G (h −1 )µν . −deth( G0

Therefore the equation of variation (1.81) can be written in the form ˜ h˜ −1 )µν ∂ν ψ) = 0. ∂µ ( −det h( ˜ This is simply the linear wave equation corresponding to the metric h: h˜ ψ = 0.

(1.86)

Now if φ is a solution of the nonlinear wave equation (1.57) and { f t } is a 1-parameter subgroup of the isometry group of Minkowski spacetime (4 , g), then for each t, (1.87) φt = φ ◦ f t

36


is also a solution. It follows that ψ=

dφ ◦ f t dt

t =0

= Xφ

(1.88)

satisfies the linear wave equation (1.86). Here X is the vectorfield generating { f t } (considered as a differential operator). The isometry group of Minkowski spacetime is the 10dimensional Poincaré group, consisting of the 4-dimensional normal Abelian subgroup of translations together with the 6-dimensional Lorentz group O(3, 1). The generators of the translations are the vectorfields Tµ =

∂ : µ = 0, 1, 2, 3 ∂xµ

(1.89)

corresponding to the rectangular coordinates x µ : µ = 0, 1, 2, 3, while the generators of the Lorenz group are the vectorfields µν = x µ

∂ ∂ − x ν µ : µ < ν = 0, 1, 2, 3; ∂xν ∂x

x µ = gµν x ν .

(1.90)

Thus, for any solution φ of the nonlinear wave equation, each of the 10 functions Tµ φ : µ = 0, 1, 2, 3; µν φ : µ < ν = 0, 1, 2, 3 satisfies the linear wave equation (1.86) with the metric h˜ being that corresponding to φ. Consider now the 1-parameter group of dilations of the Minkowski spacetime, given by: x → et x : t ∈ where x = (x 0 , x 1 , x 2 , x 3 ) is the position vector in linear, in particular rectangular coordinates. Let again φ be a given solution of the nonlinear wave equation (1.57) and let us define for each t ∈ the function φt by: φt (x) = e−t φ(et x) in terms of linear coordinates. We then have (∂µ φt )(x) = (∂µ φ)(et x) and, with σt = −∂ µ φt ∂µ φt , hence, if also:

σt (x) = σ (et x)

I µ = G(σ )∂ µ φ, µ

µ

It = G(σt )∂ µ φt ,

It (x) = I µ (et x).

(1.91)


37

µ

Therefore ∂µ I µ = 0 implies ∂µ It = 0, that is φt is, for each t, also a solution of the nonlinear wave equation. It follows that dφt = Sφ (1.92) ψ= dt t =0 satisfies the linear wave equation (1.86). Here S is the differential operator: S = D − 1, D = x µ

∂ . ∂xµ

(1.93)

Finally we remark that analogous considerations apply to the general equations of motion (1.20). This leads to the following conclusion. If ( p, s, u) is a solution of the equations of motion (1.20) and V is any one of the vectorfields: Tµ : µ = 0, 1, 2, 3, µν : µ < ν = 0, 1, 2, 3,

D

(1.94)

then ( p, ˙ s˙ , u), ˙ defined by p˙ = V p, s˙ = V s, u˙ = LV u : for V = D, u˙ = L D u + u : for V = D, is a solution of the equations of variation (1.23). Here: LV u = [V, u] is the Lie derivative of the vectorfield u with respect to the vectorfield V .

(1.95)

Chapter 2

The Basic Geometric Construction Let us choose a time function t in Minkowski spacetime, equal to the coordinate x 0 of some rectangular coordinate system. We shall denote by t an arbitrary level set of the function t. The t are parallel space-like hyperplanes. The initial data for the equations of motion (1.20) is to be given on the hyperplane 0 and consists in specification of the triplet ( p, s, u). We assume that there is a sphere S0,0 in 0 outside which the initial data coincide with those of a constant state (see (1.42), (1.43)), that is we have: p = p0 s = s0 , u 0 = 1, u i = 0 : i = 1, 2, 3.

(2.1)

By suitable translation and scaling we can then take S0,0 to be the unit sphere centered at the origin in 0 . We consider an annular interior neighborhood of S0,0 in 0 : ε

00 = {x ∈ 0 : 1 − ε0 ≤ r (x) ≤ 1}

(2.2)

where r is the Euclidean distance function from the origin in 0 and ε0 is a positive constant, subject throughout this monograph only to the condition: ε0 ≤

1 . 2

(2.3)

We define in 0 the function: u = 1−r

(2.4)

which on 0 minus the origin is a smooth function without critical points, vanishing on S0,0 and increasing inward. For each value of u in the closed interval [0, ε0], the corresponding level set S0,u of u is a sphere of radius 1 − u in the interval [1 − ε0 , 1], and we have: ε 00 = S0,u (2.5) u∈[0,ε0 ]

40

Chapter 2. The Basic Geometric Construction

In the case that the initial data is irrotational and isentropic outside S0,ε0 , the data in the exterior of S0,ε0 in 0 specify the pair (φ, ∂0 φ) in this region. This is initial data for the nonlinear wave equation (1.57) in the region in question and we have: φ = 0, ∂0 φ = k : in the exterior of S0,0 in 0 .

(2.6)

To any given initial data set as above there corresponds a unique maximal solution of the equations of motion (1.20), of the nonlinear wave equation (1.57) in the irrotational isentropic case. The notion of maximal solution or maximal development of an initial data set is the following. Given an initial data set, the local existence theorem asserts the existence of a development of this set, namely of a domain D in Minkowski spacetime, whose past boundary is the domain of the initial data, and of a solution defined in D and taking the given data at the past boundary, such that if we consider any point p ∈ D and any curve issuing at p with the property that its tangent vector at any point q − belongs to I q , the closure of the past component of the open double cone defined by h q , the acoustical metric at q, then the curve terminates in the past at a point of the domain of the initial data. The local uniqueness theorem asserts that if (D1 , ( p1 , s1 , u 1 )) and (D2 , ( p2 , s2 , u 2 )) are two developments of the same initial data set ((D1 , φ1 ) and (D2 , φ2 ) in the irrotational isentropic case), then ( p1 , s1 , u 1 ) coincides with ( p2 , s2 , u 2 ) in D1 D2 (φ1 coincides with φ2 in D1 D2 in the irrotational isentropic case). It follows that the union of all developments of a given initial data set is itself a development, the unique maximal development of the initial data set. (See [Ch] for a discussion of the above notions in the general context of equations derivable from an action principle. The notion of maximal development in the context of general relativity was introduced by ChoquetBruhat and Geroch [C-G]). We consider, in the domain of the maximal solution, the family {Cu : u ∈ [0, ε0 ]} of outgoing characteristic hypersurfaces corresponding to the family {S0,u : u ∈ [0, ε0 ]}: Cu (2.7) 0 = S0,u : ∀u ∈ [0, ε0 ]. Each bicharacteristic generator of each Cu is to extend in the domain of the maximal solution as long at it remains on the boundary of the domain of dependence of the exterior of the surface S0,u in the initial hyperplane 0 . If we denote by Wε0 the spacetime domain: Cu (2.8) Wε0 = u∈[0,ε0 ]

then the domain Mε0 of the maximal solution corresponding to the given initial data set is the union of Wε0 with the domain in Minkowski spacetime bounded by the exterior of the unit sphere S0,0 in the initial hyperplane 0 and by the outgoing characteristic hypersurface C0 corresponding to S0,0 . By the domain of dependence theorem the solution coincides in Mε0 \ Wε0 with the constant state (1.79). This implies that C0 is a complete cone, each of its bicharacteristic generators extending to infinity in the parameter t. We extend the function u to Wε0 by requiring that its level sets are precisely the outgoing characteristic hypersurfaces Cu . The function u is then a solution of the equation: (h −1 )µν ∂µ u∂ν u = 0.

(2.9)


41

We shall call such a function an acoustical function. The vectorfield Lˆ given by: Lˆ µ = −(h −1 )µν ∂ν u

(2.10)

is then a future-directed null geodesic vectorfield with respect to the Lorentzian metric h. Its integral curves are the bicharacteristic generators of each Cu . Now the parametrization of these given by Lˆ is affine; however, for reasons which shall become apparent in the following we wish the generators to be parametrized by the function t instead. For this reason we choose to work with the collinear vectorfield L = µ Lˆ

(2.11)

where the proportionality factor µ, a positive function, is chosen so that

Thus

Lt = 1.

(2.12)

1 = −(h −1 )µν ∂µ t∂ν u. µ

(2.13)

The function µ plays a fundamental role in the present monograph. For each u ∈ [0, ε0 ] there is a greatest lower bound t∗ (u) of the extent of the generators of Cu , in the parameter t, in the domain of the maximal solution. This t∗ (u) is either a positive real number or ∞. According to the above we have t∗ (0) = ∞. Let us denote t∗ε0 =

inf t∗ (u).

(2.14)

u∈[0,ε0 ]

In the following we shall confine attention to the spacetime domain Wε∗0 consisting of all the points in Wε0 whose time coordinate t is less than t∗ε0 . For each (t, u) ∈ [0, t∗ε0 ) × [0, ε0 ] we define the closed surface: St,u = Cu (2.15) t . We have:

Wε∗0 =

St,u .

(2.16)

(t,u)∈[0,t∗ε0 )×[0,ε0 ]

We define in Wε∗0 the vectorfield T by the conditions that it be tangential to the hypersurfaces t , orthogonal with respect to the metric h to the family of surfaces {St,u : u ∈ [0, ε0 ]} in each t , and that it verify: T u = 1.

(2.17)

= [L, T ].

(2.18)

Consider the commutator: By (2.12), (2.17) and also the fact that by (2.9), (2.10), (2.11), Lu = 0

(2.19)

42


as well as the fact that, by virtue of the tangentiality of T to the t , Tt = 0

(2.20)

u = t = 0.

(2.21)

we have: Therefore the vectorfield is tangential to the surfaces St,u . From (2.9), (2.10), (2.11) we have: h(L, L) = 0

(2.22)

that is, L is a null vector with respect to the acoustical metric h. Moreover, from (2.10), (2.11), (2.17), ˆ T ) = −µT u = −µ. h(L, T ) = µh( L, (2.23) Now, the requirement (see Chapter 1) that the sound cone lie within the light cone implies that the hypersurfaces t are also space-like with respect to the acoustical metric h. Therefore T is a space-like vector with respect to h and we can write: h(T, T ) = κ 2

(2.24)

where κ is a positive function. Let now N be any vector tangent to one of the characteristic hypersurfaces Cu at a point: Nu = 0. By (2.10), (2.11), this is equivalent to: h(L, N) = 0. In particular any vector X tangent to one of the surfaces St,u at a point is h-orthogonal to L: (2.25) h(L, X) = 0 : ∀X ∈ T St,u . Since, by definition, X is also h-orthogonal to T : h(T, X) = 0 : ∀X ∈ T St,u

(2.26)

it follows that at each point p on each surface St,u , the plane p spanned by L and T at p, is h-orthogonal to T p St,u , the tangent space at p to St,u , and we have: T p Wε∗0 = p ⊕ T p St,u .

(2.27)

The metric h in p is given by (2.22), (2.23), (2.24) in terms of the functions µ and κ. By virtue of the h-orthogonal decomposition (2.27) the metric h is then completely specified once we give the metric h / induced on the surfaces St,u : /(X, Y ) = h(X, Y ) : ∀X, Y ∈ T p St,u . h

(2.28)


43

For each u ∈ [0, ε0 ] the generators of Cu define a smooth one-to-one mapping of S0,u onto St,u for each t ∈ [0, t∗ε0 ). Moreover, each S0,u is diffeomorphic to the standard sphere S 2 ⊂ 3 . Therefore, given for each u ∈ [0, ε0] a diffeomorphism ϕu of S 2 onto S0,u , we can assign to any point p on any surface St,u a point ϑ of S 2 , namely the pre-image of a point q on S0,u , when p lies along the generator of Cu issuing at q. If local coordinates (ϑ 1 , ϑ 2 ) are chosen on S 2 this assignment defines local coordinates on St,u for every (t, u) ∈ [0, t∗ε0 ) × [0, ε0]. In so far as the family of diffeomorphisms {ϕu : u ∈ [0, u 0 ]} is arbitrary, these coordinates are arbitrary as they can be subjected to transformations of the form: ˜ ϑ → ϑ˜ = ϑ(u, ϑ). (2.29) The local coordinates (ϑ 1 , ϑ 2 ), together with the functions (t, u) define a complete system of local coordinates (t, u, ϑ 1 , ϑ 2 ) for Wε∗0 . We shall call these acoustical coordinates and we shall derive an expression for the acoustical metric h in Wε∗0 in these coordinates. First, the integral curves of L are the lines of constant ϑ and u, parametrized by t. Therefore the vectorfield L is given in our coordinates simply by: L=

∂ . ∂t

(2.30)

Next, by (2.17) the vectorfield T has unit u-component and by (2.20) vanishing t-component. Therefore T is given by an expression of the form: T =

∂ − ∂u

(2.31)

where is a vectorfield which is tangential to the surfaces St,u . Thus can be expanded in terms of the coordinate frame field ∂ : A = 1, 2 (2.32) ∂ϑ A and we have:

∂ . ∂ϑ A By equations (2.30), (2.31) and (2.18) we have: = A

(2.33)

[L, ] = −

(2.34)

or, in terms of components, ∂ A (2.35) = − A . ∂t By an appropriate transformation of the form (2.29) we can set = 0 along any hypersurface H with the property that each integral curve of L intersects H at a single point. In particular, we can set = 0 along any one of the hypersurfaces t . However the non-vanishing of forbids setting = 0 everywhere.

44


With / AB h

∂ ∂ = /h , A ∂ϑ ∂ϑ B

: A, B = 1, 2

(2.36)

the components of the metric h / induced by h on the surfaces St,u in the coordinate frame (2.32), the metric h, according to the above, takes in our acoustical coordinate system the form: (2.37) h = −2µdtdu + κ 2 du 2 + h/ AB (dϑ A + A du)(dϑ B + B du). We define in Wε∗0 the vectorfield B by the conditions that it be orthogonal to the hypersurfaces t with respect to the metric h and that it verify: Bt = 1

(2.38)

The hypersurfaces t being space-like relative to h, the vectorfield B is future-directed time-like relative to h. Thus, there is a positive function α such that

In fact, we have: and:

h(B, B) = −α 2 .

(2.39)

h µν B ν = −α 2 ∂µ t

(2.40)

α −2 = −(h −1 )µν ∂µ t∂ν t.

(2.41)

In particular, in the class of rectangular coordinate systems with time coordinate x 0 fixed as above, we have simply: α −2 = −(h −1 )00 = 1 + (η−2 − 1)(u 0 )2 = 1 + F(β0 )2 in general (see (1.36)),

α −2 = 1 + F(∂0 φ)2

(2.42) (2.43)

in the irrotational isentropic case (see (1.65)). The function α is the lapse function of the foliation {t } relative to the metric h. That is, the integral t2 αdt t1

along an integral curve of B, is the arc length, with respect to h, of the segment of the curve between the hypersurfaces t1 and t2 . Thus α measures the normal separation of the leaves of the foliation {t }. Note that α < 1, so the lapse function relative to the acoustical metric h of the foliation {t } is less than that relative to the Minkowski metric g. Now B at any given point p belongs to the plane p , a time-like plane relative to the metric h. Therefore B is a linear combination of L and T . In fact, in view of the conditions (2.12), (2.20) and (2.38) we have: B = L+ fT

(2.44)


45

where the coefficient f must be positive since L is outgoing. Taking the h-inner product of equation (2.44) with T , yields, in view of (2.23), (2.24) and the fact that h(B, T ) = 0, µ = f κ 2.

(2.45)

On the other hand, substituting for L from (2.44) in (2.22) yields, in view of (2.39), (2.24) and the orthogonality of B and T relative to h, 0 = h(B, B) + f 2 h(T, T ) = −α 2 + f 2 κ 2 .

(2.46)

Thus, in view of the positivity of f we have: f =

α . κ

(2.47)

Substituting in (2.45) then yields the following relation between the metric coefficients κ and µ: µ = ακ. (2.48) Consider now the metric h induced by the acoustical metric h on the hypersurfaces t . From (1.37), (1.69), (1.72), it is given in rectangular coordinates (x 1 , x 2 , x 3 ), in general, by: h i j = gi j + σ H u i u j = gi j + Hβi β j (2.49) and in the irrotational isentropic case by: h i j = g i j + H ∂i φ∂ j φ

: i, j = 1, 2, 3.

(2.50)

Here g is the metric induced by the Minkowski metric g on the t , namely the Euclidean metric: g i j = δi j (2.51) in rectangular coordinates. Note that h dominates g. Let u be the g-orthogonal projection of the velocity vectorfield u to t . The components of u in rectangular coordinates are simply u i : i = 1, 2, 3. If at a point p on t we have u( p) = 0, then h coincides with g at p. Otherwise u( p) = 0 and h has eigenvalues equal to 1 in the plane in T p t which is g-orthogonal to u, and the eigenvalue: 1 + σ |u|2g H in the linear span of u. In the irrotational isentropic case the preceding considerations take the following form. Let us denote by d f the restriction to T t of d f , the differential of a function f defined on spacetime. If at a point p ∈ t we have dφ( p) = 0, then h coincides with g at p. Otherwise dφ( p) = 0 and h has eigenvalues equal to 1 in the plane tangent to the level set of φ in t through p, and the eigenvalue 1 + ρH

46


in the line which is g-orthogonal to this plane. Here ρ is the non-negative function: ρ = |dφ|2g = (g −1 )i j ∂i φ∂ j φ =

3

(∂i φ)2

(2.52)

i=1

(in rectangular coordinates). This is the expression for ρ in the irrotational isentropic case. In the general case we have: (2.53) σ = −(g −1 )µν βµ βν and we define the function ρ by: ρ = σ |u|2g = |β|2g = (g−1 )i j βi β j =

3

(βi )2 .

(2.54)

i=1

In the general case the reciprocal h (h

−1 i j

) = (g−1 )i j −

−1

of the induced metric is given by:

σH H u i u j = (g−1 )i j − βi β j 1 + ρH 1 + ρH

(2.55)

and in the isentropic irrotational case by: (h

−1 i j

) = (g−1 )i j −

H ∂ i φ∂ j φ 1 + ρH

: i, j = 1, 2, 3.

(2.56)

Let us introduce the vectorfield Tˆ , collinear and in the same sense as T and of unit magnitude with respect to the acoustical metric h: Tˆ = κ −1 T.

(2.57)

By (2.44) and (2.36) we can express L in the form: L = B − α Tˆ .

(2.58)

In terms of the rectangular coordinate system we have: B= where, by (2.40),

∂ + V, ∂x0

V = Vi

∂ ∂xi

V i = −α 2 (h −1 )i0 = α 2 (η−2 − 1)u i u 0

in general (see (1.36)). By (2.42), substituting for

η2

(2.59)

(2.60)

from (1.72):

η2 = 1 − σ H, we obtain, in view of the definition (2.55), α2 =

1−σH 1 + ρH

(2.61)


47

in general. Substituting from (2.61) in (2.60), we obtain the following expression for V i (rectangular coordinates): Vi =

Hσ Hβ0βi ui u0 = − 1 + ρH 1 + ρH

in general, Vi = −

H ∂0φ∂i φ 1 + ρH

(2.62)

(2.63)

in the irrotational isentropic case. Writing: ∂ Tˆ = Tˆ i i ∂x

(2.64)

we arrive, in view of (2.58), (2.59) and (2.62) at the following general expression for the vectorfield L in the rectangular coordinate system: 0u i H σ u ∂ Hβ0β i ∂ ∂ ∂ i i ˆ ˆ − αT − = 0 − αT + . (2.65) L= 1 + ρ H ∂xi 1 + ρ H ∂xi ∂x0 ∂x In the irrotational isentropic case this becomes: ∂ ∂ H ∂0φ∂ i φ i ˆ L= − αT + . 1 + ρH ∂xi ∂x0

(2.66)

Now, the vectorfield T can be expressed in terms of the metric h induced on t as (2.24): h i j T j = κ 2 ∂i u. (2.67) Hence, we have: κ −2 = (h

−1 i j

) ∂i u∂ j u.

Thus κ can be viewed as the lapse function relative to h of the foliation of ε t 0 = t Wε∗0

(2.68)

(2.69)

given by the family of surfaces {St,u : u ∈ [0, ε0 ]}. Let λ be the lapse function of the same foliation relative to the Euclidean metric g: λ−2 = (g −1 )i j ∂i u∂ j u =

3

(∂i u)2 .

(2.70)

i=1

The above-mentioned comparison of the metrics h and g then implies that:

λ ≤ κ ≤ λ 1 + ρH.

(2.71)

48


We may also introduce a vectorfield S analogous to T but defined relative to the Minkowski metric g rather than the acoustical metric h. Thus S is defined by the conditions that it be tangential to the hyperplanes t , orthogonal with respect to g to the family of surfaces {St,u : u ∈ [0, ε0 ]} in each t , and that it verify: Su = 1

(2.72)

S = T −Y

(2.73)

It follows that: where Y is a vectorfield which is tangential to the surfaces St,u . Moreover, we have: g(S, S) = λ2

(2.74)

and the vectorfield S can be expressed as (2.67): gi j S j = λ2 ∂i u.

(2.75)

We shall presently derive an expression for the ratio λ/µ. Let us denote: ν = β µ ∂µ u, β µ = (g −1 )µν βν

(2.76)

where β is the 1-form defined by (1.44). Since (g −1 )µν ∂µ t∂ν u = −∂0 u and ∂ µ φ∂ ν φ∂µ t∂ν u = −νβ0 substituting in (2.13) the expression (1.36) for (h −1 )µν we obtain: 1 = ∂0 u − Fνβ0 . µ On the other hand, since

(2.77)

β µ β ν ∂µ u∂ν u = ν 2

equation (2.9) reads: 0 = (g −1 )µν ∂µ u∂ν u − Fν 2 = −(∂0 u)2 + |du|2g − Fν 2 .

(2.78)

In view of the fact that according to (2.70) |du|2g = λ−2 we then obtain: (∂0 u)2 + Fν 2 =

1 . λ2

(2.79)

Let Sˆ be the vectorfield which is collinear to and in the same sense as S and of unit magnitude with respect to the Euclidean metric. Thus Sˆ is the interior unit normal to the surfaces St,u in a given hyperplane t with respect to the Euclidean metric. By (2.74), Sˆ = λ−1 S

(2.80)


and by (2.75)

49

∂i u = gi j λ−1 Sˆ j

(2.81)

hence the definition (2.76) can be written in the form: ν = −β0 ∂0 u + λ−1 β Sˆ .

(2.82)

Substituting this expression in (2.79) the latter takes the form of a quadratic equation for the quantity (2.83) x = λ∂0 u namely: ax 2 − 2bx + c = 0

(2.84)

where the coefficients are given by: a = 1 + F(β0 )2 b = F(β0 )(β Sˆ ) c = F(β Sˆ )2 − 1

ˆ = βµ Sˆ µ . where β Sˆ = β( S)

The discriminant is: δ = b2 − ac = 1 + F((β0 )2 − (β Sˆ )2 ). Denoting by β the restriction to T t of β and by β/ the restriction to T St,u of β, we have: |β|2g = (β Sˆ )2 + |β/|2g/ where g/ is the metric induced by g on St,u . Thus, in view of the fact that from (1.44): σ = −(g −1 )µν βµ βν = (β0 )2 − |β|2g as well as the fact that, from (1.68), 1 + σ F = η−2 , we arrive at the following expression for δ: δ = η−2 + F|β/|2g/ .

(2.85)

Now, by (2.77),

λ = λ(∂0 u − Fνβ0 ). µ Substituting from (2.82), (2.83), we obtain: λ = ax − b. µ

√ By (2.84) this is equal to: ± δ. The negative root is impossible since both λ, µ > 0. We conclude that: √ λ = δ (2.86) µ with δ given by (2.85).

50


We finally calculate the Jacobian of the mapping: (t, u, ϑ 1 , ϑ 2 ) → (x 0 , x 1 , x 2 , x 3 )

(2.87)

namely the transformation from acoustical to rectangular coordinates. Since x 0 = t we have: ∂x0 ∂x0 ∂x0 = 1, = = 0 : A = 1, 2. (2.88) ∂t ∂u ∂ϑ A Also, ∂xµ = Lµ, ∂t the rectangular components of the vectorfield L. Thus, ∂xi = L i : i = 1, 2, 3. ∂t

(2.89)

Next, applying (2.31) to the rectangular coordinates x i , i = 1, 2, 3, yields: ∂xi = T i + ξi ∂u

(2.90)

ξ i = A X iA

(2.91)

where and:

∂xi : i = 1, 2, 3; A = 1, 2. (2.92) ∂ϑ A From (2.88), (2.89), (2.90) and (2.92), the Jacobian determinant of the transformation (2.87) is given by: 1 0 0 0 1 1 L T + ξ1 X1 X1 1 2 = 2 2 2 2 2. L T + ξ X1 X2 L3 T 3 + ξ 3 X 3 X 3 1 2 X iA =

We have,

where:

1 T + ξ1 X1 X1 1 2 2 = T + ξ 2 X 12 X 22 = + T 3 + ξ3 X3 X3 1 2 1 1 1 T X X 1 2 = T 2 X 12 X 22 T 3 X3 X3 1 2

and

1 1 1 ξ X X 1 2 = ξ 2 X 12 X 22 . ξ3 X3 X3 1 2

Now, by virtue of (2.91), 1 1 1 X X X A 1 2 = A · X 2A X 12 X 22 = 0. X3 X3 X3 A=1 A 1 2 2


51

Thus reduces to , which we can conveniently write as: = (T, X 1 , X 2 )

(2.93)

the triple product of the vectors T , X 1 , X 2 in the 3-dimensional Euclidean space represented by each of the hyperplanes t . The vectors X 1 , X 2 at a point of t span the tangent plane to the surface St,u through that point. Since the difference T − S = Y lies in that plane (2.73) we can also write: = (S, X 1 , X 2 ).

(2.94)

Now, the outer product X 1 ∧ X 2 of the vectors X 1 and X 2 is orthogonal to the tangent plane with respect to the Euclidean metric, therefore collinear to S and we can arrange the orientation so that it is in the same sense as S. Consequently, we have simply: = |S||X 1 ∧ X 2 |

(2.95)

where | | = | |g denotes the magnitude with respect to the Euclidean metric g. Let us denote by g/ the metric induced on the surfaces St,u by the Euclidean metric. We have, g/ = g/ AB dϑ A dϑ B

(2.96)

g/ AB = (X A , X B ).

(2.97)

where: Here ( , ) = g( , ) denotes the Euclidean inner product. We then have:

|X 1 ∧ X 2 | = |X 1 |2 |X 2 |2 − (X 1 , X 2 )2 = detg/.

(2.98)

In view of (2.74) and (2.98) we arrive at the following expression for the Jacobian determinant:

(2.99) = λ detg/. Note that by the comparison of the metrics h and g discussed above, we have:

detg/ ≤ deth/ ≤ 1 + ρ H detg/.

(2.100)

In fact we can obtain an alternative expression for the Jacobian determinant , directly from expression (2.93). For, let e be the volume form of the induced acoustical metric h on t . Then, since the determinant of the matrix of components h i j in rectangular coordinates is: (2.101) deth = 1 + ρ H √ the acoustical volume form e is 1 + ρ H times the Euclidean volume form ( , , ). Therefore (2.93) is equivalent to: e(T, X 1 , X 2 ) = √ . 1 + ρH

(2.102)

52


On the other hand, since Tˆ is the unit normal to St,u relative to h, we have: e(Tˆ , X 1 , X 2 ) = /e(X 1 , X 2 ) (2.103) √ where /e is the area form of h /. Since also /e(X 1 , X 2 ) = deth/, while T = κ Tˆ , we obtain the following alternative expression for : √ κ deth/ . (2.104) = √ 1 + ρH A main theme of the present monograph is that, provided that the initial departure from the constant state is suitably small, the mapping (2.87) extends smoothly to the future boundary of the maximal development , however the lapse function λ vanishes there, therefore the differential of the inverse mapping √ becomes singular at this boundary. On the other hand, the St,u surface area element detg/ remains positive at the boundary of the maximal development. Moreover, the functions p, s, u µ , where u µ are the rectangular components of the velocity vectorfield, or s, βµ , where βµ are the rectangular components of the 1-form β defined by (1.44), or, in the irrotational isentropic case ψµ = ∂µ φ, namely the 1st partial derivatives of the wave function with respect to the rectangular coordinates, extend smoothly to the boundary as functions of the acoustical coordinates (t, u, ϑ 1 , ϑ 2 ). However, since the transformation to the rectangular coordinates √ (x 0 , x 1 , x 2 , x 3 ) becomes singular at the future boundary, the partial derivatives of σ and the u µ , or of the βµ , of order greater than or equal to 1, or, in the irrotational isentropic case, the partial derivatives of the wave function with respect to the rectangular coordinates of order greater than or equal to 2, blow up at this boundary. The prospect of the vanishing of λ, or equivalently of κ or µ, at the future boundary of the maximal development is thus to guide all considerations which follow.

Chapter 3

The Acoustical Structure Equations In this chapter we shall formulate the basic equations which govern the geometry of the 2-parameter foliation of the spacetime manifold Wε∗0 , endowed with the acoustical metric h, given by the surfaces {St,u : t ∈ [0, t∗ε0 ), u ∈ [0, ε0 ]}. We begin with the geometry of the 1-parameter foliation of (Wε∗0 , h) given by the hypersurfaces {t : t ∈ [0, t∗ε0 )}. In the previous chapter we introduced the normal vectorfield B generating a 1-parameter group mapping these hypersurfaces onto each other, as well as the lapse function α and the induced metric h. We now define k, the second fundamental form of the t (relative always to h) by: 2αk = L B h.

(3.1)

Here L B h denotes the restriction to T t of the Lie derivative of h with respect to B. (We can say that L B h is the Lie derivative with respect to B of the induced metric h). If X, Y are any two vectors tangent to t at a point we then have: αk(X, Y ) = h(D X B, Y ) = h(DY B, X)

(3.2)

where D is the covariant derivative operator associated to h. In terms of the rectangular coordinate system we have (see (2.59)): B=

∂ + V, ∂x0

where, by (2.62), Vi = −

V = Vi

Hβ0βi 1 + ρH

∂ ∂xi

(3.3)

(3.4)

with ρ the function defined by (2.54): ρ = βi β i =

3 (βi )2 . i=1

(3.5)

54

Chapter 3. The Acoustical Structure Equations

In the irrotational isentropic case we have, by (2.63), Vi = −

H ψ0 ψi 1 + ρH

and ρ is given by (2.52): ρ = ψi ψ i =

(3.6)

3 (ψi )2

(3.7)

i=1

(rectangular coordinates). Here and in the following we denote by ψµ , µ = 0, 1, 2, 3, the partial derivatives of the wave function φ with respect to the rectangular coordinates: ψµ = ∂µ φ.

(3.8)

The components ki j , i, j = 1, 2, 3, of k in the rectangular coordinate system are given, in general, according to (3.1) by: 2αki j =

∂h i j + (LV h)i j = Bh i j + h im ∂ j V m + h m j ∂i V m ∂x0

(3.9)

where h i j are the components of the induced metric on t in rectangular coordinates, given in general by (2.49): h i j = δi j + Hβi β j (3.10) and in the isentropic irrotational case by (2.50): h i j = δi j + H ψi ψ j .

(3.11)

We shall now derive a more detailed expression for ki j in the irrotational isentropic case which will play an important role in the sequel. Differentiating (3.6) yields: dH 1 H2 ψ0 ψm ∂i σ + ψ0 ψm ∂i ρ 2 (1 + ρ H ) dσ (1 + ρ H )2 H (ψm ∂i ψ0 + ψ0 ∂i ψm ). − (1 + ρ H )

∂i V m = −

(3.12)

Now, h j m ψ m = (δ j m + H ψ j ψm )ψ m = (1 + ρ H )ψ j and: h j m ∂i ψ m = (δ j m + H ψ j ψm )∂i ψ m = ∂i ψ j +

1 H ψ j ∂i ρ 2

hence: h j m ∂i V

m

1 ψ0 =− (1 + ρ H )

1 dH ψ j ∂i σ − H 2ψ j ∂i ρ + H ∂i ψ j dσ 2

− H ψ j ∂i ψ0 . (3.13)


55

On the other hand we have: Bh i j = B(H ψi ψ j ) (3.14) dH ψi ψ j Bσ + H (ψ j ∂0 ψi + ψi ∂0 ψ j + V m (ψ j ∂m ψi + ψi ∂m ψ j )). = dσ Substituting from (3.6) and taking into account the fact that: ∂µ ψν = ∂ν ψµ = ∂µ ∂ν φ

(3.15)

we obtain, in view of (3.7), V m (ψ j ∂m ψi + ψi ∂m ψ j ) = −

1 H ψ0 (ψ j ∂i ρ + ψi ∂ j ρ). 2 (1 + ρ H )

Substituting in (3.14) then yields: dH ψi ψ j Bσ + H (ψ j ∂0 ψi + ψi ∂0 ψ j ) dσ 1 H 2ψ0 (ψ j ∂i ρ + ψi ∂ j ρ). − 2 (1 + ρ H )

Bh i j =

(3.16)

Substituting finally from (3.16) and (3.13) in (3.9) and taking into account the symmetry (3.15) we obtain the following formula for ki j in the irrotational isentropic case (rectangular coordinates): dH ψ0 (ψi ∂ j σ + ψ j ∂i σ ) 2αki j = ψi ψ j Bσ − dσ (1 + ρ H ) 2H ψ0 ∂i ψ j . − (3.17) (1 + ρ H ) We now return to the general case. The components R i j mn of the curvature tensor of the induced metric in rectangular coordinates can be calculated directly from (3.11). They satisfy the Gauss equations: R i j mn + kim k j n − k j m kin = Ri j mn

(3.18)

where Ri j mn are the corresponding components of the curvature tensor of (Wε∗0 , h) (along t ). Since t is a 3-dimensional manifold the R i j mn can be expressed in terms of the components S i j of the corresponding Ricci tensor: S i j = (h

−1 mn

)

R im j n .

(3.19)

We have, R i j mn = h im S j n + h j n S im − h j m S in − h in S j m − (1/2)(h im h j n − h j m h in )S

(3.20)

56


where S is the scalar curvature of (t , h): S = (h

−1 i j

) Si j .

(3.21)

We turn to the geometry of a given characteristic hypersurface Cu , as described by the 1-parameter foliation corresponding to the family of sections {St,u : t ∈ [0, t∗ε0 )} by the hyperplanes t . The induced metric h/ on St,u has already been introduced in Chapter 2, as has the null (relative to h) vectorfield L whose integral curves are the bicharacteristic generators of Cu , null geodesics of (Wε∗0 , h). Any vector X tangent to Cu at a point p can be uniquely decomposed into a vector collinear to L and a vector tangent to the St,u section through p which we denote by X: X = c X L + X.

(3.22)

Thus we have at each p ∈ Cu a projection of T p Cu onto T p St,u . If X, Y is a pair of vectors tangent to Cu at p, then: h(X, Y ) = h/(X, Y ). Let XA =

∂ ∂ϑ A

A = 1, 2

(3.23)

(3.24)

be the local vectorfields corresponding to the local coordinates (ϑ 1 , ϑ 2 ) introduced in the previous chapter. Given any Z ∈ T p Cu we can expand: Z = Z A X A .

(3.25)

Taking the h /-inner product with X B then yields: /(X B , Z ) = h/ AB Z A . h

(3.26)

The vectorfield L is normal in Cu to the family of sections of Cu and generates a 1-parameter group mapping these sections onto each other. We now define χ, the second fundamental form of St,u relative to Cu by: 2χ = L / L h.

(3.27)

Here L / L h denotes the restriction to T St,u of the Lie derivative of h with respect to L. (We can say that L / L h is the Lie derivative with respect to L of the induced metric h/.) If X, Y are any two vectors tangent to St,u at a point, we then have: χ(X, Y ) = h(D X L, Y ) = h(DY L, X).

(3.28)

Now, since the vectorfield Lˆ = µ−1 L is geodesic (see Chapter 2, equations (2.9)– (2.13)) , we have: D L L = µ−1 (Lµ)L. (3.29)


57

Denoting, χ AB = χ(X A , X B )

(3.30)

let us derive a propagation equation for χ AB along the generators of Cu . In view of formula (3.28) we have: Lχ AB = h(D L D X A L, X B ) + h(D X A L, D L X B ).

(3.31)

D L X A − D X A L = [L, X A ] = 0.

(3.32)

Now, Thus, by the definition of the curvature transformation of h, D L D X A L − D X A D L L = R(L, X A )L.

(3.33)

Substituting for D L L from (3.29) and taking the h-inner product with X B we obtain, in view of (3.28), (3.34) h(D L D X A L, X B ) = µ−1 (Lµ)χ AB − α AB where α AB are the curvature tensor components: α AB = R(X A , L, X B , L).

(3.35)

Recall that the curvature tensor is obtained from the curvature transformation as follows. If X, Y, Z , W are any four vectors at a point, R(W, Z , X, Y ) = h(W, R(X, Y )Z ). We turn to the second term on the right in (3.31). By (3.33) this term is h(D X A L, D X B L). Now, the fact that L is a null vectorfield with respect to the metric h, implies that for any vectorfield X the vectorfield D X L is h-orthogonal to L, therefore tangential to Cu . Thus, setting (3.36) W A = D X A L : A = 1, 2 equation (3.23) applies: h(W A , W B ) = h/(W A , W B ). By (3.26), we obtain, in view of (3.28), W A = χ AB X B ,

χ AB = (h/−1 ) BC χ AC

(3.37)

hence: /(W A , W B ) = χ AC χ BC . h This is the second term on the right in (3.31). We thus conclude that χ AB satisfies the following propagation equation along the generators of Cu : Lχ AB = µ−1 (Lµ)χ AB + χ AC χ BC − α AB .

(3.38)

58


We shall now deduce from (3.38) a propagation equation for trχ. Taking the trace of the curvature tensor we obtain the Ricci tensor: Sµν = (h −1 )κλ Rκµλν .

(3.39)

The reciprocal acoustical metric h −1 is expressed in terms of the frame L, T, X 1 , X 2 by: µ

(h −1 )µν = −α −2 L µ L ν − µ−1 (L µ T ν + T µ L ν ) + (h/−1 ) AB X A X νB .

(3.40)

This readily follows from the expressions for the inner products of the frame vectorfields with respect to h given in Chapter 2. In view of the expression (3.40) and of the symmetries of the curvature tensor, we have: µ

trα = (h /−1 ) AB Rµκνλ X A L κ X νB L λ = (h −1 )µν Rµκνλ L κ L λ = Sκλ L κ L λ or, simply, trα = S(L, L).

(3.41)

Taking account of the fact that Ltrχ = L((h /−1 ) AB χ AB ) = (h/−1 ) AB Lχ AB + χ AB L(h/−1 ) AB −1 AB = (h / ) Lχ AB − χ AB (h/−1 ) AC (h/−1 ) B D Lh/C D = (h /−1 ) AB Lχ AB − 2χ AB (h/−1 ) AC (h/−1 ) B D χC D when taking the trace of (3.38), we obtain the desired propagation equation for trχ: Ltrχ = µ−1 (Lµ)trχ − |χ|2h/ − S(L, L).

(3.42)

In terms of the acoustical coordinates (t, u, ϑ 1 , ϑ 2 ) L is given by (2.30), and the definition (3.27) becomes simply: 2χ AB =

∂h/ AB . ∂t

(3.43)

We shall now write down the Gauss and Codazzi equations of the embedding of St,u in the acoustical spacetime, that is, in the spacetime manifold endowed with the acoustical metric h. First, we consider the surface St,u as a member of the 1-parameter family {St,u : u ∈ [0, ε0 ]} defining a local foliation of the hypersurface t . The normal vectorfield which generates the 1-parameter group mapping the members of this family onto each other is the vectorfield T , introduced in the previous chapter, and the corresponding lapse function is the function κ. The second fundamental form θ of St,u relative to (t , h) is defined by: (3.44) 2κθ = L /T h where L /T h denotes the restriction to T St,u of the Lie derivative with respect to T of h. Then, if X, Y are any two vectors tangent to St,u at a point, κθ (X, Y ) = h(D X T, Y ) = h(D Y T, X).

(3.45)


59

Let us denote by k/ the restriction of k, the second fundamental form of t , to T St,u . Then, since by (2.44) and (2.47), L = α(α −1 B − κ −1 T ), we have: χ = α(k/ − θ ).

(3.46)

This allows us to express θ in terms of χ and k/. The Gauss equation of the embedding of St,u in t is expressed in terms of the local frame (X 1 , X 2 ) by: R / ABC D − θ AC θ B D + θ BC θ AD = R ABC D .

(3.47)

Here R / ABC D are the components of the curvature tensor of the induced metric h/. Since / ABC D in terms of the Gauss curvature K and the St,u is 2-dimensional we can express R components of the induced metric: R / ABC D = K (h/ AC h/ B D − h/ BC h/ AD ).

(3.48)

j

Multiplying equations (3.18) with X iA X B X Cm X nD (and summing repeated upper and lower indices) yields: R ABC D + k/ AC k/ B D − k/ BC k/ AD = R ABC D . (3.49) Substituting for R ABC D in terms of R ABC D from (3.49) in (3.47) we obtain: K (h / AC h /B D − h / BC h / AD ) + k/ AC k/ B D − k/ BC k/ AD − θ AC θ B D + θ BC θ AD = R ABC D . (3.50) In view of the symmetries of the curvature tensor we can express: R ABC D = ρ AB C D where AB are the components of the area form of (St,u , h/):

AB = deth/[ AB].

(3.51)

(3.52)

Here [AB] is the fully antisymmetric 2-dimensional symbol. Contracting (3.50) with /−1 ) B D and taking into account the fact that (1/2)(h /−1 ) AC (h (h /−1 ) B D AB C D = h/ AC we obtain:

1 (trh/ k/)2 − |k/|2h/ − (trh/ θ )2 + |θ |2h/ = ρ. (3.53) 2 This is the Gauss equation of the embedding of St,u in the acoustical spacetime. It contains the whole content of (3.50), in view of the symmetries involved. We shall presently derive the Codazzi equations of the embedding of St,u in the acoustical spacetime. Let X, Y, Z be arbitrary vectorfields tangent to a surface St,u . Extend them to the corresponding hypersurface Cu by the condition that they commute with K+

60


L. They are then tangential to each of the sections St,u , t ∈ [0, ε0 ), and by formula (3.28) we have: X (χ(Y, Z )) = D X (h(DY L, Z )) = h(D X DY L, Z ) + h(DY L, D X Z ). On the other hand, by the definition of the operator D /, (D / X χ)(Y, Z ) = X (χ(Y, Z )) − χ(D / X Y, Z ) − χ(Y, D / X Z ). Hence, / X Y, Z ) − χ(Y, D / X Z ). (D / X χ)(Y, Z ) = h(D X DY L, Z ) + h(DY L, D X Z ) − χ(D Subtracting a similar formula with X and Y interchanged and noting that: D X DY L − DY D X L = R(X, Y )L + D[X,Y ] L,

h(R(X, Y )L, Z ) = R(Z , L, X, Y )

and: D /XY − D / Y X = [X, Y ] we obtain: (D / X χ)(Y, Z ) − (D / Y χ)(X, Z ) = R(Z , L, X, Y ) +h(DY L, D X Z ) − h(D X L, DY Z ) − χ(Y, D / X Z ) + χ(X, D / Y Z ).

(3.54)

Let us define the 1-form ζ on St,u by: ζ (X) = h(D X L, T )

(3.55)

for any vector X tangent to St,u at a point. We shall find an expression for ζ below. For now we note that from (3.36), (3.37), and (2.23), we have: D X L = −µ−1 ζ(X)L + χ · X

(3.56)

for any vector X tangent to St,u at a point. Here χ · X is the vector tangent to St,u defined by: /(χ · X, Y ) = χ(X, Y ) h for any vector Y tangent to St,u at the same point. Substituting in the case of the St,u tangential vectorfield X the expression (3.56) for D X L we deduce that: /Y Z ) h(D X L, DY Z ) = µ−1 ζ(X)χ(Y, Z ) + χ(X, D

(3.57)

because −h(L, DY Z ) = h(DY L, Z ) = χ(Y, Z ). Substituting (3.57) and a similar expression with X and Y interchanged in (3.54) we arrive at the equation: (D / X χ)(Y, Z ) − (D / Y χ)(X, Z ) = R(Z , L, X, Y ) − µ−1 (ζ(X)χ(Y, Z ) − ζ(Y )χ(X, Z )). (3.58)


61

Let us set X = X A , Y = X B , Z = X C in this equation. Noting that by virtue of the symmetries of the curvature tensor we can express: R(X C , L, X A , X B ) = AB βC

(3.59)

we obtain the Codazzi equations: D / A χ BC − D / B χ AC = AB βC − µ−1 (ζ A χ BC − ζ B χ AC ).

(3.60)

Denoting by curl / χ the 1-form on St,u with components: curl / χC = where

1 AB / A χ BC − D / B χ AC ) (D 2

/−1 ) AC (h/−1 ) B D C D = ( deth/)−1 [ AB] AB = (h

the whole content of the Codazzi equations is, in view of the symmetries involved, contained in the equation: (3.61) curl / χ = β − µ−1 ζ ∧ χ. Here, we denote by ζ ∧ χ the 1-form on St,u with components: (ζ ∧ χ)C =

1 AB (ζ A χ BC − ζ B χ AC ). 2

In particular, equation (3.61) implies, is in fact equivalent to, the contraction of equations (3.60) with (h /−1 ) AC , namely the equation: div / χ − d/trχ = β ∗ − µ−1 (ζ · χ − ζ trχ).

(3.62)

Here d/trχ is the differential of the function trχ on St,u , ζ · χ is the 1-form on St,u with components: (ζ · χ) B = (h/−1 ) AC ζ A χ BC and β ∗ is the dual of β:

β B∗ = (h/−1 ) AC βC AB .

We shall now derive an expression for the 1-form ζ on St,u defined by (3.55). Substituting in this definition the expression for L just preceding (3.46), we obtain, for any vector X tangent to St,u at a point, ζ (X) = h(D X B, T ) − ακ −1 h(D X T, T ) − h(T, T )X (ακ −1 ). Recalling that h(T, T ) = κ 2 we have h(D X T, T ) = κ Xκ, hence this reduces to: ζ (X) = h(D X B, T ) − κ Xα. In view of the formula (3.2) we have: h(D X B, T ) = αk(X, T ).

62


Let us define the 1-form on St,u by: κ(X) = k(X, T )

(3.63)

for any vector X tangent to St,u at a point. The above formula for ζ can then be expressed in the form: ζ = κ(α − d/α). (3.64) We now define on each St,u section of Cu the 1-form: η = ζ + d/µ.

(3.65)

Since h(L, T ) = −µ and ζ is defined by (3.55) we have: η(X) = −h(D X T, L)

(3.66)

for every vector X tangent to St,u at a point. We denote: η A = η(X A ).

(3.67)

We can express the commutator of the vectorfields L and T , a vectorfield tangential to the surfaces St,u (see Chapter 2), in terms of the 1-forms ζ and η. We have: = A X A, Ah / AB = h(X B , D L T ) − h(X B , DT L) and, h(X B , D L T ) = −h(D L X B , T ) = −h(D X B L, T ) = −ζ B .

(3.68)

ˆ which, To compute h(X B , DT L) we express L in terms of the geodesic vectorfield L, since it is the gradient of a function (see equation (2.10)), satisfies: ˆ = h(Y, D X L) ˆ h(X, DY L)

(3.69)

for arbitrary vectors X, Y tangent to the spacetime manifold at a point, by virtue of the symmetry of the Hessian of any function. Thus, we have: ˆ = µh(X B , DT L) ˆ h(X B , DT L) = h(X B , DT (µ L)) −1 ˆ = µh(T, D X B L) = µh(T, D X B (µ L)) = h(T, D X B L) − µ−1 (X B µ)h(T, L) = ζB + X B µ = ηB .

(3.70)

Substituting (3.68) and (3.70) above yields: A = −(h/−1 ) AB (ζ B + η B ).

(3.71)

Equation (3.70) together with the facts that h(T, DT L) = h(T, D L T ) = L

1 2 κ 2


63

and h(L, DT L) = 0 (as L is null with respect to h), implies that: DT L = (h /−1 ) AB η B X A − α −1 (Lκ)L

(3.72)

(see (2.48)). Moreover, since D L T − DT L = is given by (3.71) we also obtain: /−1 ) AB ζ B X A − α −1 (Lκ)L. D L T = −(h

(3.73)

Next, let us compute D X A T . Expanding in the frame L, T, X 1 , X 2 , we can write: D X A T = a A L + b A T + c AB X B . Taking the h-inner product with L gives: −µb A = h(D X A T, L) = −η A by (3.66). Taking the h-inner product with T gives: −µa A + κ 2 b A = h(D X A T, T ) = X A

1 2 κ . 2

Finally, taking the h-inner product with X C gives, in view of (3.45), c AB h / BC = h(D X A T, X C ) = h(D X A T, X C ) = κθ AC . The above imply: D X A T = α −1 κ A L + µ−1 η A T + κθ AB (h/−1 ) BC X C

(3.74)

where we have used the fact that by (3.64) and (3.65) α −1 η − d/κ = κ. Next, we compute DT T . Since T is tangential to the hypersurfaces t we may decompose: DT T = D T T + a B. Taking the h-inner product with B we obtain, substituting from (2.44), (2.47), −α 2 a = h(DT T, B) = h(DT T, L) + ακ −1 h(DT T, T ) 1 2 −1 κ = −T µ − h(T, DT L) + ακ T 2 = −T µ − κ Lκ + αT κ = −κ(T α + Lκ)

(3.75)

64


where we have substituted from (3.72) and (2.48). From (2.67), (D T T )i = T j D j T i = κ 2 (h 2

−1 j k

) ∂k u D j {κ 2(h

−1 j k

−1 il

) ∂l u}

−1 il

−1

) ∂l u∂ j (κ 2 ) + κ 4 (h ) j k ∂k u(h 1 −1 −1 = T i T j κ −2 ∂ j (κ 2 ) + κ 4 (h )il ∂l {(h ) j k ∂ j u∂k u}. 2

= κ (h

) ∂k u(h

−1 il

) D l (∂ j u)

From (2.68) the last expression in parentheses is κ −2 , hence we obtain: 1 −1 (D T T )i = κ −2 T i T j ∂ j (κ 2 ) − (h )i j ∂ j (κ 2 ). 2 We conclude that: DT T =

1 −2 1 κ T (κ 2 )T − (h/−1 ) AB X B (κ 2 )X A . 2 2

(3.76)

Combining with the former result (3.75) for the component along B yields, in view of (2.44), (2.47), DT T = κα −2 (T α + Lκ)L +(κ −1 T κ + α −1 Lκ + α −1 T α)T 1 − (h/−1 ) AB X B (κ 2 )X A . 2

(3.77)

Note that since by (2.39) and (3.2), h(DT T, B) = −h(T, DT B) = −αk(T, T ). (3.75) yields at the same time the following propagation law for κ along the generators of Cu : 1 2 κ = −κ T α + αk(T, T ). (3.78) L 2 In the present monograph we have chosen µ instead of κ to describe the stacking of the hypersurfaces Cu . The two quantities are of course equivalent, being related by equation (2.48): µ = ακ so we could equally well have chosen to work with κ. By (3.78) we obtain the following propagation equation for µ: Lµ = κ Lα − αT α + αµk(Tˆ , Tˆ ). Here Tˆ is the unit vectorfield: (see (2.57)).

Tˆ = κ −1 T

(3.79)


65

We shall now derive an explicit form of the propagation equation for µ in the irrotational isentropic case, using the expression (3.17) for ki j in this case and the expression (2.61) for α (which is valid in general). Expanding B as in (2.44), we have, in view of (2.47), B = L + ακ −1 T.

(3.80)

Bσ = Lσ + ακ −1 T σ.

(3.81)

Thus,

Substituting in (3.17) yields: 2αψ0 ψTˆ 1 dH 2 2 ˆ ˆ αµk(T , T ) = α (ψTˆ ) − (T σ ) 2 dσ (1 + ρ H ) α H ψ0 ˆ i 1 dH (ψ ˆ )2 (Lσ ) − T (T ψi ) + µ 2 dσ T (1 + ρ H )

(3.82)

where: ψTˆ = Tˆ i ψi .

(3.83)

To calculate the terms in equation (3.79) involving derivatives of α, we deduce from (2.61) a formula for dα, the differential of α. Noting that σ + ρ = (ψ0 )2 , we obtain: −d(α 2 ) =

H (ψ0 )2 d H dσ + (dσ + α 2 dρ). 2 (1 + ρ H ) dσ (1 + ρ H )

(3.84)

Again, using the formulas σ = (ψ0 )2 − ρ, we obtain:

1 − α2 =

(ψ0 )2 H , 1 + ρH

ρ = ψ i ψi ,

H ψ0 ψ i dσ + α 2 dρ = 2ψ0 dψ0 − dψi . 1 + ρH

(3.85)

Substituting in (3.84) then yields: −αdα =

1 (ψ0 )2 d H dσ 2 (1 + ρ H )2 dσ H ψ0 H ψ0ψ i dψi . + dψ0 − (1 + ρ H ) 1 + ρH

(3.86)

66


It follows that: κ Lα − αT α =

1 (ψ0 )2 d H (T σ ) 2 (1 + ρ H )2 dσ 1 µ (ψ0 )2 d H − 2 (Lσ ) 2 α (1 + ρ H )2 dσ H ψ0 H ψ0ψ i + T ψi T ψ0 − (1 + ρ H ) 1 + ρH µ H ψ0 H ψ0ψ i Lψi . − 2 Lψ0 − α (1 + ρ H ) 1 + ρH

(3.87)

Substituting (3.87) and (3.82) in (3.79), the latter takes the form: 2 dH ψ0 − αψTˆ (T σ ) 1 + ρH dσ 2 1 µ ψ0 2 dH (Lσ ) − 2 − (αψTˆ ) 2α 1 + ρH dσ H ψ0 H ψ0 ψ i + T ψ0 − + α Tˆ i T ψi (1 + ρ H ) 1 + ρH H ψ0 ψ i µ H ψ0 Lψ0 − Lψi . − 2 1 + ρH α (1 + ρ H )

1 Lµ = 2

Recalling expression (2.66) for the vectorfield L, namely: ∂ H ψ0 ψ i ∂ i ˆ + αT L= − ∂x0 1 + ρH ∂xi and defining:

(3.89)

ψ L = L µ ψµ

we have: ψL =

(3.88)

(3.90)

ψ0 − αψTˆ . 1 + ρH

(3.91)

This is the factor in parentheses in the first term on the right-hand side in (3.88). By (3.91) the factor in parentheses in the second term is:

ψ0 1 + ρH

2 − (αψTˆ )2 = ψ L (ψ L + 2αψTˆ ).

(3.92)

Again by (3.89) the coefficient of the factor H ψ0 /(1 + ρ H ) in the third term on the right-hand side of (3.88) is: L µ (T ψµ ) = L µ T ν ∂ν ψµ = L µ T ν ∂µ ψν = T i (Lψi ) =

µ î T (Lψi ) α

(3.93)


67

using (3.15). Thus, again in view of (3.89), this coefficient combines with the coefficient of the same factor in the fourth term to: −

µ µ L (Lψµ ). α2

(3.94)

Since also, by (1.71), (2.61), H 1 =F (1 + ρ H ) α 2

(3.95)

it follows that the sum of the last two terms on the right-hand side of (3.88) is equal to: −µFψ0 L µ (Lψµ ) We conclude that (3.88) reduces to: Lµ = m + µe where: m=

dH 1 (ψ L )2 (T σ ) 2 dσ

(3.96)

(3.97)

and:

dH 1 (Lσ ) − Fψ0 L µ (Lψµ ). ψ L (ψ L + 2αψTˆ ) (3.98) 2α 2 dσ Equation (3.96) is the desired propagation equation for µ. We can then express Lκ in terms of Lµ to obtain: (3.99) Lκ = m + κe e=−

where:

m = α −1 m,

e = e − α −1 Lα.

(3.100)

Using (3.86) evaluated on L, (3.91), (3.95), as well as the expression (3.89), we find: e =

dH 1 (ψTˆ )2 (Lσ ) + Fψ0 Tˆ i (Lψi ). 2 dσ

(3.101)

The propagation equations (3.96), (3.99) play a fundamental role in this monograph. We now return again to the general case. To complete the set of connection coefficients of the frame L, T, X 1 , X 2 , we compute D X A X B . We decompose: DX A X B = D / X A X B + a AB L + b AB T. Taking the h-inner product with L gives: −µb AB = h(L, D X A X B ) = −h(D X A L, X B ) = −χ AB . Taking the h-inner product with T gives: −µa AB + κ 2 b AB = h(T, D X A X B ) = −h(D X A T, X B ) = −κθ AB .

