L. J A N O S S Y
Theory of relativity based on physical reality AKADÉMIAI KIADÓ, BUDAPEST
THEORY OF RELATIVITY BASED ON PHYSICAL REALITYUby
L. JÁNOSSY
AKADÉMIAI KIADÓ, BUDAPEST 1971
A»
® Akadémiai Kiadó, Budapest 1971 Printed in Hungary
CONTENTS
INTRODUCTION
13
Chapter I. A SURVEY OF SOMÉ EXPERIMENTÁL RESULTS RELATI N G TO THE PROPAGATION OF LIGHT A N D CONNECTED PHENOMENA
19
A. The critical velocity B. Measurement of the velocity of propagation of light signals 1. A generál remark 2. The suggestion of Galilei 3. Astronomical methods a. Observations of Römer b. The aberration of light 4. Laboratory methods a. Methods of Fizeau and of Foucault b. The experiments of Fizeau and Foucault 5. The propagation of light in refractive média a. The refractive index b. The determination of the velocity of light in a refracting médium C. Interferometric methods for the measurement of the velocity of light 1. The Michelson interferometer 2. Measurement of the phase velocity of light 3. The experiment of Fizeau D. The Doppler effect 1. Moving source 2. Observation of the Doppler effect E. The perpendicular Doppler effect 1. Experimentál observations 2. Physical interpretation of the perpendicular Doppler effect 3. Considerations in connection with the Doppler effect a. The question of the moving observer b. Acoustical and electromagnetic Doppler effects F. Somé relativistic effects 1. Variation of decay time with velocity
19 21 21 22 22 22 23 24 24 24 26 26 27 28 28 31 32 33 34 36 37 37 38 40 40 41 43 43
2. Change of mass with velocity 3. Remark on the methods of measuring the mass velocity relation Chapter II. INVESTIGATIONS CONCERNING THE CARRIER OF ELECTROMAGNETIC WAVES
44 45 48
A. The question of the ether B. Experimentál investigations 1. Rotation of the Earth 2. The Sagnac experiment 3. The experiment of Michelson and Gale C. Translational motion relatíve to the ether 1. Propagation of a spherical signal 2. Propagation of light relatíve to a moving system of reference 3. The Michelson-Morley experiment 4. The interpretation of the negatíve result of the Michelson-Morley experiment
48 50 50 50 53 53 56 57 59
5. Considerations concerning the contraction hypothesis D. Experiments of the Michelson-Morley type 1. The Trouton-Noble experiment 2. The experiment of Isaak and co-workers E. General remarks concerning the series of negatíve results 1. Generalization of negatíve experiences
64 65 65 66 69 71
Chapter III. THE PROBLEM OF MEASUREMENT A. The problem of measures 1. Representations 2. An example: the measure of electric charge 3. Distinguished representations 4. Measures of lengths B. Systems of space coordinates 1. Determination of coordinate vectors 2. Explicit determination of coordinate measures 3. Question of consistency 4. Various representations C. Problems connected with coordinate representations 1. Remark on "non-Euclidean" geometry 2. Coordinate transformations and deformations 3. Orthogonal transformations a. Definitions b. Group character of orthogonal matrices 4. Rigid bodies Chapter IV. THE LORENTZ TRANSFORMATION A. The time scale
60
72 72 72 73 76 77 79 79 81 83 83 86 86 86 89 89 90 92 94 94
1. General remarks 2. Atomic time scale 3. Systems of reference constructed with the help of light signals B. The Lorentz transformation as coordinate transformation 1. The explicit form of the Lorentz transformation 2. The physical significance of the parameters of the Lorentz matrices C. Homogeneous propagation of light 1. The concept 2. Test for homogeneous propagation of light 3. Connection between various representations a. Transformations of the propagation tensor D. The relation of systems of references obtained with light signals and with solids Chapter V. THE LORENTZ PRINCIPLE
94 97 98 100 102 104 106 106 107 108 109 110 114
A. The Lorentz transformation as deformation 114 1. Deformation operators 114 2. Lorentz deformations 116 3. Particular types of Lorentz deformations 117 B. Formulation of the Lorentz principle 120 1. Interpretation of the negatíve results of ether drift experiments in terms of the Lorentz principle 120 2. Non-orthogonal representations 123 3. General remarks on the Lorentz principle 124 C. The dynamical principle 125 1. The mechanism of the Lorentz deformation 127 a. Relaxation processes 127 b. Comparison between change of temperature and change of translational state 128 c. Deformations of unconnected systems 128 d. Length contraction of non-connected systems 130 2. Significance of subgroups of the Lorentz group 131 Chapter VI. THE INNER CONSISTENCY OF THE LORENTZ PRINCIPLE 133 A. Kinematical considerations in connection with the Lorentz principle 133 1. Addition of velocities 133 2. Addition formula and Lorentz deformations 136 B. Considerations about contraction of solids and the slowing down of clocks 137 1. The clock "paradox" 142 2. The "paradox of the twins" 146 Chapter VII. RELATIVISTIC MECHAN1CS A. Momentum and energy 1. Newton's first law