68


Hence, in view of (3.46), we obtain: b AB = µ−1 χ AB ,

a AB = α −1 k/ AB

We collect the expressions which we have obtained for the connection coefficients of the frame L, T, X 1 , X 2 in the following table: D L L = µ−1 (Lµ)L DT L = −α −1 (Lκ)L + η A X A D X A L = −µ−1 ζ A L + χ AB X B D L T = −α −1 (Lκ)L − ζ A X A DT T = κα −2 (T α + Lκ)L + (µ−1 T µ + α −1 Lκ)T 1 −1 AB / ) X B (κ 2 )X A − (h 2 D X A T = α −1 κ A L + µ−1 η A T + κθ AB X B DL X A = DX A L / X A X B + α −1 k/ AB L + µ−1 χ AB T. DX A X B = D

(3.102)

Here, /−1 ) AB ζ B , ζ A = (h

η A = (h /−1 ) AB η B , χ AB = (h/−1 ) BC χ AC ,

θ AB = (h/−1 ) BC θ AC .

Note that we have not obtained an expression for DT X A . This reflects the arbitrariness in the coordinates ϑ 1 , ϑ 2 . It shall not play any role in the following. We now investigate the connection between the derivative of χ relative to T and the derivative of η tangentially to the surfaces St,u . The proper notion of derivative of χ relative to T is that of the Lie derivative. However, as χ is defined on each surface St,u , we first extend χ to a symmetric 2-covariant tensorfield on spacetime by the conditions: χ(X, T ) = χ(X, L) = 0 for any vectorfield X. We then define L /T χ to be the restriction to St,u of LT χ, the Lie derivative with respect to T of χ thus extended. We then have: (L /T χ)(X A , X B ) = (DT χ)(X A , X B ) + χ(X A , D X B T ) + χ(X B , D X A T ) = (DT χ)(X A , X B ) + κθ BC χ AC + κθ AC χ BC

(3.103)

where we have substituted from table (3.102). To compute (DT χ)(X A , X B ) we write: (DT χ)(X A , X B ) = T (χ AB ) − χ(DT X A , X B ) − χ(X A , DT X B ).

(3.104)

Here and in the following we denote by the h-orthogonal projection to the surfaces St,u . Now, T (χ AB ) = T (h(D X A L, X B )) = h(DT D X A L, X B ) + h(D X A L, DT X B ).

(3.105)


69

We have, DT D X A L − D X A DT L − D[T ,X A ] L = R(T, X A )L hence: h(DT D X A L, X B ) = h(D X A DT L, X B ) +h(D[T ,X A ] L, X B ) + R(X B , L, T, X A ).

(3.106)

Substituting for DT L from table (3.102) we obtain: /XA X B) h(D X A DT L, X B ) = −α −1 (Lκ)χ AB + X A (η B ) − η(D / AηB = −α −1 (Lκ)χ AB + D

(3.107)

where we denote: (D / X A η)(X B ) = D / AηB . From Chapter 2 (see (2.31)) the commutator [T, X A ] is tangential to the surfaces St,u . Therefore, h(D[T ,X A ] L, X B ) = χ([T, X A ], X B ) = χ(DT X A , X B ) − χ(D X A T, X B ) (3.108) = χ(DT X A , X B ) − κθ AC χ BC where we have substituted for D X A T from Table 3.102. Also, substituting for D X A L from Table 3.102 yields: h(D X A L, DT X B ) = −µ−1 ζ A h(L, DT X B ) + χ(X A , DT X B ) = = µ−1 ζ A η B + χ(X A , DT X B )

(3.109)

in view of the fact that −h(L, DT X B ) = h(DT L, X B ) = η B . Substituting (3.107) and (3.108) in (3.106), and the result together with (3.109) in (3.105), and the result in turn in (3.104), we obtain: / A η B + µ−1 ζ A η B (3.110) (DT χ)(X A , X B ) = D −1 C −α (Lκ)χ AB − κθ AC χ B − R(X A , T, X B , L) The symmetric part of this equation is: (DT χ)(X A , X B ) =

1 1 / B η A ) + µ−1 (ζ A η B + ζ B η A ) (D / AηB + D 2 2 1 −α −1 (Lκ)χ AB − κ(θ AC χ BC + θ BC χ AC ) 2 1 − (R(X A , T, X B , L) + R(X B , T, X A , L)). 2

(3.111)

The antisymmetric part is an equation for curl / η = curl / ζ , a component of the Codazzi equations of the embedding of the hypersurfaces t in the acoustical spacetime. Since ki j , the 2nd fundamental form of the t , is expressed directly by equations (3.9) in terms

70


of the derivatives of the wave function, this equation contains no new information. Substituting finally (3.111) in (3.103) and denoting (L /T χ)(X A , X B ) = LT χ AB we obtain the desired equation: L /T χ AB =

1 1 (D / AηB + D / B η A ) + µ−1 (ζ A η B + ζ B η A ) 2 2 1 −1 −α (Lκ)χ AB + κ(θ AC χ BC + θ BC χ AC ) − γ AB 2

(3.112)

where we have defined: γ AB =

1 (R(X A , T, X B , L) + R(X B , T, X A , L)). 2

(3.113)

In the sequel we shall use, in addition to the frame field L, T, X 1 , X 2 , the frame field L, L, X 1 , X 2 , obtained by replacing T with the incoming null vectorfield L, given by: L = α −1 κ L + 2T.

(3.114)

At each point p ∈ Wε∗0 , L( p) and L( p) are, respectively, outgoing and incoming futuredirected null normals at p to the surface St,u through p, with L normalized by the condition Lt = 1 (2.12) and L normalized by the condition: h(L, L) = −2µ.

(3.115)

The reciprocal acoustical metric is expressed in terms of the null frame by: µ

(h −1 )µν = −(1/2µ)(L µ L ν + L µ L ν ) + (h/−1 ) AB X A X νB .

(3.116)

From Table 3.102 we can deduce in a straightforward manner a table for the connection coefficients of the null frame L, L, X 1 , X 2 : D L L = µ−1 (Lµ)L D L L = −L(α −1 κ)L + 2η A X A D X A L = −µ−1 ζ A L + χ AB X B D L L = −2ζ A X A D L L = (µ−1 Lµ + L(α −1 κ))L − 2µ(h/−1 ) AB X B (α −1 κ)X A D X A L = µ−1 η A L + χ AB X B DL X A = DX A L 1 1 DX A X B = D / X A X B + µ−1 χ AB L + µ−1 χ AB L. 2 2 Here,

χ AB = (h/−1 ) BC χ AC

(3.117)


71

and, in analogy with (3.46), χ = κ(k/ + θ )

(3.118)

Recapitulating, the geometry of a characteristic hypersurface Cu is described in terms of the family of sections {St,u : t ∈ [0, t∗ε0 )}, namely their induced metric h/, Gauss curvature K , and second fundamental form χ relative to Cu . The last satisfies the propagation equation (3.38) along the generators of Cu as well as the Codazzi equation (3.61) on each section. The manner of the stacking of the hypersurfaces Cu : u ∈ [0, ε0 ], forming a local foliation of the spacetime manifold, is described by the function µ which satisfies the propagation equation (3.96) along the generators of each Cu . We shall now derive expressions for the operators h and h˜ acting on functions f defined on the spacetime domain Wε∗0 . We have: h f = trw

(3.119)

where w is the Hessian of f with respect to the metric h: w = D(d f )

(3.120)

(D denotes as in the preceding the covariant derivative operator corresponding to h, and d f denotes the differential of f ). It is a symmetric 2-covariant tensorfield on Wε∗0 . In (3.119) tr denotes the trace with respect to h. In view of the expression (3.116) for h −1 we have: trw = (h −1 )µν wµν = −µ−1 w L L + (h/−1 ) AB w AB . (3.121) Let also w / be the Hessian of the restriction of f to St,u with respect to the induced metric h /: w /=D /(d/ f ) (3.122) (D / denotes as in the preceding the covariant derivative operator on St,u corresponding to h /, and d/ f denotes the differential of the restriction of f to St,u ). It is a symmetric / = h/ is given by: 2-covariant tensorfield on each St,u . Then the operator / f = trw /.

(3.123)

(Here tr is the trace with respect to h /.) In terms of the frame, we have, w AB = w(X A , X B ) = X A (X B f ) − (D X A X B ) f

(3.124)

/(X A , X B ) = X A (X B f ) − (D / X A X B ) f. w / AB = w

(3.125)

and, By the last line of Table 3.117 we then obtain:

Hence:

1 1 w AB = w / AB − µ−1 χ AB (L f ) − µ−1 χ AB (L f ). 2 2

(3.126)

1 1 (h /−1 ) AB w AB = trw / − trχ (L f ) − µ−1 trχ(L f ). 2 2

(3.127)

72


Also, by the fourth line of Table 3.117, w L L = w(L, L) = L(L f ) − (D L L) f = L(L f ) + 2ζ · d/ f.

(3.128)

Substituting (3.128) and (3.127) in (3.121) we obtain, in view of (3.119) and (3.123) the following formula for the operator h : 1 1 h f = / f − µ−1 trχ(L f ) − µ−1 trχ(L f ) 2 2 −1 −1 − µ L(L f ) − 2µ ζ · d/ f.

(3.129)

The fact that the conformal acoustical metric h˜ is related to the acoustical metric h by: h˜ µν = h µν implies that for an arbitrary function f : h˜ f = −1 h f + −2 (h −1 )µν ∂µ ∂ν f d −1 µν (h ) ∂µ σ ∂ν f. = −1 h f + −2 dσ

(3.130)

In view of the expression (3.116) for h −1 we have: 1 1 (h −1 )µν ∂µ σ ∂ν f = − µ−1 (Lσ )(L f ) − µ−1 (Lσ )(L f ) + d/σ · d/ f 2 2 Thus, setting:

1 d (Lσ ) trχ + −1 2 dσ d 1 ν= (Lσ ) trχ + −1 2 dσ ν=

(3.131) (3.132)

we obtain the following formula for the operator h˜ : / f − µ−1 L(L f ) − µ−1 (ν L f + ν L f ) h˜ f = d d/σ · d/ f. − 2µ−1 ζ · d/ f + −1 dσ

(3.133)

Consider now the functions L i and Tˆ i , i = 1, 2, 3, namely the components of the vectorfields L and Tˆ in rectangular coordinates. Recall that L 0 = 1 and Tˆ 0 = 0. We shall now derive expressions for the derivatives of these functions with respect to L and T , as well as expressions for the differentials of the restrictions of these functions to each surface St,u . These expressions shall allow us to draw the conclusion that these derivatives and differentials remain regular as µ → 0.


73

In the following we shall denote by ∇ the operator of covariant differentiation in spacetime with respect to the Minkowski metric g. Since Minkowski spacetime has a linear structure, for any tensorfield T the components of ∇T in any linear system of coordinates (in particular in rectangular coordinates) are simply the partial derivatives of the corresponding components of T . The operator of covariant differentiation in spacetime with respect to the acoustical metric h shall be denoted by D, as above. For an arbitrary vectorfield W in spacetime, we have, in an arbitrary coordinate system, ∂ W ν hν + !µλ W λ ∂xµ

Dµ W ν =

(3.134)

h

ν are the connection coefficients of the acoustical metric h where !µλ µν in the given coordinate system. On the other hand, in the same coordinate system we have:

∇µ W ν =

g ∂Wν ν λ + ! µλ W ∂xµ

(3.135)

g

ν are the connection coefficients of the Minkowskian metric g where !µλ µν in that coordinate system. Therefore, Dµ W ν = ∇µ W ν + νµλ W λ (3.136)

where is the difference of the two connections, a tensorfield, whose components in any given coordinate system are: g

h

ν ν − !µλ . νµλ =!µλ

(3.137)

Let us now fix the coordinate system to be a rectangular coordinate system of the Minkowskian metric g. We then have: g

ν !µλ = 0,

hence: ∇µ W ν =

∂Wν ∂xµ

and (3.137) reduces to: h

ν = (h −1 )νκ !µλκ νµλ =!µλ

(3.138)

where:

1 (∂µ h λκ + ∂λ h µκ − ∂κ h µλ ). (3.139) 2 For the remainder of this chapter and throughout the succeeding chapters up to Chapter 13 we shall confine ourselves to the irrotational isentropic case. The components of the acoustical metric in rectangular coordinates are then given by (1.69), and their 1st partial derivatives with respect to these coordinates are: !µλκ =

∂κ h µν =

dH ∂κ σ ∂µ φ∂ν φ + H (∂µ φ∂κ ∂ν φ + ∂ν φ∂κ ∂µ φ). dσ

(3.140)

74


Substituting in (3.139) we obtain: !µλκ =

1 dH (ψλ ψκ ∂µ σ + ψµ ψκ ∂λ σ − ψµ ψλ ∂κ σ ) + H ψκ ∂µ ψλ . 2 dσ

(3.141)

Consider now the vectorfield: L(L µ )

∂ ∂xµ

(rectangular coordinates)

(3.142)

Since L 0 = 1, we have L(L 0 ) = 0, therefore the vectorfield (3.142) is tangential to the hypersurfaces t . Hence, it can be expanded as: a L Tˆ + b L

(3.143)

where b L is an St,u - tangential vectorfield. On the other hand, the vectorfield (3.142) is also ∇ L L, the covariant derivative of L with respect to itself in the background flat Minkowski connection. Thus, ∇ L L = a L Tˆ + b L . (3.144) Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1, h(Tˆ , b L ) = 0, a L = h(∇ L L, Tˆ ).

(3.145)

b L = b LA X A

(3.146)

Writing: and taking the h-inner product with X B we obtain, since h(X A , X B ) = h/ AB , / AB = h(∇ L L, X B ). b LA h

(3.147)

Now, from (3.136), (3.138) we have: h(∇ L L, Tˆ ) = h(D L L, Tˆ ) − !αβν L α L β Tˆ ν .

(3.148)

Substituting for D L L from Table 3.102 we obtain: h(D L L, Tˆ ) = −κ −1 (Lµ) = −κ −1 m − αe

(3.149)

where we have used the propagation equation (3.96). Also, by (3.141), 1 d H −1 !αβν L α L β Tˆ ν = −κ (ψ L )2 (T σ ) + 2ψ L ψTˆ (Lσ ) + H ψTˆ L µ (Lψµ ). (3.150) 2 dσ Substituting (3.149) and (3.150) in (3.148), by virtue of equation (3.97) the terms involving κ −1 cancel and we obtain: h(∇ L L, Tˆ ) = −αe −

dH ψ L ψTˆ (Lσ ) − H ψTˆ L µ (Lψµ ). dσ

(3.151)


75

Substituting for e from (3.98) and taking account of the fact that by (3.95) and (3.91), α Fψ0 =

α −1 H ψ0 = H (α −1 ψ L + ψTˆ ) 1 + ρH

(3.152)

we obtain, in view of (3.145), aL = α

−1

ψL

1 dH µ ψ L (Lσ ) + H L (Lψµ ) . 2 dσ

(3.153)

Also, since h(D L L, X B ) = 0, we have, using (3.141), (3.154) h(∇ L L, X B ) = −!αβν L α L β X νB 1 dH 2 ψ B ψ L (Lσ ) − (ψ L )2 (d/ B σ ) − H ψ B L µ (Lψµ ) =− 2 dσ where we denote: ψ A = ψi X iA .

(3.155)

We thus obtain, in view of (3.147): 1 dH 2 ψ B ψ L (Lσ ) − (ψ L )2 (d/ B σ ) + H ψ B L µ (Lψµ ) . b LA = −(h /−1 ) AB 2 dσ (3.156) Consider next the vectorfield: L(Tˆ µ )

∂ ∂xµ

(rectangular coordinates).

(3.157)

Since Tˆ 0 = 0, we have L(Tˆ 0 ) = 0, therefore the vectorfield (3.157) is tangential to the hypersurfaces t . Hence, it can be expanded as: p L Tˆ + q L

(3.158)

where q L is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.142) is also ∇ L Tˆ , the covariant derivative of Tˆ with respect to L in the background flat Minkowski connection. Thus, (3.159) ∇ L Tˆ = p L Tˆ + q L . Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1, h(Tˆ , q L ) = 0, p L = h(∇ L Tˆ , Tˆ ).

(3.160)

q L = q LA X A

(3.161)

Writing:

76


and taking the h-inner product with X B we obtain, since h(X A , X B ) = h/ AB , q LA h / AB = h(∇ L Tˆ , X B ).

(3.162)

Since h(D L Tˆ , Tˆ ) = (1/2)L(h(Tˆ , Tˆ )) = 0, we have, using (3.141), h(∇ L Tˆ , Tˆ ) = −!αβν L α Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (Lσ ) − H ψTˆ Tˆ i (Lψi ). =− 2 dσ T Hence, in view of (3.160) we obtain: 1 dH p L = −ψTˆ ψTˆ (Lσ ) + H Tˆ i (Lψi ) . 2 dσ

(3.163)

(3.164)

Also, we have: h(∇ L Tˆ , X B ) = h(D L Tˆ , X B ) − !αβν L α Tˆ β X νB

(3.165)

h(D L Tˆ , X B ) = κ −1 h(D L T, X B ) = −κ −1 ζ B .

(3.166)

From Table 3.102,

On the other hand, by (3.141), !αβν L α Tˆ β X νB =

1 d H −1 κ ψ L ψ B (T σ ) + ψTˆ ψ B (Lσ ) − ψ L ψTˆ (d/ B σ ) 2 dσ (3.167) +H ψ B Tˆ i (Lψi ).

Substituting (3.166) and (3.167) in (3.165) and defining: ζ A = ζ A +

1 dH ψ L ψ A (T σ ) 2 dσ

(3.168)

we obtain, in view of (3.162), 1 dH ψ ˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ T − H ψ B Tˆ i (Lψi ).

q LA h / AB = −κ −1 ζ B −

(3.169)

We shall show that κ −1 ζ is regular as µ → 0. By (3.64) and (3.168) we have: κ −1 ζ A = α A − d/ A α

(3.170)

where: A = A +

1 dH ψ L ψ A (T σ ). 2ακ dσ

(3.171)


77

By the definition (3.63) and the formula (3.17) for ki j , A = ki j Tˆ i X A 1 dH ψ0 −1 ψTˆ ψ A (Bσ ) − (ψ ˆ (d/ A σ ) + κ ψ A (T σ )) = 2α dσ (1 + ρ H ) T 1 H ψ0 ˆ i − (3.172) T (d/ A ψi ). α (1 + ρ H ) j

Here we have used the fact that by (3.15): j j Tˆ i X A ∂i ψ j = Tˆ i X A ∂ j ψi = Tˆ i (X A ψi ) = Tˆ i (d/ A ψi ).

Substituting we then obtain:

Bσ = Lσ + ακ −1 T σ ψ0 1 dH − αψTˆ ψ A (T σ ) 2ακ dσ 1 + ρ H ψ0 1 dH ψTˆ ψ A (Lσ ) − (d/ A σ ) + 2α dσ 1 + ρH 1 H ψ0 ˆ i T (d/ A ψi ). − α (1 + ρ H )

A = −

(3.173)

By (3.91) the factor in parentheses in the first term on the right-hand side of (3.173) is simply ψ L , therefore in view of the definition (3.171) we obtain: 1 dH ψ0 1 H ψ0 ˆ i A = ψTˆ ψ A (Lσ ) − (d/ A σ ) − T (d/ A ψi ). (3.174) 2α dσ 1 + ρH α (1 + ρ H ) This is regular as µ → 0. It then follows from (3.170) that so is κ −1 ζ . In fact, by (3.86), −αd/ A α =

1 (ψ0 )2 d H d/ A σ 2 (1 + ρ H )2 dσ H ψ0ψ i H ψ0 d/ A ψ0 − d/ A ψi . + (1 + ρ H ) 1 + ρH

Substituting this together with (3.174) in (3.170) yields: 1 dH ψ0 ψ0 −1 κ ζA = − αψTˆ (d/ A σ ) ψTˆ ψ A (Lσ ) + 2 dσ α(1 + ρ H ) 1 + ρ H H ψ0 H ψ0 ψ i + d/ A ψi − α Tˆ i d/ A ψi . d/ A ψ0 − α(1 + ρ H ) 1 + ρH In view of (3.91) and (3.89) this reduces to, simply: 1 dH ψTˆ ψ A (Lσ ) + (ψTˆ + α −1 ψ L )ψ L (d/ A σ ) κ −1 ζ A = 2 dσ + H (ψTˆ + α −1 ψ L )L µ (d/ A ψµ ).

(3.175)

78


Consider next the vectorfield: T (L µ )

∂ ∂xµ


(3.176)

Since L 0 = 0, we have T (L 0 ) = 0, therefore the vectorfield (3.176) is tangential to the hypersurfaces t . Hence, it can be expanded as: aT Tˆ + bT

(3.177)

where bT is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.176) is also ∇T L, the covariant derivative of L with respect to T in the background flat Minkowski connection. Thus, ∇T L = aT Tˆ + bT . (3.178) Taking the h-inner product with Tˆ yields: aT = h(∇T L, Tˆ )

(3.179)

bT = bTA X A

(3.180)

and, writing and taking the h-inner product with X B yields:

We have:

bTA h / AB = h(∇T L, X B ).

(3.181)

h(∇T L, Tˆ ) = h(DT L, Tˆ ) − !αβν T α L β Tˆ ν .

(3.182)

h(DT L, Tˆ ) = Lκ

(3.183)

From Table 3.102, while from (3.141), !αβν T α L β Tˆ ν = κ

1 dH (ψTˆ )2 (Lσ ) + H ψTˆ Tˆ i (Lψi ) . 2 dσ

(3.184)

Here we have used the fact that by (3.15): Tˆ α L β ∂α ψβ = Tˆ α L β ∂β ψα = Tˆ i (Lψi ). Substituting (3.183) and (3.184) in (3.182) we obtain, in view of (3.179), 1 dH ψTˆ (Lσ ) + H Tˆ i (Lψi ) aT = Lκ − κψTˆ 2 dσ

(3.185)

and Lκ is expressed by (3.99). We have: h(∇T L, X B ) = h(DT L, X B ) − !αβν T α L β X νB

(3.186)


79

and from Table 3.102, h(DT L, X B ) = η B = ζ B + d/ B µ

(3.187)

while from (3.141), !αβν T α L β X νB =

1 dH ψ L ψ B (T σ ) + κψTˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ (3.188) + κ H ψ B Tˆ i (Lψi ).

Substituting (3.187) and (3.188) in (3.186) we obtain, in view of (3.181), 1 dH ψ L ψ B (T σ ) (3.189) 2 dσ dH 1 ψˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) + H ψ B Tˆ i (Lψi ) . −κ 2 T dσ

bTA h / AB = η B −

Consider finally the vectorfield: T (Tˆ µ )

∂ ∂xµ

(rectangular coordinates)

(3.190)

which is tangential to the hypersurfaces t . It can be expanded as: pT Tˆ + qT

(3.191)

where qT is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.190) is also ∇T Tˆ , the covariant derivative of Tˆ with respect to T in the background flat Minkowski connection. Thus, (3.192) ∇T Tˆ = pT Tˆ + qT . Taking the h-inner product with Tˆ yields: pT = h(∇T Tˆ , Tˆ )

(3.193)

qT = qTA X A

(3.194)

and, writing and taking the h-inner product with X B yields: qTA h / AB = h(∇T Tˆ , X B ).

(3.195)

Since h(DT Tˆ , Tˆ ) = (1/2)T (h(Tˆ , Tˆ )) = 0, we have, from (3.141), h(∇T Tˆ , Tˆ ) = −!αβν T α Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (T σ ) − H ψTˆ Tˆ i (T ψi ) =− 2 dσ T

(3.196)

80


hence, in view of (3.193) we obtain: pT = − We have:

1 dH (ψ ˆ )2 (T σ ) − H ψTˆ Tˆ i (T ψi ). 2 dσ T

h(∇T Tˆ , X B ) = h(DT Tˆ , X B ) − !αβν T α Tˆ β X νB

(3.197)

(3.198)

and from Table 3.102, h(DT Tˆ , X B ) = κ −1 h(DT T, X B ) = −d/ B κ

(3.199)

1 dH 2ψTˆ ψ B (T σ ) − κ(ψTˆ )2 (d/ B σ ) !αβν T α Tˆ β X νB = 2 dσ + H ψ B Tˆ i (T ψi ).

(3.200)

while from (3.141),

Substituting (3.199) and (3.200) in (3.198), we finally obtain, in view of (3.195), 1 dH ψTˆ 2 ψ B (T σ ) − κψTˆ (d/ B σ ) 2 dσ − H ψ B Tˆ i (T ψi ).

qTA h / AB = −d/ B κ −

(3.201)

We proceed to derive expressions for the differentials d/(L i ) and d/(Tˆ i ) of the restrictions of the functions L i and Tˆ i , i = 1, 2, 3, to each surface St,u . Let X be an arbitrary element of Tx St,u , the tangent plane to St,u at a point x. Then ∇ X L, the covariant derivative of the vectorfield L with respect to X in the background flat Minkowski connection is given in rectangular coordinates by: ∇ X L = X (L µ )

∂ ∂xµ


(3.202)

Since L 0 = 1, we have X (L 0 ) = 0, hence this vector is tangential to the hypersurface t . It can therefore be expanded as: ∇ X L = a/ X Tˆ + b/ X

(3.203)

where b/ X ∈ Tx St,u . Now, a / X and b/ X depend linearly on X. Therefore there is an element /a(x) ∈ Tx∗ St,u (the cotangent plane to St,u at x) and an element b/(x) ∈ L(Tx St,u , Tx St,u ) (the set of linear transformations of Tx St,u ) such that: /(x) /X = X · a a

and

b/ X = X · b/(x).

(3.204)

Thus, a /(x) is the value at x of a 1-form /a on St,u and b/(x) is the value at x of a type T11 -type tensorfield b/ on St,u . Setting X = X A (x), A = 1, 2, in (3.202), (3.203), and letting x vary over St,u , we obtain: d/ A (L i )

∂ = ∇ X A L = a/ A Tˆ + b/ A ∂xi

(3.205)


81

where: /A = a a /(X A )

and

b/ A = b/(X A ) = b/ BA X B .

(3.206)

Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1 and h(b/ A , Tˆ ) = 0, / A = h(∇ X A L, Tˆ ). a

(3.207)

Also, taking the h-inner product with X C we obtain, since h(Tˆ , X C ) = 0 and h(X B , X C ) = h / BC , b/ BA h / BC = h(∇ X A L, X C ). (3.208) We have:

h(∇ X A L, Tˆ ) = h(D X A L, Tˆ ) − !αβν X αA L β Tˆ ν .

(3.209)

h(D X A L, Tˆ ) = κ −1 ζ A .

(3.210)

From Table 3.102, By (3.141), !αβν X αA L β Tˆ ν =

1 dH ψ L ψTˆ (d/ A σ )+ ψ A ψTˆ (Lσ ) − κ −1 ψ A ψ L (T σ ) 2 dσ (3.211) +H ψTˆ L µ (d/ A ψµ ).

Substituting (3.210) and (3.211) in (3.209) and recalling the definition (3.168), we obtain, in view of (3.207), 1 dH ψ ˆ (ψ L (d/ A σ )+ ψ A (Lσ )) 2 dσ T − H ψTˆ L µ (d/ A ψµ ).

/ A = κ −1 ζ A − a

Moreover, substituting for κ −1 ζ from (3.175) this reduces to, simply: 1 dH ψ L (d/ A σ ) + H L µ (d/ A ψµ ) . / A = α −1 ψ L a 2 dσ

(3.212)

(3.213)

Also, we have: h(∇ X A L, X C ) = h(D X A L, X C ) − !αβν X αA L β X Cν

(3.214)

and from Table 3.102, h(D X A L, X C ) = χ AC

(3.215)

while from (3.141), !αβν X αA L β X Cν =

1 dH {ψ L ψC (d/ A σ )+ ψ A ψC (Lσ ) − ψ L ψ A (d/C σ )} 2 dσ (3.216) +H ψC L µ (d/ A ψµ ).

82


Substituting (3.215) and (3.216) in (3.214) we obtain, in view of (3.208), 1 dH {ψ L ψC (d/ A σ )+ ψ A ψC (Lσ ) − ψ L ψ A (d/C σ )} 2 dσ (3.217) − H ψC L µ (d/ A ψµ ).

b/ BA h / BC = χ AC −

Let again X be an arbitrary element of Tx St,u and consider ∇ X Tˆ , the covariant derivative of the vectorfield Tˆ with respect to X in the background flat Minkowski connection. This is given in rectangular coordinates by: ∂ ∇ X Tˆ = X (Tˆ µ ) µ ∂x


(3.218)

Since Tˆ 0 = 0, we have X (Tˆ 0 ) = 0, hence the vector (3.218) is tangential to t . It can therefore be expanded as: ∇ X Tˆ = p/ X Tˆ + q/ X (3.219) where q/ X ∈ Tx St,u . Since p/ X and q/ X depend linearly on X, there is an element p/(x) ∈ Tx∗ St,u and an element q/(x) ∈ L(Tx St,u , Tx St,u ) such that: p/ X = X · p/(x)

and

q/ X = X · q/(x).

(3.220)

Thus, p/(x) is the value at x of a 1-form p/ on St,u and q/(x) is the value at x of a type T11 -type tensorfield q/ on St,u . Setting X = X A (x), A = 1, 2, in (3.218), (3.219), and letting x vary over St,u , we obtain: d/ A (Tˆ i )

∂ = ∇ X A Tˆ = p/ A Tˆ + q/ A ∂xi

(3.221)

where: p/ A = p/(X A )

and

q/ A = q/(X A ) = q/ BA X B .

(3.222)

Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1 and h(q/ A , Tˆ ) = 0, p/ A = h(∇ X A Tˆ , Tˆ ).

(3.223)

Also, taking the h-inner product with X C we obtain, since h(Tˆ , X C ) = 0 and / BC , h(X B , X C ) = h q/ BA h / BC = h(∇ X A Tˆ , X C ). (3.224) Since h(D X A Tˆ , Tˆ ) = (1/2)X A (h(Tˆ , Tˆ )) = 0, we have, by (3.141), h(∇ X A Tˆ , Tˆ ) = −!αβν X αA Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (d/ A σ ) − H ψTˆ Tˆ i (d/ A ψi ) =− 2 dσ T

(3.225)

hence, in view of (3.223), also: p/ A = −

1 dH (ψ ˆ )2 (d/ A σ ) − H ψTˆ Tˆ i (d/ A ψi ). 2 dσ T

(3.226)


We have:

h(∇ X A Tˆ , X C ) = h(D X A Tˆ , X C ) − !αβν X αA Tˆ β X Cν

and from Table 3.102,

h(D X A Tˆ , X C ) = θ AC

83

(3.227) (3.228)

while from (3.141), 1 dH ψTˆ ψC (d/ A σ ) + κ −1 ψ A ψC (T σ ) − ψTˆ ψ A (d/C σ ) 2 dσ (3.229) + H ψC Tˆ i (d/ A ψi ).

!αβν X αA Tˆ β X Cν =

Substituting (3.228) and (3.229) in (3.227) we obtain, in view of (3.224), / BC = θ AC − q/ BA h

1 dH ψ ˆ { ψC (d/ A σ )− ψ A (d/C σ )} − H ψC Tˆ i (d/ A ψi ) 2 dσ T

(3.230)

where we have defined: θ AB = θ AB −

1 dH ψ A ψ B (T σ ). 2κ dσ

(3.231)

We shall show that θ is regular as µ → 0. From (3.46) we have: = −α −1 χ AB + k/AB θ AB

where: k/AB = k/ AB −

1 dH ψ A ψ B (T σ ). 2κ dσ

(3.232)

(3.233)

By the formula (3.17) for ki j , j

k/ AB = ki j X iA X B 1 dH ψ0 ψ A ψ B (Bσ ) − = ( ψ A (d/ B σ )+ ψ B (d/ A σ )) 2α dσ (1 + ρ H ) 1 H ψ0 − ω / AB α (1 + ρ H )

(3.234)

where (3.15): j

ω / AB = X iA X B ∂i ψ j = X iA (d/ B ψi ) = ω /B A. Substituting

(3.235)

Bσ = Lσ + ακ −1 T σ

we then obtain, in view of the definition (3.233), 1 dH ψ0 ψ A ψ B (Lσ ) − k/AB = ( ψ A (d/ B σ )+ ψ B (d/ A σ )) 2α dσ (1 + ρ H ) 1 H ψ0 − ω / AB . α (1 + ρ H ) This is regular as µ → 0. It then follows from (3.232) that so is θ .

(3.236)

Chapter 4

The Acoustical Curvature The acoustical structure equations involve components of the curvature tensor of the acoustical metric h in the frame (L, T, X 1 , X 2 ). These are obtained from curvature tensor components in rectangular coordinates, Rµναβ , by contraction with the appropriate frame vectorfields. Thus, the curvature components α AB occurring in the propagation equations (3.38) for χ AB are given by (see (3.35)): µ

Rµναβ X A L ν X αB L β = α AB

(4.1)

the curvature components β A occurring in the Codazzi equations (3.60) are given by (see (3.59)): µ β (4.2) Rµναβ X C L ν X αA X B = βC AB and the curvature component ρ occurring in the Gauss equation (3.53) is given by (see (3.51)): µ β (4.3) Rµναβ X A X νB X Cα X D = ρ AB C D . / Aη B + D / B η A ) we have the Moreover, in equation (3.112) relating L /T χ AB to (1/2)(D curvature components γ AB , given by (3.113): γ AB =

1 µ µ (Rµναβ X A T ν X αB L β + Rµναβ X B T ν X αA L β ). 2

(4.4)

In the present chapter we shall investigate the structure of the acoustical curvature tensor, analyzing in more detail the components above. To begin with, the curvature components in rectangular coordinates are expressed in the standard way in terms of the metric components and their partial derivatives up to 2nd order with respect to the same coordinates. We have: (2)

(1)

Rµναβ = R µναβ + R µναβ

(4.5)

(2)

where R µναβ contains and is linear in the partial derivatives of the 2nd order: (2) R µναβ =

1 (∂α ∂ν h βµ + ∂β ∂µ h αν − ∂α ∂µ h βν − ∂β ∂ν h αµ ) 2

(4.6)

86

Chapter 4. The Acoustical Curvature (1)

and R µναβ contains and is quadratic in the partial derivatives of 1st order: (1) R µναβ =

−(h −1 )κλ (!αµκ !βνλ − !βµκ !ανλ ).

(4.7)

Here (see (3.139)), λ !µνκ = h κλ !µν =

1 (∂µ h νκ + ∂ν h µκ − ∂κ h µν ). 2

(4.8)

Actually, we find it convenient to introduce, for each pair α, β, the components Hµναβ of the covariant – with respect to h – Hessian of the rectangular component h αβ , considered as a scalar function, namely: κ ∂κ h αβ . Hµναβ = ∂µ ∂ν h αβ − !µν

(4.9)

(2)

[2]

We then define R µναβ in analogy with R µναβ by replacing in (4.6) each ∂µ ∂ν h αβ by the corresponding Hµναβ : [2] R µναβ =

1 (Hανβµ + Hβµαν − Hαµβν − Hβναµ). 2

(4.10)

The decomposition: [2]

[1]

Rµναβ = R µναβ + R µναβ

(4.11)

then results, where: (1) [1] R µναβ = R µναβ

1 κ κ κ κ + (!αν ∂κ h βµ + !βµ ∂κ h αν − !αµ ∂κ h βν − !βν ∂κ h αµ ). 2

(4.12)

Using the fact that ∂κ h µν = !κµν + !κνµ we find: [1] R µναβ

=

1 −1 κλ (!κµβ + !µβκ + !βκµ )!ανλ (h ) 2 + (!κνα + !νακ + !ακν )!βµλ − (!κνβ + !νβκ + !βκν )!αµλ

−(!κµα + !µακ + !ακµ )!βνλ .

(4.13)

Denoting: ψµ = ∂µ φ,

ωµν = ∂µ ψν = ∂ν ψµ = ωνµ ,

τµ = ∂µ σ

(4.14)

1 dH (τα ψβ ψγ + τβ ψα ψγ − τγ ψα ψβ ) + H ωαβ ψγ 2 dσ

(4.15)

we have (see (3.141)): !αβγ =

Chapter 4. The Acoustical Curvature

87

hence: !αβγ + !βγ α + !γ αβ =

1 dH (τα ψβ ψγ + τβ ψα ψγ + τγ ψα ψβ ) 2 dσ +H (ωαβ ψγ + ωβγ ψα + ωαγ ψβ ).

(4.16)

Substituting in (4.13) and noting that: (h −1 )κλ ψκ ψλ = ((g −1 )κλ − Fψ κ ψ λ )ψκ ψλ = −σ (1 + σ F) = −

σ 1−σH

(4.17)

(ψ κ = (g −1 )κλ ψλ ) and: (h −1 )κλ ψλ ωκµ =

ψ κ ωκµ τµ 1 =− 1−σH 2 (1 − σ H )

(4.18)

[1]

we find the following expression for R µναβ : [1] R µναβ

σ H2 Aµναβ (1 − σ H ) 1 dH dH 1 σ +H Bµναβ − 4 (1 − σ H ) dσ dσ 1 1 dH + H H Cµναβ . − 2σ 4 (1 − σ H ) dσ

=−

(4.19)

Here, Aµναβ = ωµβ ωνα − ωµα ωνβ

(4.20)

Bµναβ = (τµ ψν − τν ψµ )(τα ψβ − τβ ψα )

(4.21)

Cµναβ = τµ ξναβ − τν ξµαβ + τα ξβµν − τβ ξαµν

(4.22)

ξµαβ = ωαµ ψβ − ωβµ ψα .

(4.23)

and: where: The expressions Aµναβ , Bµναβ , Cµναβ , all possess the algebraic properties of the Riemann tensor, namely the antisymmetry in the second as well as the first pair of indices and the cyclic property, which imply the symmetry under exchange of the first with the second pair of indices. In particular the cyclic property of Cµναβ follows from that of ξµαβ : ξµαβ + ξαβµ + ξβµα = 0. [2]

We turn to R µναβ , given by (4.10). Using (see (3.140)) ∂µ h αβ =

dH τµ ψα ψβ + H (ωµα ψβ + ωµβ ψα ), dσ

88


we deduce the following decomposition for Hµναβ : Hµναβ =

4

(i)

(4.24)

H µναβ

i=0

where:

(0) H µναβ =

dH vµν ψα ψβ dσ

(4.25)

H (wµνα ψβ + wµνβ ψα )

(4.26)

dH τµ (ωνα ψβ + ωνβ ψα ) + τν (ωµα ψβ + ωµβ ψα ) dσ

(4.27)

d2 H τµ τν ψα ψβ dσ 2

(4.28)

H (ωµα ωνβ + ωµβ ωνα ).

(4.29)

(1) H µναβ = (2) H µναβ =

(3) H µναβ = (4) H µναβ =

In equation (4.25) vµν is the covariant – relative to h – Hessian of the function σ :

We have:

vµν = Dµ Dν σ = vνµ .

(4.30)

κ τκ = Dν τµ vµν = Dµ τν = ∂µ τν − !µν

(4.31)

Also, in equation (4.26) wµνα is the covariant – relative to h – Hessian of the rectangular component ψα , considered as a scalar function:

and we have:

wµνα = Dµ Dν ψα = wνµα

(4.32)

κ ωκα = Dν ωµα . wµνα = Dµ ωνα = ∂µ ωνα − !µν

(4.33) [2]

Corresponding to the decomposition (3.33) of Hµναβ , we have a decomposition of R µναβ : [2] R µναβ =

4

[2,i] R µναβ

(4.34)

i=0

where: [2,i] R µναβ =

(i) (i) (i) 1 (i) ( H ανβµ + H βµαν − H αµβν − H βναµ ). 2

(4.35)

We find: [2,2] R µναβ =

−

1 dH Cµναβ 2 dσ

(4.36)


89

(where Cµναβ is given by (4.22)) 1 d2 H Bµναβ 2 dσ 2

(4.37)

−H Aµναβ

(4.38)

1 H (ναβ ψµ + µβα ψν + ανµ ψβ + βµν ψα ) 2

(4.39)

[2,3] R µναβ =

−

(where Bµναβ is given by (4.21)) [2,4] R µναβ =

(where Aµναβ is given by (4.20)). On the other hand, [2,1] R µναβ =

where: µαβ = wµαβ − wµβα .

(4.40)

From (4.33), in view of the second of equations (4.14), we obtain: κ κ µνα = !µα ωκν − !µν ωκα

= (h −1 )κλ (!µαλ ωκν − !µνλ ωκα ) 1 d H −1 κλ = (h ) [(τµ ψα ψλ + τα ψµ ψλ − τλ ψµ ψα )ωκν 2 dσ − (τµ ψν ψλ + τν ψµ ψλ − τλ ψµ ψν )ωκα ] + H (h −1)κλ (ωµα ωκν − ωµν ωκα )ψλ .

(4.41)

Taking account of (4.18) this becomes: 1 d H −1 κλ (h ) τλ ψµ (ψα ωκν − ψν ωκα ) 2 dσ 1 dH 1 τµ (τν ψα − τα ψν ) − 4 (1 − σ H ) dσ 1 H − (ωµα τν − ωµν τα ). 2 (1 − σ H )

µνα = −

(4.42)

Substituting in (4.39) the contribution of the first term on the right-hand side of (4.42) is seen to vanish and we obtain: [2,1] H 1 dH Bµναβ + H Cµναβ . (4.43) R µναβ = 4 (1 − σ H ) dσ The sum:

4 [2,i] R µναβ i=1

90

Chapter 4. The Acoustical Curvature [1]

combines with R µναβ , given by (4.19), to form the lower order part of the curvature tensor, which we denote by Nµναβ : [1]

Nµναβ = R µναβ +

4 [2,i] R µναβ .

(4.44)

i=1

We obtain: Nµναβ = −F Aµναβ − Here,

1 1 F2 Bµναβ − F1 Cµναβ . 2 2

1 2 dH + H dσ 2 2 σ d H 1 dH 2 + F2 = 2 (1 − σ H ) dσ dσ 2 1 F1 = (1 − σ H )

(4.45)

(4.46) (4.47)

and Aµναβ , Bµναβ , Cµναβ are given by (4.20), (4.21), (4.22), respectively. Also, recall that H . F= 1−σH [2,0] µναβ ,

The principal part of the curvature tensor is then R [2,0] µναβ

Pµναβ = R

which we denote Pµναβ :

.

(4.48)

By (4.25) and (4.35) it is given by: 1 dH (vαν ψβ ψµ + vβµ ψα ψν − vαµ ψβ ψν − vβν ψα ψµ ). (4.49) 2 dσ The decomposition of the curvature tensor into its principal part and its lower order part is then: Rµναβ = Pµναβ + Nµναβ . (4.50) Pµναβ =

Note that the principal part vanishes if and only if H is constant, in which case the original nonlinear wave equation reduces to the minimal surface equation (see Chapter 1). In the following we shall analyze in detail the curvature components α AB , β A , ρ and γ AB which enter the acoustical structure equations (see (4.1)–(4.4)). We denote the ( P)

corresponding components of the principal part by α (N)

lower order part by α

( P)

α

AB ,

(N)

AB =

(N) (N)

β A, ρ , γ

AB .

AB ,

( P)

( P) ( P) AB ,

β A, ρ , γ

Thus,

P(X A , L, X B , L)

(4.51) ( P)

P(X C , L, X A , X B ) = AB β

(4.52)

C ( P)

P(X A , X B , X C , X D ) = AB C D ρ ( P) 1 γ AB = (P(X A , T, X B , L) + P(X B , T, 2

and of the

(4.53) X A , L))

(4.54)


91

and, (N)

α

AB =

N(X A , L, X B , L)

(4.55) (N)

N(X C , L, X A , X B ) = AB β

(4.56)

C (N)

N(X A , X B , X C , X D ) = AB C D ρ 1 γ AB = (N(X A , T, X B , L) + N(X B , T, X A , L)). 2

(N)

(4.57) (4.58)

We begin with an analysis of the components of the principal part (4.49). These contain the components of the Hessian – with respect to h – of σ (see (4.30)). The com( P)

ponents of the latter which occur in α

( P) AB

( P)

( P)

β A , ρ are:

v(X A , L),

v(L, L), while in γ

AB ,

v(X A , X B )

(4.59)

there occur also the components: v(X A , T ),

v(L, T ).

(4.60)

The component v(T, T ) does not occur in the components of the curvature which enter the acoustical structure equations. Now, our approach shall yield estimates for the derivatives of σ with respect to T and L and for the derivatives – covariant with respect to h/ – of the restrictions of these to each St,u . We must therefore express (4.59) and (4.60) in terms of these. By (4.30), (4.31), we have: v(L, L) = L µ L ν Dµ ∂ν σ = L(Lσ ) − (D L L)σ Substituting for D L L from Table 3.102 we then obtain: v(L, L) = L(Lσ ) − µ−1 (Lµ)(Lσ ).

(4.61)

Next, we have: µ

v(X A , L) = X A L ν Dµ ∂ν σ = X A (Lσ ) − (D X A L)σ. Substituting for D X A L from Table 3.102 we obtain: v(X A , L) = d/ A (Lσ ) + µ−1 ζ A (Lσ ) − χ AB (d/ B σ ).

(4.62)

Next, we have: µ

v(X A , X B ) = X A X νB Dµ ∂ν σ = X A (X B σ ) − (D X A X B )σ. Substituting for D X A X B from Table 3.102 and noting that: X A (X B σ ) − (D / X A X B )σ = (D / 2 σ )(X A , X B ) = (D / 2 σ ) AB

(4.63)

92


where D / 2 σ is the Hessian – with respect to h/ – of the restriction of σ to each St,u , we obtain: / 2 σ ) AB − α −1 k/ AB (Lσ ) − µ−1 χ AB (T σ ). (4.64) v(X A , X B ) = (D Also, again by (4.30), (4.31) we have: µ

v(X A , T ) = X A T ν Dµ ∂ν σ = X A (T σ ) − (D X A T )σ. Substituting for D X A T from Table 3.102 we obtain: v(X A , T ) = d/ A (T σ ) − α −1 κ A (Lσ ) − µ−1 η A (T σ ) − κθ AB (d/ B σ ). Also, we have:

(4.65)

v(L, T ) = L µ T ν Dµ ∂ν σ = L(T σ ) − (D L T )σ

and substituting for D L T from Table 3.102 we obtain: v(L, T ) = L(T σ ) + α −1 (Lκ)(Lσ ) + ζ A (d/ A σ ).

(4.66)

Now, by (4.51), (4.49), we have: ( P)

α

AB

=

1 dH (v(X B , L)ψ L ψ A + v(X A , L)ψ L ψ B 2 dσ − v(X A , X B )(ψ L )2 − v(L, L) ψ A ψ B ).

(4.67)

We substitute in (4.67) for v(X A , L) from (4.62), expressing ζ A in terms of ζ A by (3.168), that is: 1 dH ψ L ψ A (T σ ) ζ A = ζ A − (4.68) 2 dσ where κ −1 ζ A is given by (3.175). We substitute in (4.67) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB by (3.233), that is: k/ AB = k/AB +

1 dH ψ A ψ B (T σ ) 2κ dσ

(4.69)

where k/AB is given by (3.236). Finally, we substitute for v(L, L) from (4.61), expressing Lµ by equation (3.96). In view of (3.97) we have: µ−1 (Lµ) =

dH 1 −1 µ (ψ L )2 (T σ ) + e 2 dσ

(4.70) ( P)

where e is given by (3.98). Then, in the resulting expression for α tional to the product (T σ )(Lσ ) cancel and we obtain: ( P)

α

AB =

AB ,

the terms propor-

( P) dH 1 (χ C ψ B + χ BC ψ A )(d/C σ )+ α AB µ−1 mχ AB − ψ L 2 dσ A

(4.71)

Chapter 4. The Acoustical Curvature ( P)

where α

AB ,

93

given by: ( P)

α

=

AB

1 dH [ψ L ( ψ A d/ B (Lσ )+ ψ B d/ A (Lσ )) 2 dσ −(ψ L )2 (D / 2 σ ) AB − ψ A ψ B L(Lσ ) +

1 d H −1 α ψ L ( ψ A (κ −1 ζ B )+ ψ B (κ −1 ζ A )) 2 dσ +α −1 (ψ L )2 k/AB + ψ A ψ B e (Lσ )

(4.72)

is regular as µ → 0. By (4.52), (4.49), we have: ( P)∗

β

( P)

A

= −(h /−1 ) BC AB β C 1 d H −1 BC (h / ) [(v(X A , X C ) ψ B − v(X B , X C ) ψ A )ψ L = 2 dσ +( ψ A v(X B , L)− ψ B v(X A , L)) ψC ] .

(4.73)

We substitute in (4.73) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB as in (4.69). Also, we substitute in (4.73) for v(X A , L) from (4.62), expressing ζ A in terms of ( P)∗

ζ A as in (4.68). Then, in the resulting expression for β product (T σ )(Lσ ) cancel and we obtain: ( P)∗

β

( P)∗

where β ( P)∗

β

A

=

A,

A

A,

the terms proportional to the

dH 1 ψ L (T σ )(χ AB ψ B − trχ ψ A ) = − µ−1 2 dσ ( P)∗ 1 dH B − ψ ( ψ A χ BC − ψ B χ AC )(d/C σ )+ β A 2 dσ

(4.74)

given by:

1 dH ψ L [(D / 2 σ ) AB ψ B − ( / σ ) ψ A ]+ ψ A ψ B d/ B (Lσ ) − | ψ|2 d/ A (Lσ ) 2 dσ (4.75) 1 d H −1 −ψ L ( ψ B k/AB − ψ A trk/ )+ ψ A ψ B (κ −1 ζ B ) − | ψ|2 (κ −1 ζ A ) (Lσ ) α + 2 dσ

is regular as µ → 0. By (4.53), (4.49), we have: ( P)

( P) 1 −1 AC −1 B D (h / ) (h / ) AB C D ρ (4.76) 2 1 d H −1 AC −1 B D (h / ) (h / ) [v(X C , X B ) ψ D ψ A + v(X D , X A ) ψC ψ B = 4 dσ −v(X C , X A ) ψ D ψ B − v(X D , X B ) ψC ψ A ] .

ρ =

94


We substitute in (4.76) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB as ( P)

in (4.69). Then, in the resulting expression for ρ , the terms proportional to the product (T σ )(Lσ ) again cancel and we obtain: ( P) 1 dH ρ = − µ−1 (T σ ) ψ · χ· ψ − trχ| ψ|2 + ρ 2 dσ

( P)

(4.77)

( P)

where ρ , given by: ( P)

ρ =

1 dH (D / 2 σ ) AB ψ A ψ B − ( / σ )| ψ|2 2 dσ 1 d H −1 α (Lσ ) k/AB ψ A ψ B − trk/ | ψ|2 . − 2 dσ

(4.78)

We turn finally to (4.54). By (4.49) we have: ( P) γ AB

=

1 dH {(1/2)( ψ A v(X B , T )+ ψ B v(X A , T ))ψ L 2 dσ + (1/2)( ψ A v(X B , L)+ ψ B v(X A , L))ψT − v(X A , X B )ψ L ψT − v(L, T ) ψ A ψ B }.

(4.79)

We substitute in (4.79) for v(X A , T ) and v(L, T ) from (4.65) and (4.66) respectively, expressing θ in terms of χ through (3.46), that is: θ = k/ − α −1 χ.

(4.80)

Also, we substitute in (4.79) for v(X A , L) and v(X A , X B ) from (4.62) and (4.64) respectively. We then obtain: ( P) γ AB

dH 1 (T σ )ψ L ( ψ A η B + ψ B η A ) = − µ−1 4 dσ 1 d H −1 α ψ L ψTˆ (T σ )χ AB + 2 dσ 1 d H −1 α κ(ψ L − αψTˆ )( ψ A χ BC + ψ B χ AC )(d/C σ ) + 4 dσ ( P) AB

+ γ

.

(4.81)

Here, ( P) γ AB

=

1 dH {(1/2)ψ L [ ψ A d/ B (T σ )+ ψ B d/ A (T σ )] 2 dσ + (1/2)κψTˆ [ ψ A d/ B (Lσ )+ ψ B d/ A (Lσ )] − κψ L ψTˆ (D / 2 σ ) AB − ψ A ψ B L(T σ )} +

1 dH κl AB 2 dσ

(4.82)


95

where: κl AB = (1/2)α −1 [ ψ A (ψTˆ ζ B − ψ L (κ B ))+ ψ B (ψTˆ ζ A − ψ L (κ A ))

(4.83)

−2 ψ A ψ B (Lκ) + 2ψ L ψTˆ (κk/ AB )](Lσ ) −(1/2)[ψ L ( ψ A (κk/ BC )+ ψ B (κk/ AC )) + 2 ψ A ψ B ζ C ](d/C σ ). ( P)

The γ AB are regular as µ → 0. In fact, expressing ζ A in terms of ζ A by (4.68), A in terms of A by (3.171), k/ AB in terms of k/AB by (4.69), and substituting for Lκ from equation (3.99), we find: l AB = (1/2)α −1 [ ψ A (ψTˆ (κ −1 ζ B ) − ψ L B )+ ψ B (ψTˆ (κ −1 ζ A ) − ψ L A )

(4.84)

− 2 ψ A ψ B e + 2ψ L ψTˆ k/AB ](Lσ ) −(1/2)[ψ L ( ψ A k/B C + ψ B k/A C ) + 2 ψ A ψ B (κ −1 ζ C )](d/C σ )

which shows that the l AB are themselves regular as µ → 0. We turn to the analysis of the components (4.55)–(4.58) of the lower order part ( A) ( A) ( A) AB , β A , ρ , γ AB , be the cor(B) (B) (B) (B) responding components of Aµναβ , given by (4.20). Let α AB , β A , ρ , γ AB , be the (C) (C) (C) (C) corresponding components of Bµναβ , given by (4.21). Let also α AB , β A , ρ , γ AB , be ( A)

Nµναβ of the curvature tensor, given by (4.45). Let α

the corresponding components of Cµναβ , given by (4.22). We then have: (N)

( A)

1 (B) 1 (C) − F2 α AB − F1 α AB (4.85) 2 2 (N) ( A) 1 (B) 1 (C) β A = −F β A − F2 β A − F1 β A (4.86) 2 2 (N) ( A) 1 (B) 1 (C) ρ = −F ρ − F2 ρ − F1 ρ (4.87) 2 2 (N) ( A) (B) 1 1 (C) γ AB = −F γ AB − F2 γ AB − γ AB . (4.88) 2 2 The above components involve the components of the symmetric tensor ωµν (see /, with components: (4.14)). On each St,u , we have the symmetric tensor ω α

AB

= −F α

AB

j

ω / AB = X iA X B ∂i ψ j = X iA (d/ B ψi )

(4.89)

/Tˆ , with components: the 1-forms ω / L and ω (ω / L ) A = L µ X νA ∂µ ψν = L µ (d/ A ψµ ) = X iA (Lψi ) î

(ω /Tˆ ) A = T

j X A ∂i ψ j

î

(4.90)

= T (d/ A ψi )

(4.91)

ω L L = L µ L ν ∂µ ψν = L µ (Lψµ )

(4.92)

and the functions ω L L and ω L Tˆ , where: µ

ˆν

î

ω L Tˆ = L T ∂µ ψν = T (Lψi ) (the component ωTˆ Tˆ does not occur in (4.85)–(4.88)).

(4.93)

96


We have (see (4.20)–(4.23)): ( A)

α

(B)

α

(C)

α

AB

AB =

AB =

(ω / L ) A (ω /L )B − ω / AB ω L L

(4.94)

[ψ L (d/ A σ )− ψ A (Lσ )][ψ L (d/ B σ )− ψ B (Lσ )]

(4.95)

= (d/ A σ )[ψ L (ω / L ) B − ψ B ω L L ] + (d/ B σ )[ψ L (ω / L ) A − ψ A ω L L ] −(Lσ )[2ψ L ω / AB − ψ A (ω / L ) B − ψ B (ω /L ) A ] ( A)∗

β

(B)∗

β

A=

(C)∗

β

A

(4.96)

ω / AB (ω / L ) B − trω /(ω /L ) A

(4.97)

[ ψ B (Lσ ) − ψ L (d/ B σ )][ ψ B (d/ A σ )− ψ A (d/ B σ )]

(4.98)

A=

= (Lσ )[ω / AB ψ B − trω / ψ A ] / L ) B ψ B − trω /ψ L ] +(d/ A σ )[(ω +(d/ B σ )[ω / AB ψ L − 2(ω / L ) A ψ B + (ω / L ) B ψ A ] ( A)

ρ = (1/2)[|ω /|2 − (trω /)2 ]

(B)

(4.99) (4.100)

ρ = | ψ|2 |d/σ |2 − ( ψ · d/σ )2

(4.101)

ρ = 2[ ψ · ω / · d/σ − ( ψ · d/σ )trω /].

(4.102)

(C)

Also, we have: ( A) γ AB

= (1/2)κ[(ω / L ) A (ω /Tˆ ) B + (ω / L ) B (ω /Tˆ ) A − 2ω / AB ω L Tˆ ]

(4.103)

(B) γ AB

= (1/2)[κψTˆ (d/ A σ )− ψ A (T σ )][ψ L (d/ B σ )− ψ B (Lσ )]

(4.104)

+ (1/2)[κψTˆ (d/ B σ )− ψ B (T σ )][ψ L (d/ A σ )− ψ A (Lσ )] (C) γ AB

= (1/2)(d/ A σ )κ[ψ L (ω /Tˆ ) B + ψTˆ (ω / L ) B − 2 ψ B ω L Tˆ ] /Tˆ ) A + ψTˆ (ω / L ) A − 2 ψ A ω L Tˆ ] + (1/2)(d/ B σ )κ[ψ L (ω /Tˆ ) B + ψ B (ω /Tˆ ) A − 2ψTˆ ω / AB ] + (1/2)(Lσ )κ[ ψ A (ω + (1/2)(T σ )[ ψ A (ω / L ) B + ψ B (ω / L ) A − 2ψ L ω / AB ].

(4.105)

The above formulas show that all the components (4.85)–(4.88) of the lower order part of the curvature tensor are regular as µ → 0. We now revisit the propagation equations (3.38) for χ AB . Substituting the decomposition ( P)

α AB = α

(N)

AB

+ α

AB


97 ( P)

and then substituting the formula (4.71) for α AB , and also substituting the propagation equation for µ, equation (3.96), the propagation equations for χ AB take the form: dH 1 (χ C ψ B + χ BC ψ A )(d/C σ ) Lχ AB = χ AC χ BC + eχ AB + ψ L 2 dσ A ( P)

− α

(N)

AB

− α

AB

.

(4.106)

In contrast to the original form (3.38), the right-hand side is now manifestly regular as µ → 0. Next, we revisit the Codazzi equations in the divergence form (3.62). Substituting the decomposition ( P)

βA = β

(N)

A

+ β

A

( P)∗

and then substituting the formula (4.74) for β A , and also substituting the expression for ζ A in terms of ζ A , equation (4.68), the Codazzi equations become: (div / χ) A − d/ A trχ = −µ−1 (χ AB ζ B − trχζ A ) 1 dH B ψ ( ψ A χ BC − ψ B χ AC )(d/C σ ) − 2 dσ ( P)∗

+ β

(N)∗

A

+ β

A

.

(4.107)

Since κ −1 ζ A is regular as µ → 0 (see (3.175)), so is the right-hand side. Next, we revisit the Gauss equation (3.53). Here we first substitute for θ in terms of χ and k/ from (4.80) and for k/ in terms of k/ from (4.69) to obtain: 1 1 {(trθ )2 − |θ |2 − (trk/)2 + |k/|2 } = α −2 {(trχ)2 − |χ|2 } 2 2 −α −1 (trχtrk/ − χ · k/ ) 1 dH (T σ ){trχ| ψ|2 − ψ · χ· ψ}. − µ−1 2 dσ We then substitute the decomposition ( P)

(N)

ρ=ρ + ρ ( P)

and the formula (4.77) for ρ . The Gauss equation then takes the form: K =

1 −2 α {(trχ)2 − |χ|2 } − α −1 (trχtrk/ − χ · k/ ) 2 ( P)

(N)

+ ρ + ρ which is regular as µ → 0.

(4.108)

98


Finally, we revisit equations (3.112) relating L /T χ AB to (1/2)(D / AηB + D / B η A ). Substituting the decomposition ( P) AB

γ AB = γ

(N)

+ γ

AB

( P) γ AB ,

and then substituting the formula (4.81) for and also substituting the expression for ζ A in terms of ζ A , equation (4.68), equations (3.112) become, in view also of (3.99), (4.80), and (4.69), L /T χ AB =

1 1 / B η A ) + µ−1 (ζ A η B + ζ B η A ) (D / AηB + D 2 2 1 −2 1 d H −α (T σ ) ψ L (ψ L + αψTˆ )χ AB + ( ψ A χ BC + ψ B χ AC ) ψC 2 dσ 2 1 −α −1 κ(e χ AB + χ AC χ BC ) + κ(k/AC χ BC + k/BC χ AC ) 2 1 −1 d H (ψ L − αψTˆ )( ψ A χ BC + ψ B χ AC )(d/C σ ) − α κ 4 dσ ( P) AB

− γ

(N)

− γ

AB

.

The right-hand side is now manifestly regular as µ → 0.

(4.109)

Chapter 5

The Fundamental Energy Estimate In this chapter we consider the wave equation h˜ ψ = 0

(5.1)

˜ As we have seen in the spacetime whose metric is the conformal acoustical metric h. in Chapter 1, this is the equation (1.86) satisfied by a variation of the wave function φ through solutions. The conformal acoustical metric h˜ is related to the acoustical metric h by equation (1.82), namely: (5.2) h˜ µν = h µν where the conformal factor is given by: =

G/G 0 η/η0

(5.3)

where the constants G 0 and η0 refer to the surrounding constant state. We consider, more generally, the inhomogeneous wave equation: h˜ ψ = ρ.

(5.4)

We shall be led to equations of this form, in the next chapter, when considering higher order variations. The metric h˜ is defined in the domain Mε0 of the maximal solution, discussed in Chapter 2. By virtue of the linear character of equations (5.1) and (5.4), with ρ a given function defined on Mε0 , we may assume that ψ is defined on the whole of Mε0 , as is indeed the case when ψ is a first- or higher order variation of φ. In applications to first order estimates we shall take ψ to each be one of: ψ1 = T0 φ − k; Ti φ : i = 1, 2, 3; i j φ : i < j = 1, 2, 3; Sφ.

(5.5)

For the first of the above, we appeal to the fact that if ψ is a solution of (5.1), so is ψ − c, for any constant c. Then each of the ψ1 vanishes in the exterior of the outgoing

100

Chapter 5. The Fundamental Energy Estimate

characteristic hypersurface C0 . In applications to higher order estimates we shall take each ψ to be one of: (5.6) ψn = Yi1 · · · Yin−1 ψ1 where ψ1 is any one of the above first order variations and the indices i 1 , . . . , i n−1 take values in the set {1, 2, 3, 4, 5}, with {Yi : i = 1, 2, 3, 4, 5} the set of vectorfields to be discussed in the next chapter. Each of the ψn also vanishes in the exterior of C0 . We shall therefore assume in general that ψ vanishes in the exterior of C0 . We may then confine attention to the spacetime domain Wε0 defined in Chapter 2. For any s ∈ (0, t∗ε0 ), we set: Wεs0 = {x ∈ Wε0 : x 0 ∈ [0, s]} (5.7) or, equivalently, Wεs0 = In particular we have:

St,u .

(5.8)

(t,u)∈[0,s]×[0,ε0]

s∈(0,t∗ε0 )

Wεs0 = Wε∗0

Recall that here and throughout this monograph the positive constant ε0 is subject to the restriction: 1 ε0 ≤ 2 but is otherwise arbitrary. Everything which follows depends on the geometric construction of Chapter 3. We remark here that the characteristic hypersurfaces Cu depend only on the conformal class of the metric h. Thus, the 2-parameter foliation of Wε∗0 given by the surfaces St,u likewise depends only on the conformal class of h. The geometric properties of this foliation may ˜ Since we are studying the be referred either to the metric h or to the conformal metric h. ˜ it would appear more natural if we refer these properties to wave equation in the metric h, ˜ However, our bootstrap argument is constructed relative to h, therefore we shall refer h. the said properties to h, and introduce additional assumptions referring to the conformal factor . We shall presently list the bootstrap assumptions on the geometric properties of the 2-parameter foliation which we shall need in this chapter. First we have the basic assumptions: There is a positive constant C independent of s such that in Wεs0 , A1: C −1 ≤ ≤ C A2: C −1 ≤ α/η0 A3: µ/η0 ≤ C[1 + log(1 + t)] We remark that by equation (2.43) we have: α≤1

(5.9)


101

Next, we have the assumptions: B1: C −1 (1 + t)−1 ≤ ν ≤ C(1 + t)−1 B2: |ν| ≤ C(1 + t)−1 [1 + log(1 + t)]4 B3: |χ| ˆ ≤ C(1 + t)−1 [1 + log(1 + t)]−2 B4: |χ| ˆ ≤ C(1 + t)[1 + log(1 + t)]−6 Here, 1 1 ˜ = (trχ + L log ) trχ 2 2 1 1 ˜ = (trχ + L log ). ν = trχ 2 2

ν=

(5.10) (5.11)

Moreover, χˆ and χˆ are the trace-free parts of χ and χ respectively, and the pointwise norms of tensors on St,u are with respect to the induced metric h/. Next, we have the assumptions: B5: |L log | ≤ C(1 + t)−1 [1 + log(1 + t)]−2 B6: |L log | ≤ C(1 + t)[1 + log(1 + t)]−6 and the assumptions: B7: |ζ + η| ≤ C(1 + t)−1 [1 + log(1 + t)] B8: |d/(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)] B9: |Lα| ≤ C(1 + t)−1 [1 + log(1 + t)]2 B10: |L(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)]3 B11: |L(α −1 κ)| ≤ C(1 + t)[1 + log(1 + t)]−2 as well as the assumptions: B12: |Lν + ν 2 | ≤ C(1 + t)−2 [1 + log(1 + t)]−2 B13: |Lν| ≤ C(1 + t)−2 [1 + log(1 + t)]3 B14: |d/ν| ≤ C(1 + t)−2 [1 + log(1 + t)]1/2 The next set of bootstrap assumptions are the crucial assumptions regarding the behavior of the function µ. In the following we denote by f + and f − respectively the positive and negative parts of the function f : f+ (x) = max{ f (x), 0},

f − (x) = min{ f (x), 0}

C1: µ−1 (Lµ)+ ≤ (1 + t)−1 [1 + log(1 + t)]−1 + A(t) where A(t) is a non-negative function such that: s A(t)dt ≤ C (independent of s) 0

C2: µ−1 (Lµ + Lµ)+ ≤ B(t)

102


where B(t) is a non-negative function such that: s (1 + t)−2 [1 + log(1 + t)]4 B(t)dt ≤ C (independent of s). 0

Moreover, denoting by U the region: U = {x ∈ Wε∗0 : µ < η0 /4} we have: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 in U

C3:

(5.12)

Wεs0

The final set of bootstrap assumptions concerns the existence of a function ω verifying the following conditions: D1: C −1 (1 + t) ≤ ω/η0 ≤ C(1 + t) D2: |Lω − νω| ≤ C[1 + log(1 + t)]−2 D3: |Lω| ≤ C[1 + log(1 + t)]3 D4: |d/ω| ≤ C[1 + log(1 + t)]1/2 and:

! s ! ε0

D5:

0

0

sup St,u (µ|h˜ ω|)du dt ≤ C[1 + log(1 + s)]4

The aim of the present chapter is the proof of: ˜ defined in Mε0 Theorem 5.1 Let ψ be a solution of the wave equation in the metric h, and vanishing in the exterior of C0 . Suppose that assumptions A1–A3, B1–B14, C1–C3, D1–D5 hold in Wεs0 , for some s ∈ (0, tε0 ]. Let us denote: ε0 −2 2 2 2 [η0 (Lψ) + (Lψ) + |d/ψ| ]dµh/ du. D0 = 0

S0,u

Then, there exist constants C independent of s such that: ε0 −2 −2 2 2 2 (i) sup [η0 µ(η0 + µ)(η0 (Lψ) + |d/ψ| ) + (Lψ) ]dµh/ du ≤ C D0 t ∈[0,s] 0

St,u

(ii)

St,u

(iii)

ψ 2 dµh/ ≤ Cε0 D0 s

sup u∈[0,ε0 ] 0

St,u

[η0−2 (η0 + µ)(Lψ)2 + µ|d/ψ|2 ]dµh/ dt ≤ C D0

(iv) sup [1 + log(1 + t)]−4 (1 + t)2 t ∈[0,s] ε0

× 0

(v)

St,u

sup [1 + log(1 + s)] u∈[0,ε0 ]

µ[η0−2 (Lψ + νψ)2 + |d/ψ|2 ]dµh/ −4

s 0

du ≤ C D0

(1 + t)

2

(Lψ + νψ) dµh/ dt ≤ C D0 . 2

St,u


103

Here, the pointwise norms of tensors on St,u are with respect to the induced metric h/. Moreover, there is a constant C independent of s such that: (1 + t)[1 + log(1 + t)]−1 |d/ψ|2 dµh/ dudt ≤ C D0 [1 + log(1 + s)]4 (vi) U

Wεs

0

where U is the region where µ < η0 /4 (see (5.12)). Proof. We begin with the energy-momentum-stress tensor T˜µν associated to the function ˜ ψ through the metric h: T˜µν := ∂µ ψ∂ν ψ − (1/2)h˜ µν (h˜ −1 )κλ ∂κ ψ∂λ ψ = ∂µ ψ∂ν ψ − (1/2)h µν (h −1 )κλ ∂κ ψ∂λ ψ := Tµν . We have:

(5.13)

D˜ µ T˜µν := (h˜ −1 )µλ D˜ µ T˜λν = ∂ν ψh˜ ψ

where D˜ is the covariant derivative operator associated to the metric h˜ µν . Thus, for a ˜ while for a solution of solution of 5.1 T˜µν is divergence-free with respect to the metric h, the inhomogeneous wave equation 5.4, D˜ µ T˜µν = ρ∂ν ψ.

(5.14)

We consider the future-directed, time-like with respect to the acoustical metric h, vectorfield K 0 : (5.15) K 0 = (η0−1 + α −1 κ)L + L. Also, given a function ω satisfying assumptions D1–D5 we consider the future-directed, null with respect to the acoustical metric h, vectorfield K 1 : K 1 = (ω/ν)L.

(5.16)

We denote by π˜ 0 and π˜ 1 the Lie derivatives of the conformal acoustical metric h˜ with respect to the vectorfields K 0 and K 1 respectively: π˜ 0 = L K 0 h˜

˜ π˜ 1 = L K 1 h.

(5.17)

To K 0 we associate the vectorfield P˜0 : µ P˜0 = −T˜ µν K 0ν

where:

(5.18)

T˜ µν = (h˜ −1 )µκ Tκν .

To K 1 we associate the vectorfield P˜1 , given by: µ P˜1 = −T˜ µν K 1ν − (h˜ −1 )µν (ωψ∂ν ψ − (1/2)ψ 2 ∂ν ω).

(5.19)

104


Let:

µ T˜ µν = (h˜ −1 )νλ T˜ λ = (h˜ −1 )µκ (h˜ −1 )νλ T˜κλ .

For any vectorfield X, we have, by virtue of the symmetry of T˜ µν , D˜ µ (T˜ µν X ν ) = ( D˜ µ T˜ µν )X ν + T˜ µλ (h˜ λν D˜ µ X ν ) = ρ Xψ + (1/2)T˜ µλ (h˜ λν D˜ µ X ν + h˜ µν D˜ λ X ν ) = ρ Xψ + (1/2)T˜ µλ L X h˜ µλ where we have used equation (5.14). Thus, the divergence of the vectorfield P˜0 with respect to the metric h˜ is given by: µ D˜ µ P˜0 = −ρ K 0 ψ − (1/2)T˜ µν π˜ 0,µν := Q˜ 0 .

(5.20)

Also, the divergence of the vectorfield P˜1 with respect to the metric h˜ is given by: µ D˜ µ P˜1 = −ρ(K 1 ψ + ωψ) − (1/2)T˜ µν π˜ 1,µν −ω(h˜ −1 )µν ∂µ ψ∂ν ψ + (1/2)ψ 2 ˜ ω. h

Taking into account the fact that: ˜ := h˜ µν T˜ µν = −(h˜ −1 )µν ∂µ ψ∂ν ψ trT and introducing:

π˜ 1 = π˜ 1 − 2ωh˜

(5.21)

we can write the above in the form: µ D˜ µ P˜1 = −ρ(K 1 ψ + ωψ) − (1/2)T˜ µν π˜ 1,µν + (1/2)ψ 2 h˜ ω := Q˜ 1 .

Consider now the equation

D˜ µ P˜ µ = Q˜

(5.22)

(5.23)

˜ In arbitrary local coordinates we have: for an arbitrary vectorfield P˜ and function Q. 1 1 µ µ 2 ˜ ˜ ˜ ˜

∂µ ( −deth P ) = ∂µ ( −deth˜ P˜ µ ) = −2 Dµ P µ Dµ P =

−deth˜ 2 −deth˜ where: Hence, with:

P µ = 2 P˜ µ .

(5.24)

Q = 2 Q˜

(5.25)

equation (5.23) is equivalent to the equation: Dµ P µ = Q for the vectorfield P and function Q.

(5.26)


105

We may express equation (5.26) in acoustical coordinates (t, u, ϑ 1 , ϑ 2 ) (see Chapter 3). Expanding P in the associated coordinate frame field, ∂ ∂ ∂ ∂ , , , , ∂t ∂u ∂ϑ 1 ∂ϑ 2 P = Pt

∂ ∂ ∂ + Pu + (P ϑ ) A A , ∂t ∂u ∂ϑ A

and noting that, by the expression (2.37) for the metric h in acoustical coordinates, we have:

√ −deth = µ deth/ (5.27) equation (5.26) takes the form:

∂

∂ 1 (µ deth (µ deth/ P u ) + div √ /Pt ) + / M = µQ ∂u deth / ∂t

(5.28)

where M is the vectorfield on S 2 given in the local coordinates (ϑ 1 , ϑ 2 ) by: M =µ

∂ (P ϑ ) A A . ∂ϑ A

This follows noting that: div / M=√

∂

1 ( deth/ M A ). deth/ ∂ϑ A

We integrate equation (5.28) on S 2 with respect to the measure:

dµh/ = deth/dϑ 1 dϑ 2

(5.29)

to obtain the equation: ∂ ∂ t u µP dµh/ + µP dµh/ = µQdµh/ . ∂t ∂u St,u St,u St,u

(5.30)

Replacing (t, u) by (t , u ) and integrating with respect to (t , u ) on [0, t] × [0, u], we obtain, under the hypothesis that P vanishes in the closure of the exterior of C0 , E u (t) − E u (0) + F t (u) = Qdµh . (5.31) Wut

Here E u is the “energy”: E (t) = u

tu

µP dµh/ du = t

u 0

µP dµh/ du t

St,u

(5.32)

106


and F t is the “flux”:

F (u) = t

µP dµh/ dt = u

C ut

t

St ,u

0

µP dµh/ dt . u

(5.33)

In obtaining (5.31) we have used the fact that by virtue of (5.27), Qdµh = µQdµh/ du dt .

(5.34)

In the above formulas tu is the annular region: tu = St,u

(5.35)

Wut

St ,u

[0,t ]×[0,u]

u ∈[0,u]

in the space-like hyperplane t , Cut is the characteristic hypersurface Cu truncated from above by t : Cut = St ,u (5.36) t ∈[0,t ]

and

Wut

is the spacetime domain:

Wut =

St ,u

(5.37)

(t ,u )∈[0,t ]×[0,u]

bounded by the characteristic hypersurfaces C0 and Cu and the space-like hyperplanes 0 and t . Now, the vectorfield P can also be expanded in the frame field L, L, X 1 , X 2 (see Chapter 3): PAXA P = PL L + PL L + A

From equations (2.30), (2.31), (3.114), (3.24), L=

∂ ∂t

L = α −1 κ

(5.38)

∂ ∂ ∂ +2 − A A ∂t ∂u ∂ϑ

∂ . ∂ϑ A Substituting and equating coefficients in the expansion of P yields: XA =

(5.40)

P t = P L + α −1 κ P L

(5.41)

P = 2P

(5.42)

u

and:

(5.39)

L

(P ϑ ) A = P A − 2 A P L .

We shall write down the expressions for the energy and flux integrals corresponding to the vectorfields P0 and P1 defined above.


107

We begin with the table of components of the covariant form of the energy-momentum-stress tensor Tµν (see (5.13)) in the null frame L, L, X 1 , X 2 : TL L = (Lψ)2 TL L = (Lψ)2 TL L = µ|d/ψ|2 TL A = (Lψ)(d/ A ψ) TL A = (Lψ)(d/ A ψ) T AB = (d/ A ψ)(d/ B ψ) − (1/2)h/ AB (−µ−1 (Lψ)(Lψ) + |d/ψ|2 ).

(5.43)

Here, d/ A ψ = X A ψ and the pointwise norms of tensors on St,u are with respect to the induced metric h/. We also give here, for future reference, the table of the components of the associated contravariant tensor T µν = (h −1 )µκ (h −1 )νλ Tκλ = 2 T˜ µν . From the expression 3.116 for h −1 and the above table we obtain: (Lψ)2 (Lψ)2 |d/ψ|2 LL LL T = T = 4µ2 4µ2 4µ 1 1 = − (Lψ)(d/ A ψ) T L A = − (Lψ)(d/ A ψ) 2µ 2µ 1 −1 AB (h / ) (Lψ)(Lψ) + {(d/ A ψ)(d/ B ψ) − (1/2)(h/−1 ) AB |d/ψ|2 } (5.44) = 2µ

T LL = T LA T AB where:

d/ A ψ = (h/−1 ) AB d/ B ψ.

We consider first P0 . From equations (5.24) and (5.18) we have: µ

P0 = −T µν K 0ν . Here,

(5.45)

T µν = (h −1 )µκ Tκν = T˜ µν .

The components of P0 which are h-orthogonal to the surfaces St,u are: 1 h(P0 , L) = 2µ 1 P0L = − h(P0 , L) = 2µ L

P0 = −

((η−1 + α −1 κ)TL L + TL L ) 2µ 0 ((η−1 + α −1 κ)TL L + TL L ). 2µ 0

Substituting from Table 5.43 we obtain: ((η−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 ) 2µ 0 P0L = ((η−1 + α −1 κ)µ|d/ψ|2 + (Lψ)2 ). 2µ 0 L

P0 =

(5.46)

108


Substituting these expressions in (5.41) and (5.42), and those in turn in (5.32) and (5.33) we obtain the following expressions for the energy and flux integrals associated to the vectorfield K 0 : −1 α κ(η0−1 + α −1 κ)(Lψ)2 + (Lψ)2 E0u (t) = tu 2 + (η0−1 + 2α −1 κ)µ|d/ψ|2 dµh/ du (5.47)

F0t (u) = (η0−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 dµh/ dt . (5.48) C ut

We now consider P1 . From (5.19) and (5.24) we have: µ

P1 = −{T µν K 1ν + (h −1 )µν (ωψ∂ν ψ − (1/2)ψ 2 ∂ν ω)}.

(5.49)

The components of P1 which are h-orthogonal to the surfaces St,u are: 1 h(P1 , L) = 2µ 1 P1L = − h(P1 , L) = 2µ L

P1 = −

{ων −1 TL L + ωψ(Lψ) − (1/2)ψ 2 (Lω)} 2µ {ων −1 TL L + ωψ(Lψ) − (1/2)ψ 2 (Lω)}. 2µ

Substituting from Table 5.43 yields: {ων −1 (Lψ)2 + ωψ(Lψ) − (1/2)ψ 2 (Lω)} 2µ {ων −1 µ|d/ψ|2 + ωψ(Lψ) − (1/2)ψ 2 (Lω)}. P1L = 2µ L

P1 =

Substituting these in (5.41), (5.42) and those in turn in (5.32), (5.33) yields: −1 −1 E1u (t) = ων [α κ(Lψ)2 + µ|d/ψ|2 ] tu 2

(5.50)

(5.51)

+ωψ[α −1 κ(Lψ) + (Lψ)] − (1/2)ψ 2 [α −1 κ(Lω) + (Lω)] dµh/ du

t F1 (u) = ων −1 (Lψ)2 + ωψ(Lψ) − (1/2)ψ 2 (Lω) dµh/ dt . (5.52) C ut

Consider first the flux integral (5.52). We actually define the flux integral associated to the vectorfield K 1 to be: t ων −1 (Lψ + νψ)2 dµh/ dt . (5.53) F1 (u) = C ut

We then have: F1t (u) − F1t (u)

=−

C ut

(1/2)[L(ωψ 2 ) + 2νωψ 2 ]dµh/ dt .

(5.54)


109

Now, by the definition of χ, the 2nd fundamental form of St,u relative to Cu , equation (3.27), we have: (5.55) L / L dµh/ = trχdµh/ . It follows, in view of equation (5.10) that for an arbitrary function f defined in the spacetime domain Wε∗0 we have: ∂ f dµh/ = (L f + 2ν f )dµh/ . (5.56) ∂t St,u St,u Setting f = (1/2)ωψ 2 and comparing with (5.54) we deduce: t ∂ (1/2)ωψ 2 dµh/ dt F1t (u) − F1t = − St ,u 0 ∂t =− (1/2)ωψ 2 dµh/ + (1/2)ωψ 2 dµh/ . St,u

(5.57)

S0,u

Consider next the energy integral (5.51). We actually define the energy integral associated to the vectorfield K 1 to be: −1 −1 E1u (t) = ων {α κ(Lψ + νψ)2 + µ|d/ψ|2 }dµh/ du . (5.58) u 2 t Taking into account the fact that, by (3.114), L − α −1 κ L = 2T we find: E1u (t)

− E1u (t)

=

tu

2ωψ(T ψ) 2

(5.59)

(5.60)

−(1/2)[α −1 κ Lω + Lω + 2α −1 κνω]ψ 2 dµh/ du .

Now, from the definition of θ , the 2nd fundamental form of St,u relative to t , equation (3.44), we have: (5.61) L /T dµh/ = κtrθ dµh/ . It follows that for an arbitrary function f defined on the spacetime domain Wε∗0 : ∂ f dµh/ = {T f + [κtrθ + T (log )] f }dµh/ . ∂u St,u St,u Setting f = (1/2)ωψ 2 yields: 2ωψ(T ψ) + (T ω)ψ 2 + [κtrθ + T (log )]ωψ 2 dµh/ du tu 2 u ∂ 2 du ωψ = dµ = (1/2)ωψ 2 dµh/ . h/ St,u 2 St,u 0 ∂u

(5.62)

110


Here we have used the fact that ψ vanishes on St,0 ⊂ C0 . Comparing with (5.60) we obtain: (1/2)ωψ 2 dµh/ − I (5.63) E1u (t) − E1u (t) = St,u

where I is the integral: (1/2){2T ω + α −1 κ[Lω + (ν + αtrθ )ω] + ωT (log )}ψ 2 dµh/ du . I = tu

Taking account of (5.59), equation (5.10), equations (3.46) and (3.118) which give: trχ = α(trk/ − trθ ) trχ = κ(trk/ + trθ )

(5.64)

as well as equation (5.11), we find that the coefficient of (1/2)ψ 2 in the integrant of I is Lω + νω, hence I =

tu

(1/2)(Lω + νω)ψ 2 dµh/ du

and (5.63) takes the form: u u 2 E1 (t) − E1 (t) = (1/2)ωψ dµh/ − St,u

tu

(1/2)(Lω + νω)ψ 2 dµh/ du .

(5.65)

Substituting (5.65) and (5.57) in the identity (5.31) for K 1 , we see that the surface integral St,u

(1/2)ωψ 2 dµh/

cancels and the identity takes the form: u t E1 (t) + F1 (u) = (1/2)(Lω + νω)ψ 2 dµh/ du tu − (1/2)(Lω + νω)ψ 2 dµh/ du 0u

+ E1u (0) +

Wut

Q 1 dµh

(5.66)

with E1u and F1t given by (5.58) and (5.53) respectively. Also, the identity (5.31) for K 0 reads: E0u (t) + F0t (u) = E0u (0) +

Wut

with E0u and F0t given by (5.47) and (5.48) respectively.

Q 0 dµh

(5.67)


111

By virtue of the basic assumptions A1 and A2 there is a positive constant C such that: C −1 E0u (t)

u

≤

0

St,u

≤ CE0u (t) t C −1 F0t (u) ≤ 0

[η0−2 µ(η0

+

µ)(η0−2 (Lψ)2

+ |d/ψ| ) + (Lψ) ]dµh/ du 2

2

St ,u

(5.68)

[η0−2 (η0 + µ)(Lψ)2 + µ|d/ψ|2 ]dµh/ dt

≤ CF0t (u).

(5.69)

Moreover, by virtue of the assumptions A1, A2, B1 and D1 there is a positive constant C such that: u −2 −1 u 2 2 2 C E1 (t) ≤ (1 + t) µ[η0 (Lψ + νψ) + |d/ψ| ]dµh/ du 0

≤ C −1 F1t (u) ≤

CE1u (t) t

St,u

(1 + t )2

0

St ,u

(5.70)

(Lψ + νψ)2 dµh/ dt

≤ CF1t (u).

(5.71)

We shall now show that there is a positive numerical constant C such that for all u ∈ [0, ε0 ]: ψ 2 dµh/ ≤ ε0 CE0u (t). (5.72) St,u

To prove this we make use of the acoustical coordinates(t, u, ϑ), ϑ = (ϑ 1 , ϑ 2 ). Now, on a given hyperplane t we may set = 0, taking the coordinate lines ϑ = const. on the given t to be the integral curves of the vectorfield T (see Chapter 3). We then have: T =

∂ ∂u

hence, in view of the fact that ψ vanishes on C0 , u ψ(t, u, ϑ) = (T ψ)(t, u , ϑ)du .

(5.73)

0

It follows that:

St,u

ψ 2 dµh/˜ = =

S2

ψ 2 (t, u, ϑ)dµh/˜ (t, u, ϑ)

S2

≤ ε0

S2

u

2

(T ψ)(t, u , ϑ)du dµh/˜ (t, u, ϑ) 0 u 2 (T ψ) (t, u , ϑ)du dµh/˜ (t, u, ϑ). 0

(5.74)

112


Now, in view of the fact that h /˜ = h /, (5.61) implies: L /T dµh/˜ = (κtrθ + T log )dµh/˜

(5.75)

while from (5.64) and the definitions (5.10), (5.11), we have:

Thus:

κtrθ + T log = −α −1 κν + ν.

(5.76)

L /T dµh/˜ = (−α −1 κν + ν)dµh/˜ .

(5.77)

By virtue of assumptions B1, B2 (and A1, A2) we have |α −1 κν + ν| ≤ C(1 + t)−1 [1 + log(1 + t)]4 ≤ C .

(5.78)

Integrating (5.77) along the integral curves of T on t , using (5.79) and recalling that ε0 is subject to the restriction ε0 ≤ 1/2, we obtain: C −1 ≤

dµh/˜ (t, u, ϑ) dµh/˜ (t, 0, ϑ)

: for all (u, ϑ) ∈ [0, ε0 ] × S 2 .

≤C

(5.79)

In view of (5.79), the right-hand side of (5.74) is bounded by:

u

Cε0 0

S2

(T ψ) (t, u , ϑ)dµh/˜ (t, u , ϑ) du = Cε0 2

u

(T ψ) dµh/ du . 2

St,u

0

(5.80) By (5.59),

T ψ = (1/2)((Lψ) − α −1 κ(Lψ))

hence,

(T ψ)2 ≤ (1/4)((Lψ)2 + α −2 κ 2 (Lψ)2 ).

Comparing with (5.47), the result (5.72) then follows, in view of the basic assumption A1. We now introduce the following quantities which are, by their definition, nondecreasing functions of t at each u: u

u E 1 (t)

E 0 (t) = supt ∈[0,t ]E0u (t )

(5.81)

F0t (u)

(5.82)

= supt ∈[0,t ][1 + log(1 + t )]−4 E1u (t )

t F 1 (u) = u quantities E 0 (t),

t

supt ∈[0,t ] [1 + log(1 + t )]−4 F1 (u).

(5.83) (5.84)

u E 1 (t)

are also non-decreasing functions of u at each t. In Note that the view of inequalities (5.68)–(5.72) statements (i)–(v) of Theorem 5.1 are equivalent to the statement that, under the assumptions of the theorem, ε

E 00 (s),

ε

sup F0s (u), E 1 0 (s),

u∈[0,ε0 ]

s

sup F 1 (u) ≤ C D0 .

u∈[0,ε0 ]

(5.85)


113

In view of the identities (5.66), (5.67), this will follow if we succeed in estimating, under the said assumptions, the spacetime integrals Q 0 dµh , Q 1 dµh Wut

Wut

u

u

t

in terms of the four quantities E 0 (t), F0t (u), E 1 (t), F 1 (u). From (5.20), (5.25), we have: Q 0 = −2 ρ K 0 ψ − (1/2)T µν π˜ 0,µν = Q 0,0 + Q 0,1 + Q 0,2 + Q 0,3 + Q 0,4 + Q 0,5 + Q 0,6 + Q 0,7

(5.86)

where (see Table 5.44): Q 0,0 = −2 ρ K 0 ψ Q 0,1 = −(1/2)T L L π˜ 0,L L = −(1/8)µ−2 (Lψ)2 π˜ 0,L L

(5.87) (5.88)

Q 0,2 = −(1/2)T L L π˜ 0,L L = −(1/8)µ−2 (Lψ)2 π˜ 0,L L

(5.89)

Q 0,3 = −T

LL

π˜ 0,L L = −(1/4)µ

−1

|d/ψ| π˜ 0,L L

−1

LA

2

A

Q 0,4 = −T π˜ 0,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 0,L A Q 0,5 = −T L A π˜ 0,L A = (1/2)µ−1 (Lψ)(d/ A ψ)π˜ 0,L A

(5.90) (5.91) (5.92)

and, −(1/2)T AB π˜ 0,AB = Q 0,6 + Q 0,7 where: Q 0,6 = −(1/2){d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 }π /ˆ˜ 0,AB Q 0,7 = −(1/4)µ−1(Lψ)(Lψ)trπ /˜ 0 .

(5.93) (5.94)

Here π /˜ 0 denotes the restriction to St,u of π˜ 0 , π /˜ 0,AB = π˜ 0,AB , /˜ 0 and we have made use of the trace-free nature of and π /ˆ˜ 0 denotes the trace-free part of π the factor d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 . We have (see (5.17)): π˜ 0 = π0 + (K 0 )h and the components of π0 = L K 0 h can be directly calculated from Table 3.117 of connection coefficients of the null frame L, L, X 1 , X 2 , as the vectorfield K 0 is expressed in

114


this frame by (5.15). We find in this way: π˜ 0,L L = 0 π˜ 0,L L = −4µ{L(α −1 κ) − (η0−1 + α −1 κ)L(α −1 κ)} π˜ 0,L L = −2µ{µ−1 (K 0 µ) + (K 0 log ) + 2L(α −1 κ)} π˜ 0,L A = −2(ζ A + η A ) π˜ 0,L A = 2{(η0−1 + α −1 κ)(ζ A + η A ) − µd/ A (α −1 κ)} π /˜ˆ 0,AB = 2{(η0−1 + α −1 κ)χˆ AB + χˆ AB } trπ /˜ 0 = 4{(η0−1 + α −1 κ)ν + ν}.

(5.95)

From (5.22), (5.25) we have: Q 1 = −2 ρ K 1 ψ − (1/2)T µν π˜ 1,µν + (1/2)2 ψ 2 h˜ ω

(5.96)

= Q 1,0 + Q 1,1 + Q 1,2 + Q 1,3 + Q 1,4 + Q 1,5 + Q 1,6 + Q 1,7 + Q 1,8 where (see Table 5.44): Q 1,0 = −2 ρ(K 1 ψ + ωψ) Q 1,1 = Q 1,2 = Q 1,3 = Q 1,4 = Q 1,5 = and,

−2 2 −(1/2)T π˜ 1,L ˜ 1,L L L = −(1/8)µ (Lψ) π LL −2 2 −(1/2)T π˜ 1,L L = −(1/8)µ (Lψ) π˜ 1,L L −1 −T L L π˜ 1,L /ψ|2 π˜ 1,L L = −(1/4)µ |d L LA −1 A −T π˜ 1,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 1,L A LA −1 A −T π˜ 1,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 1,L A LL

(5.97) (5.98) (5.99) (5.100) (5.101) (5.102)

−(1/2)T AB π˜ 1,AB = Q 1,6 + Q 1,7

where: Q 1,6 = −(1/2){d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 }π /ˆ˜ 1,AB Q 1,7 =

−(1/4)µ−1 (Lψ)(Lψ)trπ /˜ 1 .

(5.103) (5.104)

Also, Q 1,8 = (1/2)2 ψ 2 h˜ ω. Here

π /˜ 1

(5.105)

denotes the restriction to St,u of π˜ 1 , , π /˜ 1,AB = π˜ 1,AB

π /ˆ˜ 1 denotes the trace-free part of π /˜ 1 and we have again made use of the trace-free nature of the factor d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 .


115

By (5.21) and the fact that π˜ 1 = π1 + (K 1 )h we have:

π˜ 1 = {π1 + (K 1 log − 2ω)h}

and the components of π1 = L K 1 h can be directly calculated from Table 3.117, as the vectorfield K 1 is expressed by (5.16). We find in this way: π˜ 1,L L =0 −1 −1 −1 π˜ 1,L L = −4µ{L(ν ω) − ν ωL(α κ)} −1 −1 π˜ 1,L L = −2µ{µ K 1 µ + L(ν ω) − 2ω + K 1 log } π˜ 1,L A =0 −1 π˜ 1,L / A (ν −1 ω)} A = 2{ν ω(ζ A + η A ) − µd

π /ˆ˜ 1,AB = 2ν −1 ωχˆ AB

trπ /˜ 1 = 0.

(5.106)

We now consider the spacetime integral of Q 0 . In view of the formula (5.27), we have, for an arbitrary function f , Wut

t

µ−1 f dµh =

tu

0

f

u

dt =

0

f

C ut

du

(5.107)

where:

tu

u

f =

0

St ,u

f dµh/ du

C ut

f =

t 0

St ,u

f dµh/ dt .

(5.108)

Now, we are considering the case that the source function ρ vanishes. Thus, Q 0,0 = 0.

(5.109)

Q 0,1 = 0.

(5.110)

µ−1 |π˜ 0,L L | ≤ C(1 + t)[1 + log(1 + t)]−2

(5.111)

µ|Q 0,2 | ≤ C(1 + t)[1 + log(1 + t)]−2 (Lψ)2 .

(5.112)

By the first of formulas (5.95), also:

By assumptions B10, B11 and A1,

hence, from (5.89),

116


Writing: (Lψ)2 ≤ 2(Lψ + νψ)2 + 2(νψ)2

it then follows that:

Wut

where:

t

J0 =

|Q 0,2 |dµh ≤ C(J0 + J1 )

(1 + t )[1 + log(1 + t )]−2

and:

J1 =

t

(1 + t )[1 + log(1 + t )]−2

(νψ)2 dt

(5.114)

tu

0

tu

0

(5.113)

(Lψ + νψ)2 dt .

(5.115)

Now by assumption B1,

t

J0 ≤ C

−1

(1 + t )

[1 + log(1 + t )]

−2

tu

0

ψ

2

dt .

Substituting the estimate (5.72) we then obtain: t J0 ≤ Cε02 (1 + t )−1 [1 + log(1 + t )]−2 E0u (t )dt .

(5.116)

0

On the other hand,

t

J1 =

f 0 (t )

0

where

g(t) =

t

2

dg(t ) dt dt

(1 + t )

tu

0

(Lψ + νψ)

2

dt

(5.117)

and f 0 (t) is the function: f 0 (t) = (1 + t)−1 [1 + log(1 + t)]−2

(5.118)

Integrating by parts we obtain (g(0) = 0): J1 = f 0 (t)g(t) −

t

g(t )

0

d f 0 (t ) dt . dt

Now, by (5.69) we have: t 0

hence:

t 0

tu

St ,u

(Lψ)2 dµh/ dt ≤ CF0t (u)

(Lψ)2 dµh/ dt ≤ C

u 0

F0t (u )du .

(5.119)

(5.120)


117

Also, by inequality (5.72) and assumption B1, t 0

(νψ) dµh/ dt ≤ Cε0

t

2

tu

Let us set:

(1 + t )

t 0

sup

t

g(t) =

ψ dµh/ dt (5.121)

1/2 tu

We then have:

St ,u

(1 + t )−2 E0u (t )dt .

F(t, u) =

2

u ∈[0,u]

0

≤ Cε02

−2

(Lψ + νψ)2

.

(5.122)

(1 + t )2 F 2 (t , u)dt

(5.123)

0

while: f 0 (t)g(t) ≤ [1 + log(1 + t)]−2

t

≤

F 2 (t , u)dt

t

(1 + t )F 2 (t , u)dt

0 1/2

[1 + log(1 + t)]−4

(5.124)

0

t

(1 + t )2 F 2 (t , u)dt

1/2 .

0

Now, by (5.120) and (5.121), t F 2 (t , u)dt ≤ C 0

u 0

F0t (u )du + Cε02

t 0

On the other hand, by (5.71) we have: t (1 + t )2 F 2 (t , u)dt ≤ C 0

(1 + t )−2 E0u (t )dt .

u 0

(5.125)

F1t (u )du .

(5.126)

Substituting (5.125) and (5.126) in (5.124) then yields: 1/2 u t f 0 (t)g(t) ≤ C F0t (u )du + ε02 (1 + t )−2 E0u (t )dt 0

0

· [1 + log(1 + t)]−4

u 0

Moreover, since

F1t (u )du

1/2 .

(5.127)

d f0 −2 −2 dt ≤ C(1 + t) [1 + log(1 + t)] ,

in view of (5.123), (5.126) we have: u t t d f 0 (t ) −2 −2 t g(t ) (1 + t ) [1 + log(1 + t )] F1 (u )du dt . dt ≤ C dt 0 0 0 (5.128)

118


The bounds (5.127), (5.128) together imply:

u

J1 ≤ C 0

F0t (u )du

+

ε02

t

(1 +

0

t )−2 E0u (t )dt

1/2

1/2 F1t (u )du 0 u t +C (1 + t )−2 [1 + log(1 + t )]−2 F1t (u )du dt . · [1 + log(1 + t)]−4

u

0

(5.129)

0

Substituting the estimates (5.116) and (5.129) in (5.113) then yields: t |Q 0,2 |dµh ≤ Cε02 (1 + t )−1 [1 + log(1 + t )]−2 E0u (t )dt Wut

0

u

+C

F0t (u )du + ε02

0

0

t

(1 + t )−2 E0u (t )dt

−4 · [1 + log(1 + t)]

u 0

t

+C

−2

(1 + t )

(5.130)

[1 + log(1 + t )]

F1t (u )du

−2

0

0

u

1/2

1/2

F1t (u )du

dt .

We skip for the moment the terms Q 0,3 , Q 0,4 , Q 0,5 and we turn to the estimation of the spacetime integrals of Q 0,6 and Q 0,7 . Now π /ˆ˜ 0 is given in Table 5.95. By assumptions B3, B4, A1 and A3 (recall that −1 κ = α µ) we have: (5.131) |π /ˆ˜ 0 | ≤ C(1 + t)[1 + log(1 + t)]−6 . In view of (5.70) and (5.83) we can then estimate: t |Q 0,6 |dµh ≤ C (1 + t )−1 [1 + log(1 + t )]−6 E1u (t )dt Wut

0

≤C

0

t

u

(1 + t )−1 [1 + log(1 + t )]−2 E 1 (t )dt .

(5.132)

Also, trπ /˜ 0 is given in Table 5.95. By assumptions B1, B2, A1 and A3 we have: |trπ /˜ 0 | ≤ C(1 + t)−1 [1 + log(1 + t)]4 .

(5.133)

Writing |Lψ| ≤ |Lψ + νψ| + |νψ| we can then estimate:

Wut

|Q 0,7 |dµh ≤ C(J2 + J3 )

(5.134)


119

where: J2 =

t

f 1 (t )

|νψ||Lψ|

(5.135)

|Lψ + νψ||Lψ|

(5.136)

f 1 (t) = (1 + t)−1 [1 + log(1 + t)]4 .

(5.137)

tu

0

J3 =

t

f 1 (t )

tu

0

and: We estimate:

1/2 1/2

J2 ≤ I0 I1

(5.138)

where:

t

I0 =

(1 + t ) f 1 (t )

tu

0

t

I1 =

(1 + t )−1 f 1 (t )

(νψ)

0

2

dt

(5.139)

tu

(Lψ)2 dt .

(5.140)

By assumption B1 and the estimate (5.72) we have, with f 2 (t) = (1 + t)−1 f 1 (t) = (1 + t)−2 [1 + log(1 + t)]4

t

I0 ≤ Cε0

f 2 (t )E0u (t )dt .

0

Also, by (5.70),

t

I1 ≤ C We thus obtain: 1/2

J2 ≤ Cε0

I2 = 0

t

(5.143)

f2 (t )E0u (t )dt .

(5.144)

1 (I1 + I2 ) 2

J3 ≤ where:

t 0

We estimate:

(5.142)

f 2 (t )E0u (t )dt .

0

(1 + t ) f 1 (t )

(5.141)

(5.145)

tu

(Lψ + νψ)

Recalling the definition (5.117) we can write: t dg(t ) f 2 (t ) dt . I2 = dt 0

2

dt .

(5.146)

(5.147)

120


Now the integral I2 is similar to the integral J1 but with the function f 2 in the role of the function f0 . Since f 2 decays faster than f 0 (and the same is true for their derivatives), the integral I2 satisfies an estimate similar to (5.129):

u

I2 ≤ C 0

F0t (u )du + ε02

t 0

(1 + t )−2 E0u (t )dt

1/2

1/2 F1t (u )du 0 u t +C (1 + t )−2 [1 + log(1 + t )]−2 F1t (u )du dt . · [1 + log(1 + t)]−4

u

0

Combining the above estimates we conclude that: t |Q 0,7 |dµh ≤ C f2 (t )E0u (t )dt Wut

(5.148)

0

0

u

+C 0

F0t (u )du

(5.149)

+ ε02

0

t

(1 + t )−2 E0u (t )dt

· [1 + log(1 + t)]−4

1/2

1/2 F1t (u )du 0 u t −2 −2 +C (1 + t ) [1 + log(1 + t )] F1t (u )du dt . u

0

0

We turn to the spacetime integral of Q 1 . Since we are considering the case that the source function ρ vanishes, we have: Q 1,0 = 0.

(5.150)

Q 1,1 = Q 1,4 = Q 1,7 = 0.

(5.151)

In view of Table 5.106, also:

Consider then Q 1,2 (5.99). The coefficient π˜ 1,L L is given in Table 5.106. By assumptions B1, B10, B13, D1, D3 (and A1), 3 µ−1 |π˜ 1,L L | ≤ C(1 + t)[1 + log(1 + t)]

(5.152)

µ|Q 1,2 | ≤ C(1 + t)[1 + log(1 + t)]3 (Lψ)2 .

(5.153)

hence: Writing again, (Lψ)2 ≤ 2(Lψ + νψ)2 + 2(νψ)2 , it then follows that:

Wut

|Q 1,2 |dµh ≤ C(J4 + J5 )

(5.154)


121

where:

t

J4 =

(1 + t )[1 + log(1 + t )]3

0

t

J5 =

(1 + t )[1 + log(1 + t )]

0

tu

(νψ)2 dt

(5.155)

3

tu

(Lψ + νψ)

2

dt .

(5.156)

Now, by assumption B1 and the estimate (5.72), t J4 ≤ Cε02 (1 + t )−1 [1 + log(1 + t )]3 E0u (t )dt 0 u

≤ Cε02 E 0 (t)[1 + log(1 + t)]4 .

(5.157)

On the other hand, appealing to the definition (5.117) we can write: t dg(t ) J5 = f 3 (t ) dt dt 0

(5.158)

where f 3 is the function: f 3 (t) = (1 + t)−1 [1 + log(1 + t)]3 . Thus, the integral J5 can be estimated in a similar way to that in which the integral J1 was estimated above (see (5.117)–(5.129)), and we obtain, taking into account the fact that the function g(t) is non-decreasing, t t d f 3 (t ) d f 3 (t ) J5 = f 3 (t)g(t) − g(t ) dt ≤ g(t) f (t) + 3 dt dt dt 0 0 u ≤C F1t (u )du (5.159) 0

the factor in parentheses being bounded by a numerical constant. In conclusion, u u |Q 1,2 |dµh ≤ C F1t (u )du + Cε02 E 0 (t)[1 + log(1 + t)]4 . (5.160) Wut

0

Consider next Q 1,6 (5.103). The coefficient π /ˆ˜ 1 is given in Table 5.106. By assumptions A1, B1, B3 and D1, |π /ˆ˜ 1 | ≤ C(1 + t)[1 + log(1 + t)]−2 . Hence, by (5.70), t −2 |Q 1,6 |dµh ≤ C (1 + t )[1 + log(1 + t )] Wut

0 t

≤C 0

(5.161) tu

µ|d/ψ|

(1 + t )−1 [1 + log(1 + t )]−2 E1u (t )dt .

2

dt (5.162)

122


To estimate the spacetime integral of the term Q 1,8 , we use assumption D5 and the estimate (5.72). We have: t u 2 |Q 1,8 |dµh ≤ C sup St ,u (µ|h˜ ω|) · ψ dµh/ du dt Wut

0

≤ Cε0

0

St ,u

t 0

u 0

sup St ,u (µ|h˜ ω|)du E0u (t )dt .

Appealing to assumption D5 we then conclude that: u |Q 1,8 |dµh ≤ Cε0 [1 + log(1 + t)]4 E 0 (t).

(5.163)

Wut

We now turn to the crucial terms Q 0,3 and Q 1,3 . Consider first Q 1,3 . The coefficient π˜ 1,L L is given in Table 5.106. We have: −1 −1 π˜ 1,L L = −2µ(ων )(µ Lµ + r1 )

(5.164)

where r1 is the remainder: r1 = ω−1 (Lω − νω) − ν −1 (Lν + ν 2 ) + L log . Thus, Wut

Q 1,3 dµh =

t 0

tu

−1 −1 2 (ων )(µ Lµ + r1 )µ|d/ψ| dt . 2

(5.165)

(5.166)

Decomposing µ−1 Lµ into its positive and negative parts, µ−1 Lµ = µ−1 (Lµ)+ + µ−1 (Lµ)− we write: t Q 1,3 dµh = Wut

0

tu

(ων −1 ){µ−1 (Lµ)+ + µ−1 (Lµ)− + r1 }µ|d/ψ|2 dt . 2 (5.167)

Thus, by virtue of assumption C1 and the formula (5.58): t Q 1,3 dµh ≤ {(1 + t )−1 [1 + log(1 + t )]−1 + A(t ) + sup |r1 |}E1u dt − K (t, u) Wut

0

tu

where K (t, u) is the – non-negative – spacetime integral: −1 −1 K (t, u) = − ων µ (Lµ)− |d/ψ|2 dµh . t 2 Wu

(5.168)

(5.169)


123

Moreover, by assumptions B1, D1, B12, D2 and B5 we have: sup |r1 | ≤ C(1 + t)−1 [1 + log(1 + t)]−2 . ε

(5.170)

t 0

Consider next Q 0,3 . The coefficient π˜ 0,L L is given in Table 5.95. We have: π˜ 0,L L = −2µ(µ−1(Lµ + Lµ) + r0 )

(5.171)

where r0 is the remainder: r0 = α −2 Lµ + 2L(α −1 κ) + K 0 log = 3L(α −1 κ) + 2α −2 κ Lα + (1 + α −1 κ)L log + L log recalling that µ = ακ. Thus, t Q 0,3 dµh = Wut

0

tu

−1 (µ (Lµ + Lµ) + r0 )µ|d/ψ|2 dt . 2

(5.172)

(5.173)

Decomposing µ−1 (Lµ + Lµ) into its positive and negative parts, µ−1 (Lµ + Lµ) = µ−1 (Lµ + Lµ)+ + µ−1 (Lµ + Lµ)− we then have: t Q 0,3 dµh ≤ Wut

0

tu

−1 2 {µ (Lµ + Lµ)+ + |r0 |}µ|d/ψ| dt . 2

(5.174)

Thus, by virtue of assumption C2, the formula (5.58), as well as assumptions B1, D1, t Q 0,3 dµh ≤ (1 + t )−2 {B(t ) + sup |r0 |}E1u dt . (5.175) Wut

0

tu

Moreover, by assumptions B5, B6, B9 and B10 we have: sup |r0 | ≤ C(1 + t)[1 + log(1 + t)]−6 . ε

(5.176)

t 0

We finally have the terms Q 0,4 , Q 0,5 , Q 1,5 . Consider first Q 1,5 . From (5.102) and (5.106) we have: |Q 1,5 |dµh ≤ M1 + R1

(5.177)

Wut

where:

M1 = R1 =

Wut

Wut

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.178)

|Lψ||d/ψ||d/(ων −1 )|dµh .

(5.179)

124


We first estimate the main integral M1 . We decompose: M1 = M1 + M1

where: M1

=

M1 =

U

Uc

Wut

Wut

(5.180)

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.181)

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.182)

and U is the region defined by (5.12). According to assumption C3 we have: −(Lµ)− ≥ C −1 (1 + t )−1 [1 + log(1 + t )]−1 in U

Wut .

(5.183)

Comparing with the definition (5.169) we conclude that: −1 −1 ων µ (Lµ)− |d/ψ|2 dµh K ≥− U Wut 2 1 ≥ ων −1 (1 + t )−1 [1 + log(1 + t )]−1 µ−1 |d/ψ|2 dµh . (5.184) 2C U Wut We can thus estimate: where:

N1 =

Wut

M1 ≤ C K 1/2 N1

1/2

(5.185)

(ων −1 )(1 + t )[1 + log(1 + t )]|ζ + η|2 |Lψ|2 µ−1 dµh .

(5.186)

By virtue of assumption B7 we have: N1 ≤ C (ων −1 )(1 + t )−1 [1 + log(1 + t )]3 |Lψ|2 µ−1 dµh .

(5.187)

Wut

Thus, N1 ≤ C(N1,0 + N1,1 ) where:

N1,0 = N1,1 =

Wut

Wut

(5.188)

(ων −1 )(1 + t )−1 [1 + log(1 + t )]3 (νψ)2 µ−1 dµh

(5.189)

(ων −1 )(1 + t )−1 [1 + log(1 + t )]3 (Lψ + νψ)2 µ−1 dµh .

(5.190)

Now, by assumptions B1, D1, and A1,

t

N1,0 ≤ C 0

−1

(1 + t )

[1 + log(1 + t )]

3 0

u

ψ dµh/ du 2

St ,u

dt


125

hence, by the estimate (5.72), N1,0 ≤ ≤

Cε02

t

(1 + t )−1 [1 + log(1 + t )]3 E0u (t )dt

0 u Cε02 E 0 (t)[1 + log(1 + t)]4 .

Also, trivially, in view of the formula (5.53), u

N1,1 ≤ C

C ut

0

We conclude that:

N1 ≤ C

ων

−1

(Lψ + νψ)

(5.191)

2

u ε02 E 0 (t)[1 + log(1 + t)]4

du = C

u 0

u

+ 0

F1t (u )du .

F1t (u )du

(5.192)

(5.193)

hence from (5.185): M1

≤ C K (t, u)

Since in U c

1/2

2 u 4 ε0 E 0 (t)[1 + log(1 + t)] +

u 0

F1t (u )du

1/2

Wut it holds that µ ≥ η0 /4, 2 M1 ≤ √ ων −1 |Lψ||d/ψ||ζ + η|µ−1/2dµh η0 U c Wut 2 ≤ √ ων −1 |Lψ||d/ψ||ζ + η|µ−1/2 dµh η0 Wut 1/2 ων −1 |d/ψ|2 2 1/2 dµh ≤ √ · N1 η0 Wut (1 + t )[1 + log(1 + t )]

where N1 is given by (5.186). Thus, by formula (5.58), t 1/2 E1u (t )dt 1/2 M1 ≤ C N1 )[1 + log(1 + t )] (1 + t 0 and N1 has already been estimated in (5.193) above. We conclude that: 1/2 t E1u (t )dt M1 ≤ C 0 (1 + t )[1 + log(1 + t )] 1/2 u u · ε02 E 0 (t)[1 + log(1 + t)]4 + F1t (u )du .

.

(5.194)

(5.195)

(5.196)

(5.197)

0

We finally estimate the remainder integral R1 of (5.179). Now, assumptions B14 and D4 (together with B1, D1) imply that: ω−1 ν|d/(ων −1 )| ≤ C(1 + t)−1 [1 + log(1 + t)]1/2

(5.198)

126


hence, taking into account assumption A3, the factor µ1/2 ω−1 ν|d/(ων −1 )| is bounded in the same way as |ζ + η|, namely by C(1 + t)−1 [1 + log(1 + t)]. It follows that R1 is bounded in the same way as M1 . Collecting the above results we conclude that: 1/2 t u E (t )dt 1 |Q 1,5 |dµh ≤ C K (t, u)1/2 + 0 (1 + t )[1 + log(1 + t )] Wut 1/2 u 2 u 4 t F1 (u )du . (5.199) · ε0 E 0 (t)[1 + log(1 + t)] + 0

We now consider the term Q 0,4 . From (5.91) and (5.95) we have: |Q 0,4 |dµh ≤ M0

(5.200)

Wut

where: M0 =

Wut

Again, we decompose: where:

(5.201)

M0 = M0 + M0

(5.202)

M0 =

U

M0

|Lψ||d/ψ||ζ + η|µ−1 dµh .

=

Uc

Wut

|Lψ||d/ψ||ζ + η|µ−1 dµh

Wut

(5.203)


(5.204)

Let us define, for 0 ≤ t0 < t1 ≤ s, the region: Wut0 ,t1 = {x ∈ Wus : t0 ≤ x 0 ≤ t1 }.

(5.205)

Now, by (5.184) and assumptions B1, D1, 1 K (t, u) ≥ (1 + t )[1 + log(1 + t )]−1 µ−1 |d/ψ|2 dµh . C U Wut

(5.206)

We can thus estimate: /ψ||ζ + η|µ−1 dµh t ,t |Lψ||d U

Wu0

1

≤ C K (t1 , u)

1/2 t ,t1

≤ C K (t1 , u)1/2

t ,t1

≤ C K (t1 , u)1/2

Wu0

Wu0 t1

t0

−1

(1 + t )

2 −1

[1 + log(1 + t )]|Lψ| |ζ + η| µ 2

(1 + t )−3 [1 + log(1 + t )]3 |Lψ|2 µ−1 dµh

(1 + t )−3 [1 + log(1 + t )]3 E0u (t )dt

using assumption B7 and recalling formula (5.47).

1/2 dµh

1/2

1/2 (5.207)


127

Let us define: K (t, u) = sup [1 + log(1 + t )]−4 K (t , u)

(5.208)

t ∈[0,t ]

a non-decreasing function of t as well as u. Then, provided that there is a numerical constant C such that: 1 + log(1 + t1 ) ≤C (5.209) 1 + log(1 + t0 ) we may conclude from the above that: /ψ||ζ + η|µ−1 dµh t ,t |Lψ||d U

Wu0

(5.210)

1

≤ C K (t1 , u)1/2

t1 t0

u

(1 + t )−3 [1 + log(1 + t )]7 E 0 (t )dt

1/2 .

Let the non-negative integer N be the integral part of log t/ log 2. Then: log t = N + r, log 2

0 ≤ r < 1.

(5.211)

We set: t−1 = 0,

tn = 2n+r : n = 0, 1, . . . , N.

(5.212)

Then t N = t and we have the partition: Wut =

N

t

,t

Wun−1 n .

(5.213)

n=0

Hence, M0

=

N

M0,n

(5.214)

n=0

where, for n = 0, 1, . . . , N, = M0,n

U

t

Wun−1

,tn


(5.215)

Now with (tn−1 , tn ) in the role of (t0 , t1 ), condition (5.209) holds for a fixed numerical constant C. Therefore (5.210) applies with (tn−1 , tn ) in the role of (t0 , t1 ): ≤ C K (tn , u)1/2 An M0,n

1/2

where:

An =

tn tn−1

(5.216) u

(1 + t )−3 [1 + log(1 + t )]7 E 0 (t )dt .

(5.217)

128


Since K (tn , u) ≤ K (t, u), substituting in (5.214) we obtain: M0 ≤ C K (t, u)1/2

N

1/2

An

n=0

≤ C K (t, u)

1/2

N

1/2 (n + 1)

−2

n=0

≤ C K (t, u)1/2

N

1/2 (n + 1)2 An

N (n + 1)2 An

1/2

n=0

.

(5.218)

n=0

Now, for n = 0, 1, . . . , N and t ∈ [tn−1 , tn ], [1 + log(1 + t )]2 ≥ [1 + log(1 + tn−1 )]2 ≥ C −1 (n + 1)2 . It follows that: N N (n + 1)2 An ≤ C n=0

u

n=0 tn−1 t

=C

tn

0

(1 + t )−3 [1 + log(1 + t )]9 E 0 (t )dt u

(1 + t )−3 [1 + log(1 + t )]9 E 0 (t )dt .

Substituting in (5.218) we conclude that: t 1/2 u M0 ≤ C K (t, u)1/2 (1 + t )−3 [1 + log(1 + t )]9 E 0 (t )dt . 0

To estimate the integral M0 (5.204) we note that since µ ≥ η0 /4 in U c : 2 |Lψ||d/ψ||ζ + η|µ−1/2 dµh M0 ≤ √ η0 U c Wut 2 ≤ √ |Lψ||d/ψ||ζ + η|µ−1/2 dµh η0 Wut t 2 1/2 = √ |Lψ||d/ψ||ζ + η|µ dt . η0 0 u t

Hence: M0

1/2 1/2 t 2 2 2 2 ≤ √ µ||d/ψ| · |Lψ| |ζ + η| dt η0 0 tu tu t u 1/2 [1 + log(1 + t )]E0u (t )1/2 E1 (t ) · dt ≤C (1 + t ) 0 (1 + t )

(5.219)

(5.220)


t

≤C

−2

(1 + t )

0

[1 + log(1 + t t

·

129

0

u )]3 E 1 (t )dt

1/2

u

(1 + t )−2 [1 + log(1 + t )]3 E 0 (t )dt

1/2 .

(5.221)

From (5.200), (5.202), (5.220), (5.221) we conclude that: t 1/2 u |Q 0,4 |dµh ≤ C K (t, u)1/2 (1 + t )−3 [1 + log(1 + t )]9 E 0 (t )dt Wut

0

t

+C 0

t

·

u

(1 + t )−2 [1 + log(1 + t )]3 E 1 (t )dt −2

(1 + t )

[1 + log(1 + t

0

u )]3 E 0 (t )dt

1/2

1/2 .

We finally consider the term Q 0,5 . From (5.92) and (5.95) we have: |Q 0,5 |dµh ≤ M˜ 0 + R0

(5.222)

(5.223)

Wut

where: M˜ 0 =

R0 =

Wut

Wut

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh

(5.224)

|Lψ||d/ψ||d/(α −1 κ)|dµh .

(5.225)

Decomposing: M˜ 0 = M˜ 0 + M˜ 0 M˜ 0 = |Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh

(5.226)

M˜ 0 =

(5.228)

U

Uc

(5.227)

Wut

Wut

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh

we handle M˜ 0 in a way similar to that in which M0 was handled. By virtue of inequality (5.206) we have: /ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh t ,t |Lψ||d U

Wu0

1

≤ C K (t1 , u)

1/2 t ,t1

Wu0

(1 + t )−1 [1 + log(1 + t )]

(Lψ)2 (η0−1 + α −1 κ)2 |ζ + η|2 µ−1 dµh

1/2

130


≤ C K (t1 , u)1/2

t ,t1

Wu0

(1 + t )−3 [1 + log(1 + t )]4

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

1/2

where we have used assumptions A2, A3 and B7. It follows that under the condition 5.209 (see definition 5.208): /ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh t ,t |Lψ||d U

Wu0

1

≤ C K (t1 , u)

1/2 t ,t1

Wu0

(1 + t )−3 [1 + log(1 + t )]8

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

1/2

.

(5.229)

In reference to (5.211)–(5.213) we have: M˜ 0 =

N

M˜ 0,n

(5.230)

n=0

where, M˜ 0,n =

U

t

Wun−1

,tn

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh .

(5.231)

The estimate (5.229) then applies with (tn−1 , tn ) in the role of (t0 , t1 ): 1/2 M˜ 0,n ≤ C K (tn , u)1/2 A˜ n

where: A˜ n =

t

Wun−1

,tn

(5.232)

(1 + t )−3 [1 + log(1 + t )]8 (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh .

(5.233)

We thus obtain, as in (5.218), M˜ 0 ≤ C K (t, u)1/2

N (n + 1)2 A˜ n

1/2

n=0

and we have: N N 2 ˜ (n + 1) An ≤ C

t

,tn

(1 + t )−3 [1 + log(1 + t )]10

n−1 n=0 Wu (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

n=0

=C

Wut

(1 + t )−3 [1 + log(1 + t )]10

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh .

(5.234)


131

Now the factor (1 +t )−3 [1 +log(1 +t )]10 is bounded by a numerical constant. It follows that: N (n + 1)2 A˜ n ≤ C (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh . Wut

n=0

In view of formula (5.48) we then obtain: N 2 ˜ (n + 1) An ≤ C

u 0

n=0

F0t (u )du

(5.235)

and (5.234) then yields: M˜ 0 ≤ C K (t, u)1/2

0

u

F0t (u )du

1/2 .

(5.236)

Next we estimate M˜ 0 . Since µ ≥ η0 /4 in U c Wut , we have, by assumptions A2, A3 and B7, 2 |Lψ||d/ψ||ζ + η|(η0−1 + α −1 κ)µ−1/2dµh M˜ 0 ≤ √ η0 U c Wut ≤C (1 + t )−1 [1 + log(1 + t )]3/2|Lψ||d/ψ|(η0−1 + α −1 κ)1/2 µ−1/2 dµh . Wut

Since the factor (1 + t )−1 [1 + log(1 + t )]3/2 is bounded by a numerical constant we obtain: M˜ 0 ≤ C {(η−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 }µ−1 dµh . Wut

0

In view of formula (5.48) we then conclude that: u F0t (u )du . M˜ 0 ≤ C

(5.237)

0

Finally, the remainder integral R0 is under assumption B8 bounded by: (1 + t )−1 [1 + log(1 + t )]|Lψ||d/ψ|(η0−1 + α −1 κ)1/2 µ−1/2 dµh R0 ≤ C Wut

noting that by (5.9), (η0−1 + α −1 κ)1/2µ−1/2 = ((η0−1 + α −2 µ)/µ)1/2 ≥ α −1 ≥ 1. Since the factor (1 + t )−1 [1 + log(1 + t )] is bounded by a numerical constant we again obtain: {(η0−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 }µ−1 dµh R0 ≤ C ≤C

Wut u

0

F0t (u )du .

(5.238)

132


Putting together the above results we conclude that:

Wut

u

|Q 0,5 |dµh ≤ C K (t, u)1/2 0

F0t (u )du

1/2

u

+C 0

F0t (u )du .

(5.239)

We now focus attention on the integral identity (5.66). By (5.96) and (5.150), (5.151), (5.160), (5.162), (5.163), (5.168) (and (5.170)) and (5.199), the spacetime integral in the right-hand side of (5.66) is bounded from above by: t ˜ )E u (t )dt Q 1 dµh ≤ C M(t, u) + L(t, u) + A(t 1 Wut

0

−K (t, u) + C(K (t, u)1/2 + L(t, u)1/2 )M(t, u)1/2 .

(5.240)

Here, u u M(t, u) = E 0 (t)[1 + log(1 + t)]4 + F1t (u )du 0 t L(t, u) = (1 + t )−1 [1 + log(1 + t )]−1 E1u (t )dt

(5.241) (5.242)

0

and, ˜ = A(t) + C(1 + t)−1 [1 + log(1 + t)]−2 . A(t)

(5.243)

Note that by virtue of assumption C1 we have: t ˜ )dt ≤ C (independent of t). A(t

(5.244)

0

We now apply the inequalities: C2 1 −K + C K 1/2 M 1/2 ≤ − K + M, 2 2

C L 1/2 M 1/2 ≤

C2 1 L+ M 2 2

to the last terms on the right-hand side of (5.240). Substituting in (5.240) yields: t 3 1 ˜ )E u (t )dt . Q 1 dµh ≤ − K (t, u) + C M(t, u) + L(t, u) + A(t 1 2 2 0 Wut

(5.245)

Here and in the following we shall denote by C various numerical constants. We have: t u L(t, u) ≤ (1 + t )−1 [1 + log(1 + t )]3 E 1 (t )dt 0

1 u ≤ [1 + log(1 + t)]4 E 1 (t). 4

(5.246)


133

Also, since (see (5.84)): t

F1t (u) ≤ [1 + log(1 + t)]4 F 1 (u) defining: V1 (t, u)

we have:

u 0

= 0

u

t

F 1 (u )du

(5.247)

(5.248)

F1t (u )du ≤ [1 + log(1 + t)]4 V1 (t, u)

(5.249)

hence (see (5.241)): u

M(t, u) ≤ [1 + log(1 + t)]4 (E 0 (t) + V1 (t, u)).

(5.250)

Note that V1 (t, u) is a non-decreasing function of t at each u as well as a non-decreasing function of u at each t. Also, t t u 4 ˜ ˜ )E u A(t )E1 (t )dt ≤ [1 + log(1 + t)] A(t (5.251) 1 (t )dt . 0

0

By assumptions A1, B2, D1, D3 and the estimate (5.72), the space-like hypersurface integrals on the right-hand side of (5.66) are bounded by: u 2 2 (1/2)(Lω + νω)ψ − (1/2)(Lω + νω)ψ ≤ CE 0 [1 + log(1 + t)]4 . u tu 0 (5.252) Also, by (5.68) and (5.70) at t = 0, the remaining term on the right-hand side of (5.66) is bounded by: (5.253) E1u (0) ≤ CE0u (0). In view of (5.245)–(5.253) the integral identity (5.66) implies: 1 K (t, u) (5.254) 2 t 3 u u ˜ )E u ≤ [1 + log(1 + t)]4 E (t) + C(E 0 (t) + V1 (t, u)) + A(t 1 (t )dt . 8 1 0

E1u (t) + F1t (u) +

Keeping only the term E1u (t) on the left we have: [1 + log(1 + t)]−4 E1u (t) ≤

3 u u E (t) + C(E 0 (t) + V1 (t, u)) + 8 1

(5.255)

t 0

˜ )E u A(t 1 (t )dt .

The same holds with t replaced by any t ∈ [0, t]. Now the right-hand side of (5.255) is a non-decreasing function of t at each u. The inequality corresponding to t thus holds a

134


fortiori if we again replace t by t on the right-hand side. Taking then the supremum over all t ∈ [0, t] on the left-hand side we obtain: t 3 u u u ˜ )E u A(t E 1 (t) ≤ E 1 (t) + C(E 0 (t) + V1 (t, u)) + 1 (t )dt 8 0 which implies: u

u

E 1 (t) ≤ C(E 0 (t) + V1 (t, u)) + C

t 0

˜ )E u A(t 1 (t )dt

(5.256)

(for new constants C). This is a linear integral inequality, with respect to t, for the function u u E 1 (t). In view of (5.244) and the fact that E 0 (t) + V1 (t, u) is an non-decreasing function of t at each u, (5.256) implies: u

u

E 1 (t) ≤ C(E 0 (t) + V1 (t, u)) (for a new constant C), hence also: t u ˜ )E u (t )dt ≤ E (t) A(t 1 1 0

t

(5.257)

˜ )dt A(t

0

u

≤ C(E 0 (t) + V1 (t, u))

(5.258)

(for a new constant C). Substituting (5.257) and (5.258) on the right-hand side of (5.254) and keeping only the term F1t (u) on the left-hand side we obtain: u

[1 + log(1 + t)]−4 F1t (u) ≤ C(E 0 (t) + V1 (t, u))

(5.259)

(for a new constant C). The same holds with t replaced by any t ∈ [0, t]. Now the right-hand side of (5.259) is a non-decreasing function of t at each u. The inequality corresponding to t thus holds a fortiori if we again replace t by t on the right-hand side. Taking then the supremum over all t ∈ [0, t] on the left-hand side we obtain: t

u

F 1 (u) ≤ C(E 0 (t) + V1 (t, u)).

(5.260)

Recalling the definition (5.248) of V1 (t, u), this is a linear integral inequality, with respect t to u, for the function F 1 (u): u t u t F 1 (u) ≤ CE 0 (t) + C F 1 (u )du . (5.261) 0

u E 0 (t)

In view of the fact that is a non-decreasing function of u at each t while [0, ε0 ] is a bounded interval, (5.261) implies: t

u

F 1 (u) ≤ CE 0 (t)

(5.262)

(for a new constant C), hence also: u

V1 (t, u) ≤ Cε0 E 0 (t).

(5.263)


135

Substituting this in (5.257) we obtain: u

u

E 1 (t) ≤ CE 0 (t)

(5.264)

(for a new constant C). Also, substituting the estimates (5.257), (5.258), (5.263), in (5.254) and keeping only the term (1/2)K (t, u) on the left-hand side we obtain: u

[1 + log(1 + t)]−4 K (t, u) ≤ CE 0 (t)

(5.265)

(for a new constant C), which implies: u

K (t, u) ≤ CE 0 (t).

(5.266)

We now turn our attention to the integral identity (5.67). By (5.86) and (5.109), (5.110), (5.130), (5.132), (5.149), (5.175) (and (5.176)), (5.222), (5.239), the spacetime integral on the right-hand side of (5.67) is bounded from above by, in view of (5.249): Q 0 dµh Wut

≤

t

u

(1 + t )−2 [1 + log(1 + t )]4 B(t )E 1 (t )dt 0 t u u +C (1 + t )−1 [1 + log(1 + t )]−2 (E 1 (t ) + E 0 (t ))dt 0

+ C V0 (t, u) 1/2 t u + C V0 (t, u) + (1 + t )−1 [1 + log(1 + t )]−2 E 0 (t )dt (V1 (t, u))1/2 0 t −2 +C (1 + t ) [1 + log(1 + t )]2 V1 (t , u)dt 0

+ C K (t, u)

t

1/2

−1

(1 + t )

[1 + log(1 + t

0

u )]−2 E 0 (t )dt

+ C K (t, u)1/2 (V0 (t, u))1/2 .

(5.267)

Here we have defined:

1/2

u

V0 (t, u) = 0

F0t (u )du .

(5.268)

We are now to substitute on the right-hand side of (5.267), the estimates (5.263), (5.264), (5.266), just derived. In doing this we estimate the fourth term on the right in (5.267) by: 1/2 t u u (1 + t )−1 [1 + log(1 + t )]−2 E 0 (t )dt (5.269) CE 0 (t)1/2 V0 (t, u) + 0

t C2 δ u −1 −2 u (1 + t ) [1 + log(1 + t )] E 0 (t )dt ≤ E 0 (t) + V0 (t, u) + 2 2δ 0

136


the sixth term by: u CE 0 (t)1/2

≤

t

−1

(1 + t )

0

δ u C2 E 0 (t) + 2 2δ

t 0

[1 + log(1 + t

u )]−2 E 0 (t )dt

1/2

u

(1 + t )−1 [1 + log(1 + t )]−2 E 0 (t )dt

(5.270)

and the seventh term by: u

CE (t)1/2 (V0 (t, u))1/2 ≤

δ u C2 V0 (t, u). E 0 (t) + 2 2δ

(5.271)

The above hold for any positive constant δ (we shall choose δ below). After these substitutions, (5.267) reduces to: t 3δ u 1 1 ˜ )E u0 (t )dt Q 0 dµh ≤ E 0 (t) + C 1 + B(t V0 (t, u) + C 1 + 2 δ δ Wut 0 (5.272) (for new constants C, which are independent of δ). Here: ˜ B(t) = (1 + t)−2 [1 + log(1 + t)]4 B(t) + C(1 + t)−1 [1 + log(1 + t)]−2 . Note that by assumption C2: t

˜ )dt ≤ C B(t

(independent of t).

(5.273)

(5.274)

0

In view of (5.272) the identity (5.67) yields: t 3δ u 1 ˜ )E u0 (t )dt . B(t E0u (t) + F0t (u) ≤ E0u (0) + E 0 (t) + C 1 + V0 (t, u) + 2 δ 0 (5.275) We now set:

1 . 3 Keeping only the term E0u (t) on the left in (5.275) we then have: t 1 u ˜ )E u0 (t )dt E0u (t) ≤ E0u (0) + E 0 (t) + C V0 (t, u) + C B(t 2 0 δ=

(5.276)

(5.277)

(for new constants C). The same holds with t replaced by any t ∈ [0, t]. The right-hand side of (5.277) being a non-decreasing function of t at each u, the inequality corresponding to t holds a fortiori if we again replace t by t on the right. Taking then the supremum over all t ∈ [0, t] on the left we obtain: t 1 u u u ˜ )E u0 (t )dt B(t E 0 (t) ≤ E0 (0) + E 0 (t) + C V0 (t, u) + C 2 0


137

which implies:

u

E (t) ≤ E0u (0) + C V0 (t, u) + C

t

0

˜ )E u0 (t )dt B(t

(5.278)

(for new constants C). This is a linear integral inequality, with respect to t, for the function u E 0 (t). In view of (5.274) and the fact that V0 (t, u) is a non-decreasing function of t at each u, (5.278) implies: u E (t) ≤ C(E0u (0) + V0 (t, u)) (5.279) (for a new constant C), hence also: t ˜ )E u0 (t )dt ≤ E u0 (t) B(t 0

t

˜ )dt B(t

0

≤ C(E0u (0) + V0 (t, u))

(5.280)

(for a new constant C). Substituting (5.279) and (5.280) on the right-hand side of (5.275) (noting (5.276)) and keeping only the term F0t (u) on the left-hand side we obtain: F0t (u) ≤ C(E0u (0) + V0 (t, u))

(5.281)

(for a new constant C). Recalling the definition (5.268) of V0 (t, u), this is a linear integral inequality, with respect to u, for the function F0t (u): u t u F0t (u )du . (5.282) F0 (u) ≤ CE0 (0) + C 0

In view of the fact that E0u (0) is a non-decreasing function of u while [0, ε0] is a bounded interval, (5.282) implies: (5.283) F0t (u) ≤ CE0u (0) (for a new constant C), hence also: V0 (t, u) ≤ Cε0 E0u (0).

(5.284)

Substituting this in (5.279) we obtain: u

E 0 (t) ≤ CE0u (0).

(5.285)

Substituting finally (5.285) in (5.262), (5.264), (5.266), yields: u

t

E 1 (t), F 1 (u), K (t, u) ≤ CE0u (0).

(5.286)

In view of the inequalities (5.68)–(5.72) (and the definitions (5.81)–(5.84)) the first five conclusions of the theorem readily follow from (5.283), (5.285), (5.286), while the sixth conclusion follows from the estimate on K (t, u) of (5.286) together with the lower bound (5.206) (and the definition (5.208)). The proof of Theorem 5.1 is thus complete.

Chapter 6

Construction of the Commutation Vectorfields In the present chapter we shall construct the vectorfields Yi : i = 1, 2, 3, 4, 5 mentioned at the beginning of the previous chapter. These vectorfields are used to define the higher order variations of the wave function φ. That is, an nth order variation is of the form (see (5.6)): (6.1) ψn = Yi1 · · · Yin−1 ψ1 where ψ1 is a first order variation, namely one of the functions (5.5). Here the indices i 1 , . . . , i n−1 take values in the set {1, 2, 3, 4, 5}. Since ψ1 is a solution of the homoge˜ neous wave equation corresponding to the conformal acoustical metric h, h˜ ψ1 = 0.

(6.2)

ψn satisfies an inhomogeneous wave equation h˜ ψn = ρn

(6.3)

where the source function ρn is obtained by successively commuting each of the vectorfields Yi1 , . . . , Yin−1 with the operator h˜ . For this reason the vectorfields Yi : i = 1, 2, 3, 4, 5 shall be called commutation fields. One condition that we require of the set of commutation vectorfields is that at each point x in the spacetime domain Wε0 under consideration the corresponding set of vectors at x spans the tangent space to the spacetime manifold at x. Thus the set of all ψn for fixed n corresponding to a given ψ1 contains all derivatives of that ψ1 of order n − 1. We take the commutation field Y1 to be the vectorfield T : Y1 = T

(6.4)

Since this is transversal to the characteristic hypersurfaces Cu , we require each of the commutation fields Yi : i = 2, 3, 4, 5 to be tangential to the Cu . Moreover we require that for each u ∈ [0, ε0 ] and each point x ∈ Cu , the set of vectors {Yi (x) : i = 2, 3, 4, 5} span the tangent space to Cu at x.

140

Chapter 6. Construction of the Commutation Vectorfields

Next we take the vectorfield Y1 to be collinear to the vectorfield L whose integral curves are the bicharacteristic generators of the Cu . More precisely, we set: Y2 = Q

(6.5)

Q = (1 + t)L

(6.6)

where:

As this is transversal to the surfaces St,u , the sections of the Cu by the space-like hyperplanes t , we require each of the commutation fields Yi : i = 3, 4, 5 to be tangential to the St,u and moreover, for each t and u and each point x ∈ St,u the set of vectors {Yi (x) : i = 3, 4, 5} to span the tangent plane to St,u . More precisely, we set: Yi+2 = Ri : i = 1, 2, 3

(6.7)

where: ◦

Ri = Ri : i = 1, 2, 3

(6.8)

◦

Here the Ri are defined relative to the background Euclidean metric g on each t . They are the generators of rotations about the three rectangular coordinate axes: ◦

Ri = i j k x j

∂ 1 = i j k j k ∂xk 2

(6.9)

(rectangular coordinates), where i j k is the fully antisymmetric 3-dimensional symbol (see (1.90)). In the definition (6.8), is the orthogonal projection to the tangent plane to the surfaces St,u with respect to the induced acoustical metric h on t . Now since h˜ is a geometric differential operator on functions, defined solely ˜ its commutator with an arbitrary vectorfield Y , considered also through the metric h, as a differential operator on functions, involves just (Y ) π˜ , the Lie derivative with respect ˜ The precise formula will be given in the sequel. What we wish to note at this to Y of h. point is that control of the commutator [h˜ , Y ] depends on controlling the deformation tensor (Y ) π˜ corresponding to the commutation field Y . The (Y ) π˜ refer to the conformal ˜ Since h˜ = h, these are related to the (Y ) π, the deformation tensors acoustical metric h. relative to the acoustical metric h, by: (Y )

π˜ =

(Y )

π + (Y )h

(6.10)

For any pair Z 1 , Z 2 of vectors at a point we have: (Y )

π(Z 1 , Z 2 ) = h(D Z 1 Y, Z 2 ) + h(D Z 2 Y, Z 1 )

(6.11)


141

The components of the deformation tensors of the commutation fields T and Q in the null frame L, L, X 1 , X 2 can then be directly computed from Table 3.117. We find: (T ) (T ) (T )

π˜ L L = 0 π˜ L L = 4µT (α −1 κ) π˜ L L = −2(T µ + µT log )

(T )

π˜ L A = −(ζ A + η A ) π˜ L A = −α −1 κ(ζ A + η A ) (T ) ˜ˆ π / AB = (χˆ AB − α −1 κ χˆ AB )

(T )

tr

(T ) ˜

π / = 2(ν − α −1 κν)

(6.12)

and: (Q)

π˜ L L = 0

(Q)

π˜ L L = 4µ{Q(α −1 κ) − α −1 κ}

(Q)

π˜ L L = −2{Qµ + µQ log + µ}

(Q)

π˜ L A = 0

(Q)

π˜ L A = 2(1 + t)(ζ A + η A ) (Q) ˆ˜ π / AB = 2(1 + t)χˆ AB (Q) ˜ tr π / = 4(1 + t)ν

(6.13)

/˜ the restriction of (Y ) π˜ to St,u , and by (Y ) π /ˆ˜ the trace-free part Here we denote by (Y ) π (Y ) ˜ of π /. Noting that the definition (6.8) of the commutation fields Ri : i = 1, 2, 3 is intrinsic to each space-like hypersurface t , we shall derive expressions for the components of (Ri ) π in the frame L, T, X , X , taking advantage of the fact that T, X , X is a frame 1 2 1 2 field for each t . For the following we refer to Table 3.102. We have: (Ri )

π L L = 2h(D L Ri , L) = −2h(Ri , D L L) = −2µ−1 (Lµ)h(Ri , L) = 0

(6.14)

L being h- orthogonal to the Ri , since L is h- orthogonal to the St,u while the Ri are tangential to the St,u . For the same reason T is h- orthogonal to the Ri , hence: (Ri )

πT T = 2h(DT Ri , T ) = −2h(Ri , DT T )

Noting that since the Ri are tangential to the St,u we can expand Ri = RiA X A , we obtain (see Table 3.102):

(Ri )

πT T = 2κ Ri κ

(6.15)

142


Next, (Ri )

π LT = h(D L Ri , T ) + h(DT Ri , L) = −h(Ri , D L T ) − h(Ri , D L T ) = −η A RiA + ζ A RiA

hence, by (3.65):

(Ri )

Next, we have:

(Ri )

π LT = −Ri µ

(6.16)

π L A = h(D L Ri , X A ) + h(D X A Ri , L)

Now, h(D X A Ri , L) = −h(Ri , D X A L) = −χ AB RiB On the other hand, from the definition (6.8), ◦

◦

D L Ri = (D L ) Ri +(D L Ri ) Noting that for any vectorfield Z h(Z , X A ) = h(Z , X A ), we obtain:

◦

◦

h(D L Ri , X A ) = h((D L ) Ri , X A ) + h(D L Ri , X A ) Thus, (Ri )

Next,

◦

◦

π L A = h((D L ) Ri , X A ) + h(D L Ri , X A ) − χ AB RiB (Ri )

(6.17)

πT A = h(DT Ri , X A ) + h(D X A Ri , T )

By (3.45), h(D X A Ri , T ) = −h(Ri , D X A T ) = −h(Ri , D X A T ) = −κθ AB RiB Using also the fact that by definition (6.8): ◦

◦

DT Ri = (DT ) Ri +(DT Ri ) we obtain: (Ri )

◦

◦

πT A = h((DT ) Ri , X A ) + h(DT Ri , X A ) − κθ AB RiB

Finally, we have: (Ri )

and:

π AB = h(D X A Ri , X B ) + h(D X B Ri , X A ) ◦

◦

D X A Ri = (D X A ) Ri +(D X A Ri )

(6.18)


143

Thus, (Ri )

◦

◦

π AB = h((D X A ) Ri , X B ) + h((D X B ) Ri , X A ) ◦

◦

+h(D X A Ri , X B ) + h(D X B Ri , X A ) ◦

(6.19) ◦

◦

The formulas (6.17)–(6.19) involve (D) Ri as well as D Ri . Since the Ri are tangential to the t we can expand: ◦

Ri = RiA X A + λi Tˆ , for some functions λi . Since

Tˆ = κ −1 T

(6.20)

T = 0,

for any vectorfield Z we have: (D Z )T = D Z (T ) − (D Z T ) = −(D Z T ) Setting Z equal to L, T, X A : A = 1, 2 successively, we obtain, using Table 3.102, (D L )T = ζ A X A (DT )T = κ(d/ A κ)X A (D X A )T = −κθ AB X B

(6.21)

Since X A = X A , for any vectorfield Z we have: (D Z )X A = D Z (X A ) − (D Z X A ) = D Z X A − (D Z X A ) In particular, from Table 3.102, (D L )X A = −µ−1 ζ A L Next, we can expand: DT X A − (DT X A ) = a A L + b A T Taking the h- inner products with L and T successively yields: h(DT X A , L) = −µb A

and

h(DT X A , T ) = −µa A + κ 2 b A

On the other hand we have: h(DT X A , L) = −h(X A , DT L) = −η A and: h(DT X A , T ) = −h(X A , DT T ) = κd/ A κ

(6.22)

144


Substituting and solving for a A and b A yields: a A = −α −1 d/ A κ + α −2 η A

b A = µ−1 η A

and

We thus obtain: (DT )X A = (−α −1 d/ A κ + α −2 η A )L + µ−1 η A T

(6.23)

Finally, we have: (D X A )X B = D X A X B − (D X A X B ) Since /XA X B, (D X A X B ) = D the last entry of Table 3.102 gives: (D X A )X B = α −1 k/ AB L + µ−1 χ AB T

(6.24)

◦

Substituting in (D) Ri the expansion (6.20), that is: ◦

(D) Ri = RiA (D)X A + κ −1 λi (D)T, and using the formulas (6.21)–(6.24) for (D)T and (D)X A , we obtain the following ◦

results for the expressions in (6.17)–(6.19) involving (D) Ri : ◦

h((D L ) Ri , X A ) = κ −1 λi ζ A ◦

h((DT ) Ri , X A ) = λi d/ A κ ◦

h((D X A ) Ri , X B ) = −λi θ AB

(6.25)

◦

The expressions (6.17)–(6.19) involve also D Ri . Now, in an arbitrary coordinate system we have: ◦ Dµ Riν =

◦

∂ Riν h ν ◦λ + !µλ Ri ∂xµ

h

ν are the connection coefficients of the acoustical metric h where !µλ µν in the given coordinate system. On the other hand, in the same coordinate system we have: ◦ ∇µ Riν =

◦

◦ g ∂ Riν ν + !µλ Riλ µ ∂x

g

ν are the connection coefficients of the Minkowskian metric g where !µλ µν in that coordinate system. Therefore, ◦

◦

◦

Dµ Riν = ∇µ Riν +νµλ Riλ

(6.26)


145

where is the difference of the two connections, a tensorfield, whose components in any given coordinate system are: g

h

ν ν − !µλ νµλ =!µλ

(6.27)

Let us now fix the coordinate system to be a rectangular coordinate system of the Minkowskian metric g. We then have: g

ν !µλ = 0,

hence: ◦ ∇µ Riν = ◦

◦

∂ Riν ∂xµ

and the components Riν are given by (6.9): ◦

◦

Ri0 = 0, Thus,

◦

Rin = i j n x j ◦

∇0 Riν = 0,

∇µ Ri0 = 0,

(6.28) ◦

∇m Rin = imn

(6.29)

(In the above formulas the Greek indices take the values 0,1,2,3 while the Latin indices take the values 1,2,3.) In rectangular coordinates of g (6.27) reduces to: h

ν = (h −1 )νκ !µλκ νµλ =!µλ

(6.30)

where, as in Chapter 3 (see equation (3.139)), 1 (∂µ h λκ + ∂λ h µκ − ∂κ h µλ ) 2 According to equation (3.141) we have: !µλκ =

1 dH (∂µ σ ψλ ψκ + ∂λ σ ψµ ψκ − ∂κ σ ψµ ψλ ) + H ψκ ∂µ ψλ 2 dσ

!µλκ =

(6.31)

From (6.26), (6.29)–(6.31) we conclude that the expressions ◦

◦

◦

h(D L Ri , X A ), h(DT Ri , X A ), h(D X A Ri , X B ), which appear in the formulas (6.17), (6.18), (6.19) respectively, are given by: ◦ µ

◦

h(D L Ri , X A ) = h µν L λ Dλ Ri X νA = =

◦ µ +!λµν Ri )X νA ◦ h mn L l ilm X nA + H ψ A Rim (Lψm ) ◦ ◦ 1 dH {(Lσ )( Rim ψm ) ψ A + ψ L [( Ri +

(6.32)

◦ µ L λ (h µν ∇λ Ri

2 dσ

◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

146

Chapter 6. Construction of the Commutation Vectorfields ◦ µ

◦

h(DT Ri , X A ) = h µν T λ Dλ Ri X νA

(6.33)

◦ µ T λ (h µν ∇λ Ri

◦ µ +!λµν Ri )X νA ◦ h mn T l ilm X nA + H ψ A Rim (T ψm ) ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [( Ri +

= =

2 dσ

◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

◦ µ

◦

h(D X A Ri , X B ) = h µν X λA Dλ Ri X νB = = +

(6.34)

◦ µ X λA (h µν ∇λ Ri

◦ µ +!λµν Ri )X νB ◦ h mn X lA ilm X nB + H ψ B Rim (d/ A ψm ) ◦ ◦ 1 dH {(d/ A σ )( Rim ψm ) ψ B + ( Ri σ ) ψ A

2 dσ

◦

ψ B − (d/ B σ )( Rim ψm ) ψ A }

The above formulas are expressed in rectangular coordinates of the Euclidean metric g on t . We now substitute in (6.17) from (6.25) and (6.32) to obtain: (Ri )

π L A = −χ AB RiB + ilm L l h mn X nA

◦ +κ −1 λi ζ A + H ψ A Rim (Lψm ) ◦ ◦ 1 dH {(Lσ )( Rim ψm ) ψ A + ψ L [( Ri +

2 dσ

(6.35) ◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

Next we substitute in (6.18) from (6.25) and (6.33) to obtain: (Ri )

πT A = −κθ AB RiB + ilm T l h mn X nA

◦ +λi d/ A κ + H ψ A Rim (T ψm ) ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [( Ri +

2 dσ

(6.36) ◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

Finally, we substitute in (6.19) from (6.25) and (6.34). Since h mn = δmn + H ψm ψn (rectangular coordinates of the Euclidean metric g on t ), and ilm δmn = iln is antisymmetric in l, n while X lA X nB + X lB X nA is symmetric, only the second part of h mn contributes to ilm h mn (X lA X nB + X lB X nA ), hence: ilm h mn (X lA X nB + X lB X nA ) = H ilm ( ψ A X lB + ψ B X lA )ψm


147

We thus obtain: (Ri )

π AB = −2λi θ AB

◦

+H ilm ( ψ A X lB + ψ B X lA )ψm + H ( ψ A X lB + ψ B X lA ) Rim ∂l ψm dH ◦ ( Ri σ ) ψ A ψ B + (6.37) dσ In the formulas (6.35), (6.36), (6.37) for ◦ Ri

(Ri ) π

L A,

(Ri ) π

RiA X A σ

(Ri ) π

T A,

AB ,

respectively,

κ −1 λi T σ

we are to substitute for σ the expression + corresponding to the expansion (6.20). Now, the coefficient λi in the expansion (6.20) can only be assumed to be bounded as µ → 0. Therefore, to obtain (Ri ) π L A , (Ri ) πT A , (Ri ) π AB , bounded as µ → 0, we must show that the coefficient of λi in each of the formulas (6.35), (6.36), (6.37), is in fact bounded as µ → 0. The coefficient of λi in (6.37) is: −2θ AB +

1 dH ψ A ψ B (T σ ) = −2θ AB κ dσ

is bounded as µ → 0. In fact, we have (see (3.232)): (see (3.231)) where θ AB = −α −1 χ AB + k/AB θ AB

(6.38)

where k/AB , defined by (see (3.233)): k/AB = k/ AB −


(6.39)

and given by (3.236), is bounded as µ → 0. The coefficient of λi in (6.37) is then: 2(α −1 χ AB − k/AB ) and this is bounded as µ → 0. The formula (6.37) takes the form: (Ri )

π AB = 2λi (α −1 χ AB − k/AB ) +H ilm ( ψ A X lB + ψ B X lA )ψm

◦

+H { ψ A (d/ B ψm )+ ψ B (d/ A ψm )} Rim dH ψ A ψ B (Ri σ ) + dσ

(6.40)

The coefficient of λi in (6.36) is: d/ A κ +

1 dH ψ ˆ ψ A (T σ ) 2 dσ T

This coefficient is indeed bounded as µ → 0. Recalling from Chapter 2 (see (2.57)) that T = κ Tˆ

(6.41)

148


let us define the functions y i by setting: Tˆ i = −

xi + yi 1 − u + η0 t

(6.42)

Using (6.42) and the fact that: ◦

ilm x l h mn X nA = h mn Rim X nA = h/ AB RiB we re-write first two terms in (6.36) in the form: B l n −κθ AB Ri + ilm T h mn X A = −κ θ AB +

/h AB 1 − u + η0 t

(6.43)

RiB + κilm y l h mn X nA

Then formula (6.36) takes the form: / AB h (Ri ) RiB + κilm y l h mn X nA πT A = −κ θ AB + (6.44) 1 − u + η0 t ◦ 1 dH +λi d/ A κ + ψTˆ ψ A (T σ ) + H ψ A Rim (T ψm ) 2 dσ ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [(Ri σ ) ψ A − (d/ A σ )( Rim ψm )]} + 2 dσ Finally, the coefficient of λi in (6.35) is: κ −1 ζ A +

1 dH ψ L ψ A (T σ ) = κ −1 ζ A 2κ dσ

(see (3.168)) where κ −1 ζ is bounded as µ → 0. In fact, we have (see (3.170)): κ −1 ζ A = α A − d/ A α

(6.45)

where A , defined by (see (3.171)): A = A +

1 dH ψ L ψ A (T σ ) 2ακ dσ

(6.46)

and given by (3.174), is bounded as µ → 0. Let us also define the functions z i by setting: Li =

η0 x i + zi 1 − u + η0 t

(6.47)

Comparing (6.42) and (6.47) with equation (2.66) of Chapter 2, we conclude that the functions z i are related to the functions y i by: z i = −αy i +

H ψ0ψi (α − η0 )x i + 1 − u + η0 t 1 + ρH

(6.48)


149

Using (6.47) and (6.43) we re-write the first two terms in (6.35) in the form: η0 h/ AB B l n −χ AB Ri + ilm L h mn X A = − χ AB − RiB + ilm z l h mn X nA 1 − u + η0 t The formula (6.35) takes the form: / AB η0 h (Ri ) RiB + ilm z l h mn X nA π L A = − χ AB − 1 − u + η0 t

(6.49)

◦

+λi (κ −1 ζ A ) + H ψ A Rim (Lψm ) ◦ ◦ 1 dH + {(Lσ )( Rim ψm ) ψ A + ψ L [(Ri σ ) ψ A − (d/ A σ )( Rim ψm )]} 2 dσ In the remainder of the present chapter we shall show how the (Y ) π, ˜ the deformation tensors of the commutation fields Y , are to be controlled in terms of χ, the second fundamental form of the surfaces St,u with respect to the characteristic hypersurfaces Cu , the function µ, lapse function of the characteristic foliation, and the ψµ = ∂µ φ, the derivatives of the wave function with respect to the translations of the underlying Minkowski spacetime. The estimates of the present chapter are only preliminary and are intended to give the reader a feeling for the sizes of the various quantities and their interdependence. The actual estimates shall be derived in Chapters 10, 11, and 12. In the following the basic bootstrap assumptions A1, A2, A3 of Chapter 5 are assumed to hold. For convenience we restate these assumptions below. The set of three basic bootstrap assumptions shall be referred to as assumptions A. There is a positive constant C independent of s such that in Wεs0 , A1: C −1 ≤ ≤ C A2: C −1 ≤ α/η0 A3: µ/η0 ≤ C[1 + log(1 + t)] We recall that:

∂φ (6.50) ∂xµ (rectangular coordinates). To obtain the estimates which follow we shall make use of the following bootstrap assumptions on the ψµ and their first derivatives. ψµ =

In the following, we denote by δ0 a positive constant, which is by definition less than or equal to unity. This constant is used to keep track of the relative size of various quantities. It is to be chosen suitably small at various points in the remainder of this monograph. On the other hand, no smallness condition is assumed on ε0 , other than the condition: 1 ε0 ≤ 2 There is a positive constant C independent of s such that in Wεs0 , E1: |ψ0 − k|, |ψi | ≤ Cδ0 (1 + t)−1 E2: |T ψµ | ≤ Cδ0 (1 + t)−1 ,

|Lψµ |, |d/ψµ | ≤ Cδ0 (1 + t)−2

150


The constant k in the first of E1 is the positive constant characterizing the surrounding constant state (see Chapter 1, last paragraph). We remark here that the second of E2 shall, in the actual estimates of Chapters 10, 11, and 12, follow from the bound: |Qψµ | ≤ Cδ0 (1 + t)−1 while the third shall follow from the bounds: |Ri ψµ | ≤ Cδ0 (1 + t)−1

: i = 1, 2, 3

We shall also make use of the following bootstrap assumptions on the first derivatives of the function µ on the hypersurfaces t and on the quadratic form χ in the tangent plane at each point on each of the surfaces St,u . There is a positive constant C independent of s such that in Wεs0 , |T µ| ≤ Cδ0 [1 + log(1 + t)], |d/µ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] η0 h / ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] F2: χ − 1−u+η t

F1:

0

Here and in the following, η0 denotes the sound speed in the surrounding constant state. In addition to the above bootstrap assumptions, we make the following assumption on the initial data.

There is a positive constant C such that on 00 we have: I0:

|κ − 1| ≤ Cδ0

Under the assumptions A, E, F, and I0, the estimates which we shall derive shall hold in Wεs0 with constants which are independent of s. First, on the basis of assumptions A and E alone we shall derive estimates on k/ and , components of k, the second fundamental form of the hypersurfaces t with respect to the acoustical metric h, and on the first derivatives of α, the lapse function of the foliation of the acoustical spacetime by the t . We begin by noting that since σ is given by: σ = (ψ0 )2 −

3 (ψi )2 i=1

the assumptions E1, E2 imply the following bounds on σ and its first derivatives: |σ −k 2 | ≤ Cδ0 (1+t)−1 ,

|T σ | ≤ Cδ0 (1+t)−1 , |Lσ |, |d/σ | ≤ Cδ0 (1+t)−2 (6.51)

Recall that Tˆ is the unit normal to the surfaces St,u in t relative to the induced acoustical metric h. Since for each i = 1, 2, 3, |Tˆ i | is bounded by the Euclidean magnitude of the vector Tˆ which is in turn bounded by its magnitude with respect to h, namely by unity, we have the following elementary bound on Tˆ i , i = 1, 2, 3: |Tˆ i | ≤ 1

(6.52)

|ψTˆ | ≤ Cδ0 (1 + t)−1

(6.53)

It follows by assumption E1 that:


151

Next, we have the following elementary bound on L i , i = 1, 2, 3: |L i | ≤ 1

(6.54)

This is deduced as follows. We first define the t -tangent vector: L = Li

∂ ∂xi

(6.55)

Then we have:

∂ +L (6.56) ∂x0 Now in the tangent space at each point in spacetime, the sound cone is contained in the light cone. Thus the vector L is future-directed, time-like or null with respect to the background Minkowskian structure. Hence the Euclidean magnitude of L is bounded by L 0 = 1. The bound then follows as |L i | is in turn bounded by the Euclidean magnitude of L. The bound (6.54) together with assumption E1 implies: L=

|ψ L − k| ≤ Cδ0 (1 + t)−1

(6.57)

The induced acoustical metric h on t dominates the Euclidean metric g (see Chap−1 ter 2). Hence h is dominated by g −1 . In fact in rectangular coordinates of the Euclidean metric we have: (g−1 )i j = δi j

g i j = δi j ,

h i j = δi j + H ψi ψ j ,

(h

−1 i j

) = δi j −

H ψi ψ j (1 + ρ H )

(6.58)

−1

Note also that h , the reciprocal of the induced acoustical metric on the t is expressed in terms of the frame Tˆ , X 1 , X 2 by: (h

−1 i j

j ) = Tˆ i Tˆ j + (h/−1 ) AB X iA X B

(6.59)

It follows by virtue of assumption E1 that: /−1 ) AB ψ A ψ B = (h /−1 ) AB X iA X B ψi ψ j ≤ (h | ψ|2 = (h j

≤

−1 i j

3

) ψi ψ j

(ψi )2 ≤ Cδ02 (1 + t)−2

i=1

that is:

| ψ| ≤ Cδ0 (1 + t)−1

(6.60)

According to (6.39), k/ is given by: k/ AB = k/AB +


(6.61)

152


and k/ is given by (3.236). For a bilinear form w in the tangent space to the surface St,u at a point, the magnitude |w| of w is given by: |w|2 = (h /−1 ) AC (h/−1 ) B D w AB wC D By (6.60) and the bounds (6.51), the second term on the right in (6.61), multiplied by κ, is bounded in magnitude by Cδ03 (1 + t)−3 . The estimate of the first term in the expression (3.236) for k/ is similar, the bound in magnitude being by Cδ02 (1 + t)−3 , using again (6.60), assumption E1 and the bounds (6.51). To estimate the second term, the square of the magnitude of ω / AB = X iA (d/ B ψi ) is: /−1 ) AC (h /−1 ) B D ω / AB ω /C D = (h/−1 ) AC (h/−1 ) B D X iA X C (d/ B ψi )(d/ D ψ j ) |ω /|2 = (h j

≤ (h

−1 i j

) (h /−1 ) B D (d/ B ψi )(d/ D ψ j ) ≤

3

|d/ψi |2 ≤ Cδ02 (1 + t)−4

(6.62)

i=1

using assumption E2. Hence (using also E1) the second term in the expression (3.236) is bounded in magnitude by Cδ0 (1 + t)−2 . We thus obtain the estimate: |k/ | ≤ Cδ0 (1 + t)−2

(6.63)

Combining with the previous estimate for the second term on the right in (6.61), multiplied by κ, and taking also account of the fact that by virtue of assumptions A we have:

we conclude that:

κ = α −1 µ ≤ C[1 + log(1 + t)]

(6.64)

κ|k/| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(6.65)

According to (6.46) is given by: A = A +

1 dH ψ L ψ A (T σ ) 2ακ dσ

(6.66)

and is given by (3.174). For a linear form v in the tangent space to St,u at a point, the magnitude |v| of v is given by: |v|2 = (h/−1 ) AB v A v B By (6.53), (6.60), assumption E1, and the bounds (6.51), we conclude that the first term on the right in (3.174) is bounded in magnitude by Cδ02 (1+t)−3 . Also, by (6.52) and assumption E2 the second term on the right in (3.174) is bounded in magnitude by Cδ0 (1 + t)−2 . We thus obtain the estimate: (6.67) | | ≤ Cδ0 (1 + t)−2 By assumption E1 and the estimates (6.60) and (6.51), the second term on the right in (6.66), multiplied by κ, is bounded in magnitude by Cδ02 (1 + t)−2 . Taking account of (6.64) we then conclude that κ|| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(6.68)


153

Next we shall estimate, on the basis of assumptions A and E, the function α and its first derivatives. Now, the function α is expressed by equation (2.43) as: α −2 = 1 + F(ψ0 )2

(6.69)

In the surrounding constant state we have: α = η0 Thus, assumption E1 implies: |α − η0 | ≤ Cδ0 (1 + t)−1

(6.70)

Differentiating (6.69) tangentially to St,u we obtain: −2α −3 d/α =

dF (ψ0 )2 d/σ + 2Fψ0 d/ψ0 dσ

Using assumptions E1, E2 and the bounds (6.51) yields: |d/α| ≤ Cδ0 (1 + t)−2

(6.71)

Differentiating (6.69) with respect to T , −2α −3 T α =

dF (ψ0 )2 T σ + 2Fψ0 T ψ0 dσ

−2α −3 Lα =

dF (ψ0 )2 Lσ + 2Fψ0 Lψ0 dσ

and with respect to L,

and using assumptions E1, E2 and the bounds (6.51) yields: |T α| ≤ Cδ0 (1 + t)−1 ,

|Lα| ≤ Cδ0 (1 + t)−2

(6.72)

The estimates (6.71) and (6.67) yield an estimate for κ −1 ζ through the expression (6.45): (6.73) |κ −1 ζ | ≤ Cδ0 (1 + t)−2 while the estimate (6.68) together with (6.71) (and (6.64)) yields the following estimate for ζ : (6.74) |ζ | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] The assumptions E1, E2 and A, allow us to bound also Lµ, given by equation (3.96): Lµ = m + µe (6.75) where m and e are given by (3.97) and (3.98) respectively. By (6.57) and (6.51): |m| ≤ Cδ0 (1 + t)−1

(6.76)

154


while by (6.57), (6.53), (6.51), (6.54), and assumptions E1 and E2:

Hence, we obtain: In view of the relation κ =

α −1 µ

|e| ≤ Cδ0 (1 + t)−2

(6.77)

|Lµ| ≤ Cδ0 (1 + t)−1

(6.78)

and the second of (6.72) we also obtain: |Lκ| ≤ Cδ0 (1 + t)−1

(6.79)

Integrating this inequality along the integral curves of L, generators of the Cu , we obtain: |κ − κ0 | ≤ Cδ0 log(1 + t)

(6.80)

Here the value of κ0 at a point in Wε∗0 is the value of κ at the corresponding point on 0 along the same integral curve of L. Combining (6.80) with the assumption I0 on the initial data, we obtain the following estimate in Wεs0 : |κ − 1| ≤ Cδ0 [1 + log(1 + t)]

(6.81)

which improves the estimate (6.64). We now combine assumptions F1 with (6.71) and the first of (6.72) to estimate the first derivatives of κ on the hypersurfaces t : |T κ| ≤ Cδ0 [1 + log(1 + t)],

|d/κ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.82)

Next, we combine assumptions F1 with the estimate (6.74) for ζ to obtain an estimate for η through equation (3.65), that is,

We obtain:

η = ζ + d/µ

(6.83)

|η| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.84)

We now combine assumption F2 with the estimate (6.65) and the bounds (6.51) to obtain estimates for χ, ˆ trχ, ˆ and the functions ν, ν, defined by (5.10), (5.11): 1 d log trχ + Lσ 2 dσ 1 d log Lσ ν= trχ + 2 dσ ν=

(6.85) (6.86)

If δ0 is assumed suitably small, assumption F2 implies: trχ ≥ C −1 (1 + t)−1

(6.87)


155

This restriction on the size of δ0 is imposed from now on. According to equations (3.46), (3.118), χ = α(k/ − θ )

(6.88)

χ = κ(k/ + θ )

(6.89)

We thus have: trχ = 2κtrk/ − α −1 κtrχ

and

χˆ = 2κ k/ˆ − α −1 κ χˆ

The estimate (6.65) together with (6.87), assumption F2 and the basic assumptions A then yield: ˆ ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 |trχ| ≤ C(1 + t)−1 [1 + log(1 + t)], |χ|

(6.90)

Similarly, we obtain: ˆ ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 κ|trθ | ≤ C(1 + t)−1 [1 + log(1 + t)], κ|θ|

(6.91)

Finally, combining assumption F2 and (6.87) with the bounds (6.51) we obtain: C −1 (1 + t)−1 ≤ ν ≤ C(1 + t)−1 , ν − η0 (1 + η0 t)−1 ≤ C(1 + t)−2 [1 + log(1 + t)] (6.92) and combining the first of the estimates (6.90) with the bounds (6.51) we obtain: |ν| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.93)

The above results allow us to bound the deformation tensors of the vectorfields T and Q, given by Tables 6.12 and 6.13 respectively. / L and (Y ) π / L the In the following, for any commutation field Y we denote by (Y ) π 1-forms on each surface St,u with components: (Y )

π / L (X A ) =

(Y )

πL A,

(Y )

π / L (X A ) =

(Y )

πL A

/˜ L and (Y ) π /˜ L . and similarly for (Y ) π By the first of the estimates (6.82) and (6.72) together with the bounds (6.51) we obtain, in reference to the Table 6.12, µ−1 |

(T )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.94)

|

(T )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.95)

By the estimates (6.74) and (6.84) (and the basic assumptions A), | −1 µ |

(T ) ˜

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] (T ) ˜ π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.96) (6.97)

156


By the second of the estimates (6.90) and assumption F2 (and the basic assumptions A), | (T ) π (6.98) /ˆ˜ | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 Also, by the estimates (6.92) and (6.93), |tr

(T ) ˜

π /| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.99)

By (6.79) and the second of the estimates (6.72), and by (6.78) together with the bounds (6.51), we obtain, in reference to Table 6.13, µ−1 |

(Q)

π˜ L L | ≤ C[1 + log(1 + t)]

(6.100)

|

(Q)

π˜ L L | ≤ C[1 + log(1 + t)]

(6.101)

By the estimates (6.74) and (6.84), |

(Q) ˜

π / L | ≤ Cδ0 [1 + log(1 + t)]

(6.102)

Finally, by assumption F2 and the estimate (6.92) we obtain: (Q) ˆ˜

π /| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] |tr (Q)π /˜ | ≤ C |

We can obtain a more precise estimate on tr (Q) π /˜ by writing (see (6.13)), 1 − η0 (Q) ˜ π / − 4 = −4 + 4( − 1)(1 + t)ν tr 1 + η0 t η0 +4(1 + t) ν − 1 + η0 t

(6.103) (6.104)

(6.105)

Then by (6.92), together with the fact that since (k 2 ) = 1 by virtue of the first of (6.51) we have | − 1| ≤ Cδ0 (1 + t)−1 we obtain: |tr

(Q) ˜

π / − 4| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.106) (6.107)

To bound the deformation tensors of the vectorfields Ri we must derive estimates for the functions λi , defined by equation (6.20), which enter the expressions (6.40), (6.44), (6.49), for the components (Ri ) π AB , (Ri ) πT A , (Ri ) π L A , respectively, as well as the functions y i , which enter (6.44) and are defined by equation (6.42), and the functions z i , which enter (6.49) and are defined by equation (6.47).


157

Let us denote by r the Euclidean radial coordinate: " # 3 # r = $ (x i )2

(6.108)

i=1

The estimates which follow require upper and lower bounds for the function r on each surface St,u . We derivethese bounds as follows. First, since the function r achieves its maximum value in t Wε∗0 at the outer boundary t C0 = St,0 where it is equal to 1 + η0 t, we have the upper bound: r ≤ 1 + η0 t

: in Wε∗0

(6.109)

This suffices for the time being. Another upper bound for r will be derived later. We turn to the lower bound. We have: 3 3 xi T i x i Tˆ i =κ Tr = r r i=1

(6.110)

i=1

In view of the fact that the Euclidean magnitude of the vector Tˆ is bounded by unity, this implies: |T r | ≤ κ (6.111) Hence, by the estimate (6.81), |T r | ≤ 1 + Cδ0 [1 + log(1 + t)] (6.112) Integrating (6.112) along the integral curves of T from t C0 = St,0 where r = 1+η0 t, to St,u , yields the lower bound: r ≥ 1 − u + η0 t − Cδ0 u[1 + log(1 + t)]

(6.113)

In view of the fact that, in Wε∗0 , u takes values in [0, ε0 ], taking δ0 suitably small we conclude that: (6.114) r ≥ C −1 (1 + η0 t) We proceed to derive a suitable estimate for the coefficient functions λi . According to (6.20) we have: ◦ (6.115) λi = h( Ri , Tˆ ) As we remarked earlier Tˆ is the unit normal to the surfaces St,u in t relative to the induced acoustical metric h. From (3.11) we have: ◦

◦

◦

h( Ri , Tˆ ) = Ri , Tˆ + H Rim ψm ψTˆ

(6.116)

Here and in the following we shall denote by , the inner product with respect to the Euclidean metric on t : , = g( , )

158


We shall also denote by the magnitude of tensors at a point in t with respect to the Euclidean metric g, reserving the notation | | for the corresponding magnitude with respect to the induced acoustical metric h (or simply the absolute value of real numbers or functions at a point). Since the metric h dominates the Euclidean metric g, we have: Tˆ ≤ 1

(6.117)

Also, from (6.28), (6.109), ◦

Ri =

r 2 − (x i )2 ≤ r ≤ 1 + η0 t

(6.118)

From (6.117), (6.118) and assumption E1 we have: ◦

| Rim ψm ψTˆ | ≤ Cδ02 (1 + t)−1

(6.119)

We remark that |Ri |, the magnitude of Ri with respect to the metric h is bounded ◦

◦

by | Ri |, the corresponding magnitude of Ri . Since the maximal eigenvalue of h with respect to the Euclidean metric is 1 + ρ H (see (6.58)), we have: ◦

|Ri |2 ≤ (1 + ρ H ) Ri 2

(6.120)

In particular, by (6.118) and assumptions E1 |Ri | ≤ C(1 + t)

(6.121)

◦

Let us introduce the functions λi by: ◦

◦

λi = Ri , Tˆ

(6.122)

By (6.115), (6.116), we then have: ◦

◦

λi =λi +H Rim ψm ψTˆ

(6.123) ◦

Thus, in view of (6.119) we shall arrive at an estimate for λi once we bound λi . Our ◦

approach is to derive an ordinary differential equation for λi along the integral curves of the vectorfield L, generators of Cu , and use the following additional assumption on the initial data. ε

There is a positive constant C such that on 00 : I1:

◦

| λi | ≤ Cδ0

Now, we have:

◦

L λi =

m

◦

(L Rim )Tˆ m +

m

◦

Rim L Tˆ m

(6.124)


159

According to equation (2.66) of Chapter 2: ∂ ∂ H ψ0ψ i i ˆ L= − αT + 0 1 + ρ H ∂xi ∂x

(6.125)

Hence, by (6.140) and in view of the fact that according to (6.28), ◦

Rim = i j m x j , we have:

◦

L Rim = −

H ψ0ψ j i j m 1 + ρH

(6.126)

Moreover, L Tˆ m is given by equation (3.159) of Chapter 3: L Tˆ m = p L Tˆ m + q Lm

(6.127)

where p L is given by (3.164) and: q Lm = q LA X m A

(6.128)

is given through (3.169). Substituting (6.126) and (6.127) in (6.124) we obtain, in view of the definition (6.122), ◦

◦

◦

L λi = p L λi + Ri , q L −

H ψ0 i j m ψ j Tˆ m 1 + ρH

(6.129)

Now, according to (3.164) p L = −ψTˆ

1 dH ψ ˆ (Lσ ) + H Tˆ i (Lψi ) 2 dσ T

(6.130)

Using (6.53), (6.51), and assumption E2, we obtain: | p L | ≤ Cδ02 (1 + t)−3

(6.131)

According to (3.169): 1 dH ψ ˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ T − H ψ B Tˆ i (Lψi )

/ AB = −κ −1 ζ B − q LA h

(6.132)

Using the estimate (6.73) together with (6.53), (6.60), (6.51), and assumption E2, we obtain: (6.133) |q L | ≤ Cδ0 (1 + t)−2 In view also of (6.118) it follows that: ◦

◦

◦

| Ri , q L | ≤ Ri q L ≤ Ri |q L | ≤ Cδ0 (1 + t)−1

(6.134)

160


Finally, the last term on the right in (6.129) is also bounded by: Cδ0 (1 + t)−1 by virtue of assumption E1. In view of the above results, equation (6.129) implies the following ordinary differential inequality along the integral curves of L: ◦

◦

|L λi | ≤ Cδ02 (1 + t)−3 | λ +Cδ0 (1 + t)−1

(6.135)

ε

Integrating this inequality from 00 and taking into account the assumption I1 on the initial data, we obtain the following estimate on Wεs0 : ◦

| λi | ≤ Cδ0 [1 + log(1 + t)]

(6.136)

which, through (6.12) implies, in view of assumption E1, that on Wεs0 : |λi | ≤ Cδ0 [1 + log(1 + t)]

(6.137)

This is the desired estimate for the functions λi . ◦

The derivatives of the functions λi with respect to L have been estimated above. We ◦

shall presently derive expressions for the remaining first derivatives of the λi , which show ◦

that all first order derivatives of the λi can be directly estimated. We have: ◦

T λi =

◦

(T Rim )Tˆ m +

m

◦

Rim T (Tˆ m )

(6.138)

m

According to (6.28),

◦

Rim = i j m x j , hence:

◦

Tˆ Rim = i j m Tˆ j and:

◦

(T Rim )Tˆ m = i j m Tˆ j Tˆ m = 0

(6.139) (6.140)

m

that is, the first term on the right in (6.138) vanishes. The equation in question then simplifies to: ◦ ◦ Rim T (Tˆ m ) (6.141) T λi = m

Now

T (Tˆ m )

is given by (3.192): T (Tˆ m ) = pT Tˆ m + qTm

(6.142)

where pT is given by (3.197), while qTm = qTA X m A

(6.143)


161

(see (3.194)) is given through (3.201). Substituting (6.142) in (6.141) we obtain, in view of the definition (6.122), ◦

◦

◦

T λi = pT λi + Ri , qT Next, we have:

(6.144)

◦

j

d/ A Rim = i j m d/ A x j = i j m X A hence:

◦

j d/ A λi = i j m X A Tˆ m +

◦

Rim d/ A Tˆ m

(6.145)

(6.146)

m

Moreover, d/ A Tˆ m is given by equation (3.221) of Chapter 3: d/ A Tˆ m = p/ A Tˆ m + q/mA

(6.147)

where p/ A is given by (3.226) and: q/mA = q/ BA X m B

(6.148)

is given through (3.230). Substituting (6.147) in (6.146) we obtain, in view of the definition (6.122), ◦

◦

◦

ˆj d/ A λi = p/ A λi −i j m X m / A A T + Ri , q

(6.149)

Note that for any vector V in Euclidean space, we have: ◦ Ri , V 2 = r 2 V 2

(6.150)

i

where is the Euclidean projection operator to the Euclidean coordinate spheres: ij = δ ij − r −2 x i x j This fact readily follows using the identities: i j m ikn = δ j k δmn − δ j n δkm ,

(6.151)

i

i m ni = nm

i

Taking in (6.150) V to be Tˆ we obtain, in view of the definition (6.122), Tˆ 2 = r −2

◦

(λi )2

(6.152)

i

The estimate (6.136) together with the lower bound (6.114) then yields: Tˆ ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.153)

162


We shall now use (6.153) to obtain another upper bound for r , complementing the bound (6.109). Let us denote by N the Euclidean outward unit normal to the Euclidean coordinate spheres: xi ∂ (6.154) N= r ∂xi Then (6.110) takes the form: T r = κN, Tˆ (6.155) We decompose:

Tˆ = NN, Tˆ + Tˆ

(6.156)

This is the Euclidean orthogonal decomposition of Tˆ relative to the Euclidean coordinate spheres. Hence, (6.157) Tˆ 2 = N, Tˆ 2 + Tˆ 2 and, since:

h i j Tˆ i Tˆ j = 1,

we have:

h i j = δi j + H ψi ψ j ,

Tˆ 2 = 1 − H (ψTˆ )2

(6.158)

1 − N, Tˆ 2 = H (ψTˆ )2 + Tˆ 2

(6.159)

Substituting in (6.157) yields:

The estimates (6.53) and (6.153) then imply: 1 − N, Tˆ 2 ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.160)

If δ0 is suitably small, this does not exceed 1/2, say. Thus the Euclidean angle between Tˆ and N lies either in the sector [0, π/4] or in the sector [3π/4, π]. Now, Tˆ is the inward unit normal to the surfaces St,u in t with respect to the induced acoustical metric h, while St,0 is itself a coordinate sphere, the inner boundary of the exterior Euclidean region. Thus Tˆ = −N and the angle is π on St,0 . It follows by continuity that the angle in question lies in the sector [3π/4, π], where N, Tˆ < 0, 1 − N, Tˆ > 1. Then (6.160) implies that: 0 ≤ 1 + N, Tˆ ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.161)

Going back now to (6.155), writing: T r + 1 = κ(1 + N, Tˆ ) − (κ − 1)

(6.162)

and using the estimates (6.161) and (6.81) we obtain: |T r + 1| ≤ Cδ0 [1 + log(1 + t)] Since r = 1 + η0 t on St,0 , we have, on St,u :

u

r − 1 − η0 t + u = 0

(T r + 1)du

(6.163)

(6.164)


163

where the integration is along an integral curve of T , from St,0 to St,u . Using the estimate (6.163) we then obtain: |r − 1 − η0 t + u| ≤ Cδ0 u[1 + log(1 + t)]

(6.165)

This estimate gives an upper bound as well as a lower bound for r . The lower bound coincides with the lower bound (6.113), but the upper bound does not coincide with the upper bound (6.109). It is stronger than the upper bound (6.109) when Cδ0 [1 + log(1 + t)] < 1, weaker otherwise. We note finally that the estimate (6.165) implies: 1 1 − ≤ Cδ0 u(1 + t)−2 [1 + log(1 + t)] (6.166) r 1 − u + η0 t Consider now the vectorfield: y = Tˆ + N = Tˆ + (1 + N, Tˆ )N

(6.167)

The estimates (6.153) and (6.161) imply: y ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.168)

Let y i : i = 1, 2, 3 be the rectangular components of the vectorfield y . Comparing the definitions (6.42), (6.154), and (6.167), we see that the functions y i are expressed in terms of the y i by: 1 1 − y i = y i − x i (6.169) r 1 − u + η0 t The bounds (6.168), (6.166), together with the elementary bound (6.109), imply the following estimate on Wεs0 : |y i | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.170)

Using this estimate together with (6.70) and assumption E1, we then obtain through (6.47) the following estimate on Wεs0 : |z i | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.171)

We turn to estimate the deformation tensors of the vectorfields Ri . Since L = α −1 κ L + 2T , expressions (6.14)–(6.16) are equivalent to: (Ri )

πL L = 0

(Ri )

π L L = −2Ri µ

(Ri )

π L L = 4µRi (α

(6.172) (6.173) −1

κ)

(6.174)

The bootstrap assumption F1 and the estimates (6.71) and (6.82) imply: (Ri )

π L L | ≤ Cδ0 [1 + log(1 + t)]

(6.175)

−1 (Ri )

π L L | ≤ Cδ0 [1 + log(1 + t)]

(6.176)

| µ

|

164


Next, we estimate the St,u 1-form (Ri ) π / L , whose components are (Ri ) π L A , given by (6.49). We have five terms on the right-hand side, each of which represents an St,u 1-form. By assumption F2 and (6.121) the first term is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)] The square magnitude of the second term is: (h /−1 ) AB h mn h pq X nA X B ilm z l ir p z r q

Here all repeated indices except i are taken to be summed. Substituting the expansion (6.59) we conclude that this does not exceed: h mp ilm z l ir p z r Substituting from (6.58) this becomes: l =i

 (z l )2 + H 

3

2 ilm z l ψm 

l,m=1

which does not exceed: (1 + Hρ)

(z l )2 l =i

From assumption E1 and the estimate (6.171) we then conclude that the second term on the right in (6.49) is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)] By the estimates (6.73) and (6.137), the third term is bounded in magnitude by: Cδ02 (1 + t)−2 [1 + log(1 + t)] By assumptions E1, E2 and the bound (6.118), the fourth term is bounded in magnitude by: Cδ02 (1 + t)−2 Lastly, by assumption E1, and the bounds (6.51), (6.118), (6.121), the fifth term is bounded in magnitude by: Cδ02 (1 + t)−2 Combining the above results we obtain the estimate: |

(Ri )

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.177)

/T whose components (Ri ) πT A are given by Next we estimate the St,u 1-form (Ri ) π (6.44). We have again five terms on the right-hand side, each of which represents an St,u


165

1-form. In regard to the first term we express (see (6.38)): 1 dH / h h/ = κ θ AB + − ψ A ψ B (T σ ) κ θ AB + 1 − u + η0 t 1 − u + η0 t 2 dσ η0 h/ (α − η0 )h/ = −α −1 κ χ − − α −1 κ 1 − u + η0 t 1 − u + η0 t 1 dH ψ A ψ B (T σ ) +κk/AB − 2 dσ By assumption F2 and the estimates (6.70), (6.63), (6.60) and (6.51), the first term is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)]2 The second term is handled in a similar way to that in which the second term on the right in (6.49) was handled. Thus, its magnitude does not exceed ) (1 + Hρ) (y l )2 l =i

which by the estimate (6.170) and assumption E1 is bounded by: Cδ0 (1 + t)−1 [1 + log(1 + t)]2 The third term on the right in (6.44) is: 1 dH ψTˆ ψ A (T σ ) λi d/ A κ + 2 dσ From the estimate (6.137) together with the bounds (6.51), (6.53), (6.60), (6.82), we conclude that this term is bounded in magnitude by: Cδ02 (1 + t)−1 [1 + log(1 + t)]2 By assumptions E1, E2 and the bound (6.118), the fourth term ◦

H ψ A Rim (T ψm ), is bounded in magnitude by:

Cδ02 (1 + t)−1

Lastly, by assumption E1, and the bounds (6.64), (6.51), (6.118), (6.121), the fifth term is bounded in magnitude by: Cδ02 (1 + t)−2 Combining the above results we obtain the estimate: |

(Ri )

π /T | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]2

(6.178)

166


Since L = α −1 κ L + 2T , it follows from (6.177) and (6.178) that: |

(Ri )

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]2

Finally, we estimate the quadratic form point. The components of this quadratic form, (Ri )

(Ri ) π / in

π /(X A , X B ) =

(Ri )

(6.179)

the tangent space to St,u at each

π AB ,

are given by equation (6.40). Here we have four terms on the right. By assumption F2 and the estimates (6.63) and (6.137), the trace-free part of the first term is bounded in magnitude by: Cδ02 (1 + t)−2 [1 + log(1 + t)] while by assumption F2, (6.87), and the estimates (6.63) and (6.137) the trace of this term is bounded in absolute value by: Cδ0 (1 + t)−1 [1 + log(1 + t)] The second term on the right in (6.40) is of the form: H ( i w AB + i w B A ) where: i

w AB = X kA X lB ψk i ωl ,

i

ωl = ilm ψm

The square of the magnitude of the second term is thus: 2H 2(h /−1 ) AC (h /−1 ) B D ( i w AB i wC D + i w AB i w DC ) and, writing

n /−1 )mn (h /−1 ) AB X m A X B = (h

and substituting the expansion (6.59) we have, in view of the fact that h by g−1 ,

−1

is dominated

(h /−1 ) AC (h /−1 ) B D i w AB i wC D = (h/−1 )kp ψk ψ p (h/−1 )lq i ωl i ωq −1

−1

≤ (h )kp ψk ψ p (h )lq i ωl i ωq   ≤ (ψk )2 ( i ωl )2 ≤ (ψk )2  (ψl )2  k

l

l =i

k

Similarly, 2 (h /−1 ) AC (h /−1 ) B D i w AB i w DC = (h/−1 )kq ψk i ωq   ≤ (h /−1 )kp ψk ψ p (h /−1 )lq i ωl i ωq ≤ (ψk )2  (ψl )2  k

l =i


167

We thus conclude by assumption E1 that the second term on the right in (6.40) is bounded in magnitude by: Cδ02 (1 + t)−2 By assumptions E1, E2 and the bound (6.118), the third term ◦

H Rim (X nA ψn (d/ B ψm ) + X nB ψn (d/ A ψm )), is bounded in magnitude by:

Cδ02 (1 + t)−2

Lastly, by assumption E1, and the bounds (6.51) and (6.121), the fourth term is bounded in magnitude by: Cδ03 (1 + t)−3 Combining the above results we conclude that the following estimates hold: |

(Ri )

trπ /| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] | (Ri ) π /ˆ | ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.180) (6.181)

˜ which refer to the metric Since according to (6.10) the deformation tensors (Y ) π, ˜h = h, are related to the deformation tensors (Y ) π, which refer to the metric h, by (Y )

π˜ =

(Y )

π+

d (Y σ )h dσ

and by the bounds (6.51) and (6.121) we have |Ri σ | ≤ Cδ0 (1 + t)−1 , (6.172) and the estimates (6.175)–(6.177), (6.179)–(6.181) imply the following for the deformation tensors (Ri ) π: ˜ (Ri )

π˜ L L = 0

(6.182)

(Ri )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.183)

−1 (Ri )

≤ Cδ0 [1 + log(1 + t)]

| µ

|

| | |

π˜ L L | (Ri ) ˜ π /L | (Ri ) ˜ π /L | (Ri ) ˜

≤ Cδ0 (1 + t)

−1

[1 + log(1 + t)]

≤ Cδ0 (1 + t)

−1

[1 + log(1 + t)]

(6.184) (6.185) 2

(6.186)

trπ /| ≤ Cδ0 (1 + t) [1 + log(1 + t)] | (Ri ) π /˜ˆ | ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.187)

−1

In concluding the present chapter we shall estimate the functions (Y )

δ=

1 tr˜ 2

(Y )

π˜ − µ−1 Y µ − 2−1 Y

(6.188) (Y ) δ,

defined by: (6.189)

168


These functions play an important role in the next chapter. Here tr˜ denotes the trace with ˜ Since respect to the metric h. h˜ −1 = −1 h −1 we have:

tr˜

(Y )

π˜ = −1 tr

(Y )

π˜ = −1 (−µ−1

(Y )

π˜ L L + tr

(Y ) ˜

π /)

Hence, (Y )

1 δ = − −1 µ−1 (Y ) π˜ L L − µ−1 Y µ 2 d −1 1 (Y ) ˜ tr Yσ + π /−2 2 dσ

From Tables 6.12 and 6.13 we then find: 1 (T ) tr δ = −1 2 1 (Q) tr δ = −1 2

d Tσ dσ d (Q) ˜ Qσ + 1 π /− dσ (T ) ˜

π /−

(6.190)

(6.191) (6.192)

Hence, by the estimates (6.99), (6.104) and the bounds (6.51) we obtain: |

(T )

δ| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.193)

|

(Q)

δ| ≤ C

(6.194)

Finally, from the formulas (6.173) and (6.10), (Ri )

π˜ L L = −2Ri µ − 2µRi

We thus have: (Ri )

δ = −1

1 tr 2

(Ri ) ˜

π /−

d Ri σ dσ

(6.195)

The estimate (6.187) together with the bounds (6.51) then yields: |

(Ri )

δ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.196)

Chapter 7

Outline of the Derived Estimates of Each Order We begin with the following proposition. Proposition 7.1 Let ψ be a solution of the inhomogeneous wave equation h˜ ψ = ρ and let Y be an arbitrary vectorfield. Then ψ = Y ψ satisfies the inhomogeneous wave equation: h˜ ψ = ρ where the source function ρ is related to the source function ρ by: ˜ ρ = div

(Y )

1 J˜ + Yρ + tr˜ 2

(Y )

πρ ˜

Here (Y ) J˜ is the commutation current associated to ψ and Y , a vectorfield with components, with respect to an arbitrary local frame field, (Y ) ˜µ

J =

and

(Y ) π ˜

1 ˜ −1 µα ˜ −1 νβ ((h ) (h ) + (h˜ −1 )να (h˜ −1 )µβ − (h˜ −1 )µν (h˜ −1 )αβ ) 2

(Y )

π˜ αβ ∂ν ψ

is the Lie derivative of the metric h˜ with respect to the vectorfield Y .

Proof. Let f s be the local 1-parameter group of diffeomorphisms generated by Y . We denote by f s∗ the corresponding pullback. We then have: f s∗ h˜ ( f s∗ ψ) = f s∗ (h˜ ψ) = f s∗ ρ

(7.1)

170

Chapter 7. Outline of the Derived Estimates of Each Order

Now, in an arbitrary system of local coordinates, h˜ ψ =

and: fs∗ h˜ ( f s∗ ψ) =

1 −deth˜

1 −det f s∗ h˜

−1 µν ˜ ˜ ν ψ) −deth∂ ∂µ ((h )

−1 µν ˜ ˜ ν ( f s∗ ψ)) −det f s∗ h∂ ∂µ ((( f s∗ h) )

(7.2)

(7.3)

Let us differentiate the expression (7.3) with respect to s at s = 0. In view of the facts that: d ˜ f s∗ h = LY h˜ = (Y ) π, ˜ ds s=0 d 1 −det f s∗ h˜ = −deth˜ tr˜ (Y ) π, ˜ ds 2 s=0 d ˜ −1 )µν (( f s∗ h) = −(h˜ −1 )µα (h˜ −1 )νβ (Y ) π˜ αβ ds s=0 and:

d fs∗ ψ ds

= s=0

d ψ ◦ fs ds

= Yψ s=0

we obtain: 1 d 1 ˜ ν ψ) fs∗ h˜ ( f s∗ ψ) =−

˜ µ ((h˜ −1 )µν −deth∂ (7.4) tr˜ (Y ) π∂ ds 2 −deth˜ s=0 1 ˜ +

∂µ −deth(−( h˜ −1 )µα (h˜ −1 )νβ (Y ) π˜ αβ −deth˜ ˜ ν (Y ψ) +(1/2)(h˜ −1)µν tr˜ (Y ) π)∂ ˜ ν ψ + (h˜ −1 )µν −deth∂ On the other hand, by (7.1), d d d = = = Yρ ˜ ( f s∗ ψ) f s∗ ρ ρ ◦ fs ds fs∗ h ds ds s=0 s=0 s=0

(7.5)

Comparing (7.4) and (7.5) and in view of the expression (7.2) and a similar expression with Y ψ in the role of ψ, the proposition follows. Let X be an arbitrary vectorfield. In an arbitrary system of local coordinates the divergence of X with respect to the acoustical metric h is expressed by: divX = Dµ X µ = √

∂ √ ( −deth X µ ) −deth ∂ x µ 1


171

while its divergence with respect to the conformal acoustical metric h˜ is expressed by: 1 ∂ ˜ ( −deth˜ X µ ) divX = D˜ µ X µ =

µ ∂ x ˜ −deth

Since h˜ µν = h µν , we have:

√ −deth˜ = 2 −deth,

D˜ µ X µ = −2 Dµ (2 X µ ) (Y ) J˜,

If we apply this to the vectorfield

˜ div where:

(Y )

we obtain:

J˜ = −2 div

(Y )

(7.6)

J = 2

(Y )

(Y )

J

(7.7)

J˜

(7.8)

With respect to an arbitrary local frame field we have: (Y ) µ

J =

1 −1 µα −1 νβ ((h ) (h ) + (h −1 )να (h −1 )µβ − (h −1 )µν (h −1 )αβ ) 2

Setting

(Y ) µν

π˜

we can write: (Y ) µ

J =

= (h −1 )µα (h −1 )νβ

(Y ) µν

π˜

(Y )

1 − (h −1 )µν tr 2

π˜ αβ

(Y )

π˜ ∂ν ψ

(Y )

π˜ αβ ∂ν ψ (7.9) (7.10) (7.11)

We now consider the nth order variations ψn of the wave function φ, as defined in the beginning of the previous chapter, by applying to a first order variation ψ1 a string of commutation vectorfields Yi : i = 1, 2, 3, 4, 5 of length n − 1 (see equation (6.1)). We shall use Proposition 7.1 to derive a recursion formula for the corresponding source functions ρn , h˜ ψn = ρn (7.12) (see equation (6.3)), starting from the fact that the source function ρ1 corresponding to a first order variation vanishes (see equation (6.2)): ρ1 = 0

(7.13)

Now, an nth order variation ψn results by applying one of the commutation vectorfields to an n − 1th order variation ψn−1 . Denoting by Y one of the Yi : i = 1, 2, 3, 4, 5, we have: (7.14) ψn = Y ψn−1 Thus Proposition 7.1 directly applies to ψn−1 and ψn yielding the following relation between ρn and ρn−1 : ˜ ρn = div

(Y )

1 Jñ−1 + Yρn−1 + tr˜ 2

(Y )

πρ ˜ n−1

(7.15)

172


Here,

(Y ) J˜ n−1

(Y ) ˜µ Jn−1

=

is the commutation current associated to ψn−1 and Y :

1 ˜ −1 µα ˜ −1 νβ ((h ) (h ) + (h˜ −1 )να (h˜ −1 )µβ − (h˜ −1 )µν (h˜ −1 )αβ ) 2

(Y )

π˜ αβ ∂ν ψn−1 (7.16) Equation (7.15) is a recursion formula for the sources ρn , but it is not quite in the form which can be used in our estimates. To obtain the appropriate form we consider instead the re-scaled sources: (7.17) ρñ = 2 µρn Then by (7.15), (7.16), and (7.7)–(7.11), the ρñ satisfy the recursion formula: ρñ = Here: (Y ) µ Jn−1

and

(Y ) δ

(Y )

σn−1 + Y ρñ−1 +

(Y )

δ ρñ−1

(Y )

=

σn−1 = µdiv (Y ) Jn−1 1 −1 µν (Y ) (Y ) µν π˜ − (h ) tr π˜ ∂ν ψn−1 2

(7.18) (7.19) (7.20)

are the functions defined by equation (6.189): (Y )

δ=

1 tr˜ 2

(Y )

π˜ − µ−1 Y µ − 2−1 Y

(7.21)

Moreover, by (7.13), ρ˜1 = 0

(7.22)

Since ψn is a solution of the inhomogeneous wave equation (7.12), the argument of Theorem 5.1 can be applied to ψn to yield estimates analogous to those of that theorem with ψn in the role of ψ, provided that we can appropriately estimate the error integrals contributed by the source function ρn . These are the spacetime integrals Q 0,0,n dµh , Q 1,0,n dµh (7.23) Wut

Wut

where Q 0,0,n and Q 1,0,n are the terms Q 0,0 and Q 1,0 , associated to the vectorfields K 0 and K 1 respectively, given by (5.87) and (5.97) with ψn and ρn in the role of ψ and ρ: Q 0,0,n = −2 ρn K 0 ψn Q 1,0,n = −2 ρn (K 1 ψn + ωψn )

(7.24) (7.25)

Now, by virtue of equation (5.27) we have: dµh = µdtdudµh/ Thus, in view of (7.17) the error integrals (7.23) take the form: − ρñ (K 0 ψn )dt du dµh/

(7.26)

(7.27)

Wut

and −

Wut

ρñ (K 1 ψn + ωψn )dt du dµh/

(7.28)


173

We shall first consider the contribution of the term (Y ) σn−1 in ρñ (see (7.18)–(7.20)) to the above error integrals. Let V be an arbitrary vectorfield defined in the spacetime domain Wε0 . We decompose: /, where V / = V = V A X A V = V LL + V LL + V is tangential to the surfaces St,u . We have: V L = −(1/2µ)h(V, L) V L = −(1/2µ)h(V, L) V A = (h/−1 ) AB h(V, X B ) The divergence of the vectorfield V is then expressed as: divV = (D L V ) L + (D L V ) L + (D X A V ) A Replacing V above by D L V , D L V , D X A V , respectively, we obtain: (D L V ) L = −(1/2µ)h(D L V, L) (D L V ) L = −(1/2µ)h(D L V, L) (D X A V ) A = (h/−1 ) AB h(D X A V, X B ) Moreover, substituting the decomposition of V we obtain, appealing to Table 3.117, h(D X A V, X B ) = V L h(D X A L, X B ) + V L h(D X A L, X B ) + h(D X A V /, X B ) /XAV /, X B ) = χ AB V L + χ AB V L + h/(D hence: (D X A V ) A = trχ V L + trχ V L + div / V / We thus obtain: divV = −(1/2µ)(h(D L V, L) + h(D L V, L)) + trχ V L + trχ V L + div / V / We can express the first term on the right-hand side in terms of: VL = h(V, L),

VL = h(V, L)

Appealing again to Table 3.117 we obtain: h(D L V, L) = L(h(V, L)) − h(V, D L L) = L(VL ) + h(V, 2ζ A X A ) = L(VL ) + 2ζ A V A h(D L V, L) = L(h(V, L)) − h(V, D L L) = L(VL ) − h(V, −L(α −1 κ)L + 2η A X A ) = L(VL ) + L(α −1 κ)VL − 2η A V A

174


Substituting and noting that according to equation (3.65) we have: η A − ζ A = d/ A µ we obtain the following formula for the divergence of an arbitrary spacetime vectorfield V : / (µV /) −2µdivV = L(VL ) + L(VL ) − 2div +L(α −1 κ)VL + trχ VL + trχ VL

(7.29)

We shall apply the above formula to the commutation current (Y ) Jn−1 , given by (7.20). We introduce the vectorfields (Y ) Z˜ and (Y ) Z˜ , associated to the commutation vectorfield Y , and tangential to the surfaces St,u , by the conditions that: h( Z˜ , V ) =

(Y )

π(L, ˜ V ),

h( Z˜ , V ) =

(Y )

π(L, ˜ V )

(7.30)

for any vector V ∈ T Wε0 . In terms of the null frame we have: (Y )

where:

(Y )

Z˜ A =

Z˜ = (Y )

Z˜ A X A ,

(Y )

Z˜ =

π˜ L B (h /−1 ) AB ,

(Y )

A Z˜ =

(Y )

(Y )

A Z˜ X A

(Y )

π˜ L B (h/−1 ) AB

(7.31)

Note that the St,u - tangential vectorfields (Y ) Z˜ and (Y ) Z˜ correspond, through the in/ L and (Y ) π / L , respectively, which were introduced in duced metric h /, to the 1-forms (Y ) π the previous chapter. Taking into account these definitions as well as the fact that: tr we bring the components of (Y ) (Y )

(Y )

π˜ = −(1/µ)

(Y ) J n−1

Jn−1,L = −(1/2)tr Jn−1,L

µ

π˜ L L + tr

(Y ) ˜

π /

(Y ) ˜ (Y )

Z˜ · d/ψn−1 (Y ) ˜ Z · d/ψn−1 (Y )

π˜ L L (Lψn−1 )

A Z˜ A (Lψn−1 ) − (1/2) (Y ) Z˜ (Lψn−1 ) +(1/2)( (Y ) π˜ L L − µtr (Y ) π /˜ )(h/−1 ) AB d/ B ψn−1 +µ (Y ) π /˜ BC (h/−1 ) AB (h/−1 )C D d/ D ψn−1

= −(1/2)

(7.32)

to the following form:

π /(Lψn−1 ) + (Y ) ˜ = −(1/2)tr π /(Lψn−1 ) + −(1/2µ)

(Y ) A /Jn−1

(Y )

(Y )

(7.33)

We note here the absence of a term involving (1/µ) (Y ) π˜ L L in the expressions for (Y ) J (Y ) J n−1,L , n−1,L , despite the presence of such a term in (7.32). This is related to the fact that the operator g on a 2-dimensional manifold M with a metric g is conformally covariant. Here, in the role of such a 2-dimensional manifold we have the 2-dimensional distribution of time-like planes spanned by the vectors L and L at each point. This distribution is not integrable, the obstruction to integrability being [L, L] = 2,

= −(h/−1 ) AB (ζ B + η B )X A

(7.34)


175

(see (3.71) and Table 3.117). However, the conformal covariance is still reflected by the fact that the restriction of the commutation current to the plane spanned by L and L depends only on the trace-free, relative to this plane, part of the restriction of π˜ to the plane, therefore not on π˜ L L . Applying formula (7.29) to (Y ) Jn−1 we obtain: (Y )

σn−1 = −(1/2)L(

(Y )

−(1/2)L(α

−1

Jn−1,L ) − (1/2)L( κ)

(Y )

(Y )

Jn−1,L ) + div / (µ

Jn−1,L − (1/2)trχ

(Y )

(Y )

/Jn−1 )

Jn−1,L − (1/2)trχ

(Y )

(7.35) Jn−1,L

The first line on the right in (7.35) contains the principal terms, the derivatives of products of components of (Y ) π˜ with derivatives of ψn−1 , while the second line contains lower order terms of the form of triple products of connection coefficients of the null frame with components of (Y ) π˜ and derivatives of ψn−1 . To estimate the contribution of (Y ) σn−1 to the error integrals we shall decompose: (Y )

σn−1 =

(Y )

σ1,n−1 +

(Y )

σ2,n−1 +

(Y )

σ3,n−1

(7.36)

where (Y ) σ1,n−1 is to contain all the terms which are products of components of (Y ) π˜ with 2nd derivatives of ψn−1 , (Y ) σ2,n−1 is to contain all the terms which are products of 1st derivatives of (Y ) π˜ with 1st derivatives of ψn−1 , and (Y ) σ3,n−1 is to contain only lower order terms. Now, in view of the expressions (7.33), the first two terms on the right in (7.35) contain L( (Y ) Z˜ · d/ψn−1 ) and L( (Y ) Z˜ · d/ψn−1 ). In decomposing each of these into a term which is a product of a component of (Y ) π˜ with a 2nd derivative of ψn−1 , a term which is a product of a 1st derivative of (Y ) π˜ with a 1st derivative of ψn−1 , and a lower order term, we shall make use of the following lemma. Lemma 7.1 Let V be a vectorfield defined on the spacetime domain Wε∗0 and tangential to the surfaces St,u . Also, let f be a function on Wε∗0 . We then have: L(V · d/ f ) = V · d/ L f + L / L V · d/ f L(V · d/ f ) = V · d/ L f + L / L V · d/ f − (V · d/(α −1 κ))L f Here, we denote: L / L V = L L V = [L, V ],

L / L V = L L V = [L, V ]

Proof. Since V is tangential to the St,u we have: V · d/ f = V · d f Now, the exterior derivative commutes with the Lie derivative. Hence: L(V · d f ) = L L V · d f + V · d L f

(7.37)

L(V · d f ) = L L V · d f + V · d L f

(7.38)

and:

176


We decompose: / L V + (L L V ) L L + (L L V ) L L LL V = L and we have: (L L V ) L = −(1/2µ)h(L L V, L) (L L V ) L = −(1/2µ)h(L L V, L) Taking account of the fact that V is h-orthogonal to L and L, and appealing to Table 3.117, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = −h(V, D L L) − h(DV L, L) = 2ζ (V ) − 2ζ (V ) = 0 and, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = h(D L V, L) = −h(V, D L L) = 0 We conclude that /L V LL V = L

(7.39)

that is, L L V is tangential to the surfaces St,u . (This follows also by considering the fact that the 1-parameter group "s generated by L maps St,u onto St +s,u while the 1-parameter group generated by V maps St,u onto itself.) We decompose: / L V + (L L V ) L L + (L L V ) L L LL V = L and we have: (L L V ) L = −(1/2µ)h(L L V, L) (L L V ) L = −(1/2µ)h(L L V, L) Taking again account of the fact that V is h- orthogonal to L and L, and appealing to Table 3.117, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = h(D L V, L) = −h(V, D L L) = 2µV (α −1 κ) and, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = −h(V, D L L) − h(DV L, L) = −2η(V ) + 2η(V ) = 0 We conclude that:

LL V = L / L V − (V (α −1 κ))L

In view of (7.39), (7.40) and the formulas (7.37), (7.38), the lemma follows.

(7.40)

Chapter 7. Outline of the Derived Estimates of Each Order (Y ) Z ˜,

Applying Lemma 7.1 to the case V = L(

(Y )

Z˜ · d/ψn−1 ) =

(Y )

Z˜ · d/ψn−1 ) =

f = ψn−1 , we obtain:

/L Z˜ · d/ Lψn−1 + L

(Y )

while applying the lemma to the case V = L(

177

(Y ) Z˜ ,

(Y )

Z˜ · d/ψn−1

(7.41)

f = ψn−1 , yields:

/ L (Y ) Z˜ · d/ψn−1 Z˜ · d/ Lψn−1 + L − (Y ) Z˜ · d/(α −1 κ)Lψn−1 (Y )

(7.42)

We are now ready to substitute the expressions given by Table 7.33 for the components of the commutation current (Y ) Jn−1 into the expression (7.35) for (Y ) σn−1 . There results a decomposition of the form (7.36), with (Y ) σ1,n−1 , (Y ) σ2,n−1 , and (Y ) σ3,n−1 given by: (Y )

(Y ) ˜

π /(L Lψn−1 + ν Lψn−1 )

σ1,n−1 = (1/2)tr

−1 (Y )

+(1/2)(µ π˜ L L )L 2 ψn−1 − (Y ) Z˜ · d/ Lψn−1 − (Y ) Z˜ · d/ Lψn−1 (Y )

+(1/2) (Y )

σ2,n−1 = (1/4)L(tr

π˜ L L / ψn−1 + µ

(Y ) ˆ˜

π /· D / 2 ψn−1

(Y ) ˜

π /)Lψn−1 + (1/4)L(tr

+(1/4)L(µ

(7.43)

(Y ) ˜

π /)Lψn−1

−1 (Y )

π˜ L L )Lψn−1 ˜ Z · d/ψn−1 − (1/2)L −(1/2)L /L / L (Y ) Z˜ · d/ψn−1 −(1/2)div / (Y ) Z˜ Lψn−1 − (1/2)div / (Y ) Z˜ Lψn−1 /ˆ˜ ) · d/ψ +(1/2)d/ (Y ) π˜ · d/ψ + div / (µ (Y ) π (Y )

LL

n−1

n−1

(7.44)

and: (Y )

σ3,n−1 =

(Y ) L σ3,n−1 Lψn−1

+

(Y ) L σ3,n−1 Lψn−1

+

(Y )

/σ3,n−1 · d/ψn−1

(7.45)

where: (Y ) L σ3,n−1

π / + (1/4)trχ(µ−1

(Y )

Z˜ · d/(α

(Y )

π˜ L L )

−1

κ) = −(1/4)(L log )tr (Y ) π /˜ +(1/2)

(Y ) L σ3,n−1 (Y ) /σ3,n−1

(Y ) ˜

= (1/4)trχtr

(7.46) (7.47)

(Y ) ˜

= −(1/2)(tr π /) −(1/2)(trχ + L(α −1 κ))

The above expressions for (Y ) σ1,n−1 , (Y ) σ2,n−1 , Note the presence of the lower order term (1/2)tr

(Y ) ˜

(Y )

Z˜ − (1/2)trχ

(Y ) σ 3,n−1 ,

π /ν Lψn−1

(Y )

Z˜

(7.48)

are as mentioned earlier.

178


in (Y ) σ1,n−1 , contributing to the first term in (7.43). This lower order term is subtracted from the second term of the second line in (7.35), yielding, in view of definition (5.10), ν = (1/2)(trχ + L log ), the coefficient (Y ) σ3,n−1 of Lψn−1 in (Y ) σ3,n−1 . Then (Y ) σ3,n−1 consists of the second line in (7.35), after the subtraction just mentioned, the commutator term: L

(1/4)tr

(Y ) ˜

π /[L, L]ψn−1 = −(1/4)tr

(Y ) ˜

π /(L(α −1 κ)Lψn−1 + 2 · d/ψn−1 )

which results from re-writing the second term of the first line in (7.35), and the contribution of the last term in (7.42). We shall begin our estimates of the contribution of (Y ) σn−1 to the error integrals (7.27), (7.28), with the estimates of the partial contribution of (Y ) σ1,n−1 . In these estimates we shall make use of the following assumption on the components of the deformation tensors of the commutation fields. G0: There is a positive constant C independent of s such that in Wεs0 , for all five commutation fields Y , we have: (Y )

µ

−1 (Y )

|

|

π˜ L L = 0 π˜ L L | ≤ C[1 + log(1 + t)]

(Y )

≤ C[1 + log(1 + t)]

| |

≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C[1 + log(1 + t)]

π˜ L L | (Y ) ˜ π /L | (Y ) ˜ π /L |

| |

(Y ) ˆ˜

(Y )

π /| ≤ C(1 + t)−1 [1 + log(1 + t)] trπ /˜ | ≤ C

This assumption holds by virtue of the results of Chapter 6. We shall also make use of the following assumptions concerning the set of rotation fields {Ri : i = 1, 2, 3}. H0: There is a positive constant C independent of s such that for any function f differentiable on each surface St,u we have: |d/ f |2 ≤ C(1 + t)−2

(Ri f )2

i

H1: There is a positive constant C independent of s such that for any differentiable 1-form ξ on each surface St,u we have: |D /ξ |2 ≤ C(1 + t)−2

i

|L / R i ξ |2


179

Assumption H1 implies in particular, taking ξ = d/ f , where f is any function twice differentiable on each St,u , that: |D / 2 f |2 ≤ C(1 + t)−2 |d/ Ri f |2 i

H2: There is a positive constant C independent of s such that for any differentiable traceless symmetric 2-covariant tensorfield ϑ on each surface St,u we have: |D /ϑ|2 ≤ C(1 + t)−2 |L / Ri ϑ|2 i u (t), F t (u), E u (t), F t (u), the integrals (5.47), In the following we denote by E0,n 0,n 1,n 1,n (5.48), (5.58), (5.53), respectively, of Chapter 5, with ψn in the role of ψ: u α −1 κ(η0−1 + α −1 κ)(Lψn )2 + (Lψn )2 E0,n (t) = tu 2 +(η0−1 + 2α −1 κ)µ|d/ψn |2 dµh/ du (7.49)

t F0,n (u) = (η0−1 + α −1 κ)(Lψn )2 + µ|d/ψn |2 dµh/ dt (7.50) u (t) = E1,n t (u) = F1,n

C ut

tu

C ut

−1 −1 ων {α κ(Lψn + νψn )2 + µ|d/ψn |2 }dµh/ du 2

(7.51)

ων −1 (Lψn + νψn )2 dµh/ dt

(7.52)

In each of the above the sum is over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. The assumptions A1, A2, B1, D1, of Chapter 5, imply the analogues of (5.68)–(5.71): u −2 −2 −1 u 2 2 2 [η0 µ(η0 + µ)(η0 (Lψn ) + |d/ψn | ) + (Lψn ) ]dµh/ du C E0,n (t) ≤ 0

St,u

u ≤ CE0,n (t) t C −1 F0,n (u) ≤

t 0

St ,u

[η0−2 (η0 + µ)(Lψn )2 + µ|d/ψn |2 ]dµh/ dt

t ≤ CF0,n (u) u (t) ≤ (1 + t)2 C −1 E1,n

u 0

St,u

u ≤ CE1,n (t) t −1 t 2 (1 + t ) C F1,n (u) ≤ 0

t ≤ CF1,n (u)

(7.53)

µ[η0−2 (Lψn + νψn )2 + |d/ψn |2 ]dµh/ du

St ,u

(7.54)

(7.55)

(Lψn + νψn )2 dµh/ dt (7.56)

180


Again, in each of the above the sum is over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. Moreover, we consider, as in (5.81)–(5.84) the following quantities which are, by their definition, non-decreasing functions of t at each u: u

Note

u E 0,n (t) = supt ∈[0,t ]E0,n (t )

(7.57)

t F0,n (u)

(7.58)

u u E 1,n (t) = supt ∈[0,t ] [1 + log(1 + t )]−4 E1,n (t ) t t F 1,n (u) = supt ∈[0,t ] [1 + log(1 + t )]−4 F1,n (u) u that the quantities E 0,n (t), E 1,n are also non-decreasing functions of u The contribution of (Y ) σn−1 to the error integrals (7.27) and (7.28) is

−

Wut

(K 0 ψn )

(Y )

and −

Wut

(K 1 ψn + ωψn )

(7.59) (7.60) at each t.

σn−1 dt du dµh/ (Y )

σn−1 dt du dµh/

(7.61)

(7.62)

respectively. We shall first consider the partial contribution of (Y ) σ1,n−1 , given by (7.43), to each of these integrals, and we begin with the partial contribution to the integral (7.61), which is associated to the vectorfield (see (5.15)) K 0 = (η0−1 + α −1 κ)L + L namely, the error integral:

−

Wut

(K 0 ψn )

(Y )

σ1,n−1 dt du dµh/

(7.63)

(7.64)

The first term in (7.43), (1/2)tr

(Y ) ˜

π /(L Lψn−1 + ν Lψn−1 )

involves Lψn−1 . Now, by (3.114), (6.6), L is expressed in terms of the commutation fields by: (7.65) L = α −1 κ(1 + t)−1 Q + 2T Thus we have: (1/2)(L Lψn−1 + ν Lψn−1 ) = (LT ψn−1 + νT ψn−1 ) +(1/2)α −1 κ(1 + t)−1 (L Qψn−1 + ν Qψn−1 )

(7.66)

+(1/2)(1 + t)−1 (L(α −1 κ) − (1 + t)−1 α −1 κ)Qψn−1 The leading term is that on the first line on the right. We shall estimate below the contribution of this term, namely the integral (K 0 ψn )tr (Y ) π (7.67) − /˜ (LT ψn−1 + νT ψn−1 )dt du dµh/ Wut


181

The contributions of the terms on the second and third lines in (7.65) are easier to estimate because of the presence of the beneficial factor (1 + t)−1 (the third line is a lower order term). By the last of assumptions G0, the integral (7.67) is bounded in absolute value by: C(M L + M L ) where:

L

M = ML =

Wut

Wut

(η0−1 + α −1 κ)|Lψn ||LT ψn−1 + νT ψn−1 |dt du dµh/

(7.68)

|Lψn ||LT ψn−1 + νT ψn−1 |dt du dµh/

(7.69)

Since T is one of the commutation fields, we have, by (7.56), (7.60), for all u ∈ [0, ε0 ], t (1 + t )2 (LT ψn−1 + νT ψn−1 )2 dµh/ dt ≤ CF 1,n (u)[1 + log(1 + t)]4 (7.70) C ut

To estimate M L we apply an argument similar to that of Chapter 5, following the definitions (5.211)–(5.213). That is, we define the non-negative integer N to be the integral part of log t/ log 2: log t = N + r, 0 ≤ r < 1 (7.71) log 2 and we set: t−1 = 0,

tm = 2m+r : m = 0, 1, . . . , N

(7.72)

Then t N = t and we have the partition: Wut =

N

t

Wum−1

,tm

(7.73)

m=0

Hence, ML =

N

L

Mm

(7.74)

m=0

where: L

Mm =

t ,tm Wum−1

|Lψn ||LT ψn−1 + νT ψn−1 |dt du dµh/

(7.75)

We have: L Mm

≤

t

Wum−1

,tm

(1 + t )

·

t

Wum−1

3/2

,tm

1/2

(LT ψn−1 + νT ψn−1 ) dt du dµh/

−3/2

(1 + t )

2

(Lψn ) dt du dµh/ 2

1/2 (7.76)

182


Now, for t ∈ [tm−1 , tm ] it holds that:

−1/2 1 (1 + t )−1/2 ≤ (1 + tm−1 )−1/2 = 1 + tm 2 −1/2 1 1 + tm ≤ = 21/2 (1 + tm )−1/2 2 2

Therefore, writing (1 + t )3/2 = (1 + t )−1/2 · (1 + t )2 , we obtain: (1 + t )3/2 (LT ψn−1 + νT ψn−1 )2 dt du dµh/ tm−1 ,tm Wu ≤ 21/2(1 + tm )−1/2 (1 + t )2 (LT ψn−1 + νT ψn−1 )2 dt du dµh/ t

Wum

and,

t Wum

(1 + t )2 (LT ψn−1 + νT ψn−1 )2 dt du dµh/ u

=

2

t C um

0

(1 + t ) (LT ψn−1 + νT ψn−1 ) dµh/ dt 2

du

u tm tm F1,n (u )du ≤ C F 1,n (u )[1 + log(1 + tm )]4 du 0 0 u t ≤C F 1,n (u )du [1 + log(1 + tm )]4 ≤C

u

0

hence:

3/2

t Wum−1

(1 + t ) ,tm

2

(LT ψn−1 + νT ψn−1 ) dt du dµh/ ≤ C Am

u 0

t

F 1,n (u )du (7.77)

where: Am = (1 + tm )−1/2 [1 + log(1 + tm )]4 = (1 + 2m+r )−1/2 [1 + log(1 + 2m+r )]4 On the other hand, −3/2 2 (1 + t ) (Lψ ) dt du dµ ≤ n h/ t ,tm Wum−1

Wut

(1 + t )−3/2 (Lψn )2 dt du dµh/

t

≤C 0

u (1 + t )−3/2 E0,n (t )dt

(7.78)

It follows, in view of (7.76) that: u 1/2 t 1/2 t L 1/2 u F 1,n (u )du (1 + t )−3/2 E0,n (t )dt Mm ≤ C Am 0

0

and, by (7.74), N 1/2 L M ≤C Am m=0

u 0

t F 1,n (u )du

1/2 0

t

u (1 + t )−3/2 E0,n (t )dt

1/2 (7.79)


183

Now, N

1/2 Am

≤

m=0

N

1/2 (m + 1)

−2

m=0

≤C

N

1/2 (m + 1) Am 2

m=0 1/2

N

≤ C ,

(m + 1)2 Am

m=0

because the series

∞

(m + 1)2 Am

m=0

is convergent, the asymptotic form of the mth term being: m2 2(m+r)/2 We conclude that:

u

ML ≤ C 0

t

F 1,n (u )du

[(m + r ) log 2]4

1/2

t 0

u (1 + t )−3/2 E0,n (t )dt

1/2 (7.80)

We turn to the integral M L ((7.68)). We have (taking into account assumptions A): 1/2 −1 ML ≤ (η0 + α −1 κ)(Lψn )2 dt du dµh/ (7.81) Wut

· ≤C 0

u

1/2 Wut

−1 (η0−1 + α −1 κ)(LT ψn−1 + νT ψn−1 )2 dt du dµh/

1/2 t F0,n (u )du ·

1/2 Wut

[1 + log(1 + t )](LT ψn−1 + νT ψn−1 )2 dt du dµh/

The integral in the last factor can be expressed as the sum: N [1 + log(1 + t )](LT ψn−1 + νT ψn−1 )2 dt du dµh/ t ,tm m−1 m=0 Wu

Since for t ∈ [tm−1 , tm ] it holds that: (1 + t )−2 [1 + log(1 + t )] ≤ (1 + tm−1 )−2 [1 + log(1 + tm )] ≤ 4(1 + tm )−2 [1 + log(1 + tm )] we obtain, in view of (7.70), [1 + log(1 + t )](LT ψn−1 + νT ψn−1 )2 dt du dµh/ tm−1 ,tm Wu ≤ 4(1 + tm )−2 [1 + log(1 + tm )] (1 + t )2 (LT ψn−1 + νT ψn−1 )2 dt du dµh/ t

Wum

184


≤ (1 + tm ) ≤ C Bm 0

−2

u

[1 + log(1 + tm )]

t

C um

0 u

tm F 1,n (u )du

where:

2

u

≤ C Bm 0

(1 + t ) (LT ψn−1 + νT ψn−1 ) dµh/ dt 2

du

t

F 1,n (u )du

Bm = (1 + tm )−2 [1 + log(1 + tm )]5

In view of the fact that the series

∞

Bm

m=0

is convergent, we then obtain, summing over m, 2 [1 + log(1 + t )](LT ψn−1 + νT ψn−1 ) dt du dµh/ ≤ C Wut

Substituting in (7.81) then yields: u 1/2 L t M ≤C F0,n (u )du 0

u 0

u 0

t F 1,n (u )du

t

F 1,n (u )du

(7.82)

1/2 (7.83)

We have thus completed the estimate of the integral (7.67). Next we shall estimate the contribution of the second term in (7.43) (1/2)(µ−1

(Y )

π˜ L L )L 2 ψn−1

to the error integral (7.64). Writing L 2 ψn−1 = L((1 + t)−1 Qψn−1 ) = (1 + t)−1 L Qψn−1 − (1 + t)−2 Qψn−1 the contribution in question takes the form: −(1/2) (K 0 ψn )(µ−1 (Y ) π˜ L L )(1 + t )−1 (L Qψn−1 )dt du dµh/ +(1/2)

Wut

Wut

(K 0 ψn )(µ−1

(Y )

π˜ L L )(1 + t )−2 (Qψn−1 )dt du dµh/

(7.84)

By the second of assumptions G0 and (7.63) the first of these integrals is bounded in absolute value by: L C(I1 + I1L ) where: L

I1 = I1L

=

Wut

Wut

|Lψn |(1 + t )−1 [1 + log(1 + t )]|L Qψn−1 |dt du dµh/

(7.85)

(η0−1 + α −1 κ)|Lψn |(1 + t )−1 [1 + log(1 + t )]|L Qψn−1 |dt du dµh/ (7.86)


We estimate: L I1

≤C

185

1/2 −2

(Lψn ) (1 + t ) 2

Wut

t

≤C

[1 + log(1 + t )] dt du dµh/ 2

1/2

×

Wut

(L Qψn−1 ) dt du dµh/ 2

−2

(1 + t )

[1 + log(1 + t

0

u )]2 E0,n (t )dt

and: I1L

≤C 0

u

1/2 0

u

t F0,n (u )du

1/2 (7.87)

t F0,n (u )du

(7.88)

Also, by the second of assumptions G0 and (7.63), the second of the integrals (7.84) is bounded by: C |Lψn ||Qψn−1 |(1 + t )−2 [1 + log(1 + t )]dt du dµh/ Wut

+C

Wut

(η0−1 + α −1 κ)|Lψn ||Qψn−1 |(1 + t )−2 [1 + log(1 + t )]dt du dµh/ 1/2 −2

2

≤C

Wut

(Lψn ) (1 + t )

Wut

(Qψn−1 ) (1 + t ) +α

−1

Wu

≤C

κ)(Lψn ) dt du dµh/ 2

+α

−1

−4

κ)(Qψn−1 ) (1 + t ) 2

−2

(1 + t )

[1 + log(1 + t

0

·

[1 + log(1 + t )]dt du dµh/

Wu t

1/2

(η0−1 t

·

1/2

(η0−1 t

+C

1/2 −2

2

[1 + log(1 + t )]dt du dµh/

·

u )]E0,n (t )dt

[1 + log(1 + t )] dt du dµh/ 2

1/2

1/2 t u ε02 (1 + t )−2 [1 + log(1 + t )]E0,n (t )dt 0

u

+C 0

t F0,n (u )du

1/2

ε02

t 0

−4

(1 + t )

[1 + log(1 + t

(7.89)

u )]3 E0,n (t )dt

Here we have made use of inequality (5.72) with ψn in the role of ψ, namely: u (t) ψn2 dµh/ ≤ ε0 CE0,n St,u

1/2

(7.90)

186


the sum being over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. Next we shall estimate the contribution of the third and fourth terms in (7.43) −

(Y )

Z˜ · d/ Lψn−1 −

(Y )

Z˜ · d/ Lψn−1

(7.91)

to the error integral (7.64). Expressing L and L in terms of the commutation fields T and Q, L = α −1 κ(1 + t)−1 Q + 2T L = (1 + t)−1 Q (7.91) takes the form: Z˜ · d/T ψn−1 − (1 + t)−1 (α −1 κ −(1 + t)−1 (Y ) Z˜ · d/(α −1 κ)Qψn−1 −2

(Y )

(Y )

Z˜ +

(Y )

Z˜ ) · d/ Qψn−1 (7.92)

We shall derive below an estimate of the contribution of the first line in (7.92) to the error integral (7.64). The contribution of the second line, being a lower order term, is easier to handle. By the fourth and fifth of assumptions G0, the coefficients of d/T ψn−1 and d/ Qψn−1 in (7.92) are bounded in magnitude by: C(1 + t)−1 [1 + log(1 + t)] /˜ L |, | (Y ) Z˜ | = | (Y ) π /˜ L |.) Also, T ψn−1 and Qψn−1 are among (Note that | (Y ) Z˜ | = | (Y ) π the ψn . It follows that the contribution of the first line in (7.92) to the error integral (7.64) is bounded in absolute value by a constant multiple of a sum of two terms of the form: |K 0 ψn ||d/ψn |(1 + t )−1 [1 + log(1 + t )]dt du dµh/ (7.93) Wut

Substituting (7.63), this is in turn bounded by: |Lψn ||d/ψn |(1 + t )−1 [1 + log(1 + t )]dt du dµh/ Wut

+

Wut

(7.94)

(η0−1 + α −1 κ)|Lψn ||d/ψn |(1 + t )−1 [1 + log(1 + t )]dt du dµh/

We estimate the first integral of (7.94) by the product: 1/2 −2 2 2 (1 + t ) [1 + log(1 + t )] (Lψn ) dt du dµh/ Wut

1/2 Wut

Now, the integral in the first factor is bounded by: t u C (1 + t )−2 [1 + log(1 + t )]2 E0,n (t )dt 0

|d/ψn | dt du dµh/ 2


while the integral in the second factor decomposes into: 2 |d / ψ | dt du dµ + |d/ψn |2 dt du dµh/ n h/ U

Uc

Wut

187

(7.95)

Wut

where U is the region defined by (5.12): U = {x ∈ Wε∗0 : µ < η0 /4} Since 4µ/η0 ≥ 1 in U c , we can estimate: 4 2 |d / ψ | dt du dµ ≤ µ|d/ψn |2 dt du dµh/ n h/ t c t η 0 Wu Wu U u t ≤C F0,n (u )du

(7.96)

0

To estimate the integral over U Wut we make use of the spacetime integral K n (t, u) defined by (5.169), with ψn in the role of ψ: −1 −1 ων µ (Lµ)− |d/ψn |2 dµh K n (t, u) = − Wut 2 According to (5.206) (which relies on assumption C3 as well as B1, D1) we have: 1 K n (t, u) ≥ (1 + t )[1 + log(1 + t )]−1 |d/ψn |2 dt du dµh/ (7.97) C U Wut Recalling the definition (5.208), that is, in the present context, K n (t, u) = sup [1 + log(1 + t )]−4 K n (t , u) t ∈[0,t ]

and noting that for t ∈ [tm−1 , tm ] it holds that: (1 + t )−1 [1 + log(1 + t )] ≤ 2(1 + tm )−1 [1 + log(1 + tm )], we then obtain:

U

W

tm−1 ,tm

|d/ψn |2 dt du dµh/ ≤ C Am K n (tm , u)

where: Am = (1 + tm )−1 [1 + log(1 + tm )]5 = (1 + 2m+r )−1 [1 + log(1 + 2m+r )]5 Hence, in view of the fact that the series ∞ m=0

Am

(7.98)

188


is convergent (and the fact that K n (t, u) is a non-decreasing function of t) we conclude that: N 2 |d/ψn | dt du dµh/ = |d/ψn |2 dt du dµh/ U

m=0 U

Wut

≤C

N

W tm−1 ,tm

Am

K n (t, u) ≤ C K n (t, u)

(7.99)

m=0

Combining with (7.96) yields: |d/ψn |2 dt du dµh/ ≤ C K n (t, u) + Wut

u 0

t F0,n (u )du

(7.100)

Thus, the first of the integrals (7.94) is bounded in absolute value by: 1/2 1/2 t u −2 2 u t (1 + t ) [1 + log(1 + t )] E0,n (t )dt K n (t, u) + F0,n (u )du C 0

0

(7.101) Moreover, in view of (7.50), assumptions A, and the estimate (7.100), the second of the integrals (7.94) is bounded in absolute value by: 1/2 Wut

(1 + t )−2 [1 + log(1 + t )]2 (1 + α −1 κ)2 (Lψn )2 dt du dµh/

1/2

2

·

Wut

u

≤C 0

|d/ψn | dt du dµh/

t F0,n (u )du

1/2

u

K n (t, u) + 0

t F0,n (u )du

1/2 (7.102)

This concludes the estimate of the contribution of (7.91), the third and fourth terms in (7.43), to the error integral (7.64). Finally, we estimate the contribution of the last two terms in (7.43), (1/2)

(Y )

π˜ L L / ψn−1 + µ

(Y ) ˆ˜

π /· D / 2 ψn−1

to the error integral (7.64). By the third and sixth of assumptions G0, this contribution is bounded in absolute value by a constant multiple of |K 0 ψn ||D / 2ψn−1 |[1 + log(1 + t )]dt du dµh/ (7.103) Wut

By assumption H1, we have: |D / 2 ψn−1 | ≤ C(1 + t)−1

i

|d/ Ri ψn−1 | ≤ C(1 + t)−1

|d/ψn |

(7.104)


189

since the Ri ψn−1 are among the ψn . Therefore (7.103) is bounded by a constant multiple of a sum of three terms of the form: |K 0 ψn ||d/ψn |(1 + t )−1 [1 + log(1 + t )]dt du dµh/ Wut

This is precisely the same form as (7.93), to which the previous estimate applies. We collect the above results, in simplified form, in the following lemma. Lemma 7.2 We have: (K 0 ψn ) (Y ) σ1,n−1 dt du dµh/ Wut t u u t ≤C (1 + t )−3/2 E0,n (t )dt + F0,n (u )du 0 0 1/2 u 1/2 t +C F 1,n (u )du + K n (t, u) 0

t

× 0

u (1 + t )−3/2 E0,n (t )dt +

0

u

t F0,n (u )du

1/2

We now consider the partial contribution of (Y ) σ1,n−1 to the error integral (7.62), which is associated to the vectorfield K 1 . In view of (5.16), namely K 1 = (ω/ν)L the error integral (7.62) is: (ω/ν)(Lψn + νψn ) − Wut

(Y )

(7.105)

σn−1 dt du dµh/

(7.106)

and we are here considering the partial contribution of (Y ) σ1,n−1 to this, namely: − (ω/ν)(Lψn + νψn ) (Y ) σ1,n−1 dt du dµh/ (7.107) Wut

Now, by virtue of assumptions B1 and D1 of Chapter 5 we have: C −1 (1 + t)2 ≤ ω/ν ≤ C(1 + t)2 So, the integral (7.107) is bounded in absolute value by a constant multiple of (1 + t )2 |Lψn + νψn || (Y ) σ1,n−1 |dt du dµh/ Wut

and our task is to estimate this integral.

(7.108)

(7.109)

190


We begin with the contribution of the first term in (7.43). Expressing this term using the expansion (7.66), we shall estimate explicitly the contribution of the leading term, namely the integral: Wut

(1 + t )2 |Lψn + νψn ||tr

(Y ) ˜

π /||LT ψn−1 + νT ψn−1 |dt du dµh/

(7.110)

However, by the last of assumptions G0 and the fact that T ψn−1 is one of the ψn , this is simply bounded by: u t F1,n (u )du (7.111) C 0

To estimate the contribution of the second term in (7.43) to the integral (7.109), we write: L 2 ψn−1 = (1 + t)−1 L Qψn−1 − (1 + t)−2 Qψn−1 = (1 + t)−1 (L Qψn−1 + ν Qψn−1 ) − (1 + t)−2 (1 + (1 + t)ν)Qψn−1 In view of the second of assumptions G0 and assumption B1 of Chapter 5, as well as the fact that Qψn−1 is one of the ψn , the contribution of the second term in (7.43) to (7.109) is then bounded by: C

Wut

+C

u

≤C 0

(1 + t )[1 + log(1 + t )]|Lψn + νψn |2 dt du dµh/ Wut

[1 + log(1 + t )]|Lψn + νψn ||ψn |dt du dµh/

t F1,n (u )du

u

+C 0

t F1,n (u )du

(7.112) 1/2

ε02

t

−2

(1 + t )

[1 + log(1 + t

0

u )]2 E0,n (t )dt

1/2

Here we have used (7.90) in estimating the second integral. To estimate the contribution of the third and fourth terms in (7.43), namely of (7.91), to the integral (7.109), we use the expression (7.92). We shall estimate explicitly the contribution of the first line of (7.92), which contains the principal terms. The remarks which we made in considering the contribution of these terms to the integral (7.64) lead in the present case to the conclusion that the contribution of the same terms to the integral (7.109) is bounded by a constant multiple of a sum of two terms of the form: Wut

(1 + t )[1 + log(1 + t )]|Lψn + νψn ||d/ψn |dt du dµh/

(7.113)


191

This is bounded by: ×

1/2 2

Wut

Wut

2

(1 + t ) |Lψn + νψn | dt du dµh/ 1/2

(7.114)

[1 + log(1 + t )]2 |d/ψn |2 dt du dµh/

Now, the integral in the first factor of (7.114) is bounded by: u t F1,n (u )du C 0

The integral in the second factor decomposes into: 2 2 [1 + log(1 + t )] |d / ψ | dt du dµ + n h/ U

Uc

Wut

Wut

[1 + log(1 + t )]2 |d/ψn |2 dt du dµh/ (7.115)

In U c we have 4µ/η0 ≥ 1; we can thus estimate, in view of (7.55), [1 + log(1 + t )]2 |d/ψn |2 dt du dµh/ Uc

Wut

4 ≤ [1 + log(1 + t )]2 µ|d/ψn |2 dt du dµh/ η0 Wut t u ≤C (1 + t )−2 [1 + log(1 + t )]2 E1,n (t )dt

(7.116)

0

On the other hand, the integral over U is by (7.97) a fortiori bounded by: [1 + log(1 + t )]2 |d/ψn |2 dt du dµh/ ≤ C K n (t, u) U

(7.117)

Wut

Combining with (7.116), we obtain: [1 + log(1 + t )]2 |d/ψn |2 dt du dµh/ Wut

t u ≤ C K n (t, u) + (1 + t )−2 [1 + log(1 + t )]2 E1,n (t )dt

(7.118)

0

We conclude that (7.113) is bounded by:

u

C 0

t F1,n (u )du

1/2

t

K n (t, u) + 0

−2

(1 + t )

[1 + log(1 + t

u )]2 E1,n (t )dt

1/2

(7.119)

192


Finally, we estimate the contribution of the last two terms in (7.43) to the error integral (7.109). By the third and sixth of assumptions G0 and (7.104) this contribution is bounded in absolute value by a constant multiple of a sum of three terms of the form (7.113), which has already been estimated. We collect the above results in the following lemma. Lemma 7.3 We have: (K 1 ψn + ωψn ) (Y ) σ1,n−1 dt du dµh/ Wut 1/2 u u t t ≤C F1,n (u )du + C F1,n (u )du 0

0

t

· K n (t, u) +

−2

(1 + t )

2

[1 + log(1 + t )]

0

u u (E1,n (t ) + ε02 E0,n (t ))dt

1/2

We now consider the partial contribution of (Y ) σ2,n−1 , given by (7.44) to each of the error integrals (7.61), (7.62). To estimate this contribution we must make an assumption on the Lie derivatives, with respect to the commutation fields, of the components of the deformation tensors of the commutation fields. This assumption is simply that these derivatives satisfy the same bounds as the components themselves, as given by assumption G0. That is, we assume: G1: There is a positive constant C independent of s such that in Wεs0 , for all five commutation fields we have: (Y )

Y( |Y (µ

−1 (Y )

|Y (

(Y )

|L /Y ( |L /Y (

π˜ L L ) = 0 π˜ L L )| ≤ C[1 + log(1 + t)]

π˜ L L )| (Y ) ˜ π / L )| (Y ) ˜ π / L )|

≤ C[1 + log(1 + t)] ≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C[1 + log(1 + t)]

|L /Y ( (Y ) π /ˆ˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] /˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] |Y ( (Y ) trπ Here in each line Y occurs twice. In each of these occurrences Y is meant to range independently over the set of the five commutation fields {Yi : i = 1, 2, 3, 4, 5}. The last of the assumptions G1 seems stronger than what corresponds to the last of assumptions G0, however we recall from Chapter 6 (see (6.99), (6.187)) that we actually have /˜ | ≤ C(1 + t)−1 [1 + log(1 + t)] : i = 2 | (Yi ) trπ for all commutation fields except for Y2 = Q, while for Q we have ((6.107)): |tr

(Q) ˜

π / − 4| ≤ C(1 + t)−1 [1 + log(1 + t)]


193

We see then that the last of assumptions G1 is in fact in accordance with the other assumptions. In the following we shall also make use of the bounds: u (Lψn−1 )2 dµh/ ≤ Cε0 E0,n (t)

St,u

S

u (Lψn−1 )2 dµh/ ≤ Cε0 (1 + t)−2 E0,n (t)

t,u

u |d/ψn−1 |2 dµh/ ≤ Cε0 (1 + t)−2 E0,n (t)

St,u

(7.120)

These bounds follow from inequality (7.90), the first two by expressing L and L in terms of the commutation fields T and Q, and the last by virtue of assumption H0. We consider now the error integral: (K 0 ψn ) (Y ) σ2,n−1 dt du dµh/ (7.121) − Wut

The first term in (7.44), (Y ) ˜

π /)Lψn−1 ,

(1/4)L(tr involves L(tr

(Y ) π /˜ ).

Writing L = (1 + t)−1 Q, we have, from the last of assumptions G1, |L(tr

(Y ) ˜

π /)| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.122)

It follows that the contribution of the first term to the integral (7.121) is bounded in absolute value by: L C(I2 + I2L ) where:

L

I2 = I2L

=

Wut

|Lψn ||Lψn−1 |(1 + t )−2 [1 + log(1 + t )]dt du dµh/

Wut

(η0−1 + α −1 κ)|Lψn ||Lψn−1 |(1 + t )−2 [1 + log(1 + t )]dt du dµh/ (7.124)

Using the first of (7.120) we can estimate: 1/2 L I2

(7.123)

t

≤

u

0

0

0

St ,u

(Lψn ) dµh/ du

0

u

1/2

2

St ,u

(Lψn−1 ) dµh/ du

· (1 + t )−2 [1 + log(1 + t )]dt u 1/2 2 u 1/2 ε0 E0,n (t ) E0,n (t ) (1 + t )−2 [1 + log(1 + t )]dt

t

≤C

2

= Cε0 0

t

u (1 + t )−2 [1 + log(1 + t )]E0,n (t )dt

(7.125)

194


and (taking account also of assumptions A): I2L

1/2 (η0−1 t

≤

+α

Wu

−1

κ)(Lψn ) dt du dµh/ 2

(7.126) 1/2

(η0−1 t

·

+α

−1

κ)(Lψn−1 ) (1 + t )

Wu

u

≤ Cε0 0

−4

t F0,n (u )du

2

1/2

t

−4

(1 + t )

[1 + log(1 + t )] dt du dµh/

[1 + log(1 + t

0

2

u )]3 E0,n (t )dt

1/2

The second term in (7.44), (1/4)L(trπ /˜ )Lψn−1 , /˜ ). Writing L = 2T +α −1 κ(1+t)−1 Q, we have, from the last of assumptions involves L(trπ G1, /˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] |L(trπ

(7.127)

It follows that the contribution of the second term to the integral (7.121) is bounded in absolute value by: L

C(I3 + I3L ) where:

L

I3 = I3L =

Wut

Wut

|Lψn ||Lψn−1 |(1 + t )−1 [1 + log(1 + t )]dt du dµh/

(7.128)

(η0−1 + α −1 κ)|Lψn ||Lψn−1 |(1 + t )−1 [1 + log(1 + t )]dt du dµh/ (7.129)

Using the second of (7.120) we can estimate: L I3

≤

t 0

u

0

≤C 0

(Lψn ) dµh/ du 2

St ,u

1/2 0

u

1/2

(Lψn−1 ) dµh/ du 2

St ,u

· (1 + t )−1 [1 + log(1 + t )]dt 1/2 t 1/2 2 u u (t ) (t ) ε0 (1 + t )−2 E0,n E0,n

· (1 + t )−1 [1 + log(1 + t )]dt

= Cε0 0

t

u (1 + t )−2 [1 + log(1 + t )]E0,n (t )dt

(7.130)


and (taking account also of assumptions A): I3L ≤

Wut

195

1/2

(η0−1 + α −1 κ)(Lψn )2 dt du dµh/

·

(7.131) 1/2

Wut

u

≤ Cε0 0

(η0−1 + α −1 κ)(Lψn−1 )2 (1 + t )−2 [1 + log(1 + t )]2 dt du dµh/ t F0,n (u )du

1/2

t 0

u (1 + t )−4 [1 + log(1 + t )]3 E0,n (t )dt

1/2

The third term in (7.44), (1/4)L(µ−1 involves L(µ−1 tions G1,

(Y ) π ˜

L L ). Writing

|L(µ−1

(Y )

(Y )

π˜ L L )Lψn−1 ,

L = (1 + t)−1 Q, we have, from the second of assump-

π˜ L L )| ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.132)

It follows that the contribution of the third term to the integral (7.121) is also bounded in absolute value by: L C(I3 + I3L ) The fourth and fifth terms in (7.44), −(1/2)L /L involve L /L

(Y ) Z˜

and L /L

(Y )

(Y ) Z ˜.

Z˜ · d/ψn−1 − (1/2)L /L

(Y )

Z˜ · d/ψn−1 ,

Now, from (7.30) or (7.31), we have:

(Y ) ˜

/· π /L = h

(Y )

Z˜ ,

(Y ) ˜

(Y )

π / L = h/ ·

Z˜

hence: L /Y L /Y Since

(Y ) ˜

/Y π / L = h/ · L (Y ) ˜ π / L = /h · L /Y (Y )

π˜ = (

(Y )

Z˜ + (Y ) ˜ Z+ (Y )

(Y )

π /· (Y ) π /·

(Y ) (Y )

Z˜ Z˜

(7.133)

π + Y (log )h),

and by assumptions E1, E2 of Chapter 6 we have (see (6.51)):

which implies:

|Y σ | ≤ C(1 + t)−1

(7.134)

|Y log | ≤ C(1 + t)−1

(7.135)

for all five commutation fields Y , we then obtain, by virtue of assumptions G0, G1, |L /Y |L /Y

Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)] (Y ) ˜ Z | ≤ C[1 + log(1 + t)] (Y )

(7.136)

196


These imply, writing L = 2T + α −1 κ(1 + t)−1 Q, L = (1 + t)−1 Q, that |L /L

(Y )

˜ |L Z|, /L

(Y )

Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.137)

It follows that the contribution of the fourth and fifth terms in (7.44) to the error integral (7.121) is bounded in absolute value by: L

C(I4 + I4L ) where:

L

I4 = I4L =

|Lψn ||d/ψn−1 |(1 + t )−1 [1 + log(1 + t )]dt du dµh/

(7.138)

(η0−1 + α −1 κ)|Lψn ||d/ψn−1 |(1 + t )−1 [1 + log(1 + t )]dt du dµh/

(7.139)

Wut

Wut

Using the last of (7.120) we can estimate: L I4

≤

t 0

u

2

0

t

≤C 0

St ,u

(Lψn ) dµh/ du

1/2

u

1/2

0

2

St ,u

|d/ψn−1 | dµh/ du

· (1 + t )−1 [1 + log(1 + t )]dt 1/2 u 1/2 2 u ε0 (1 + t )−2 E0,n E0,n (t ) (t ) (1 + t )−1 [1 + log(1 + t )]dt

t

= Cε0

u (1 + t )−2 [1 + log(1 + t )]E0,n (t )dt

0

(7.140)

and (taking account also of assumptions A): 1/2

I4L ≤

Wut

(η0−1 + α −1 κ)(Lψn )2 dt du dµh/

·

(7.141) 1/2

Wut

u

≤ Cε0 0

(η0−1 + α −1 κ)|d/ψn−1 |2 (1 + t )−2 [1 + log(1 + t )]2 dt du dµh/ t F0,n (u )du

1/2

t 0

u (1 + t )−4 [1 + log(1 + t )]3 E0,n (t )dt

1/2

/ (Y ) Z˜ , respectively. The sixth and seventh terms in (7.44) involve div / (Y ) Z˜ and div ˜ ˜ Now, hypothesis H1 applied to the 1-forms π /L , π / L on each St,u yields: |D /

(Y )

˜ 2 = |D Z| /

(Y ) ˜

|D /

(Y )

˜ 2 = |D / Z|

(Y ) ˜

π / L |2 ≤ C(1 + t)−2

|L / Ri

(Y ) ˜

|L / Ri

(Y ) ˜

π / L |2

i

π / L | ≤ C(1 + t) 2

−2

i

π / L |2

(7.142)


197

(the Ri are tangential to the St,u ). The fourth and fifth of assumptions G1 then imply: |D /

(Y )

|D /

(Y )

Z˜ | ≤ C(1 + t)−2 [1 + log(1 + t)] Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.143)

Z˜ | ≤ C(1 + t)−2 [1 + log(1 + t)] Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.144)

thus also: |div /

(Y )

|div /

(Y )

It then follows (compare with (7.122) and (7.127)) that the contribution of the sixth term in (7.44), −(1/2)div / (Y ) Z˜ Lψn−1 , to the integral (7.121) is bounded in absolute value by: L

C(I2 + I2L ) while the contribution of the seventh term, (Y )

−(1/2)div /

Z˜ Lψn−1 ,

is bounded in absolute value by: L

C(I3 + I3L ) The eighth term in (7.44), (1/2)d/ involves d/ |d/

(Y )

(Y ) π ˜

LL.

(Y )

π˜ L L · d/ψn−1

By hypothesis H0 and the third of assumptions G1,

π˜ L L | ≤ C(1 + t)−1

) (Ri

(Y ) π ˜

LL)

2

≤ C(1 + t)−1 [1 + log(1 + t)]

(7.145)

i

It follows that the contribution of the eighth term to (7.121) is bounded, like the contributions of the fourth and fifth terms, in absolute value by: L

C(I4 + I4L ) Finally, the ninth (last) term in (7.44), div / (µ involves: div / (µ

(Y ) ˆ˜

(Y ) ˆ˜

π /) · d/ψn−1

π /) = µdiv /

(Y ) ˆ˜

π / + d/µ ·

(Y ) ˆ˜

π /

198


Now, hypothesis H2 applied to the traceless symmetric tensorfield yields: |D / (Y ) π |L / Ri (Y ) π /ˆ˜ |2 ≤ C(1 + t)−2 /ˆ˜ |2

(Y ) π /ˆ˜

on each St,u (7.146)

i

(the Ri are tangential to the St,u ). The sixth of assumptions G1 then yields: |D /

(Y ) ˆ˜

thus also: |div /

π /| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.147)

(Y ) ˆ˜

π /| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.148)

Taking into account the second of assumptions F1 of Chapter 6 (together with the basic bootstrap assumption A3), we then obtain: |div / (µ

(Y ) ˆ˜

π /)| ≤ C(1 + t)−2 [1 + log(1 + t)]2

(7.149)

Comparing the eighth and ninth terms and (7.145) and (7.149) we conclude that the contribution of the ninth (last) term in (7.44) to the error integral (7.121) is a fortiori bounded in absolute value by: L C(I4 + I4L ) We collect the above results, in simplified form, in the following lemma. Lemma 7.4 We have: (Y ) (K 0 ψn ) σ2,n−1 dt du dµh/ Wut t u (1 + t )−2 [1 + log(1 + t )]E0,n (t )dt + ≤ Cε0 0

We now consider the error integral (K 1 ψn + ωψn ) −

(Y )

Wut

u 0

t F0,n (u )du

σ2,n−1 dt du dµh/ ,

or (see (7.106)), −

Wut

(ω/ν)(Lψn + νψn )

(Y )

σ2,n−1 dt du dµh/

By (7.108) this integral is bounded in absolute value by a constant multiple of (1 + t )2 |Lψn + νψn || (Y ) σ2,n−1 |dt du dµh/ Wut

and our task is to estimate this integral.

(7.150)

(7.151)


199

In view of (7.122) the contribution of the first term in (7.44) to the integral (7.151) is bounded in absolute value by:

u

|Lψn + νψn ||Lψn−1 |[1 + log(1 + t )]dµh/ dt du

C 0

Cu

u

≤C

0

1/2 Cu

(1 + t )2 (Lψn + νψn )2 dµh/ dt

1/2

· Cu

u

≤C 0

(1 + t )−2 [1 + log(1 + t )]2 (Lψn−1 )2 dµh/ dt

1/2 t F1,n (u )

u

≤ Cε0

0

t F1,n (u )du

0

t

u ε0 E0,n (t )(1 + t )−2 [1 +

1/2

t 0

du

log(1 + t )] dt 2

(7.152)

1/2

u (1 + t )−2 [1 + log(1 + t )]2 E0,n (t )dt

du

1/2

Here we have used the first of the inequalities (7.120). In view of (7.127) the contribution of the second term in (7.44) to the integral (7.151) is bounded in absolute value by:

u

|Lψn + νψn ||Lψn−1 (1 + t )|[1 + log(1 + t )]dµh/ dt

C 0

Cu

u

≤C

0

Cu

2

(1 + t ) (Lψn + νψn ) dµh/ dt

u

0

u

≤ Cε0 0

2

2

[1 + log(1 + t )] (Lψn−1 ) dµh/ dt

t (u ) F1,n

du

1/2

Cu

1/2 2

·

≤C

1/2

t F1,n (u )du

t 0

du

u ε0 E0,n (t )(1 + t )−2 [1 + log(1 + t )]2 dt

1/2

t 0

−2

(1 + t )

[1 +

(7.153) 1/2

u log(1 + t )]2 E0,n (t )dt

du

1/2

Here we have used the second of the inequalities (7.120). In view of (7.132) the contribution of the third term in (7.44) to the integral 7.151 is also bounded in the same way.

200


In view of (7.137) the contribution of the fourth and fifth terms in (7.44) to (7.151) is bounded in absolute value by: u

|Lψn + νψn ||d/ψn−1 |(1 + t )|[1 + log(1 + t )]dµh/ dt du

C Cu

0

u

≤C

1/2 2

Cu

0

(1 + t ) (Lψn + νψn ) dµh/ dt

Cu

u

0

[1 + log(1 + t )] |d/ψn−1 | dµh/ dt

t (u ) F1,n

u

≤ Cε0 0

1/2

· ≤C

2

1/2

t F1,n (u )du

t

2

2

du

u ε0 E0,n (t )(1 + t )−2 [1 + log(1 + t )]2 dt

0 1/2 t 0

−2

(1 + t )

[1 +

(7.154) 1/2

u log(1 + t )]2 E0,n (t )dt

du

1/2

Here we have used the last of the inequalities (7.120). In view of the bounds (7.144) the contribution of the sixth term in (7.44) to (7.151) is bounded in absolute value as in (7.152) while that of the seventh term is bounded in absolute value as in (7.153). Finally, in view of (7.145) and (7.149) the contributions of the eighth and ninth terms in (7.44) to the integral (7.151) are bounded in absolute value as in (7.154). We collect the above results in the following lemma. Lemma 7.5 We have: (Y ) (K 1 ψn + ωψn ) σ2,n−1 dt du dµh/ Wut u 1/2 t 1/2 t −2 2 u ≤ Cε0 F1,n (u )du (1 + t ) [1 + log(1 + t )] E0,n (t )dt 0

0

Finally, we consider the partial contribution of (Y ) σ3,n−1 , given by (7.45) to the error integrals (7.61) and (7.62). Now, assumptions E1, E2, F1, F2, of Chapter 6 imply (see (6.90), (6.74), (6.84), (6.72), (6.79)): |L log | ≤ C(1 + t)−2 |trχ| ≤ C(1 + t)−1 ,

|trχ | ≤ C(1 + t)−1 [1 + log(1 + t)]

|| ≤ C(1 + t)−1 [1 + log(1 + t)] |L(α −1 κ)| ≤ C(1 + t)−1 ,

|d/(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.155) (7.156) (7.157) (7.158)


201

Together with assumptions G0 the above in turn imply that the coefficients (Y ) σ L 3,n−1 ,

(Y ) σ L 3,n−1 ,

given by (7.46), (7.47), (7.48), respectively, satisfy the bounds: | | |

(Y ) L σ3,n−1 | (Y ) L σ3,n−1 | (Y ) /σ3,n−1 |

≤ C(1 + t)−2 ≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.159)

Comparing with (7.122), (7.127), (7.137) it follows that the contributions of the second term in (7.45) to the error integrals (7.61), (7.62), are bounded as the corresponding contributions of the first term in (7.44), the contributions of the first term in (7.45) are bounded as the corresponding contributions of the second term in (7.44), and the contributions of the third term in (7.45) are bounded as the corresponding contributions of the fourth and fifth terms in (7.44). Thus, the contribution of (Y ) σ3,n−1 to each of the error integrals (7.61) and (7.62) is bounded in the same way as the corresponding contribution of (Y ) σ2,n−1 . Lemmas 7.2–7.5 together with the conclusion just reached yield the following lemma. Lemma 7.6 We have: t u (Y ) −3/2 u t ≤ C (K ψ ) σ dt du dµ (1 + t ) E (t )dt + F (u )du 0 n n−1 h/ 0,n 0,n Wut 0 0 1/2 u 1/2 t +C F 1,n (u )du + K n (t, u) 0

t

× 0

u (1 + t )−3/2 E0,n (t )dt +

u 0

t F0,n (u )du

1/2

and: (Y ) (K 1 ψn + ωψn ) σn−1 dt du dµh/ Wut u 1/2 t t {K n (t, u)}1/2 ≤C F1,n (u )du + C F1,n (u )du 0

1/2 t + C F1,n (u )du

t 0

1/2 u u dt (1 + t )−2 [1 + log(1 + t )]2 E1,n (t ) + ε02 E0,n

Recalling (see (7.18), (7.27), (7.28)) that the contributions of (Y ) σn−1 (through ρñ ) to the error integrals (7.23) are precisely the integrals estimated in the above lemma, and comparing with (5.240) and (5.267), with ψn in the role of ψ, we conclude that the contributions in question can be absorbed into the corresponding right-hand sides.

Chapter 8

Regularization of the Propagation Equation for d/trχ. Estimates for the Top Order Angular Derivatives of χ In the present chapter we shall deal with the problem of obtaining estimates for χ and its derivatives with respect to the set of rotation vectorfields {Ri : i = 1, 2, 3} of order n, given energy estimates of order n +1 for the ψµ : µ = 0, 1, 2, 3, the partial derivatives of the wave function φ with respect to the rectangular coordinates. That is, we shall estimate , defined in Chapter 7, associated the former in terms of the quantities E0,n+1 and E1,n+1 to n + 1st order variations ψn+1 . These variations are obtained by applying to the ψµ a string of commutation vectorfields of length n. We call the derivatives with respect to the set of rotation vectorfields angular derivatives. We are thus requiring estimates for the nth order angular derivatives of χ in terms of estimates for the n + 1st order derivatives of the ψµ . The propagation equations (3.38) for χ AB involve on the right-hand side the curva( P)

ture components α AB , whose principal part α AB , given by (4.71), (4.72), contains the 2nd derivatives of σ , hence the 2nd derivatives of the ψµ . Now, the propagation equations are ordinary differential equations along the generators of the characteristic hypersurfaces Cu . Thus, in integrating these equations to obtain estimates for χ, no regularity is expected to be gained except along the generators. Consequently, in the estimates obtained in this way there will be a loss of one degree of differentiability along the St,u sections, and we can only estimate the n − 1st order angular derivatives of χ in terms of estimates for the n +1st order derivatives of ψµ . Such estimates, that is of the next-to-the-top order angular derivatives of χ in terms of the top order energies, are in fact also needed in our approach, and shall be derived in Chapter 12. In the present chapter however, we concentrate on the main problem, which is that of deriving estimates for the top order angular derivatives of χ in terms of the top order energies. To accomplish this aim we must avoid the loss of one degree of differentiability along the St,u sections.

Chapter 8. The Propagation Equation for d/trχ

204

Now a propagation equation of the type we are considering does not lead to such a loss of differentiability when – and only when – the principal part on the right-hand side can be put into the form of a derivative with respect to L. This is in fact the case for trace equation (3.42), the principal part on the right-hand side of which is contained in the Ricci tensor component S(L, L). The fact that the principal part of S(L, L) is of the form of a derivative with respect to L shall be demonstrated with the help of the following proposition. We denote, as in the previous chapters, τµ = ∂µ σ,

ωµν = ∂µ ψν = ∂ν ψµ = ωνµ ,

ωµλ = (g −1 )νλ ωµν

Proposition 8.1 The function σ satisfies the following inhomogeneous wave equation in the acoustical metric h: d h σ = −−1 a − 2b dσ where a and b are the functions: a = (h −1 )µν τµ τν

and

b = (h −1 )µν ωµα ωνα

Proof. Each ψα , α = 0, 1, 2, 3 is the derivative of the wave function φ with respect to a translation of the underlying Minkowski spacetime, therefore in accordance with the results of Chapter 1 each of these functions is a solution of the linear wave equation (1.86) ˜ with respect to the conformal acoustical metric h: h˜ ψα = 0

(8.1)

It follows that σ being given by: σ = −(g −1 )αβ ψα ψβ , satisfies the equation: h˜ σ = −2(h˜ −1 )µν (g −1 )αβ ∂µ ψα ∂ν ψβ

(8.2)

the coefficients (g −1 )αβ , the components of the reciprocal Minkowski metric in rectangular coordinates, being constant. Moreover, the fact that the conformal acoustical metric h˜ is related to the acoustical metric h by: h˜ µν = h µν implies that for an arbitrary function f : h˜ f = −1 h f + −2 (h −1 )µν ∂µ ∂ν f and we have:

(8.3)

d ∂µ σ dσ Considering equation (8.2) together with the relation (8.3) in the case f = σ , the proposition follows. ∂µ =


205

We now consider the Ricci tensor Sµν = (h −1 )κλ Rκµλν ( P) µν

and its lower order part S

(N) µν ,

( P) µν

+ S

and decompose it into its principal part S

Sµν = S

(N) µν ,

(8.4)

according to the corresponding decomposition (4.50) of the curvature tensor: ( P) S µν =

(N) S µν =

(h −1 )κλ Pκµλν ,

(h −1 )κλ Nκµλν

(8.5)

Now, the principal part Pκµλν of the curvature tensor is given by (4.49): Pκµλν =

1 dH (ψκ ψν vλµ + ψλ ψµ vκν − ψµ ψν vλκ − ψλ ψκ vµν ) 2 dσ

where vµν is the covariant – relative to h – Hessian of σ (see (4.30), (4.31)): vµν = Dµ τν = Dµ Dν σ = vνµ Taking into account the fact that: (h −1 )κλ ψκ ψλ = ((g −1 )κλ − Fψ κ ψ λ )ψκ ψλ σ = −σ − Fσ 2 = − 1−σH we obtain: ( P) S µν =

1 dH 2 dσ

Here,

ψµ ξν + ψν ξµ +

σ vµν − ψµ ψν trv 1−σH

(8.6) (8.7)

ξµ = (h −1 )κλ ψκ vµλ

(8.8)

trv = h σ

(8.9)

and, is expressed by Proposition 8.1 in terms of lower order terms. We now consider the Ricci tensor component S(L, L) = trα (see (3.41)) which appears in the propagation equation (3.42) for trχ. In analogy with (3.41) we have: ( P)

(N)

( P)

S (L, L) = tr α ,

(N)

and α

AB

(N)

S (L, L) = tr α

is given by (4.85), (4.94)–(4.96). From (8.7), ( P)

( P) µν

S (L, L) = S

Lµ Lν

(8.10)


206

is given by: ( P)

S (L, L) =

1 dH 2 dσ

where

2ψ L ξ L +

σ v(L, L) − ψ L2 trv 1−σH

(8.11)

ξ L = ξµ L µ ,

so that by the definition (8.8): ξ L = (h −1 )κλ ψκ (L µ vµλ ) = (h −1 )κλ ψκ D L τλ

(8.12)

Thus, ξ L is up to lower order terms equal to Ln where n is the function: n = (h −1 )κλ ψκ τλ

(8.13)

Also (see (4.61)) v(L, L) is up to a lower order term equal to L(Lσ ). We then conclude from (8.11), in view of the fact that by (8.9) and Proposition 8.1, trv = −−1

d a − 2b dσ

(8.14)

( P)

is of lower order, that in fact S (L, L) is equal, up to lower order terms, to the L derivative of a function f , where: dH σ 1 f = τL (8.15) ψL n + dσ 2 (1 − σ H ) (N)

(N)

Since S (L, L) = tr α is (by definition) of lower order, it follows that we can express S(L, L) in the form: S(L, L) = L f + g (8.16) where the function g, which is actually defined by this equation, is of lower order, namely of the order of the 1st derivatives of the ψµ . The function g decomposes into the sum of four functions: g=

4 (i) g

(8.17)

i=1

where:

dH dH ψL ξL − L ψL n dσ dσ (2) 1 dH σ 1 dH σ g = v(L, L) − L τL 2 dσ 1 − σ H 2 dσ 1 − σ H (3) 1 dH 2 g =− ψ trv 2 dσ L (1)

g =

(4)

(N)

g = tr α

(8.18) (8.19) (8.20) (8.21)


207

(1)

The function g is the sum of three terms: (1)

g =

dH d2 H dH ψ L (ξ L − Ln) − n L(ψ L ) − ψ L n τL dσ dσ dσ 2

(8.22)

To calculate Ln we write: Ln = L((h −1 )αβ τα ψβ ) = L µ ∂µ ((h −1 )αβ τα ψβ ) = τα ψβ L µ ∂µ (h −1 )αβ + (h −1 )αβ ψβ L µ ∂µ τα + (h −1 )αβ τα L µ ∂µ ψβ Expressing ∂µ τα in terms of Dµ τα = vµα and writing ∂µ ψβ = ωµβ , this becomes: Ln = (h −1 )αβ ψβ L µ vµα + (h −1 )αβ τα L µ ωµβ ν +(h −1 )αβ ψβ L µ !µα τν + τα ψβ L µ ∂µ (h −1 )αβ

(8.23)

The last two terms are equal to: α L µ τα ψβ (h −1 )νβ !µν + ∂µ (h −1 )αβ

β = −L µ τα ψβ (h −1 )να (h −1 )λβ !µνλ = −L µ τα ψβ (h −1 )να !µν

Using the formula (3.141), that is: !µνλ =

1 dH (τµ ψν ψλ + τν ψµ ψλ − τλ ψµ ψν ) + H ψλ ωµν 2 dσ

as well as (8.6), the last is found to be equal to: 1 dH σ σH 2 (τ L n + ψ L a) + ψ L n + (h −1 )αβ τα L µ ωµβ 2 dσ 1 − σ H 1−σH Substituting in (8.23) we then obtain: 1 dH νL + Ln = ξ L + 1−σH 2 dσ Here,

σ (τ L n + ψ L a) + ψ L n 2 1−σH

νµ = (h −1 )αβ τα ωµβ ,

ν L = νµ L µ

(8.24)

(8.25)

To calculate L(ψ L ) we write: L(ψ L ) = L(L α ψα ) = L µ ∂µ (L α ψα ) = (L µ ∂µ L α )ψα + L µ L α ωµα Now an expression for L(L α ) has already been obtained in Chapter 3. However here it is simpler to calculate directly ψα L(L α ): α Lν ) ψα L(L α ) = ψα L µ (Dµ L α − !µν = ψα µ−1 (Lµ)L α − (h −1 )αλ !µνλ L µ L ν


208

Substituting for !µνλ from the formula above and using (8.6), the last is found to be equal to: dH σH 1 σ −1 ωL L + ψL τ L + nψ L µ (Lµ)ψ L + 1−σH dσ 1 − σ H 2 Hence we obtain: L(ψ L ) =

dH ωL L + µ−1 (Lµ)ψ L + ψ L 1−σH dσ

σ 1 τ L + nψ L 1−σH 2

(8.26)

Substituting (8.24) and (8.26) in (8.22) we obtain the following expression for the (1)

function g :

(1)

g=−

(1) dH dH −1 nψ L nψ L + µ (Lµ) + g dσ dσ

(8.27)

where: (1)

dH 1 (ν L ψ L + nω L L ) (1 − σ H ) dσ dH 2 d2 H σ (aψ L + 3nτ L )ψ L − nτ L ψ L − 2(1 − σ H ) dσ dσ 2

g =−

(8.28)

Also, using (4.61) we obtain, in reference to (8.19), (2)

g =−

1 σ d H −1 1 d µ (Lµ)τ L − 2 1 − σ H dσ 2 dσ

σ dH 1 − σ H dσ

τ L2

Moreover, substituting (8.14) in (8.20) we obtain: (3) d 1 dH 2 g= ψ L −1 a + 2b 2 dσ dσ

(8.29)

(8.30)

Substituting in the definitions of the functions a and b (see Proposition 8.1) the expansion (3.116) of the reciprocal acoustical metric in terms of the null frame L, L, X 1, X 2: (8.31) (h −1 )κλ = −(1/2µ)(L κ L λ + L κ L λ ) + (h/−1 ) AB X κA X λB and recalling that τµ = ∂µ σ and ωµα = ∂µ ψα , we obtain the following expressions: a = −µ−1 (Lσ )(Lσ ) + |d/σ |2 b = −µ

−1

α

(8.32) α

(Lψ )(Lψα ) + d/ψ · d/ψα

(8.33)

which show that these functions are both O(µ−1 ) as µ → 0. Substituting in definition (8.13) the expansion (3.40) of the reciprocal acoustical metric h −1 in terms of the frame L, T, X 1 , X 2 : (h −1 )κλ = −α −2 L κ L λ − µ−1 (L κ T λ + T κ L λ ) + (h/−1 ) AB X κA X λB

(8.34)


209

we obtain the following expressions for the function n: n = −µ−1 ψ L (T σ ) − α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ

(8.35)

which shows that n = O(µ−1 ) as µ → 0. Substituting also the expansion (8.34) in definition (8.25) we obtain the following expression for the component ν L : / L · d/σ ν L = −µ−1 ω L L (T σ ) − α −2 (ω L L + αω L Tˆ )(Lσ ) + ω

(8.36)

which shows that ν L = O(µ−1 ) as µ → 0, as well. (See equations (4.89)–(4.93) in Chapter 4.) (2) (3)

In view of the above results, it follows from (8.27)–(8.30) that the functions g , g (1)

(1)

as well as g are all O(µ−1 ) as µ → 0, however g = O(µ−2 ) as µ → 0. On the other (4)

hand, g , given by (8.21) is O(1) as µ → 0. Now, by (8.35) we have: dH dH dH 2 nψ L = −µ−1 ψ L (T σ ) + ψ L −α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ dσ dσ dσ The first term on the right is −2µ−1 m, where m, given by equation (3.97), is the first term in Lµ as given by equation (3.96). We thus have: dH nψ L = −2µ−1 (Lµ) + 2e˜ dσ

(8.37)

where,

1 dH ψ L −α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ (8.38) 2 dσ and e is given by equation (3.98). On the other hand, according to definition (8.15) we have the following alternative expression for the left-hand side of (8.37): e˜ = e +

σ dH 1 dH nψ L = f − τL dσ 2 (1 − σ H ) dσ

(8.39)

In view of expressions (8.37) and (8.39), the first term on the right in (8.27) can be written in the form: dH dH −1 nψ L nψ L + µ (Lµ) (8.40) − dσ dσ dH 1 σ dH τ L − 2e˜ nψ L = µ−1 (Lµ) f − µ−1 (Lµ) 2 (1 − σ H ) dσ dσ Thus, defining: (1)

(1)

dH g =g − dσ

σ 1 −1 µ (Lµ) τ L + 2enψ ˜ L 2 (1 − σ H )

(8.41)


210 (1)

we have g = O(µ−1 ) as µ → 0, and: (1)

(1)

g = µ−1 (Lµ) f + g

(8.42)

fˇ = µf

(8.43)

Setting finally: and:

gˇ = µ

4 (i) g + g

(1)

(8.44)

i=2

the functions fˇ, gˇ are both bounded as µ → 0 and we have, by (8.16), (8.17) and (8.42): µS(L, L) = L fˇ + gˇ

(8.45)

We now substitute the expression (8.45) into the propagation equation (3.42) for trχ, multiplied by µ, and bring the term L fˇ to the left-hand side. We thus obtain the following regularized form of this equation: 1 ˆ 2 − gˇ L(µtrχ + fˇ) = 2(Lµ)trχ − µ(trχ)2 − µ|χ| 2

(8.46)

Here, decomposing χ into its trace-free part χˆ and its trace, we write: |χ|2 =

1 (trχ)2 + |χ| ˆ 2 2

(8.47)

Let us introduce the St,u 1-form:

We have:

x 0 = µd/trχ + d/ fˇ

(8.48)

x 0 = d/(µtrχ + fˇ) − (d/µ)trχ

(8.49)

We shall obtain a propagation equation for x 0 , by differentiating equation (8.46) tangentially to the St,u . We shall make use of the following lemma. Lemma 8.1 For an arbitrary function φ we have: L / L (d/φ) = d/(Lφ) Proof. This can most readily be established by using the (L, T, X 1 , X 2 ) frame. If we evaluate each side on L or T , then both sides vanish by definition. If we evaluate on X A , A = 1, 2, then the left-hand side is: L L (d/φ) · X A = L(d/φ · X A ) − d/φ · [L, X A ] = L(X A φ)


211

for, d/φ · X A = X A φ and [L, X A ] = 0. On the other hand, the right-hand side is: d/(Lφ) · X A = X A (Lφ) The two sides are then equal, in view of the fact that [L, X A ] = 0. Taking φ = µtrχ + fˇ in Lemma 8.1 we obtain, in view of (8.49) and (8.46), the following propagation equation for x 0 : 1 ˆ 2 ) − g0 (8.50) trχ − 2µ−1 (Lµ) d/ fˇ − µd/(|χ| L / L x 0 + trχ − 2µ−1 (Lµ) x 0 = 2 where:

1 g0 = d/gˇ − trχd/( fˇ + 2Lµ) + (d/µ)(Ltrχ + |χ|2 ) 2 In (8.51) we may substitute for Lµ from equation (3.96): Lµ = m + µe

(8.51)

(8.52)

By (8.37), (8.39) and (8.52), dH nψ L = −2µ−1 m + 2(e˜ − e) dσ σ dH 1 (Lσ ) = f − 2 (1 − σ H ) dσ Thus, defining: e = e − e˜ −

σ dH 1 (Lσ ) 4 (1 − σ H ) dσ

(8.53)

fˇ is given by, in view of (8.43), fˇ = −2(m + µe)

(8.54)

In (8.51) we may substitute for Ltrχ + |χ|2 from the original propagation equation (3.42), Ltrχ + |χ|2 = µ−1 (Lµ)trχ − trα, in which we write:

( P)

(N)

trα = tr α +tr α ( P)

and substitute for tr α from the trace of equation (4.71), that is: ( P)

tr α = µ−1 mtrχ − ψ L

( P) dH ψ · χ · d/σ + tr α dσ

In view of (8.52) we then obtain: Ltrχ + |χ|2 = f 0

(8.55)


212

where f 0 is the function: f 0 = etrχ + ψ L

( P) (N) dH ψ · χ · d/σ − tr α −tr α dσ

(8.56)

( P)

The function f 0 is bounded as µ → 0. The term tr α is obtained by taking the trace of equation (4.72): ( P)

tr α =

1 dH 2ψ L ψ · d/(Lσ ) − ψ L2 / σ − | ψ|2 L(Lσ ) 2 dσ

+ 2α −1 ψ L ψ · (κ −1 ζ ) + α −1 ψ L2 trk/ + | ψ|2 e (Lσ )

(8.57)

We conclude from the above that the St,u 1-form g0 defined by (8.51) is of the order of the 2nd derivatives of the ψµ and bounded as µ → 0. By (8.52), (8.54) and (8.55) it is given by: ˇ (8.58) g0 = d/gˇ − µtrχd/eˇ + (d/µ)( f 0 − trχ e) where we have defined: eˇ = e − e

(8.59)

To control the higher order angular derivatives of trχ, we introduce the St,u 1-forms: (i1 ...il )

xl = µd/(Ril . . . Ri1 trχ) + d/(Ril . . . Ri1 fˇ)

(8.60)

Thus, for a given positive integer l we have the multi-indices (i 1 , . . . , i l ) of length l, where each i k ∈ {1, 2, 3}, k = 1, . . . , l. To the multi-index (i 1 , . . . , i l ) there corresponds the string (Ri1 , . . . , Ril ) of rotation vectorfields. In deriving propagation equations for the xl , we shall make use of the following lemma. Lemma 8.2 Let Y be an arbitrary St,u - tangential vectorfield on the spacetime domain Wε∗0 . We have: [L, Y ] = (Y ) Z where (Y ) Z is an St,u - tangential vectorfield, associated to Y , and defined by the condition that for any vector V ∈ T Wε∗0 : h(

(Y )

Z, V ) =

(Y )

π(L, V )

In terms of the (L, T, X 1 , X 2 ) frame (or the null frame (L, L, X 1 , X 2 )), (Y )

Z=

(Y )

Z A X A,

(Y )

ZA =

(Y )

π L B (h/−1 ) AB

Proof. Let "s be the 1-parameter group generated by L and #s the 1-parameter group generated by Y . Then #s maps each surface St,u onto itself, while "s maps St,u onto St +s,u . Hence #−s ◦ "−s ◦ #s ◦ "s (8.61)


213

maps each St,u onto itself. Thus, the orbit by (8.61) of any given point q ∈ St,u lies in St,u . But this orbit must coincide to O(s 2 ) as s → 0 with the integral curve of the commutator [L, Y ] through q, parametrized by s 2 . It follows that [L, Y ] = (Y ) Z is a vectorfield which is tangential to the surfaces St,u , consequently we can expand: (Y )

Z=

(Y )

ZBXB

Taking the h- inner product with X A we obtain: / AB h

(Y )

Z B = h(

(Y )

Z , X A)

= h([L, Y ], X A ) = h(D L Y, X A ) − h(DY L, X A ) = (Y ) π L A − h(D X A Y, L) − h(DY L, X A ) Now, since h(L, X A ) = 0 we have: −h(DY L, X A ) = h(L, DY X A ) hence, substituting, / AB h

(Y )

ZB =

(Y )

π L A + h(L, [Y, X A ])

=

(Y )

πL A,

because the vectorfield [Y, X A ] is St,u -tangential, hence h-orthogonal to L, both Y, X A being St,u - tangential vectorfields. The lemma is thus proved. The following lemma, essentially a corollary of Lemma 8.2, is also used in the derivation of the propagation equations for the xl . Lemma 8.3 Let Y be an arbitrary St,u - tangential vectorfield on the spacetime domain Wε∗0 and let ξ be an arbitrary St,u 1-form on Wε∗0 . We have: /Y ξ − L /Y L /L ξ = L / (Y ) Z ξ L /L L Proof. Since both L, Y are tangential to the hypersurfaces Cu we can restrict ourselves to a given Cu . In defining L / L ξ, L /Y ξ we are considering the extension of ξ to T Cu by the condition ξ(L) = 0 We have: (L L ξ )(L) = L(ξ(L)) − ξ([L, L]) = 0 therefore: L /L ξ = LL ξ However, (LY ξ )(L) = Y (ξ(L)) − ξ([Y, L]) = ξ([L, Y ]) = ξ(

(8.62) (Y )

Z)


214

by Lemma 8.2. Since L /Y ξ is defined by restricting LY ξ to T St,u and then extending to /Y ξ )(L) = 0, it follows that, on the manifold Cu , T Cu by the condition (L L /Y ξ = LY ξ − ξ(

(Y )

Z )dt

(8.63)

in view of the fact that Lt = 1, d/t = 0. Consider now the evaluation of L /L L /Y ξ − L /Y L /L ξ on the frame vectorfields X A . This evaluation is: /Y ξ − LY L / L ξ )(X A ) (L L L

(8.64)

Substituting (8.62) and (8.63), (8.64) becomes: (L L LY ξ − L(ξ(

(Y )

Z ))dt − ξ(

(Y )

Z )L L (dt) − LY L L ξ )(X A )

(8.65)

Now, dt (X A ) = d/t (X A ) = 0, while: L L (dt) = d(Lt) = 0 therefore (8.65) reduces to: (L L LY ξ − LY L L ξ )(X A ) = (L[L ,Y ] ξ )(X A )

(8.66)

and by Lemma 8.2 this is: (L

(Y ) Z

ξ )(X A )

The lemma thus follows. Moreover, in the proof of the next proposition, as well as in the proofs of several of the propositions and lemmas which follow, we shall make use of a general elementary proposition on linear recursions, the proof of which is by a simple application of the principle of induction. Proposition 8.2 Let (yn : n = 1, 2, . . . ) be a given sequence in a space X and (An : n = 1, 2, . . . ) a given sequence of operators in X. Suppose that (x n : n = 0, 1, 2, . . . ) is a sequence in X satisfying the recursion: x n = An x n−1 + yn Then for each n = 1, 2, . . . we have: x n = An . . . A1 x 0 +

n−1 m=0

An . . . An−m+1 yn−m


To present the propagation equation for we introduce the functions:

215 (i1 ...il ) x

l

in as simple a form as possible,

fˇl = Ril . . . Ri1 fˇ (i1 ...il ) h l = Ril . . . Ri1 |χ| ˆ 2 (i1 ...il )

(8.67) (8.68)

Proposition 8.3 For each non-negative integer l and each multi-index (i 1 , . . . , i l ), the St,u 1-form (i1 ...il ) xl satisfies the propagation equation: L / L (i1 ...il ) xl + trχ − 2µ−1 (Lµ) (i1 ...il ) xl 1 trχ − 2µ−1 (Lµ) d/ (i1 ...il ) fˇl − µd/ (i1 ...il ) h l − (i1 ...il ) gl = 2 where the St,u 1-form (i1 ...il )

(i1 ...il ) g l

is given by:

gl = L / Ril . . . L / R i1 g 0 −

l−1

L / Ril . . . L / Ril−k+1 L / (Ril−k )

(i1 ...il−k−1 ) Z

xl−k−1

k=0

+

l−1

L / Ril . . . L / Ril−k+1

(i1 ...il−k )

yl−k

k=0

where the St,u 1-form g0 is given by (8.51), or, more explicitly, by (8.58). Here, for each j = 1, . . . , l, (i1 ...i j ) y j is the St,u 1-form: (i1 ...i j )

y j = (Ri j µ) (i1 ...i j −1 ) a j −1 + µRi j trχ − Ri j Lµ + 1 + (Ri j trχ)d/ 2

where

(i1 ...i j −1 ) a

(i1 ...i j −1 )

j −1

(i1 ...i j −1 )

(Ri j )

Z µ d/(Ri j −1 . . . Ri1 trχ)

fˇj −1

is the St,u 1-form:

a j −1 = L / L d/(Ri j −1 . . . Ri1 trχ) + trχd/(Ri j −1 . . . Ri1 trχ) + d/

(i1 ...i j −1 )

h j −1

Proof. The propagation equation of the proposition reduces for l = 0 to the propagation equation (8.50) already established. Thus, by induction, assuming that the propagation equation of the proposition holds with l replaced by l − 1, that is, assuming that: L / L (i1 ...il−1 ) xl−1 + trχ − 2µ−1 (Lµ) (i1 ...il−1 ) xl−1 1 −1 = (8.69) − 2µ (Lµ) d/ (i1 ...il−1 ) fˇl−1 − µd/ (i1 ...il−1 ) h l−1 − (i1 ...il−1 ) gl−1 2


216

holds for some St,u 1-form (i1 ...il−1 ) gl−1 , what we must show is that a propagation equation of the form given by the proposition holds true for l, where (i1 ...il ) gl is an St,u 1-form related to (i1 ...il−1 ) gl−1 by a certain recursion relation. This recursion relation shall then determine (i1 ...il ) gl , for each l, from the St,u 1-form g0 , given by (8.58). We begin by re-writing the term 2µ−1 (Lµ) (i1 ...il−1 ) xl−1 − d/ (i1 ...il−1 ) fˇl−1 in (8.69) as: 2(Lµ)d/(Ril−1 . . . Ri1 trχ) (see definitions (8.60), (8.67)) obtaining the equation: L /L

(i1 ...il−1 )

xl−1 + trχ

(i1 ...il−1 )

xl−1 1 = 2(Lµ)d/(Ril−1 . . . Ri1 trχ) + trχd/ 2

(8.70) (i1 ...il−1 )

fˇl−1 − µd/

(i1 ...il−1 )

h l−1 −

(i1 ...il−1 )

gl−1

We now apply L / Ril to this equation. Taking into account the fact that by Lemma 8.1 applied to the St,u - tangential vectorfield Ril and to the functions Ril−1 . . . Ri1 trχ,

(i1 ...il−1 )

fˇl−1 ,

(i1 ...il−1 )

h l−1 ,

we have: L / Ril d/(Ril−1 . . . Ri1 trχ) = d/(Ril . . . Ri1 trχ) L / Ril d/(

(i1 ...il−1 )

fˇl−1 ) = d/(

(i1 ...il )

fˇl )

L / Ril d/(

(i1 ...il−1 )

h l−1 ) = d/(

(i1 ...il )

hl )

(see definitions (8.67), (8.68)), we obtain: L / Ril L /L

(i1 ...il−1 )

xl−1 + trχL / Ril

(i1 ...il−1 )

(i1 ...il−1 )

xl−1

= 2(Lµ)d/(Ril . . . Ri1 trχ) + 2(Ril Lµ)d/(Ril−1 . . . Ri1 trχ) 1 1 + trχd/ (i1 ...il ) fˇl + (Ril trχ)d/ (i1 ...il−1 ) fˇl−1 2 2 − µd/ (i1 ...il ) h l − (Ril µ)d/ (i1 ...il−1 ) h l−1 −L / Ril

(i1 ...il−1 )

gl−1

(8.71)

Next, we apply Lemma 8.3 setting Y = Ril , ξ = /L L / Ril L

xl−1 + (Ril trχ)

(i1 ...il−1 )

xl−1 = L /L L / Ril

(i1 ...il−1 )

(i1 ...il−1 ) x

l−1 ,

xl−1 − L / (Ril )

to express: (i1 ...il−1 )

Z

xl−1

Now, definition (8.60) with l replaced by l − 1 reads: (i1 ...il−1 )

xl−1 = µd/(Ril−1 . . . Ri1 trχ) + d/(Ril−1 . . . Ri1 fˇ)

(8.72)


217

Applying L / Ril to this we obtain, in view of definition (8.60) and Lemma 8.1, L / Ril

(i1 ...il−1 )

xl−1 =

(i1 ...il )

xl + (Ril µ)d/(Ril−1 . . . Ri1 trχ)

(8.73)

Applying L / L to (8.73) and expressing: L Ril µ = Ril Lµ +

(Ril )

Zµ

using the fact that according to Lemma 8.2, [L, Ril ] =

(Ril )

Z

(8.74)

xl−1 = L / L (i1 ...il ) xl + (Ril µ)L / L d/(Ril−1 . . . Ri1 trχ) + Ril Lµ + (Ril ) Z µ d/(Ril−1 . . . Ri1 trχ)

(8.75)

yields: / Ril L /L L

(i1 ...il−1 )

We then substitute (8.75) in (8.72) and the result in (8.71). Using (8.73) to re-write the second term on the left-hand side in (8.71), a propagation equation for (i1 ...il ) xl of the form given by the proposition results, with (i1 ...il ) gl expressed in terms of (i1 ...il−1 ) gl−1 by the recursion formula: (i1 ...il )

gl = L / Ril

(i1 ...il−1 )

gl−1 − L /

(Ri ) l Z

(i1 ...il−1 )

xl−1 +

(i1 ...il )

yl

(8.76)

The proposition then follows by applying Proposition 8.2 to this recursion, with the space of St,u 1-forms in the role of the space X, the operators L / Ril in the role of the operators (i ...i ) (i ...i ) ...i ) (i l l 1 1 1 l−1 An , and gl , yl − L / (Ril ) xl−1 , in the role of x n , yn , respectively. Z

We remark here that the St,u 1-form sition 8.3 reduces for j = 1 to

(i1 ...i j −1 ) a

j −1 defined in the statement of Propo-

a0 = Ld/trχ + trχd/trχ + d/h 0

(8.77)

Applying d/ to equation (8.55) we obtain, in view of Lemma 8.1 and the fact that |χ|2 =

1 (trχ)2 + h 0 , 2

simply: a0 = d/ f 0

(8.78)

/ Ri1 to (8.77) to obtain, in view of (8.78), the following For j ≥ 2, we apply L / Ri j −1 . . . L expression for

(i1 ...i j −1 ) a

j −1 :

(i1 ...i j −1 )

a j −1 = d/(Ri j −1 . . . Ri1 f0 ) +

(i1 ...i j −1 )

b j −1

(8.79)


218

where: (i1 ...i j −1 )

b j −1 = [L /L , L / Ri j −1 . . . L / Ri1 ]d/trχ +

j −1

j −1

m=1 k1 0

(8.284)

In any case we have: δ1 ≤ Cδ0

(8.285)

In the following we denote: b(t, s) =

[1 + log(1 + t)] (s − t) , (1 + t) (1 + s)

c(t) =

[1 + log(1 + t)] (1 + t)2

(8.286)

We also denote: µˆ s,m (t) =

min

(u,ϑ)∈[0,ε0]×S 2

µˆ s (t, u, ϑ)

(8.287)

From Proposition 8.6 we have, in the present case, µˆ s (t, u, ϑ) ≥ 1 − δ1 log(1 + t) − Cδ0 b(t, s)

(8.288)

for all (u, ϑ) ∈ [0, ε0 ] × S 2 , hence: µˆ s,m (t) ≥ 1 − δ1 log(1 + t) − Cδ0 b(t, s)

(8.289)


252

On the other hand, evaluating the expression for µˆ s (t, u, ϑ) given in the statement of Proposition 8.6 at a minimum (u m , ϑm ) of the function Eˆ s (u, ϑ) on [0, ε0 ] × S 2 , we obtain: µˆ s,m (t) ≤ µˆ s (t, u m , ϑm ) = 1 − δ1 log(1 + t) + Q 0,s (t, u m , ϑm ) hence: µˆ s,m (t) ≤ 1 − δ1 log(1 + t) + Cδ0 b(t, s)

(8.290)

Moreover, again from Proposition 8.6 we have:

∂ µˆ s − ∂t hence:

−

(t, u, ϑ) ≤ −

∂ µˆ s − ∂t

( Eˆ s )− (u, ϑ) − ( Qˆ 1,s )− (t, u, ϑ) (1 + t)

−

(t, u, ϑ) ≤

δ1 + Cδ0 c(t) (1 + t)

(8.291)

Consider now the integral Ia (t). Let us set: 1

t1 = e 2aδ1 − 1

(8.292)

We have two subcases to consider according as to whether t is ≤ t1 or > t1 . Subcase 2a) t ≤ t1 . Since t ≤ t, where t is the variable of integration in the integral Ia (t), in this subcase we have t ≤ t1 , that is: 1 − δ1 log(1 + t ) ≥ 1 −

1 2a

(8.293)

hence, by (8.289): µˆ s,m (t ) ≥ 1 −

1 a

(8.294)

provided that: 1 (8.295) 2C where C is the constant appearing in (8.289). This is a smallness condition on δ0 of the form (8.283). In view of (8.279), the lower bound (8.294) implies: δ0 a ≤

2 µm (t ) ≥1− η0 a hence also:

2 a provided again that a smallness condition on δ0 of the form (8.283) holds. µm (t ) ≥ 1 −

(8.296)


253

Now, by (8.291) with t replaced by t and (8.294) we have: 1 −1 δ1 M(t ) ≤ 1 − + Cδ0 c(t ) a (1 + t )

(8.297)

The estimates (8.296) and (8.297) then yield: 2 −a t δ1 1 −1 + Cδ 1− Ia (t) ≤ 1 − c(t ) dt 0 a a (1 + t ) 0 Here,

t 0

1 δ1 dt = δ1 log(1 + t) ≤ δ1 log(1 + t1 ) = (1 + t ) 2a

while,

t

Cδ0 c(t )dt ≤ C δ0 ≤

0

1 2a

(8.298)

(8.299)

(8.300)

provided that a smallness condition on δ0 of the form (8.283) holds, the function c being integrable on [0, ∞). In view of the fact that the coefficient in (8.298): 2 −a 1 −1 → e2 , as a → ∞ 1− 1− a a it follows that for t ≤ t1 :

C (8.301) a In particular this holds for t = t1 . Since, in general, µm (t) ≤ 1 by definition, the inequality (8.301) implies the inequality of the lemma in Subcase 2a). Ia (t) ≤

Subcase 2b) t > t1 . In this subcase we write: t µ−a Ia (t) = Ia (t1 ) + m (t )M(t )dt

(8.302)

t1

Substituting (8.301) with t replaced by t1 we obtain: t C µ−a Ia (t) ≤ + m (t )M(t )dt a t1

(8.303)

Consider now the upper bound (8.290) evaluated at t = s. Since b(s, s) = 0, this reads: µˆ s,m (s) ≤ 1 − δ1 log(1 + s) (8.304) On the other hand, µ > 0 in the maximal domain Wε0 , in particular in Wεs0 . It follows that: 1 s ≤ t∗ where t∗ = e δ1 − 1 (8.305)


254

Let us introduce the variable:

τ = log(1 + t )

(8.306)

and correspondingly write: τ = log(1 + t),

σ = log(1 + s)

and:

1 , 2aδ1 Then in the integral in (8.303) we have: τ1 = log(1 + t1 ) =

τ∗ = log(1 + t∗ ) =

(8.307) 1 δ1

τ1 ≤ τ ≤ τ ≤ σ ≤ τ∗

(8.308)

(8.309)

Expressing b(t, s) (see (8.286)) in terms of the new variables we have: b(t, s) = Since, 0≤

(1 + τ ) (eσ − eτ ) eτ eσ

(8.310)

(eσ − eτ ) = 1 − e−(σ −τ ) ≤ σ − τ, eσ

it follows that:

(1 + τ ) (σ − τ ) eτ Substituting (8.311) in (8.289) with t replaced by t we obtain: 0 ≤ b(t, s) ≤

(1 + τ ) µˆ s,m (t ) ≥ 1 − δ1 τ − Cδ0 (σ − τ ) τ e Cδ0 (1 + τ ) = 1 − δ1 σ + δ1 1 − (σ − τ ) δ1 eτ We have:

1 (1 + τ ) (1 + τ1 ) 1 − 2aδ 1 ≤ = 1 + e eτ1 2aδ1 eτ

(8.311)

(8.312)

(8.313)

Hence, under the smallness condition Cδ0 ≤

1 2a

on δ0 (of the same form as (8.283)):

1 1 Cδ0 (1 + τ ) − 1 ≤ 1+ e 2aδ1 τ δ1 2aδ1 2aδ1 e

Now, in view of the fact that 1 1 −1 lim 1+ e x = 0, x→0+ x x

(8.314)


255

there is a positive constant K (a), depending on a, such that: 1 1 1 1 −1 : ∀x ≤ 1+ e x ≤ x x a K (a) Here we set: x = 2aδ1 ≤ 2aCδ0 We then conclude that the smallness condition: δ0 ≤ on δ0 , implies that:

1 2Ca K (a)

1 1 1 − 1 1+ e 2aδ1 ≤ 2aδ1 2aδ1 a

(8.315)

(8.316)

It then follows from (8.314) that: 1−

Cδ0 (1 + τ ) 1 ≥1− τ δ1 a e

(8.317)

Therefore, in reference to (8.312), taking into account the fact that: 1 − δ 1 σ ≥ 1 − δ 1 τ∗ = 0

(8.318)

we obtain: Cδ0 (1 + τ ) 1 − δ1 σ + δ1 1 − (σ − τ ) δ1 eτ 1 ≥ 1 − δ1 σ + 1 − δ1 (σ − τ ) a 1 1 ≥ 1− (1 − δ1 σ ) + 1 − δ1 (σ − τ ) a a 1 = 1− (1 − δ1 τ ) a

(8.319)

From (8.312) we then conclude that the following lower bound holds: 1 µˆ s (t , u, ϑ) ≥ µˆ s,m (t ) ≥ 1 − (1 − δ1 τ ) a : ∀(u, ϑ) ∈ [0, ε0] × S 2 , ∀t ≥ t1

(8.320)

We now consider the bound (8.291). Since (see (8.286), (8.306)) (1 + t )c(t ) =

(1 + τ ) eτ

(8.321)


256

the bound (8.291) with t replaced by t reads: Cδ0 (1 + τ ) ∂ µˆ s −(1 + t ) (t , u, ϑ) ≤ δ1 1 + ∂t − δ1 eτ

(8.322)

If the smallness condition (8.315) on δ0 holds, this implies: 1 ∂ µˆ s −(1 + t ) (t , u, ϑ) ≤ δ1 1 + ∂t − a : ∀(u, ϑ) ∈ [0, ε0 ] × S 2 , ∀t ≥ t1

The bounds (8.320) and (8.323) together yield: 1 1 + δ1 a (1 + t )M(t ) ≤ 1 (1 − δ1 τ ) 1− a Also, by (8.279), taking Cδ0 ≤ a −1 , and the lower bound (8.320), 2 µm (t ) (1 − δ1 τ ) ≥ 1− η0 a 2 µm (t ) ≥ 1 − (1 − δ1 τ ) a

hence also:

(8.323)

(8.324)

(8.325)

(8.326)

We now consider the integral on the right in inequality (8.303). In terms of the variable τ we have (see (8.306)–(8.308)): τ t −a µm (t )M(t )dt = µ−a m (t )(1 + t )M(t )dτ τ1

t1

thus, the bounds (8.324) and (8.326) yield: τ t 1 + a1 1 −a µm (t )M(t )dt ≤ (1 − δ1 τ )−a−1 δ1 dτ a 1 2 1 − t1 τ1 1− a a Since:

τ

τ1

(1 − δ1 τ )−a−1 δ1 dτ =

1 1 [(1 − δ1 τ )−a − (1 − δ1 τ1 )−a ] ≤ (1 − δ1 τ )−a a a

while the coefficient:

1+ 1−

1 a 1 a

(8.327)

1 a → e2 , as a → ∞ 2 1− a


we conclude that:

t t1

257

µ−a m (t )M(t )dt ≤

C (1 − δ1 τ )−a a

(8.328)

On the other hand, by the upper bound (8.290) and (8.311), (1 + τ ) µˆ s,m (t) ≤ 1 − δ1 τ + Cδ0 (σ − τ ) τ e Cδ0 (1 + τ ) (σ − τ ) = 1 − δ1 σ + δ1 1 + δ1 eτ

(8.329)

Now by (8.313) with τ replaced by τ and (8.316), 1 Cδ0 (1 + τ ) ≤ δ1 eτ a provided that the smallness condition (8.315) on δ0 holds. Hence, Cδ0 (1 + τ ) 1 δ1 1+ ≤ 1 + δ1 eτ a By virtue of (8.331) we obtain, in reference to (8.329), Cδ0 (1 + τ ) 1 − δ1 σ + δ1 1 + (σ − τ ) δ1 eτ 1 δ1 (σ − τ ) ≤ 1 − δ1 σ + 1 + a 1 1 ≤ 1+ (1 − δ1 σ ) + 1 + δ1 (σ − τ ) a a 1 = 1+ (1 − δ1 τ ) a Substituting in (8.329) we then obtain the following upper bound: 1 µˆ s,m (t) ≤ 1 + (1 − δ1 τ ) a

(8.330)

(8.331)

(8.332)

(8.333)

which together with (8.279) (and a smallness condition on δ0 of the form (8.283)) yields: 2 µm (t) ≤ 1+ (8.334) (1 − δ1 τ ) η0 a hence also:

2 µm (t) ≤ 1 + (1 − δ1 τ ) a

Therefore: µ−a m (t) ≥

1 1+

2 a

a (1 − δ1 τ )−a

(8.335)

(8.336)


258

Since the reciprocal of the coefficient: 2 a 1+ → e2 , as a → ∞ a we conclude that: µ−a m (t) ≥

1 (1 − δ1 τ )−a C

Comparison of (8.328) with (8.337) yields the conclusion: t C −a µ−a µ (t) m (t )M(t )dt ≤ a m t1

(8.337)

(8.338)

In view of (8.303), (8.338) and the fact that µm (t) ≤ 1 the inequality of the lemma follows in Subcase 2b) as well. Case 2) is thus concluded. The proof of the lemma is now complete. The proof of Lemma 8.11 yields the following corollaries. Corollary 1 Under the assumptions of Lemma 8.11, there is a positive constant C independent of s such that the following holds. Suppose that at some (u, ϑ) ∈ [0, ε0 ] × S 2 we have Eˆ s (u, ϑ) < 0. Then we have: µˆ s (t, u, ϑ) ≤C µˆ s (t , u, ϑ) for all t ∈ [0, t] and all t ∈ [0, s]. Proof. The argument of Case 2) of the proof of Lemma 8.11 leading to upper and lower bounds on µˆ s,m (t) applies with [0, ε0 ] × S 2 replaced by any closed subset thereof, in particular the single point (u, ϑ) in which case we simply have: Eˆ s,m = Eˆ s (u, ϑ) = −δ1 , δ1 > 0,

µˆ s,m (t) = µˆ s (t, u, ϑ)

Then the lower and upper bounds (8.289) and (8.290) become: 1 − δ1 log(1 + t) − Cδ0 b(t, s) ≤ µˆ s (u, ϑ) ≤ 1 − δ1 log(1 + t) + Cδ0 b(t, s) (8.339) Fixing then a = 2 and setting, in accordance with (8.292), 1

t1 = e 4δ1 − 1

(8.340)

we have to consider the two subcases according as to whether t is ≤ t1 or > t1 . In the first subcase (8.294) holds, that is, in the present circumstances, µˆ s (t , u, ϑ) ≥

1 2

(8.341)


259

while by the upper bound in (8.339), µˆ s (t, u, ϑ) ≤

3 2

(8.342)

thus the inequality of the corollary holds in the first subcase. In the second subcase, we have the upper bound (8.333), which in the present circumstances is: µˆ s (t, u, ϑ) ≤

3 (1 − δ1 τ ) 2

(8.343)

If then t ≤ t1 , the lower bound (8.341) holds, so the inequality of the corollary holds, while if t > t1 , then the lower bound (8.320) holds, which under the present circumstances is: 1 (8.344) µˆ s (t , u, ϑ) ≥ (1 − δ1 τ ) 2 thus, in view of the fact that (1 − δ1 τ ) ≤ (1 − δ1 τ ), the inequality of the corollary again holds. Corollary 2 Under the assumptions of Lemma 8.11, there is a positive constant C independent of s and a such that: −a µ−a m (t ) ≤ Cµm (t) holds for all t ∈ [0, t] and all t ∈ [0, s]. Proof. We have the two cases of Lemma 8.11 to consider. Case 1) Eˆ s,m ≥ 0. In this case, the lower bound (8.281) holds with t replaced by t : µm (t ) ≥ 1 − Cδ0

(8.345)

On the other hand we have in general, by definition, µm (t) ≤ 1 It follows that:

µ−a m (t ) ≤ (1 − Cδ0 )−a µ−a m (t)

(8.346)

and this is bounded by a constant independent of a provided that: Cδ0 ≤

1 a

Case 2) Eˆ s,m < 0. In this case we define δ1 as in (8.284) and t1 as in (8.292) and we consider the subcases 2a) t ≤ t1 and 2b) t ≥ t1 separately. In subcase 2a) the lower bound (8.296) holds. We thus have: µ−a 2 −a m (t ) (8.347) ≤ 1− a µ−a m (t) which is bounded by a constant independent of a (for a ≥ 4).


260

In subcase 2b) the lower bound (8.326) as well as the upper bound (8.335) hold, hence: −a −a 1 − a2 1 − a2 1 − δ1 τ a µ−a m (t ) ≤ (8.348) ≤ 1 − δ1 τ µ−a 1 + a2 1 + a2 m (t) (for, τ ≤ τ ) which is bounded by a constant independent of a (for a ≥ 4), as required. Corollary 3 The assumptions of Lemma 8.11 imply the assumption AS. Proof. We consider a given (u, ϑ) ∈ [0, ε0 ] × S 2 as in the proof of Corollary 1. Again, we have the two cases Eˆ s (u, ϑ) ≥ 0 and Eˆ s (u, ϑ) < 0 to consider. In the first case we have µˆ s (t, u, ϑ) ≥ 1 − Cδ0 as in Case 1 of the proof of Lemma 8.11 (see (8.277)). On the other hand from Proposition 8.6 we have: ∂ µˆ s Eˆ s (u, ϑ) 1 (t, u, ϑ) = + Qˆ 1,s (t, u, ϑ) ≤ Cδ0 ∂t (1 + t) (1 + t) Therefore: −2µ−1

∂µ ∂ µˆ s 1 = −2µ ˆ −1 ≥ −Cδ0 s ∂t ∂t (1 + t)

On the other hand, assumption F2 implies, as we have seen, trχ ≥ C −1

1 (1 + t)

The assumption AS thus follows in Case 1) if δ0 is suitably small. In the second case we set as in the proof of Corollary 1, Eˆ s (u, ϑ) = −δ1 , δ1 > 0, a = 2 and define t1 by (8.340). Then we have to consider the subcases t ≤ t1 and t > t1 . In the first subcase we have (see (8.341)) µˆ s (t, u, ϑ) ≥

1 2

while from Proposition 8.6 ∂ µˆ s [1 + log(1 + t)] (t, u, ϑ) ≤ Qˆ 1,s (t, u, ϑ) ≤ Cδ0 ∂t (1 + t)2 Therefore: −2µ−1

∂µ ∂ µˆ s [1 + log(1 + t)] = −2µˆ −1 ≥ −Cδ0 s ∂t ∂t (1 + t)2

and the assumption AS follows in subcase 2a) if δ0 is suitably small.


261

In the second subcase we have from Proposition 8.6, (1 + t)

∂ µˆ s (t, u, ϑ) = −δ1 + (1 + t) Qˆ 1,s (t, u, ϑ) ∂t Cδ0 (1 + τ ) δ1 ≤ −δ1 1 − ≤− τ δ1 e 2

if δ0 is suitably small, according to (8.315) with a = 2. It follows that in subcase 2b) −2µ−1

∂µ >0 ∂t

and the assumption AS a fortiori holds. We now return to (8.187). What we wish to do is to obtain an estimate for the L 2 norm of (i1 ...il ) Fl (t) on [0, ε0] × S 2 . In general, if φ is a function defined on Wε∗0 and we consider φ(t, u), the corresponding function on S 2 depending on t and u, then the L 2 ε norm of φ on t 0 is, by definition: ) φ L 2 ( ε0 ) =

ε

t

) =

t 0

φ 2 dµh/ du

[0,ε0 ]×S 2

(φ(t, u))2 dµh/(t,u)du

(8.349)

On the other hand, the L 2 norm of φ(t, .) on [0, ε0 ] × S 2 is: ) φ(t) L 2 ([0,ε0 ]×S 2 ) =

[0,ε0 ]×S 2

(φ(t, u))2 dµh/(0,0)du

/(0, 0) being the standard metric on S 2 . We have: h √ √ deth /(t, u) deth/(0, u) dµh/(t,u) = √ dµh/(0,0) = A(t, u) √ dµh/(0,0) deth /(0, 0) deth/(0, 0)

(8.350)

(8.351)

ε

Now, by the definition of θ (see Chapter 3), we have on 00 , noting that vanishes there, 1 ∂

deth/ κtrθ = √ deth/ ∂u The assumptions on the initial data imply: κtrθ − 2 ≤ Cδ0 1−u

ε

: on 00

ε

: on 00

(8.352)

(8.353)

It then follows integrating (8.352) that: e

−Cδ0

√ deth/(0, u) (1 − u) ≤ √ ≤ eCδ0 (1 − u)2 deth/(0, 0) 2

(8.354)


262

Combining with (8.185) we obtain from (8.351): e−Cδ0 (1 − u + η0 t)2 dµh/(0,0) ≤ dµh/(t,u) ≤ eCδ0 (1 − u + η0 t)2 dµh/(0,0)

(8.355)

In view of (8.349), (8.350) it follows that: C −1 (1 + t)φ(t) L 2 ([0,ε0 ]×S 2 ) ≤ φ L 2 ( ε0 ) ≤ C(1 + t)φ(t) L 2 ([0,ε0]×S 2 ) t

(8.356)

Note moreover that if ξ is an St,u 1-form defined on Wε∗0 and we consider ξ(t, u), the corresponding 1-form on S 2 depending on t and u, then setting φ = |ξ |, φ(t, u) = |ξ(t, u)|, the above apply to the functions φ and φ(t, u) and we have: φ L 2 ( ε0 ) = ξ L 2 ( ε0 ) , t

φ(t) L 2 ([0,ε0 ]×S 2 ) = |ξ(t)| L 2 ([0,ε0]×S 2 )

t

(8.357)

hence, by (8.356), C −1 (1 + t)|ξ(t)| L 2 ([0,ε0 ]×S 2 ) ≤ ξ L 2 ( ε0 ) ≤ C(1 + t)|ξ(t)| L 2 ([0,ε0 ]×S 2 ) (8.358) t

0

1

2

We now consider the terms (i1 ...il ) M l (t, u), (i1 ...il ) M l (t, u), (i1 ...il ) M l (t, u) on the right-hand side of (8.187), given by (8.188), (8.189), (8.190) respectively. First, by the basic bootstrap assumption A3 and (8.358) at t = 0,

(i1 ...il )

0

M l (t) L 2 ([0,ε0 ]×S 2 ) ≤ C[1 + log(1 + t)]2

We turn to

(i1 ...il )

xl (0) L 2 ( ε0 )

(8.359)

0

1

(i1 ...il )

M l (t, u). We partition [0, ε0 ] × S 2 , into the open set:

Vs− = {(u, ϑ) ∈ [0, ε0 ] × S 2 : Eˆ s (u, ϑ) < 0}

(8.360)

and its complement, the closed set: Vs+ = {(u, ϑ) ∈ [0, ε0 ] × S 2 : Eˆ s (u, ϑ) ≥ 0}

(8.361)

We then have:

(i1 ...il )

1

M l (t)2L 2 ([0,ε0 ]×S 2 )

= Now, for fixed t,

(i1 ...il )

(i1 ...il ) 1

2 M l (t) L 2 (Vs− ) +

(i1 ...il ) 1

2 M l (t) L 2 (Vs+ )

(8.362)

1

M l (t) restricted to Vs− is a definite integral, with respect to t , 1

of a function of t with values in the space L 2 (Vs− ). Thus, the norm of (i1 ...il ) M l (t) in L 2 (Vs− ) does not exceed the integral of the L 2 (Vs− )-norm of the said function, with respect to t : t 1 1 (i1 ...il ) M l (t) L 2 (Vs− ) ≤ (i1 ...il ) N l (t, t ) L 2 (Vs− ) dt (8.363) 0


263

where: 1

(i1 ...il )

N l (t, t , u) µ(t, u) 2 3 −1 ∂µ = (1 − u + η0 t ) −2µ (t , u) |d/ µ(t , u) ∂t −

(8.364) (i1 ...il )

fˇl (t , u)|

We have: (i1 ...il )

1

N l (t, t ) L 2 (Vs− ) µ(t) 2 3 −1 ∂µ ≤ (1 + η0 t ) max max −2µ (t ) |d/ µ(t ) ∂t − Vs− Vs−

(8.365) (i1 ...il )

fˇl (t )| L 2 (Vs− )

Now, by Corollary 1 to Lemma 8.11 we have: µ(t) max ≤C µ(t ) Vs−

(8.366)

while by definition (8.273), ∂µ (t ) ≤ 2M(t ) max −2µ−1 ∂t − Vs−

(8.367)

We thus obtain, substituting in (8.365),

(i1 ...il )

1

N l (t, t ) L 2 (Vs− ) ≤ C(1 + t )3 M(t )|d/

(i1 ...il )

fˇl (t )| L 2 ([0,ε0]×S 2 )

(8.368)

Let us define: (i1 ...il )

Pl (t) = (1 + t)2 |d/

Suppose that, for non-negative quantities (i1 ...il )

Pl (t) ≤

(i1 ...il )

fˇl (t)| L 2 ([0,ε0]×S 2 )

(i1 ...il )

(0) (1) P l , (i1 ...il ) P l ,

(0) Pl

(i1 ...il )

(i1 ...il )

(t) +

Defining then the non-decreasing non-negative quantities (i1 ...il )

(0)

P l,a = sup {µam (t )

(i1 ...il )

t ∈[0,t ]

(i1 ...il )

(1)

P l,a = sup {(1 + t )1/2 µam (t ) t ∈[0,t ]

(1) Pl

we have: (t)

(i1 ...il ) (0) Pl

(0)

P l,a ,

(t )}

(i1 ...il )

(8.369)

(1) Pl

(8.370) (i1 ...il )

(1)

P l,a , by: (8.371)

(t )}

(8.372)


264

Then, for t ∈ [0, t] we have: (i1 ...il )

Pl (t ) ≤

µ−a m (t )

(0)

(i1 ...il )

−1/2 (i1 ...il )

P l,a (t) + (1 + t )

(1)

P l,a (t)

(8.373)

Substituting in (8.368) we obtain, in view of the definition (8.369), (i1 ...il )

1

N l (t, t ) L 2 (Vs− )

≤ C (1 + t)

(i1 ...il )

(0)

(8.374)

(1)

P l,a (t) + (1 + t)1/2 P l,a (t) µ−a m (t )M(t )

for all t ∈ [0, t]. Substituting this estimate in (8.363) we then obtain:

(i1 ...il )

1

M l (t) L 2 (Vs− ) ≤ C (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)

1/2

(1)

P l,a (t) Ia (t) (8.375)

At this point we apply Lemma 8.11 to conclude that:

(i1 ...il )

1

M l (t) L 2 (Vs− )

≤ Ca

−1

(1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)

1/2

(1)

P l,a (t) µ−a m (t)

(8.376)

In analogy with (8.363) and (8.365), with Vs+ in the role of Vs− , we have: t 1 1 (i1 ...il ) (i1 ...il ) N l (t, t ) L 2 (Vs+ ) dt (8.377) M l (t) L 2 (Vs+ ) ≤ 0

and: (i1 ...il )

1

N l (t, t ) L 2 (Vs+ ) µ(t) 2 −1 ∂µ −2µ ≤ (1 + η0 t )3 max max (t ) |d/ µ(t ) ∂t − Vs+ Vs+

(8.378) (i1 ...il )

fˇl (t )| L 2 (Vs+ )

To estimate the second and third factors on the right in (8.378) we can apply the argument of Case 1) of Lemma 8.11, replacing [0, ε0] × S 2 by its closed subset Vs+ . By the bound (8.278) we then have: [1 + log(1 + t )] −1 ∂µ (t ) ≤ Cδ0 (8.379) max −2µ ∂t − (1 + t )2 Vs+ and, in view of the lower bound (8.277), µ(t) max ≤ C[1 + log(1 + t)] µ(t ) Vs+

(8.380)


265

Substituting the estimates (8.379) and (8.380) as well as the definition (8.369) in (8.378), and the result in (8.377) we obtain: t 1 [1 + log(1 + t )] (i1 ...il ) Pl (t )dt (i1 ...il ) M l (t) L 2 (Vs+ ) ≤ Cδ0 [1 + log(1 + t)]2 (1 + t ) 0 (8.381) hence, by (8.373),

(i1 ...il )

1

M l (t) L 2 (Vs+ ) ≤ Cδ0 [1 + log(1 + t)]2

t 0

(i1 ...il )

(0)

(1)

P l,a (t)+ P l,a (t) ·

[1 + log(1 + t )] −a µm (t )dt (1 + t )

(8.382)

To estimate the integral in (8.382) we follow the proof of Corollary 2 of Lemma 8.11. We have the two cases Eˆ s,m ≥ 0 and Eˆ s,m < 0 to consider. In the first case the lower bound (8.345) holds, hence if 1 δ0 a ≤ , C we have: 1 −a µm (t )−a ≤ 1 − ≤C (8.383) a and we obtain: t [1 + log(1 + t )] −a µm (t )dt ≤ C[1 + log(1 + t)]2 (8.384) (1 + t ) 0 In the second case we define, as before, δ1 > 0 by Eˆ s,m = −δ1 , and t1 according to (8.292). Then for t ≤ t1 the lower bound (8.296) holds, hence an estimate of the form (8.384) holds for all t ≤ t1 , in particular if we set t = t1 . It thus remains for us to estimate, when t > t1 , the integral: t [1 + log(1 + t )] −a µm (t )dt (8.385) (1 + t ) t1 Here we have t > t1 , so the lower bound (8.326) holds, under the smallness condition (8.315) on δ0 . Thus, the integral (8.385) is bounded by a constant multiple of the integral, in terms of the variable τ (see (8.306)–(8.308)), τ τ (1 − δ1 τ )1−a (1 + τ )(1 − δ1 τ )−a dτ ≤ (1 + τ ) (1 − δ1 τ )−a dτ ≤ (1 + τ ) · δ1 (a − 1) τ1 τ1 −1 1 =2 1− τ1 (1 + τ )(1 − δ1 τ )1−a (8.386) a The last equality is by virtue of the first equation of (8.308). By the upper bound (8.335) we then conclude that the last is bounded by a constant multiple of [1 + log(1 + t)]2 µ1−a m (t)


266

We conclude that, in general, t [1 + log(1 + t )] −a µm (t )dt ≤ C[1 + log(1 + t)]2 µ1−a m (t) (1 + t ) 0

(8.387)

which, substituted in (8.382), yields:

(i1 ...il )

1

M l (t) L 2 (Vs+ )

(i1 ...il )

≤ Cδ0 [1 + log(1 + t)]4

(0)

(i1 ...il )

P l,a (t) +

(8.388)

(1)

P l,a (t) µ1−a m (t)

Combining finally the estimates (8.376) and (8.388) (see (8.362)) and taking into account the assumption that the product aδ0 ≤ C −1 , we obtain:

(i1 ...il )

1

M l (t) L 2 ([0,ε0]×S 2 )

≤ Ca −1 (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)1/2

(i1 ...il )

(1)


2

(i1 ...il )

We turn to

(i1 ...il )

(8.389)

M l (t, u). We have:

2

M l (t)2L 2 ([0,ε0 ]×S 2 ) =

(i1 ...il )

2

M l (t)2L 2 (Vs− ) +

(i1 ...il )

2

M l (t)2L 2 (Vs+ ) (8.390)

and, in analogy with (8.363) and (8.377),

(i1 ...il )

2

M l (t) L 2 (Vs− ) ≤

and,

(i1 ...il )

t

i1 ...il )

N l (t, t ) L 2 (Vs− ) dt

i1 ...il )

N l (t, t ) L 2 (Vs+ ) dt

0

2

M l (t) L 2 (Vs+ ) ≤

t

2

0

(8.391)

2

(8.392)

where: (i1 ...il )

2

N l (t, t , u) =

µ(t, u) µ(t , u)

2

3

(1 − u + η0 t )

1 trχ(t , u) |d/ 2

(i1 ...il )

fˇl (t , u)| (8.393)

We have: (i1 ...il )

2

N l (t, t ) L 2 (Vs− ) µ(t) 2 1 trχ(t ≤ (1 + η0 t )3 max max ) |d/ µ(t ) 2 Vs− Vs−

(8.394) (i1 ...il )

fˇl (t )| L 2 (Vs− )


267

Now, by Corollary 1 to Lemma 8.11, (8.366) holds, while by bootstrap assumption F2, 1 trχ(t ) ≤ C(1 + t )−1 max (8.395) [0,ε0 ]×S 2 2 We thus obtain, substituting in (8.394),

2

(i1 ...il )

N l (t, t ) L 2 (Vs− ) ≤ C(1 + t )2 |d/

(i1 ...il )

fˇl (t )| L 2 ([0,ε0 ]×S 2 )

(8.396)

or, in terms of the definition (8.369),

2

(i1 ...il )

N l (t, t ) L 2 (Vs− ) ≤ C

P l,a (t) + (1 + t )−1/2

≤C

Pl (t )

(0)

(i1 ...il )

≤C

(i1 ...il )

(i1 ...il )

(0)

(i1 ...il )

−1/2 (i1 ...il )

P l,a (t) + (1 + t )

(1)

P l,a (t) µ−a m (t )

(1)


(8.397)

Here, in the last two steps we have appealed to (8.373) and to Corollary 2 of Lemma 8.11. Substituting the estimate (8.397) in (8.391) yields:

(i1 ...il )

2

M l (t) L 2 (Vs− )

≤ C (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)1/2

(1)

(i1 ...il )

(8.398)


In analogy with (8.394) we have: (i1 ...il )

2

N l (t, t ) L 2 (Vs+ ) µ(t) 2 1 trχ(t max ) |d/ ≤ (1 + η0 t )3 max µ(t ) 2 Vs+ Vs+

To estimate

µ(t) max ) µ(t Vs+

=

max

(u,ϑ)∈Vs+

(8.399) (i1 ...il )

µˆ s (t, u, ϑ) µˆ s (t , u, ϑ)

fˇl (t )| L 2 (Vs+ )

we appeal to Proposition 8.6 to obtain: 1 + Cδ0 + Eˆ s (u, ϑ) log(1 + t) µ(t) max (8.400) ≤ max µ(t ) Vs+ (u,ϑ)∈Vs+ 1 − Cδ0 + Eˆ s (u, ϑ) log(1 + t ) √ √ We have two cases to distinguish √ according as to whether t is < t or ≥ t. In the √ second case we have 1 + t ≥ 1 + t ≥ 1 + t, hence: 1 Eˆ s (u, ϑ) log(1 + t ) ≥ Eˆ s (u, ϑ) log(1 + t) 2

: ∀(u, ϑ) ∈ Vs+


268

and from (8.400) we obtain:

µ(t) max µ(t ) Vs+

: t ≥

≤2

√ t

(8.401)

provided that: 1 3 On the other hand, in the first case we have, in any case: µ(t) max ≤ C[1 + log(1 + t)] µ(t ) Vs+ Cδ0 ≤

: t
ε1 η0 (1 + t) ε

t 0

: for all t ∈ [0, s]

(12.59)

provided that δ0 is suitably small. Also, fixing any positive constant ε2 < 1, the estimate (12.46) implies that: sup y < 1 − ε2 ε

: for all t ∈ [0, s0 ]

(12.60)

t 0

provided that δ0 is suitably small. Let now s 0 be the maximal value of s0 ∈ [0, s] such that hypothesis H0 holds on s Wε00 . We shall show that in fact s 0 = s. For, suppose that s 0 < s. Since we can take s s0 = s 0 in Lemma 12.1, the estimate (12.46) holds on Wε00 , hence also inequality (12.60) holds for all t ∈ [0, s 0 ], in particular at t = s 0 . However, (12.60) being a strict inequality, it must, by continuity, hold in a suitably small interval [s 0 , s∗ ) ⊂ [s 0 , s]. This contradicts the maximality of s 0 . We conclude that hypothesis H0 holds on Wεs0 . From Lemma 12.1 with s0 = s and the estimates which follow we then obtain the following proposition. Q Proposition 12.3 Let assumptions E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 and max λi L ∞ ( ε0 ) ≤ Cδ0 0

i

0

674

Chapter 12. Recovery of the Acoustical Assumptions

Then if δ0 is suitably small, hypothesis H0 holds on Wεs0 . Moreover, we have, on Wεs0 : 1 + η0 t − u − Cδ0 u[1 + log(1 + t)] ≤ r ≤ 1 + η0 t + min {0, −u + Cδ0 u[1 + log(1 + t)]} y ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] and, for all t ∈ [0, s], the estimates: κ − 1 L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

◦

max λi L ∞ ( ε0 ) , max λi L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

i

and:

t

i

max y i L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

i

In the sequel we shall also make use of the following proposition, which is analogous to Proposition 12.2. Proposition 12.4 Consider a surface St,u for which the following hold: inf r > 0

St,u

sup y < 1 St,u

Then for every symmetric 2-covariant St,u tensorfield ϑ we have: 2

|ϑ| ≤ 1 − sup y 2

St,u

−2 inf r

−4 3

St,u

(ϑ(Ri , R j ))2

i, j =1

pointwise on St,u . Proof. We have: 3

  3 3 (ϑ(Ri , R j ))2 = (Ri )a (Ri )c  (R j )b (R j )d  ϑab ϑcd

i, j =1

i=1

(12.61)

j =1

We substitute from (12.49) for the first two factors on the right. In fact, noting that, from (12.40), (12.45), x b = r N b = r (y b − Tˆ b ), and Tˆ b ab = 0, we have: x b ab = r y b ab

(12.62)

3 (Ri )a (Ri )c = r 2 (δbd − y b y d )ab cd

(12.63)

(12.49) takes the form:

i=1


675

Substituting then from (12.63) for the first two factors on the right in (12.61), we obtain: 3

(ϑ(Ri , R j ))2 = r 4 (δa c − y a y c )(δb d − y b y d )ϑa b ϑc d

i, j =1

= r 4 ϑ2 − 2i y ϑ2 + (ϑ(y , y ))2

(12.64)

where i y ϑ is the St,u 1-form with rectangular components:

(i y ϑ)b = y a ϑa b

(12.65)

Consider the expression: ϑ2 − 2i y ϑ2 + (ϑ(y , y ))2 in the right-hand side of (12.64), at a given point. It is invariant under the orthogonal group. By a suitable proper orthogonal transformation of the rectangular coordinates we can arrange so that y 2 = y 3 = 0 at the given point. Then this expression becomes: 2 1 − (y 1 )2 (ϑ11 )2 + (ϑ22 )2 + (ϑ33 )2 + 2 1 − (y 1 )2 (ϑ12 )2 + (ϑ13 )2 + 2(ϑ23 )2 2 (ϑ11 )2 + (ϑ22 )2 + (ϑ33 )2 + 2(ϑ12 )2 + 2(ϑ13 )2 + (ϑ23 )2 ) ≥ 1 − (y 1 )2 2 = 1 − y 2 ϑ2 We conclude that:

2 ϑ2 − 2i y ϑ2 + (ϑ(y , y ))2 ≥ 1 − y 2 ϑ2

(12.66)

Thus, (12.64) implies: 3

2 (ϑ(Ri , R j ))2 ≥ r 4 1 − y 2 ϑ2

(12.67)

i, j =1 ε

Taking into account the fact that, since the induced acoustical metric h on t 0 dominates the Euclidean metric, we have: |ϑ|2 =

3

(h

−1 ac

a,b,c,d=1

) (h

−1 bd

) ϑab ϑcd ≤

3

(ϑab )2 = ϑ2

(12.68)

a,b=1

the proposition then follows. We now turn to the propagation equation (4.106) of Chapter 4 for the symmetric 2-covariant St,u tensorfield χ: ( P) (N) dH 1 ( ψ ⊗ (χ $ · d/σ ) + (χ $ · d/σ )⊗ ψ))− α − α L / L χ = χ · χ $ + eχ + ψ L 2 dσ (12.69)

676

Chapter 12. Recovery of the Acoustical Assumptions ( P)

Here, α is given by (4.72):

( P)

( P P)

( P N)

α = α + α

(12.70)

( P P)

where α is the principal part: ( P P) 1 dH α = ψ L ( ψ ⊗ d/(Lσ ) + d/(Lσ )⊗ ψ) − ψ L2 D / 2 σ − ( ψ⊗ ψ)(L)2 σ 2 dσ

(12.71)

( P N)

and α is the lower order part: ( P N) 1 d H −1 α = α ψ L ( ψ ⊗ (κ −1 ζ ) + (κ −1 ζ )⊗ ψ) + α −1 ψ L2 k/ + ( ψ⊗ ψ)e (Lσ ) 2 dσ (12.72) (N)

Also, in (12.69), α is the lower order term given by (4.85) and (4.94)–(4.96): (N)

( A) 1 (B) 1 (C) α = −F α − F2 α − F1 α 2 2

(12.73)

where: ( A)

α =ω /L ⊗ ω /L − ωL L ω /

(12.74)

(B)

α = (ψ L (d/σ ) − (Lσ ) ψ) ⊗ (ψ L (d/σ ) − (Lσ ) ψ)

(12.75)

(C)

α = (d/σ ) ⊗ (ψ L ω / L − ω L L ψ) + (ψ L ω / L − ω L L ψ) ⊗ (d/σ ) −(Lσ )(2ψ L ω /− ψ ⊗ ω /L − ω / L ⊗ ψ)

(12.76)

Taking account of the fact that L / L h/ = 2χ the propagation equation (12.69) is seen to be equivalent to the following propagation equation for the symmetric 2-covariant St,u tensorfield: η0 h/ 1 − u + η0 t 1 χ · a + χ ˜· a + b L / L χ = χ · χ $ + 2 1 where a is the T1 type St,u tensorfield: χ = χ −

a = eI + ψ L

dH (d/σ )$ ⊗ ψ dσ

(12.77)

(12.78)

and b is the symmetric 2-covariant St,u tensorfield: b=

( P) (N) 1 η0 dH η0 eh/ + ψL ( ψ ⊗ d/σ + d/σ ⊗ ψ) − α − α 1 − u + η0 t 2 1 − u + η0 t dσ (12.79)


677

Q QQ Lemma 12.2 Let assumptions E{2} , E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 ,

max λi L ∞ ( ε0 ) ≤ Cδ0 i

0

and:

0

χ L ∞ ( ε0 ) ≤ Cδ0 0

s Wε00

Moreover, let hypothesis H1 hold on for some s0 ∈ (0, s]. Then, provided that δ0 is suitably small, we have, for all t ∈ [0, s0 ]: χ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

Proof. The assumptions of the lemma contain those of Proposition 12.3. Therefore, under the present assumptions the conclusions of Proposition 12.3 hold. We shall first estimate the coefficients a and b in the propagation equation (12.77). The function e is given by equation (3.98): e=− Now

1 dH (Lσ ) − Fψ0 ω L L ψ L (ψ L + 2αψTˆ ) 2α 2 dσ

(12.80)

ω L L = L µ (Lψµ ) = Lψ0 + L i (Lψi )

is like ψ L − k = L µ ψµ − k = ψ0 − k + L i ψi , but with Lψ0 , Lψi , in the role of ψ0 − k, ψi , respectively. Thus, as assumption E{0} implies the second of the estimates (12.7), Q assumption E{0} implies: ω L L L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2

(12.81)

t

The estimates (12.7), the second of (12.9), together with (12.81), imply: e L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2

(12.82)

t

This estimate, together with the estimate (12.23), which now holds for all t ∈ [0, s], since by Proposition 12.3 hypothesis H0 holds on Wεs0 , and the estimates (12.19) and the second of (12.7), yields: a L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t


(12.83)

Using the same estimates we deduce that the first term on the right in (12.79) is bounded ε in L ∞ (t 0 ) by: Cδ0 (1 + t)−3 : for all t ∈ [0, s] (12.84) ε

while the second term on the right in (12.79) is bounded in L ∞ (t 0 ) by: Cδ02 (1 + t)−4


(12.85)

678

Chapter 12. Recovery of the Acoustical Assumptions (N)

Next we consider the term α . According to inequality (6.62) of Chapter 6: |ω /|2 ≤

3

|d/ψi |2

(12.86)

i=1

By hypothesis H0 and assumption E{1} this implies: ω / L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t


(12.87)

In view of the estimates (12.20), (12.81), (12.87), the second of (12.9), (12.23) (which holds for all t ∈ [0, s]), the second of (12.7), (12.19), we deduce from expressions (12.74)–(12.76) the estimates: ( A)

α L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 , t

(B)

α L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 , t

(C)

α L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 ,

(12.88)

t

for all t ∈ [0, s]. It follows that: (N)

α L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 t

( P)


(12.89)

( P N)

We turn to the term α . We consider first the lower order part α , given by (12.72). Let us recall the fact that k/ is given by (see (10.195)) ψ0 1 H ψ0 1 dH ψ⊗ ψ(Lσ ) − ( ψ ⊗ d/σ + d/σ ⊗ ψ) − ω / (12.90) k/ = 2α dσ 1 + ρH α 1 + ρH Using the estimate (12.87) together with the second of (12.9), (12.23) (which holds for all t ∈ [0, s]), and (12.19), we obtain: k/ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s].

(12.91)

This, together with the estimate (12.24), which now holds for all t ∈ [0, s] (since by Proposition 12.3 hypothesis H0 holds on Wεs0 ), the estimate (12.82), as well as the second of (12.9), the second of (12.7), and (12.19), then yield: ( P N)

α

L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 t


(12.92)

( P P)

QQ Q Finally, we consider the principal part α , given by (12.71). By assumptions E{0} , E{0} , E{0} , we have:

(L)2 σ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t


(12.93)


679

By virtue of hypothesis H0, we have, pointwise: |d/ Lσ | ≤ C(1 + t)−1 max |Ri Lσ |

: on Wεs0

(12.94)


(12.95)

i

Q hence, by assumptions E{1} , E{1} , we have:

d/ Lσ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

To estimate D / 2 σ we must appeal to hypothesis H1 as well as hypothesis H0. First, by hypothesis H1 applied to the St,u 1-form d/σ , we have, pointwise: / Ri d/σ | |D / 2 σ | ≤ C(1 + t)−1 max |L i

s

: on Wε00

Then, since L / Ri d/σ = d/ Ri σ , we have, by hypothesis H0, |L / Ri d/σ | ≤ C(1 + t)−1 max |R j Ri σ | j

: on Wεs0

Thus, combining, we obtain, pointwise: |D / 2 σ | ≤ C(1 + t)−2 max |R j Ri σ | i, j

s

: on Wε00

(12.96)

hence, by assumption E{2} : D / 2 σ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.97)

The estimates (12.93), (12.95), (12.97), together with the previous estimates, the second of (12.7) and (12.19), imply: ( P P)

α L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.98)

The estimates (12.84), (12.85), (12.89), (12.92) and (12.98), yield: b L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.99)

Let now ϑ, ϑ , be a pair of symmetric 2-covariant St,u tensorfields. Their pointwise inner product, with respect to the induced acoustical metric h/, is expressed relative to an arbitrary local frame field by: (ϑ, ϑ ) = (h/−1 ) AC (h/−1 ) B D ϑ AB ϑC D

(12.100)

In terms of the corresponding T11 type tensorfields ϑ $ , ϑ $ , ϑ$ = h /−1 · ϑ = ϑ · h/−1 , we have:

ϑ $ = h/−1 · ϑ = ϑ · h/−1 ,

(ϑ, ϑ $ ) = tr(ϑ $ · ϑ $ )

(12.101)

680


Thus:

/ L ϑ $ ) · ϑ $ ) + tr(ϑ $ · (L / L ϑ $ )) L(ϑ, ϑ ) = tr((L

Since

(12.102)

/−1 ) = −2χ $$ = −2h/−1 · χ · h/−1 L / L (h

we have:

L / L ϑ $ = (L / L ϑ)$ − 2χ $ · ϑ,

L / L ϑ $ = (L / L ϑ )$ − 2χ $ · ϑ

It follows that: / L ϑ) + (ϑ, L /L ϑ ) L(ϑ, ϑ ) = (ϑ , L −2tr(ϑ $ · χ $ · ϑ $ ) − 2tr(ϑ $ · χ $ · ϑ $ )

(12.103)

Taking in particular ϑ = ϑ we obtain: |ϑ|L|ϑ| =

1 L(ϑ, ϑ) = (ϑ, L / L ϑ) − 2tr(ϑ $ · χ $ · ϑ $ ) 2

Writing χ$ = we have: tr(ϑ $ · χ $ · ϑ $ ) =

(12.104)

η0 I + χ $ 1 − u + η0 t η0 |ϑ|2 + tr(ϑ $ · χ $ · ϑ $ ) 1 − u + η0 t

and (12.104) takes the form: |ϑ|L|ϑ| = (ϑ, L / L ϑ) −

2η0 |ϑ|2 + 2tr(ϑ $ · χ · ϑ $ ) 1 − u + η0 t

(12.105)

Now, the following inequality holds: |tr(ϑ $ · χ $ · ϑ $ )| ≤ |χ ||ϑ|2

(12.106)

To see this, we work in an orthonormal (relative to h/) frame (E 1 , E 2 ) of eigenvectors of χ at a point. Let λ1 , λ2 be the corresponding eigenvalues. We then have: tr(ϑ $ · χ $ · ϑ $ ) = λ1 (ϑ11 )2 + (λ1 + λ2 )(ϑ12 )2 + λ2 (ϑ22 )2 Since |ϑ|2 = (ϑ11 )2 + 2(ϑ12 )2 + (ϑ22 )2 it follows that:

|tr(ϑ $ · χ · ϑ $ )| ≤ max{|λ1 |, |λ2 |}|ϑ|2

In view of the fact that: max{|λ1 |, |λ2 |} ≤ the inequality (12.106) then follows.

(λ1 )2 + (λ2 )2 = |χ |


681

Taking into account the inequality (12.106) as well as the fact that |(ϑ, L / L ϑ)| ≤ |ϑ||L / L ϑ| we deduce from (12.105): / L ϑ| L (1 − u + η0 t)2 |ϑ| ≤ (1 − u + η0 t)2 2|χ ||ϑ| + |L

(12.107)

We now apply this to the case ϑ = χ . In view of the facts that:

|χ · χ $ | = |(χ $ )2 | = (λ1 )4 + (λ2 )4 ≤ (λ1 )2 + (λ2 )2 = |χ |2 , |χ · a| = |χ ˜· a| ≤ |a||χ | we then obtain the inequality: L (1 − u + η0 t)2 |χ | ≤ (1 − u + η0 t)2 (3|χ | + |a|)|χ | + |b|

(12.108)

For the sake of clarity, in the following argument we shall be specific about the numerical constants. Let P(t) be the property: P(t) :

χ L ∞ ( ε0 ) ≤ C0 δ0 (1 + t )−2 [1 + log(1 + t )] : for all t ∈ [0, t] t

Choosing C0 suitably large, we have, by the assumptions of the lemma on the initial conditions, χ L ∞ ( ε0 ) < C0 δ0 (12.109) 0

It follows by continuity that P(t) is true for sufficiently small positive t. Let then t0 be the least upper bound of the set of values of t ∈ [0, s0 ] for which P(t) holds. Then by t continuity P(t0 ) is true. Hence, in Wε00 , we have, in view of the estimate (12.83): 3|χ | + |a| ≤ (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)]

t

: in Wε00

(12.110)

Substituting in (12.108) and taking into account the estimate (12.99) we obtain the following inequality along the integral curves of L: L (1 − u + η0 t)2 |χ | ≤ (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)] (1 − u + η0 t)2 |χ | +Cδ0 (1 + t)−1

(12.111)

Consider a given integral curve of L. Setting, along the given integral curve: x(t) = (1 − u + η0 t)2 |χ |

(12.112)

the inequality (12.111) takes the form: dx ≤ fx +g dt

(12.113)

682


where: f (t) = (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)],

g(t) = Cδ0 (1 + t)−1

(12.114)

Integrating from t = 0 yields: x(t) ≤ e

!t 0

f (t )dt

t

x(0) +

e−

! t 0

f (t )dt

g(t )dt

(12.115)

0

Since

x(0) ≤ χ L ∞ ( ε0 )

(12.116)

0

while x(t) ≥

η 2 0

2

(1 + t)2 |χ |

(as u ≤ ε0 ≤ 1/2 and η0 < 1)

taking into account the facts that: t ∞ f (t )dt ≤ f (t )dt = 2(3C0 + C)δ0 , 0

0

t

(12.117)

g(t )dt = Cδ0 log(1 + t)

0

(12.118) we deduce from (12.115): η 2

0 ε (1 + t)2 |χ | ≤ e2(3C0+C)δ0 χ L ∞ ( ε0 ) + Cδ0 log(1 + t) : on t 0 (12.119) 0 2 ε

Taking the supremum on t 0 then yields: χ L ∞ ( ε0 ) ≤ t

2 η0

2

e2(3C0+C)δ0 χ L ∞ ( ε0 ) + Cδ0 log(1 + t) (1+t)−2 (12.120) 0

This holds for all t ∈ [0, t0 ]. Let us now fix C0 large enough so that: C 0 δ0 ≥

8 η02

χ L ∞ ( ε0 ) and C0 > 0

8 η02

C

(12.121)

(the first is possible by the assumptions of the lemma on the initial conditions). Then, provided that δ0 satisfies the smallness condition: δ0 ≤

log 2 2(3C0 + C)

(12.122)

the inequality (12.120) implies: χ L ∞ ( ε0 ) < C0 δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all t ∈ [0, t0 ]

(12.123)

It follows by continuity that P(t) is true for some t > t0 . This contradicts the definition of t0 , unless t0 = s0 . Consequently, the property P(s0 ) is true, in fact the strict inequality (12.123) holds for all t ∈ [0, s0 ], which implies the conclusion of the lemma.


683

Let us now investigate hypothesis H1. We consider L / Ri ξ for an arbitrary St,u 1-form ξ . The components of this in rectangular coordinates are given by: / Ri ξ )a + ξm (D / R i )m (L / Ri ξ )a = (D a, Hence, we have:

|L / R i ξ |2 =

i

i

+2

(D / Ri ξ )a = (Ri )m (D /ξ )ma

(12.124)

|D / R i ξ |2 −1 (h )ab (D / Ri ξ )a ξn (D / Ri )nb i

−1 + (h )ab ξm ξn (D / R i )m / Ri )nb a (D

(12.125)

i

Consider the first term on the right in (12.125). We have: −1 |D / R i ξ |2 = (h )ab (D / Ri ξ )a (D / R i ξ )b i

i

= (h

−1 ab

)

m n /ξ )ma (D (Ri ) (Ri ) (D /ξ )nb

(12.126)

i

Substituting from (12.63) and noting that: n m /ξ )ma (D /ξ )nb = (D /ξ )ca (D /ξ )db c d (D

we obtain:

|D / Ri ξ |2 = r 2 (h

−1 ab

) (δcd − y c y d )(D /ξ )ca (D /ξ )db

(12.127)

i

Consider the symmetric 2-covariant St,u tensorfield ϑ with rectangular components: ϑcd = (h ◦

◦

−1 ab

) (D /ξ )ca (D /ξ )db

◦

(12.128) ε

Let µ1 , µ2 , µ3 , be the eigenvalues of ϑ, considered as a tensorfield on t 0 , relative to the background Euclidean metric. Since ϑ is positive semi-definite, we have: ◦

µi ≥ 0 : i = 1, 2, 3 It follows that:

◦

max µi ≤ i

where

◦

◦

µi =tr ϑ

(12.129) (12.130)

i ◦

tr ϑ = δcd ϑcd

(12.131)

is the trace of ϑ with respect to the background Euclidean metric. The inequality (12.130) implies: ◦

y c y d ϑcd ≤ y 2 tr ϑ

(12.132)

684


hence, in reference to (12.127), (h

◦ ) (δcd − y c y d )(D /ξ )ca (D /ξ )db = (δcd − y c y d )ϑcd ≥ 1 − y 2 tr ϑ (12.133)

−1 ab

Recalling from Chapter 6 the fact that (see (6.58)): δcd = (h we have:

−1 cd

)

H ψc ψd 1 + ρH

+

◦

tr ϑ = δcd ϑcd ≥ (h

(12.134)

−1 cd

) ϑcd = trϑ.

(12.135)

Substituting in (12.133), assuming that y ≤ 1, and noting that: trϑ = (h

−1 cd

) (h

−1 ab

) (D /ξ )ca (D /ξ )db = |D /ξ |2

(12.136)

(12.127) is seen to imply:

|D / Ri ξ |2 ≥ r 2 (1 − y 2 )|D /ξ |2 .

(12.137)

i

Consider next the third term on the right in (12.125). Here, we must first obtain a suitable expression for D / Ri . Let (X A : A = 1, 2) be an arbitrary local frame field for St,u . We have (see Chapter 6): ◦

◦

/(D h / X A Ri , X B ) = h(D X A Ri , X B ) and D X A Ri = (D X A ) Ri +(D X A Ri ) hence: ◦

◦

/(D h / X A Ri , X B ) = h((D X A ) Ri , X B ) + h(D X A Ri , X B )

(12.138)

Now, according to the last of equations (6.25): ◦

h((D X A ) Ri , X B ) = −λi θ AB

(12.139)

while according to equation (6.34): ◦

◦

h(D X A Ri , X B ) = h mn X lA ilm X nB + H ψ B ( Ri )m (d/ A ψm ) (12.140) ◦ ◦ 1 dH 1 dH {(d/ A σ ) ψ B − (d/ B σ ) ψ A } (( Ri )m ψm ) ( Ri σ ) ψ A ψ B + + 2 dσ 2 dσ ◦

Expressing the factor Ri σ in the third term on the right in (12.140) as: ◦

Ri σ = Ri σ + κ −1 λi T σ


685

adding (12.139), (12.140), and recalling that (see (6.38)): −θ AB +

1 dH ψ A ψ B T σ = −θ AB = α −1 χ AB − k/AB 2κ dσ

we obtain:

◦

/(D h / X A Ri , X B ) = ((νi )b ) AB + ((τi )b ) AB where:

(12.141)

◦

((νi )b ) AB = h mn X lA ilm X nB

(12.142) ◦

are the components in the frame (X A : A = 1, 2) of the 2-covariant St,u tensorfields νi , whose rectangular components are given by: ◦

(νi )ln = ll nn h mn il m

(12.143)

((τi )b ) AB are the components in the frame (X A : A = 1, 2) of the 2-covariant St,u tensorfield: ◦

(12.144) (τi )b = λi (α −1 χ − k/ ) + H ( Ri )m (d/ψm )⊗ ψ ◦ 1 dH + (Ri σ ) ψ⊗ ψ + ((d/σ )⊗ ψ− ψ ⊗ (d/σ ))(( Ri )m ψm ) 2 dσ ◦

◦

Introducing the T11 type St,u tensorfields νi , τi , corresponding to (νi )b , (τi )b , respectively: ◦

◦

νi = (νi )b · h/−1

τi = (τi )b · h/−1

(12.145)

we conclude from (12.141) that D / Ri is given by: ◦

D / Ri =νi +τi

(12.146)

◦

Now, by (12.143), the rectangular components of νi are given by: ◦

◦

(νi )lk = (νi )ln (h Since

−1 nk

)

= ll nn il m h mn (h

nn h mn (h

−1 nk

)

= km

−1 nk

)

(12.147)

this reduces to, simply: ◦

(νi )lk = km ll il m

(12.148)

and the rectangular components of D / Ri are given by:

(D / Ri )lk = km ll il m + (τi )lk

(12.149)

686


Using the expression (12.149) we obtain, in view of (12.50): n a b (D / R i )m / Ri )nb = m a (D m n b (δa b δm n − δa n δb m ) i

+

n b a (τi )m (τi )nb m a n b ib n + m a ia m i

i

n + (τi )m a (τi )b

(12.150)

i

Consider then the third term on the right in (12.125). Substituting (12.150) we find: −1 −1 (h )ab ξm ξn (D / R i )m / Ri )nb = δa b (h )ab aa bb ξ 2 − |ξ |2 (12.151) a (D i

+2

−1 (h )ab (τi · ξ )a (vi )b + |τi · ξ |2 i

i

where the vi are the St,u 1-forms with rectangular components:

(vi )b = bb ib n ξn Now, since (h

−1 ab

) aa bb = (h

−1 a b

)

(12.152)

− Tˆ a Tˆ b

(12.153)

and, from equations (6.58) of Chapter 6, δa b = h a b − H ψa ψb we obtain: δa b (h

−1 ab

(12.154)

) aa bb = 2 − H | ψ|2

(12.155)

therefore, in view of (12.58) the sum of the first two terms on the right in (12.151) is not less than: (12.156) (1 − H | ψ|2 )ξ 2 The third term on the right in (12.151) is bounded in absolute value by: ) ) 2 2 |τi · ξ ||vi | ≤ 2 |τi ||ξ ||vi | ≤ 2|ξ | |τi | |vi |2 i

i

i

i

Using (12.50) we obtain: −1 |vi |2 = (h )ab aa bb ia m ib n ξm ξn i

i

= (h

−1 ab

) aa bb (δa b δm n − δa n δb m )ξm ξn

= δa b (h

−1 ab

) aa bb ξ 2 − |ξ |2

(12.157)


hence, by (12.155),

687

|vi |2 = (2 − H | ψ|2 )ξ 2 − |ξ |2

(12.158)

i

Now, by (12.134), ξ 2 = δab ξa ξb = (h

−1 ab

) ξa ξb +

H H (ψa ξa )(ψb ξb ) ≤ |ξ |2 + ρξ 2 1 + ρH 1 + ρH

hence: ξ 2 ≤ (1 + ρ H )|ξ |2

(12.159)

In view of (12.159), equation (12.158) implies: ) |vi |2 ≤ (1 + ρ H )|ξ |

(12.160)

i

By (12.157) and (12.160), the third term on the right in (12.151) is bounded in absolute value by: ) 2(1 + ρ H ) |τi |2 |ξ |2 (12.161) i

Since the fourth term on the right in (12.151) is non-negative, we conclude from (12.156), (12.161), that the third term on the right in (12.125) is not less than:   ) 2 1 − H | ψ| − 2(1 + ρ H ) |τi |2  |ξ |2 (12.162) i

provided that H |ψ|2 ≤ 1. Finally, we consider the second term on the right in (12.125): −1 −1 (h )ab (D / Ri ξ )a ξn (D / Ri )nb = 2 (h )ab (Ri )m (D /ξ )ma ξn (D / Ri )nb 2 i

(12.163)

i

From (12.48) and (12.149) we obtain, using (12.50), k n b n (Ri )m (D / Ri )nb = m m ikm x (n b ib n + (τi )b ) i

(12.164)

i

n b m k = (δkb δm n − δkn δm b )m m n b ib n + m x

=

n b b m n n m k m n n (b x ) − b (n x ) + m x

ikm (τi )nb

i

ikm (τi )nb

i

Consider the first term on the right in (12.164). We have, in view of (12.154),

bb x b = r bb N b = r δb c bb N c = r (h b c − H ψb ψc )bb N c

688


By (12.45) and the fact that h b c bb Tˆ c = 0, we have:

h b c bb N c = h b c bb y c Hence, substituting, we obtain:

bb x b = r bb (h b c y c − H ψ N ψb )

(12.165)

ψ N = N i ψi

(12.166)

where: Introducing the St,u -tangential vectorfield: /y = · y

(12.167)

and noting that bb h b c = bc h b b , (12.165) takes the form:

bb x b = r (h b b y/b − H ψ N ψb )

(12.168)

Consider next the second term on the right in (12.164). Here we have:

nn x n = r nn N n = r nn y n or, simply:

nn x n = r y/n

(12.169)

In view of (12.168), (12.169), equation (12.164) reduces to: n k (Ri )m (D / Ri )nb = r m /b − H ψ N ψb ) − m /n + m ikm (τi )nb b y n n (h b b y m N i

i

(12.170) Hence, from (12.163), the second term on the right in (12.125) is given by: −1 ab /ξ )(ξ · y/ ) + (h ) (ηi )a (τi · ξ )b 2r ζn ξn − tr(D

(12.171)

i

where ζ is the St,u 1-form with rectangular components: ζn = (D /ξ )n a (y/a − H ψ N ( ψ$ )a )

(12.172)

and the ηi are the St,u 1-forms with rectangular components: (ηi )a = N k ikm (D /ξ )m a Now, by (12.134), for any pair of St,u 1-forms ζ , ξ we have: ζn ξn = δm n ζm ξn = (h

−1 m n

)

ζm ξn +

H ψm ψn ζm ξn 1 + ρH

(12.173)


689

hence, denoting: ψ · ζ = ψm ζm , we have: |ζn ξn | ≤ |ζ ||ξ | +

ψ · ξ = ψm ξm H |ψ · ζ ||ψ · ξ | 1 + ρH

By (12.159): (ψ · ζ )2 ≤ ρζ 2 ≤ (1 + ρ H )|ζ |2, (ψ · ξ )2 ≤ ρξ 2 ≤ (1 + ρ H )|ξ |2, hence: |ψ · ζ ||ψ · ξ | ≤ (1 + ρ H )|ζ ||ξ |, and substituting above we obtain, for any pair of St,u 1-forms ζ , ξ : |ζn ξn | ≤ (1 + ρ H )|ζ ||ξ |

(12.174)

In the present case, where ζ is given by (12.172), recalling that (see (12.18)): √ |ψ N |, | ψ| ≤ ρ we have:

|ζ | ≤ |D /ξ |(|y/ | + ρ H )

(12.175)

hence by (12.174) the first term in parenthesis in (12.171) is bounded in absolute value by: (12.176) (1 + ρ H )|D /ξ ||ξ |(|y/| + ρ H ) The second term in parenthesis in (12.171) is bounded in absolute value by: √ 2|D /ξ ||ξ ||y/ |

(12.177)

where we have used the fact that for any 2-covariant St,u tensorfield M we have: 1 (trM)2 ≤ |M|2 2 Finally, the third term in parenthesis in (12.171) is bounded in absolute value by: ) ) 2 |ηi ||τi ||ξ | ≤ |ξ | |τi | |ηi |2 (12.178) i

i

i

Moreover, using (12.50) we obtain: −1 |ηi |2 = (h )ab N k N l ikm iln (D /ξ )m a (D /ξ )n b i

i

= (h

−1 ab

) N k N l (δkl δm n − δkn δlm )(D /ξ )m a (D /ξ )n b

= (h

−1 ab

≤ (h

−1 ab

) ((D /ξ )n a (D /ξ )n b − N m N n (D /ξ )m a (D /ξ )n b ) ) (D /ξ )n a (D /ξ )n b ≤ (1 + ρ H )|D /ξ |2

(12.179)

690


The last inequality is proved as follows. By (12.134) we have: (h

−1 ab

) (D /ξ )n a (D /ξ )n b = |D /ξ |2 +

H |ψ · D /ξ |2 1 + ρH

where ψ · D /ξ is the St,u 1-form with rectangular components: /ξ )n a (ψ · D /ξ )a = ψn (D By (12.128), (12.130): |ψ · D /ξ |2 = (h

◦

−1 ab

) ψm ψn (D /ξ )m a (D /ξ )n b = ψm ψn ϑm n ≤ ρ tr ϑ

Now, by (12.134), (12.135), denoting ψm ψn ϑm n = ψ · ϑ · ψ, ◦

tr ϑ = trϑ +

H ρH ◦ tr ϑ ψ · ϑ · ψ ≤ trϑ + 1 + ρH 1 + ρH

hence:

◦

tr ϑ ≤ (1 + ρ H )trϑ

(12.180)

In view of (12.136) we then conclude that: /ξ |2 |ψ · D /ξ |2 ≤ ρ(1 + ρ H )|D Substituting this above yields: (h

−1 ab

) (D /ξ )n a (D /ξ )n b ≤ (1 + ρ H )|D /ξ |2

(12.181)

which is the last inequality in (12.179). Combining the above results we conclude that the second term on the right in (12.125) is bounded in absolute value by:   )   √

|τi |2 |D /ξ ||ξ | (12.182) 2r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H   i

Combining the results (12.137), (12.182), (12.162), for the three terms on the righthand side of (12.125), yields the following proposition. Proposition 12.5 Consider a surface St,u for which the following hold: sup y ≤ 1

sup H | ψ|2 ≤ 1

St,u

St,u

Then for every St,u 1-form ξ we have:



|L / Ri ξ |2 ≥ r 2 (1 − y 2 )|D /ξ |2 + 1 − H | ψ|2 − 2(1 + ρ H )

i

− 2r pointwise on St,u .

  

)

 |τi |2  |ξ |2

i

√

(1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H

) i

|τi |2

  

|D /ξ ||ξ |


691

As a corollary of Proposition 12.5 we have the following. Corollary 12.5.a Suppose that: sup |ψ0 − k|, sup ε t 0

ε t 0

sup y ≤ Cδ0 ε

and

t 0

√ ρ ≤ Cδ0

sup

)

ε

t 0

|τi |2 ≤ Cδ0

i

for some fixed positive constant C. Then if δ0 is suitably small, for any St,u 1-form ξ , ε differentiable on the St,u , we have, pointwise on t 0 :

|L / R i ξ |2 ≥

i

1 2 r |D /ξ |2 + |ξ |2 2

If also: inf r ≥ ε1 η0 (1 + t) ε

t 0

ε

(see Corollary 12.2.a) then hypothesis H1 holds on t 0 . In fact, for any St,u 1-form ξ , ε differentiable on the St,u , we have, pointwise on t 0 : |L / R i ξ |2 |D /ξ |2 ≤ C(1 + t)−2 i

where C is the constant: C=

2 ε12 η02

Proof. The assumptions imply: 1 y ≤ , 2

ρ H + 2(1 + ρ H )

)

|τi |2 ≤

i

1 4

and: (1 + ρ H )(ρ H +

1 + ρ H y ) +

2(1 + ρ H )y +

1 + ρH

) i

|τi |2 ≤

1 4

if δ0 is suitably small. In view of (12.18), the first two imply that the coefficients of /ξ |2 , |ξ |2 in the inequality of Proposition 12.5 are not less than 3/4, while since r 2 |D

|y/ | ≤ |y | ≤ 1 + ρ H y ε

(for any vector V tangent to t 0 we have: |V |2 ≤ (1 + ρ H )V 2 ), the coefficient of −2r |D /ξ ||ξ | in the same inequality is not greater than 1/4. The first statement of the

692


corollary then follows (for, 2r |D /ξ ||ξ | is not greater than r 2 |D /ξ |2 + |ξ |2 ), and the second statement is an immediate consequence. Let us now return to Lemma 12.2 and let s 0 be the maximal value of s0 ∈ [0, s] s such that hypothesis H1 holds on Wε00 . We shall show that in fact s 0 = s. For, suppose that s 0 < s. Now the first assumption of Corollary 12.5.a holds for all t ∈ [0, s] by virtue of assumption E{0} , while the second and fourth likewise hold for all t ∈ [0, s] by virtue of Proposition 12.3. Also, from the definition (12.144), the estimate for λi of Proposition 12.3, the estimate (12.91), hypothesis H0, assumption E{1} , and the bound (12.28), we obtain: max sup |(τi )b − λi α −1 χ| ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)] i

ε

t 0


(12.183)

ε

Thus, if for some t ∈ [0, s], χ satisfies on t 0 the estimate: χ L ∞ ( ε0 ) ≤ C0 (1 + t)−1

(12.184)

t

for some fixed positive constant C0 , then we have: max τi L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

i

(12.185)

ε

so on t 0 the third assumption of Corollary 12.5.a holds as well, hence hypothesis H1 ε holds on t 0 . On the other hand, taking s0 = s 0 in Lemma 12.2, yields the conclusion that the estimate (12.186) χ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

holds for all t ∈ [0, s 0 ], in particular at t = s 0 . But, if δ0 is suitably small this implies that: 1 χ L ∞ ( ε0 ) ≤ C0 (1 + t)−1 : at t = s 0 t 2 therefore by continuity (12.184) must hold in a suitably small interval [s 0 , s∗ ] ⊂ [s 0 , s]. But then all assumptions of Corollary 12.5.a hold for all t ∈ [0, s∗ ], hence hypothesis H1 holds on Wεs0∗ , contradicting the maximality of s 0 . We conclude that hypothesis H1 holds on Wεs0 . From Lemma 12.2 with s0 = s we then obtain the following proposition. Q QQ Proposition 12.6 Let assumptions E{2} , E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 ,

max λi L ∞ ( ε0 ) ≤ Cδ0 i

0

and:

0

χ L ∞ ( ε0 ) ≤ Cδ0 0

Then, if δ0 is suitably small, hypothesis H1 holds on Wεs0 . Moreover, we have, for all t ∈ [0, s]: χ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t


693

In view of the fact that the assumptions of Proposition 12.6 contain those of Proposition 12.3, it follows that under the assumptions of Proposition 12.6, the assumptions of Proposition 10.1 hold for l = 1. Therefore the conclusions of Proposition 10.1 and all its corollaries hold for l = 1. / Ri ϑ for an arbitrary Let us now investigate hypotheses H2 and H2 . We consider L symmetric 2-covariant St,u tensorfield ϑ. The components of this in rectangular coordinates are given by: (L / Ri ϑ)ab = (D / Ri ϑ)ab + ϑmb (D / R i )m / R i )m a + ϑam (D b, m /ϑ)mab (D / Ri ϑ)ab = (Ri ) (D Hence, we have: |L / Ri ϑ|2 = |D / Ri ϑ|2 i

+2

(h

i −1 ac

) (h

(12.187)

(12.188)

−1 bd

) (ϑmb (D / R i )m / R i )m / Ri ϑ)cd a + ϑam (D b )(D

i

−1 −1 + (h )ac (h )bd (ϑmb (D / R i )m / R i )m / Ri )nc + ϑcn (D / Ri )nd ) a + ϑam (D b )(ϑnd (D i

Consider the first term on the right in (12.188). We have: −1 −1 |D / Ri ϑ|2 = (h )ac (h )bd (D / Ri ϑ)ab (D / Ri ϑ)cd i

i

= (h

−1 ac

) (h

−1 bd

)

(12.189)

(Ri ) (Ri ) m

n

/ϑ)ncd (D /ϑ)mab (D

i

Substituting from (12.63) and noting that: n m /ϑ)mab (D /ϑ)ncd = (D /ϑ)kab (D /ϑ)lcd k l (D

we obtain:

|D / Ri ϑ|2 = r 2 (h

−1 ac

) (h

−1 bd

) (δkl − y k y l )(D /ϑ)kab (D /ϑ)lcd

(12.190)

i

Consider the symmetric 2-covariant St,u tensorfield ϕ with rectangular components: ϕkl = (h

−1 ac

) (h

−1 bd

) (D /ϑ)kab (D /ϑ)lcd

(12.191)

Since ϕ is positive semi-definite, we have, as in (12.132), (12.135), ◦

y k y l ϕkl ≤ y 2 tr ϕ and

◦

tr ϕ ≥ trϕ

(12.192)

Assuming that y ≤ 1 and noting that: trϕ = (h

−1 kl

) (h

−1 ac

) (h

−1 bd

) (D /ϑ)kab (D /ϑ)lcd = |D /ϑ|2

(12.193)

694


(12.190) is then seen to imply: |D / Ri ϑ|2 ≥ r 2 (1 − y 2 )|D /ϑ|2

(12.194)

i

Consider next the third term on the right in (12.188). This is the sum of two terms: 2III1 + 2III2

where: III1 = (h

−1 ac

) (h

−1 bd

) ϑmb ϑnd

(D / R i )m / Ri )nc a (D i

III2 = (h

−1 ac

) (h

−1 bd

) ϑam ϑnd

(12.195)

(D / R i )m / Ri )nc b (D

(12.196)

i

Substituting (12.150) in (12.195) we find: III1 = (h

−1 ac

−1

) (h )bd ϑm b ϑn d (δa c aa cc δm n − na m c ) −1 −1 (h )ac (h )bd (wi )ab (τi · ϑ)cd + |τi · ϑ|2 +2 i

(12.197)

i

where the wi are the 2-covariant St,u tensorfields with rectangular components:

(wi )ab = aa ia m ϑm b

(12.198)

and: (τi · ϑ)ab = (τi )m a ϑmb Substituting from (12.155) in the first term on the right in (12.197) and in view of (12.68), we obtain that this term is (2 − H | ψ|2 )δm n (h

−1 bd

) ϑm b ϑn d − |ϑ|2 ≥ (1 − H | ψ|2 )|ϑ|2

(12.199)

provided that H | ψ|2 ≤ 1. The second term on the right in (12.197) is bounded in absolute value by: ) ) 2 2 |wi ||τi · ϑ| ≤ 2|ϑ| |wi | |τi |2 (12.200) i

i

i

Using (12.50) we obtain: −1 −1 |wi |2 = (h )ac (h )bd aa cc ia m ic n ϑm b ϑn d i

i

= (h = (h

−1 ac

) (h

−1 bd

−1 bd

) aa cc (δa c δm n − δa n δc m )ϑm b ϑn d

) δa c (h

−1 ac

) aa cc δm n ϑm b ϑn d − |ϑ|2


695

hence, by (12.155), −1 |wi |2 = (2 − H | ψ|2 )δm n (h )bd ϑm b ϑn d − |ϑ|2

(12.201)

i

Let M be the symmetric 2-covariant St,u tensorfield with rectangular components: Mm n = (h We have: δm n (h

−1 bd

) ϑm b ϑn d ◦

−1 bd

) ϑm b ϑn d =tr M,

(12.202)

|ϑ|2 = trM

(12.203)

Now M is positive semi-definite, therefore the analogue of (12.130) holds for M. It follows that (see (12.132)): ◦

ψ · M · ψ = ψm ψn Mm n ≤ ρ tr M

(12.204)

It then follows by (12.134) that: ◦

tr M = trM + hence:

ρH ◦ H ψ · M · ψ ≤ trM + tr M 1 + ρH 1 + ρH ◦

tr M ≤ (1 + ρ H )trM

(12.205)

In view of (12.203) and (12.205), (12.201) implies: ) |wi |2 ≤ (1 + ρ H )|ϑ|

(12.206)

i

By (12.200) and (12.206), the second term on the right in (12.197) is bounded in absolute value by: ) |τi |2 |ϑ|2 (12.207) 2(1 + ρ H ) i

We conclude from (12.199), (12.207), that:   ) III1 ≥ 1 − H | ψ|2 − 2(1 + ρ H ) |τi |2  |ϑ|2 + |τi · ϑ|2 i

(12.208)

i

provided that H | ψ|2 ≤ 1. Substituting (12.150) in (12.196) we find: III2 = (h

−1 ac

−1

) (h )bd ϑam ϑn d (δb c δm n bb cc − nb m (12.209) c ) −1 −1 −1 −1 +2 (h )ac (h )bd (τi ˜· ϑ)ab (wi )cd + (h )ac (h )bd (τi ˜· ϑ)ab (τi · ϑ)cd i

i

696


where:

(τi ˜· ϑ)ab = (τi · ϑ)ba

Since (h

−1 ac

) cc = aa (h

−1 a c

)

,

(h

−1 bd

) bb = dd (h

−1 d b

)

we have, in view of (12.154), δb c (h

−1 ac

) (h

−1 bd

) bb cc = δb c (h

−1 a c

)

(h

−1 d b

)

= (h b c − H ψb ψc )(h = (h

−1 a d

)

aa dd

−1 a c

)

aa dd − H (h

(h

−1 d b

−1 d b

)

)

aa dd

ψb (h

−1 a c

)

ψc aa dd

Consequently, the first term on the right in (12.209) is given by: δm n ((h

−1 a d

)

◦

ϑm a ϑn d − H ξm ξn ) − (trϑ)2 =tr M − H ξ 2 − (trϑ)2

where ξ is the St,u 1-form with rectangular components: ξm = ψb (h

−1 b d

)

ϑd m

(12.210)

and M is as in (12.202). Since ◦

tr M ≥ trM = |ϑ|2 and (see (12.159)): ξ 2 ≤ (1 + ρ H )|ξ |2 ≤ (1 + ρ H )ρ|ϑ|2 ,

(12.211)

it follows that the first term on the right in (12.209) is not less than:

Since:

(1 − ρ H (1 + ρ H ))|ϑ|2 − (trϑ)2

(12.212)

|τi ˜· ϑ| = |τi · ϑ|

(12.213)

the second term on the right in (12.209) is bounded in absolute value in the same way as the second term on the right in (12.197) (see (12.207)). Finally, again by (12.213), the third term on the right in (12.209) is bounded in absolute value by: |τi · ϑ|2 (12.214) i

We conclude that:  III2 ≥ 1 − ρ H (1 + ρ H ) − 2(1 + ρ H )

) i

 |τi |2  |ϑ|2 − (trϑ)2 −

|τi · ϑ|2

i

(12.215)


697

Combining (12.208) and (12.215) yields:  III1 + III2 ≥ 2 − H | ψ|2 − ρ H (1 + ρ H ) − 4(1 + ρ H )

)

 |τi |2  |ϑ|2 − (trϑ)2

i

(12.216) Consider finally the second term on the right in (12.188). This is: 4II where: II = (h

−1 ac

) (h

−1 bd

) ϑmb

k (D /ϑ)kcd (D / i R)m a (Ri )

(12.217)

i

Substituting from (12.170) we obtain: −1 −1 /ϑ)d II = r (h )bd ϑm b ιm d − (h )bd (ϑ · y/ )b (trD +

−1 −1 (h )ac (h )bd (τi · ϑ)ab (i )cd

(12.218)

i

Here ι is the 2-covariant St,u tensorfield with rectangular components: /ϑm cd (y/c − H ψ N ( ψ$ )c ) ιm d = (D

(12.219)

and the i are the symmetric 2-covariant St,u tensorfields with rectangular components: /ϑ)k cd (i )cd = N l ilk (D

(12.220)

Also, in (12.218), (trD /ϑ)d = (h

−1 ac

) (D /ϑ)acd ,

(ϑ · y/ )b = ϑbm y/m

Now, by (12.134) we have: −1 −1 bd −1 bd (h )m n + (h ) ϑm b ιm d = (h ) ≤ |ϑ||ι| +

H ψm ψn ϑm b ιn d 1 + ρH

H |ψ · ϑ||ψ · ι| 1 + ρH

(12.221)

where ψ · ϑ, ψ · ι, are the St,u 1-forms with rectangular components: (ψ · ϑ)a = ψm ϑm a ,

(ψ · ι)a = ψm ιm a

By (12.202)–(12.205): |ψ · ϑ|2 = (h

−1 ab

) ψm ϑm a ψn ϑn b = ψm ψn Mm n ◦

= ψ · M · ψ ≤ ρ tr M ≤ ρ(1 + ρ H )trM = ρ(1 + ρ H )|ϑ|2

(12.222)

698


Defining also the symmetric 2-covariant St,u tensorfield K with rectangular components: K m n = (h we have: |ψ · ι|2 = (h

−1 ab

) ιm a ιm b

(12.223)

−1 ab

) ψm ιm a ψn ιn b = ψm ψn K m n

hence, by virtue of the fact that K is positive semi-definite: ◦

|ψ · ι|2 = ψ · K · ψ ≤ ρ tr K ≤ ρ(1 + ρ H )trK = ρ(1 + ρ H )|ι|2

(12.224)

From (12.222), (12.224) we obtain: |ψ · ϑ||ψ · ι| ≤ ρ(1 + ρ H )|ϑ||ι|

(12.225)

Substituting this in (12.221) we obtain: −1 bd (h ) ϑm b ιm d ≤ (1 + ρ H )|ϑ||ι| Since also:

(12.226)

|ι| ≤ |D /ϑ|(|y/ | + ρ H )

(12.227)

we conclude that the first term in parenthesis in (12.218) is bounded in absolute value by: (1 + ρ H )|ϑ||D /ϑ|(|y/| + ρ H )

(12.228)

Next, since: 1 |trD /ϑ|2 ≤ |D /ϑ|2 2 the second term in parenthesis in (12.218) is bounded in absolute value by: √ 2|y/ ||ϑ||D /ϑ|

(12.229)

Finally the third term in parenthesis in (12.218) is bounded in absolute value by: ) ) 2 |ϑ||τi ||i | ≤ |ϑ| |τi | |i |2 (12.230) i

i

i

Moreover, using (12.50) we obtain: −1 −1 |i |2 = (h )ac (h )bd N k N l ikm iln (D /ϑ)m ab (D /ϑ)n cd i

i

= (h = (h ≤ (h

−1 ac

−1 bd

−1 ac

−1 bd

−1 ac

−1 bd

) (h ) (h ) (h

) N k N l (δkl δm n − δkn δlm )(D /ϑ)m ab (D /ϑ)n cd

) ((D /ϑ)m ab (D /ϑ)m cd − N m N n (D /ϑ)m ab (D /ϑ)n cd ) ) (D /ϑ)m ab (D /ϑ)m cd ≤ (1 + ρ H )|D /ϑ|2

The last inequality is proved as follows. By (12.134) we have: (h

−1 ac

) (h

−1 bd

) (D /ϑ)m ab (D /ϑ)m cd = |D /ϑ|2 +

H |ψ · D /ϑ|2 1 + ρH

(12.231)


699

where ψ · D /ϑ is the symmetric 2-covariant St,u tensorfield with rectangular components: (ψ · D /ϑ)ab = ψm (D /ϑ)m ab Recalling the definition (12.191), |ψ · D /ϑ|2 = (h

−1 ac

) (h

◦

−1 bd

) ψm ψn (D /ϑ)m ab (D /ϑ)n cd = ψm ψn ϕm n ≤ ρ tr ϕ

by the analogue of (12.130) for ϕ, which holds by virtue of the fact that ϕ is positive semi-definite. By (12.134), (12.135), denoting ψm ψn ϕm n = ψ · ϕ · ψ, ◦

tr ϕ = trϕ +

ρH ◦ H ψ · ϕ · ψ ≤ trϕ + tr ϕ 1 + ρH 1 + ρH

hence:

◦

tr ϕ ≤ (1 + ρ H )trϕ

(12.232)

In view of (12.193) we then conclude that: |ψ · D /ϑ|2 ≤ ρ(1 + ρ H )|D /ϑ|2 Substituting this above yields: (h

−1 ac

) (h

−1 bd

) (D /ϑ)m ab (D /ϑ)m cd ≤ (1 + ρ H )|D /ϑ|2

(12.233)

which is the last inequality in (12.231). Combining the above results we conclude that:   )  

√ |II| ≤ r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H |τi |2 |D /ϑ||ϑ| (12.234)   i

Combining the results (12.194), (12.234), (12.216), for the three terms on the right in (12.188), yields the following proposition. Proposition 12.7 Consider a surface St,u for which the following hold: sup y ≤ 1

sup H | ψ|2 ≤ 1

St,u

St,u

Then for every symmetric 2-covariant St,u tensorfield ϑ we have: |L / Ri ϑ|2 ≥ r 2 (1 − y 2 )|D /ϑ|2   i ) + 2 2 − H | ψ|2 − ρ H (1 + ρ H ) − 4(1 + ρ H ) |τi |2  |ϑ|2 − 2(trϑ)2 i

 ) 

√ − 4r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H |τi |2 |D /ϑ||ϑ|    

i

pointwise on St,u .

700


As a corollary of Proposition 12.7 we have the following. Corollary 12.7.a Suppose that: sup |ψ0 − k|, sup ε t 0

ε t 0

sup y ≤ Cδ0 ε

and

t 0

√ ρ ≤ Cδ0

sup

)

ε

t 0

|τi |2 ≤ Cδ0

i

for some fixed positive constant C. Then if δ0 is suitably small, for any symmetric 2ε covariant St,u tensorfield ϑ, differentiable on the St,u , we have, pointwise on t 0 : i

1 |L / Ri ϑ|2 ≥ r 2 |D /ϑ|2 + 2|ϑ|2 − 2(trϑ)2 2

If also: inf r ≥ ε1 η0 (1 + t) ε

t 0

ε

(see Corollary 12.2.a) then hypotheses H2 and H2 hold on t 0 . In fact, for any symmetric ε 2-covariant St,u tensorfield ϑ, differentiable on the St,u , we have, pointwise on t 0 : 2 −2 2 2 |D /ϑ| ≤ C(1 + t) |L / Ri ϑ| + 2|ϑ| i

where C is the constant: C=

2 ε12 η02

and if the tensorfield ϑ is trace-free, we have the stronger inequality: |L / Ri ϑ|2 |D /ϑ|2 ≤ C(1 + t)−2 i

Proof. The assumptions imply: 1 y ≤ , 2

ρ H (2 + ρ H ) + 4(1 + ρ H )

)

|τi |2 ≤

i

and: (1 + ρ H )(ρ H +

1 + ρ H y ) +

2(1 + ρ H )y +

1 + ρH

1 2

) i

|τi |2 ≤

1 4

if δ0 is suitably small. In view of (12.18), the first two imply that the coefficients of r 2 |D /ϑ|2 and |ϑ|2 in the inequality of Proposition 12.7 are not less than 3/4 and 3 respectively, while the coefficient of −2r |D /ϑ||θ | in the same inequality is not greater than


701

1/2. Since 2r |D /ϑ||ϑ| is not greater than (1/2)r 2|D /ϑ|2 + 2|ϑ|2 , the first statement of the corollary then follows. The trace-free case of the second statement is an immediate consequence, while the general case follows noting that (trϑ)2 ≤ 2|ϑ|2 . We remark that the assumptions of Corollary 12.7.a coincide with the assumptions of Corollary 12.5.a. Having shown that the assumptions of Corollary 12.5.a follow from those of Proposition 12.6, we conclude that under the assumptions of Proposition 12.6 hypotheses H2 and H2 hold on Wεs0 . In the sequel we shall make use of the following proposition, which is based on Propositions 12.2 and 12.4. Proposition 12.8 Let the assumptions of Proposition 12.6 hold and let X be an arbitrary St,u -tangential vectorfield defined on Wεs0 and differentiable on the St,u . Then, for any St,u 1-form ξ we have, pointwise: −1 |L / X ξ | ≤ C(1 + t) / Ri ξ | + |ξ | + |ξ | max |L / Ri X| |X| max |L i

i

Also, for any symmetric 2-covariant St,u tensorfield ϑ we have, pointwise: −1 |X| max |L |L / X ϑ| ≤ C(1 + t) / Ri ϑ| + |ϑ| + |ϑ| max |L / Ri X| i

i

Proof. Consider first the case of a St,u 1-form ξ . By Proposition 12.2 it holds, pointwise, that: |(L / X ξ )(Ri )| (12.235) |L / X ξ | ≤ C(1 + t)−1 i

Now, we have: (L / X ξ )(Ri ) = X (ξ(Ri )) − ξ([X, Ri ]) = X · d/(ξ(Ri )) + ξ(L / Ri X)

(12.236)

hence: |(L / X ξ )(Ri )| ≤ |X||d/(ξ(Ri ))| + |ξ ||L / Ri X|

(12.237)

By hypothesis H0: |d/(ξ(Ri ))| ≤ C(1 + t)−1

|R j (ξ(Ri ))|

(12.238)

R j (ξ(Ri )) = (L / R j ξ )(Ri ) + ξ([R j , Ri ])

(12.239)

/ R j ξ ||Ri | + |ξ ||[R j , Ri ]| |R j (ξ(Ri ))| ≤ |L

(12.240)

j

and we have: hence: Now, by Corollary 10.1.e with l = 1: |Ri |, |[R j , Ri ]| ≤ C(1 + η0 t)

(12.241)

702


It follows that:

/ R j ξ | + |ξ | |R j (ξ(Ri ))| ≤ C(1 + t) |L

(12.242)

Substituting this in (12.238) we obtain: |d/(ξ(Ri ))| ≤ C max |L / R j ξ | + |ξ |

(12.243)

j

Substituting (12.243) in turn in (12.237) we obtain: |(L / X ξ )(Ri )| ≤ C |X| max |L / R j ξ | + |ξ | + |ξ ||L / Ri X| j

(12.244)

Substituting finally (12.244) in (12.235) yields the statement of the proposition in regard to St,u 1-forms. Consider next the case of a symmetric 2-covariant St,u tensorfield ϑ. By Proposition 12.4 it holds, pointwise, that: |(L / X ϑ)(Ri , R j )| (12.245) |L / X ϑ| ≤ C(1 + t)−2 i, j

Now, we have: (L / X ϑ)(Ri , R j ) = X (ϑ(Ri , R j )) − ϑ([X, Ri ], R j ) − ϑ(Ri , [X, R j ]) = X · d/(ϑ(Ri , R j )) + ϑ(L / Ri X, R j ) + ϑ(Ri , L / R j X)

(12.246)

hence: / Ri X| + |Ri ||L / R j X| |(L / X ϑ)(Ri , R j )| ≤ |X||d/(ϑ(Ri , R j ))| + |ϑ| |R j ||L / Rk X| (12.247) ≤ |X||d/(ϑ(Ri , R j ))| + C(1 + t)|ϑ| max |L k

in view of the first of the bounds (12.241). By hypothesis H0: |d/(ϑ(Ri , R j ))| ≤ C(1 + t)−1 |Rk (ϑ(Ri , R j ))|

(12.248)

k

and we have: / Rk ϑ)(Ri , R j ) + ϑ([Rk , Ri ], R j ) + ϑ(Ri , [Rk , R j ]) Rk (ϑ(Ri , R j )) = (L

(12.249)

hence, in view of the bounds (12.241): |Rk (ϑ(Ri , R j ))| ≤ C(1 + t)2 |L / Rk ϑ| + |ϑ|

(12.250)

Substituting this in (12.248) we obtain: / Rk ϑ| + |ϑ| |d/(ϑ(Ri , R j ))| ≤ C(1 + t) max |L k

(12.251)


703

Substituting (12.251) in turn in (12.247) we obtain: |(L / X ϑ)(Ri , R j )| ≤ C |X| max |L / Rk ϑ| + |ϑ| + |ϑ| max |L / Rk X| k

k

(12.252)

Substituting finally (12.252) in (12.245) yields the statement of the proposition in regard to symmetric 2-covariant St,u tensorfields. In the sequel we shall also make use of the following commutation lemma, which is analogous to Lemma 8.3. Lemma 12.3 Let Y be an arbitrary St,u -tangential vectorfield and ϑ an arbitrary 2covariant St,u tensorfield, defined on Wεs0 . Then we have: /Y ϑ − L /Y L /L ϑ = L / (Y ) Z ϑ L /L L Proof. Following the proof of Lemma 8.3, we may restrict ourselves to a given Cu . In defining L / L ϑ, L /Y ϑ we are considering the extension of ϑ to T Cu by the condition ϑ(L, W ) = ϑ(W, L) = 0 for any W ∈ T Cu . Thus, with W an arbitrary vectorfield tangential to Cu we have: (L L ϑ)(L, W ) = L(ϑ(L, W )) − ϑ([L, L], W ) − ϑ(L, [L, W ]) = 0 ([L, W ] is tangential to Cu ). Similarly, (L L ϑ)(W, L) = 0. It follows that: L /L ϑ = LL ϑ

(12.253)

However, with X an arbitrary St,u -tangential vectorfield, we have, by Lemma 8.2: (LY ϑ)(L, X) = Y (ϑ(L, X)) − ϑ([Y, L], X) − ϑ(L, [Y, X]) = ϑ([L, Y ], X) = ϑ( (Y ) Z , X) ([Y, X] is St,u -tangential). Similarly, (LY ϑ)(X, L) = ϑ(X,

(Y ) Z ).

Also,

(LY ϑ)(L, L) = Y (ϑ(L, L)) − ϑ([Y, L], L) − ϑ(L, [Y, L]) = 0 /Y ϑ is defined by restricting LY ϑ to T St,u and then ([Y, L] is tangential to Cu ). Since L extending to T Cu by the condition (L /Y ϑ)(L, W ) = (L /Y ϑ)(W, L) = 0 : for any W ∈ T Cu , it follows that, on the manifold Cu , L /Y ϑ = LY ϑ − (ϑ · where ϑ ·

(Y ) Z

and

(Y ) Z

(ϑ · (

(Y )

(Y )

(Y )

Z ) ⊗ dt − dt ⊗ (

(Y )

Z · ϑ)

(12.254)

: for all X ∈ T St,u

(12.255)

· ϑ are the St,u 1-forms defined by: Z )(X) = ϑ(X,

Z · ϑ)(X) = ϑ(

(Y )

(Y )

Z ),

Z , X)

704


This is so because (12.254) holds when evaluated on a pair (X 1 , X 2 ) of St,u -tangential vectorfields, by virtue of the definition of L /Y ϑ and the fact that X 1 t = X 2 t = 0. When evaluated on (L, X), where X is any St,u tangential vectorfield, the left-hand side vanishes, and the right-hand side likewise vanishes by virtue of the definition of (Y ) Z · ϑ and the fact that Xt = 0 while Lt = 1. Similarly, when evaluated on (X, L), where X is any St,u tangential vectorfield, the left-hand side vanishes, and the right-hand side likewise vanishes by virtue of the definition of ϑ · (Y ) Z . Finally, when evaluated on (L, L) both sides vanish, recalling that (LY ϑ)(L, L) = 0. Consider now the evaluation of L /L L /Y ϑ − L /Y L /L ϑ on the St,u frame vectorfields X A . This evaluation is: (L L (L /Y ϑ) − LY (L / L ϑ))(X A , X B )

(12.256)

Substituting from (12.253), (12.254) this becomes: (L L LY ϑ − L L (ϑ · (Y ) Z ) ⊗ dt − dt ⊗ L L ( = (L L LY ϑ − LY L L ϑ)(X A , X B )

(Y )

Z · ϑ) − LY L L ϑ)(X A , X B ) (12.257)

for, L L dt = d(Lt) = 0 and dt (X A ) = dt (X B ) = 0. Therefore (12.256) reduces to: (L L LY ϑ − LY L L ϑ)(X A , X B ) = (L[L ,Y ] ϑ)(X A , X B ) = (L

(Y ) Z

ϑ)(X A , X B )

(12.258)

The lemma thus follows. Let us now define, given a positive integer l and a multi-index (i 1 . . . i l ), the symmetric 2-covariant St,u tensorfield: (i1 ...il ) χl

=L / Ril . . . L / R i1 χ

(12.259)

We shall derive from the propagation equation (12.77), using the above lemma, a propagation equation for (i1 ...il ) χ . We first apply Lemma 11.23 taking X in that lemma to be the space of symmetric 2-covariant St,u tensorfields, A1 , . . . , An to be the operators L / R i1 , . . . , L / Rin and B to be the operator L / L to obtain: / R i1 , L / L ]χ = [L / Ril . . . L

l−1

L / Ril . . . L / Ril−k+1 [L / Ril−k , L / L ]L / Ril−k−1 . . . L / R i1 χ

(12.260)

k=0

/ Ril−k−1 . . . L / Ri1 χ , we obtain: By Lemma 12.3 applied to the case Y = Ril−k , ϑ = L [L / Ril−k , L / L ]L / Ril−k−1 . . . L / Ri1 χ = −L / (Ril−k ) L / Ril−k−1 . . . L / R i1 χ Z

(12.261)


705

Substituting in (12.260) then yields: L /L

(i1 ...il ) χl

=

l−1

L / Ril . . . L / Ril−k+1 L / (Ril−k ) L / Ril−k−1 . . . L / R i1 χ Z

k=0

+L / Ril . . . L / Ri1 (L /L χ )

(12.262)

We shall express the last term using the propagation equation (12.77). First, writing: χ · χ $ = χ · h/−1 · χ

(12.263)

we express: / Ri1 (χ · χ $ ) = L / Ril . . . L where the remainder (i1 ...il )

(i1 ...il ) χl

(i1 ...il ) $ χl

· χ $ + χ ·

(i1 ...il )

+

rl

(12.264)

(i1 ...il ) r

l is given by: (L / R )s1 χ · (L / R )s2 h/−1 · (L / R )s 3 χ

rl =

(12.265)

partitions

|s1 |+|s2 |+|s3 |=l;|s1 |,|s3 | =l

where the sum is over all ordered partitions {s1 , s2 , s3 } of the set {1, . . . , l} into three ordered subsets s1 , s2 , s3 , such that |s1 |, |s3 | = l. Next, we have: / Ri1 (χ · a) = L / Ril . . . L where

(i1 ...il ) s l

(i1 ...il ) χl

·a+

(i1 ...il )

sl

(12.266)

is the 2-covariant St,u tensorfield: (i1 ...il ) sl = (L / R )s1 χ · (L / R )s 2 a

(12.267)

partitions |s1 |+|s2 |=l,|s1 | =l

We denote by

(i1 ...il ) s˜ l

the transpose of

L / Ril . . . L / Ri1 (L /L χ ) =

1 2

(i1 ...il ) s . l

(i1 ...il ) χl

· a +

From equation (12.77) we then obtain: (i1 ...il ) ˜ χl · a

+

(i1 ...il )

bl

(12.268)

where a is the T11 type St,u tensorfield: a = a + 2χ $ and

(i1 ...il ) b l

(12.269)

is the symmetric 2-covariant St,u tensorfield:

(i1 ...il )

bl = L / Ril . . . L / R i1 b +

1 2

(i1 ...il )

sl +

(i1 ...il )

s˜l +

(i1 ...il )

rl

(12.270)

Equation (12.262) with the last term given by (12.268) constitutes the propagation equation for (i1 ...il ) χl .

706


Given a non-negative integer l let us denote by X / l the statement that there is a constant C independent of s such that for all t ∈ [0, s]: X / l : max L / Ril . . . L / Ri1 χ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] i1 ...il

t

We then denote by X / [l] the statement: X / [l]

: X / 0 and . . . and X / l

The following lemma relates X / [l−1] to X / [l−1] in the context of Proposition 10.1. Lemma 12.4 Let hypothesis H0 and the estimate (10.30) of Chapter 10 hold. Let also Q / [l−1] and X / [l−1] hold, for some positive integer l. Then the bootstrap assumptions E / [l] , E the assumption X / [l−1] also holds. / 0 , the lemma is trivial for l = 1. We proceed by Proof. Since X / 0 coincides with X induction. Let then the lemma hold with l replaced by l − 1, for some l ≥ 2. That is, let Q / [l−1] , E / [l−1] , hypothesis H0 and the estimate (10.30), imply X / [l−2] . X / [l−2] together with E But then the assumptions of Proposition 10.1 with l replaced by l − 1 hold. Therefore the conclusion of Proposition 10.1 and those of all its corollaries hold with l replaced by l − 1. In particular, the conclusion of Corollary 10.1.d holds with l replaced by l − 1, that is: max

(Ri )

π /∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] : for all t ∈ [0, s] (12.271) t

i

Now for any positive integer n we have: / R i1 χ = L / R in . . . L / R i1 χ + L / R in . . . L

η0 L /R . . . L / R i2 1 − u + η0 t in

(Ri1 )

π /

(12.272)

hence: max L / R in . . . L / Ri1 χ L ∞ ( ε0 )

i1 ...in

(12.273)

t

≤ max L / R in . . . L / Ri1 χ L ∞ ( ε0 ) + C(1 + t)−1 max i1 ...in

t

j

(R j )

π /∞,[n−1], ε0 t

Therefore, taking n = l − 1, X / [l−1] together with (12.271) imply X / [l−1] . This completes the inductive step and the proof of the lemma. Proposition 12.9 Let the assumptions of Proposition 12.6 hold. Let, in addition, the Q QQ assumptions E{l+2} , E{l+1} , E{l} , hold on Wεs0 for some non-negative integer l, and let the initial data satisfy: χ ∞,[l], ε0 ≤ Cl δ0 0

Then there is a constant Cl independent of s such that for all t ∈ [0, s] we have: χ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

that is,

X / [l]

holds on

Wεs0 .


707

Proof. For l = 0 the proposition reduces to Proposition 12.6. We proceed by induction. Let then the proposition hold with l replaced by l − 1, for some l ≥ 1. Then X / [l−1] holds s s on Wε0 , hence by Lemma 12.4, X / [l−1] holds on Wε0 as well. Thus the assumptions of Proposition 10.1 all hold. Therefore the conclusion of Proposition 10.1 and those of all its corollaries hold. By Proposition 12.6 and the estimate (12.83) we have: a L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t


(12.274)

We shall presently estimate (i1 ...il ) bl (see (12.270)) using the corollaries of Proposition 10.1 together with the assumptions of the present proposition for l. First, by Corollaries 10.1.b, 10.1.g: ψTˆ ∞,[l], ε0 , ψ L − k∞,[l], ε0 , ψ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 t

t

t


(12.275)

Next, since ω L L is like ψ L − k but with Lψ0 , Lψi , in the role of ψ0 − k, ψi , respectively, ω / L is like ψ but with Lψi in the role of ψi , and ω / is like ψ but with d/ψi in the role of ψi , Q by virtue of assumptions E{l} , E{l+1} we obtain: ω L L ∞,[l], ε0 , ω / L ∞,[l], ε0 , ω /∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t

t


(12.276)

The same assumptions imply: Lσ ∞,[l], ε0 , d/σ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t


(12.277)

The estimates (12.275)–(12.277) yield, through (12.80), (12.17), (12.90): e∞,[l], ε0 , κ −1 ζ ∞,[l], ε0 , k/ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t

t


(12.278)

The estimates (12.275)–(12.277) also yield, through (12.73)–(12.76): (N)

α ∞,[l], ε0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] t

(12.279)

Moreover, using the estimates (12.278) we obtain, through (12.72): ( P N)

α

∞,[l], ε0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] t

(12.280)

( P P)

We turn to α , given by (12.71). Here we must estimate (L)2 σ , d/ Lσ , as well as QQ Q 2 , E{l} imply: D / σ in ∞,[l], ε0 norm. First, the assumptions E{l} t

(L)2 σ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−3 : for all t ∈ [0, s] t

(12.281)

708


Q Also, the assumptions E{l+1} , E{l+1} , together with hypothesis H0 imply:

d/ Lσ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−3 : for all t ∈ [0, s] t

(12.282)

In the case of D / 2 σ we have, by Lemma 10.9 (see (10.427)): L / R ik . . . L / R i1 D /2σ = D / 2 Ri k . . . Ri 1 σ +

(i1 ...ik )

ck

(12.283)

where: (i1 ...ik )

ck = −

k−1

L / R ik . . . L / Rik−m+1 (

(Rik−m )

π /1 · d/(Rik−m−1 . . . Ri1 σ ))

(12.284)

m=0

Here, k ≤ l. In the term corresponding to m ∈ {0, . . . , k − 1} in the sum in (12.284), (Rik−m ) π /1 receives at most m angular derivatives. Now, since the assumptions of the present proposition include those of Proposition 12.6, by the remark following the proof of Corollary 12.7.a, under the present assumptions hypotheses H2 and H2 (as well as H0 and H1) hold on Wεs0 . Taking also into account the inductive hypothesis, which as we have already noted, implies that X / [l−1] holds on Wεs0 , the assumptions of Lemma 10.11 all hold with l − 1 in the role of l. Therefore we have: max i

(Ri )

π /1 ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] (12.285) t

It follows that, for k ≤ l − 1: max max

(i1 ...ik )

k≤l−1 i1 ...ik

ck L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] : for all t ∈ [0, s] t

(12.286) For k = l, on the other hand, we write: (i1 ...il )

cl = −L / Ril . . . L / R i2 ( −

l−2

(Ri1 )

π /1 · d/σ )

L / Ril . . . L / Ril−m+1 (

(Ril−m )

π /1 · d/(Ril−m−1 . . . Ri1 σ )) (12.287)

m=0

The sum on the right in (12.287) is bounded as in (12.286). Moreover, expanding the first term on the right in (12.287): / R i2 ( L / Ril . . . L

(Ri1 )

π /1 · d/σ ) = (L / Ril . . . L / Ri2 (Ri1 ) π /1 ) · d/σ (L / R )s1 (Ri1 ) π /1 · d/(R)s2 σ +

(12.288)

partitions |s1 |+|s2 |=l−1,s2 =∅

the sum in (12.288) is likewise bounded as in (12.286). Consequently:

(i1 ...il )

cl + (L / Ril . . . L / R i2

(Ri1 )

π /1 ) · d/σ L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]


t

(12.289)

Chapter 12. Recovery of the Acoustical Assumptions (Ri1 )

We shall now estimate L / Ril . . . L / R i2 (Ri1 )

709

π /1 pointwise. Since

π /1 =

(Ri1 )

(π /1 )b · h/−1

we have, by Lemma 10.11 with l reduced to l − 1 and Corollary 10.1.d, L / Ril . . . L / R i2

/ R i2 To estimate L / Ril . . . L

lemma ϑ = (10.434)):

(Ri1 )

(Ri1 )

π /1 − (L / Ril . . . L / R i2

Cl δ02 (1 +

≤ (Ri1 )

(Ri1 )

t)

−3

(π /1 )b ) · h/−1 L ∞ ( ε0 ) t

[1 + log(1 + t)]

2


(12.290)

(π /1 )b pointwise we apply Lemma 10.10 taking in that

π / and replacing {1, . . . , k} by {2, . . . , l}. In view of the fact that (see (Ri1 )

ˇ/ (π /1 )b = D

(Ri1 )

π /

we then obtain: (Ri1 )

(π /1 )b

ˇ/L / R i2 =D / Ril . . . L

(Ri1 )

L / Ril . . . L / R i2

(12.291) π /−

l−2

L / Ril . . . L / Ril−m+1 (

(Ril−m )

π /1 · L / Ril−m−1 . . . L / R i2

(Ri1 )

π /)

m=0

In the sum on the right in (12.291) we have angular derivatives of (Ril−m ) π /1 of order at most m ≤ l − 2, so Lemma 10.11 with l reduced to l − 1 applies, and angular derivatives of (Ri1 ) π / of order at most l − 2, so Corollary 10.1.d applies. We thus obtain that the sum ε on the right in (12.291) is bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2


(12.292)

On the other hand, by hypothesis H2 the first term on the right in (12.291) is bounded pointwise by: /R j L / Ril . . . L / Ri2 (Ri1 ) π /| (12.293) C(1 + t)−1 max |L j

Now, according to equation (10.206) of Chapter 10 we have: (Ri )

π / − 2λi α −1 χ = 2λi (α −1 η0 (1 − u + η0 t)−1 h/ − k/ ) +H i j m ( ψ ⊗ d/x j + d/x j ⊗ ψ)ψm +H ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)i j m x j dH ( ψ⊗ ψ)(Ri σ ) + dσ

(12.294)

By Corollary 10.1.a: max λi ∞,[l], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

i


(12.295)

and: max d/x i ∞,[l], ε0 ≤ C i

t


(12.296)

710


These, together with the estimates (12.275) and the third of (12.278), imply:

(Ri )

π /−2λi α −1 χ ∞,[l], ε0 ≤ Cl δ0 (1+t)−1 [1+log(1+t)] : for all t ∈ [0, s] (12.297) t

Moreover, we have, pointwise on Wεs0 : |L /R j L / Ril . . . L / Ri2 (λi α −1 χ )| ≤ |λi α −1 ||L /R j L / Ril . . . L / R i2 χ | +Cl λi α −1 ∞,[l], ε0 χ ∞,[l−1], ε0 t

t

≤ Cδ0 [1 + log(1 + t)]|L /R j L / Ril . . . L / R i2 χ | +Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

(12.298)

by (12.295) and X / [l−1] (the inductive hypothesis). It follows that: max |L /R j L / Ril . . . L / R i2 j

(Ri1 )

π /| ≤ Cδ0 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

−1

+Cl δ0 (1 + t) [1 + log(1 + t)] : pointwise on Wεs0

(12.299)

Going back to (12.291), the bounds (12.292) and (12.299) imply: |L / Ril . . . L / R i2

(Ri1 )

(π /1 )b | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

−2

+Cl δ0 (1 + t) [1 + log(1 + t)] : pointwise on Wεs0

(12.300)

hence, by (12.290), also: |L / Ril . . . L / R i2

(Ri1 )

π /1 | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

+Cl δ0 (1 + t)

−2

[1 + log(1 + t)]

: pointwise on Wεs0

(12.301)

In view of (12.289) we then conclude that: |

(i1 ...il )

cl | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

+Cl δ02 (1 +

t)

: pointwise on

−4

[1 + log(1 + t)]

Wεs0

(12.302)

Using hypotheses H1 and H0 and assumption E{l+2} we then obtain through (12.283) for k = l: |L / Ril . . . L / R i1 D / 2 σ | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

−3

+Cl δ0 (1 + t) : pointwise on Wεs0

(12.303)


711

We also obtain through (12.283) for k ≤ l − 1, using assumption E{l+1} and the bound (12.286): D / 2 σ ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−3 t


(12.304)

( P P)

Returning to the expression (12.71) for α , (12.275) and the estimates (12.281), (12.282), imply: 0 0 0( P P) 1 d H 2 2 0 0 α + ψ D / σ0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] (12.305) 0 0 ε 2 dσ L ∞,[l],t 0 On the other hand, by the estimates (12.303), (12.304) and (12.275): 1 d H 2 2 L ψ D / R i1 / σ / Ril . . . L 2 dσ L ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / Ri2 χ | + Cl δ0 (1 + t)−3 j

: pointwise on

Wεs0

(12.306)

Combining the results (12.279), (12.280), (12.305), (12.306), we conclude through (12.79) (using also (12.275), (12.277) and the first of (12.278)) that: / Ri1 b| ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | |L / Ril . . . L j

−3


(12.307)

The first of the estimates (12.278), together with (12.275), (12.277) and Corollary 10.1.d, implies: a∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t


(12.308)

This together with the inductive hypothesis X / [l−1] yields, through (12.267):

(i1 ...il )

sl L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] t


(12.309)

Also, X / [l−1] together with Corollary 10.1.d yields, through (12.265):

(i1 ...il )

rl L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 t


(12.310)

Combining the bounds (12.307), (12.309) and (12.310), we obtain, through (12.270): |

(i1 ...il )

bl | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ | j

−3


(12.311)

712


The estimates (12.274) and (12.311), imply, through (12.268): / Ri1 (L / L χ )| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] max | |L / Ril . . . L

( j1 ... jl )

χ|

j1 ... jl

+Cl δ0 (1 + t)−3 : pointwise on Wεs0

(12.312)

What remains to be estimated is the sum on the right in (12.262). Consider a given term in this sum, corresponding to k ∈ {0, . . . , l − 1}. Let us recall at this point Lemma 8.5. Now this lemma holds with ξ being an St,u tensorfield of any type. We may thus / Ril−k−1 . . . L / Ri1 χ in apply Lemma 8.5 with the symmetric 2-covariant St,u tensorfield L the role of ξ (and the index set {l − k + 1, . . . , l} in the role of the index set {1, . . . , l}) to express the term in question in the form: L / Ril . . . L / Ril−k+1 L / +

k

(Ri ) l−k Z

L / Ril−k−1 . . . L / R i1 χ = L / (Ril−k )

l

L /

p=1 m 1

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Algebraic Topology (EMS Textbooks in Mathematics)

Lectures on Algebraic Topology (EMS Series of Lectures in Mathematics)

Lectures on Differential Geometry (Ems Series of Lectures in Mathematics)

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Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

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Basic Noncommutative Geometry (Ems Series of Lectures in Mathematics)

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