148 148 148
2. Elastic collisions 3. Inelastic collisions B. Equivalence of mass and energy 1. Remark on the mechanism of increase of mass with energy C. Distant collisions 1. Experimentál evidence D. Mechanical laws in terms of four-vectors and tensors 1. Newton's laws 2. The energy-momentum tensor Chapter VIII. THE ELECTROMAGNETIC FIELD
162
A. Maxwell's equations 1. Another formulation B. Solutions of Maxwell's equations 1. Gauge transformation 2. Retarded potentials 3. Advanced potentials 4. Wandering waves C. Maxwell's equations in terms of four-tensors 1. Retarded four-potential 2. The motion of light signals in terms of Maxwell's equations D. Maxwell's equations and the Lorentz principle 1. The field of a point charge Chapter IX. RELATIVISTIC EFFECTS OF THE
149 153 154 155 155 157 157 159 160
162 163 165 167 168 169 171 172 174 177 179 181
ELECTROMAGNETIC
FIELD
183
A. Effects of the first order 1. Effective field strengths 2. The field of dipoles B. Transformation properties of four-currents 1. The electric field of a moving current C. Further effects of the first order 1. Doppler effect and aberration 2. Frequencies of the Doppler effect 3. Effect of aberration 4. Intensities in the Doppler effect 5. Observation of the effect of aberration of star light 6. Propagation of light in a refracting médium a. Dispersion 7. The experiment of Fizeau D. Effects of the second order 1. Action of a charge upon itself 2. Mass defect E. Relativistic mechanics of a continuum
183 183 184 186 187 189 189 190 191 192 193 195 197 199 202 202 205 206
1. Interpretation of the Trouton-Noble experiment F. Transient phenomena
206 207
Chapter X. THEORY OF GRAVITATION
211
A. Observational facts B. Statement of the problem of the theory of gravitation 1. Mathematical formulation of the problem 2. Experimentál criteria for homogeneous regions a. An example 3. Construction of straight systems of references a. Locally homogeneous regions b. Criteria for homogeneous regions 4. Almost straight system of reference 5. Similar regions C. The generalized Lorentz principle 1. The Lorentz principle formulated in terms of curved coordinates 2. Generalization to inhomogeneous regions a. A physical example 3. The Lorentz principle valid for small physical systems a. First approximation b. A second approximation 4. The ambiguity in the formulation of the Lorentz principle Chapter XI. APPLICATIONS OF THE GENERALIZED PRINCIPLE
211 212 214 214 216 216 217 218 222 225 226 227 228 228 230 230 231 233
LORENTZ 235
A. Geodetic orbits 235 1. Definition 335 2. Lorentz invariance of geodetic orbits 236 a. Geodetic orbits and the Lorentz principle 237 B. Equation of motion in a gravitational field 237 1. Variational principles 238 a. Deviations from geodetic orbits 239 2. The physical contents of the variational principle 239 C. Connection between a gravitational field and the propagation of light 242 1. The equations of motions in a gravitational field 242 2. Integrals of the equations of motions 244 a. Perihelion motion 245 b. The deflection of light in the vicinity of the Sun 246 c. The red shift of spectral lines 247 D. Connection between the sources of gravitation and the propagation tensor g 249 1. Einstein's equations of gravitation 250 2. Energy momentum considerations 251
3. The Schwarzschild solution of the gravitational equations 4. The relativistic effects in the field given by Schwarzschild a. The planetary motion b. Deflection of light E. Electromagnetic field and gravitation 1. An invariant formulation 2. Question of electromagnetic polarization of the ether 3. Remark on the consistency of the generalized theory of electromagnetic fields F. Energy and momentum relations of the gravitational field 1. The gravitational force 2. Another aspect of the gravitational equations 3. The mechanism of the gravitational force Chapter XII. COSMOLOGICAL PROBLEMS A. The physical significance of invariant formulation of physical laws 1. Tensors and distinguished measures 2. The physical significance of the tensor g 3. A normál form of the propagation tensor B. Physical contents of particular representations 1. Stationary representations 2. The energy momentum distribution C. Cosmological problems 1. The results of astronomical observations 2. The solution of Friedmann D. Analysis of Friedmann's solution 1. The recession of the Galaxies a. The measures of intergalactical distances b. Doppler effect E. Mach's principle a. The Thirring effect Appendix I. TENSOR ANALYSIS IN HOMOGENEOUS REGIONS A. Systems of reference 1. The Lorentz system 2. Straight systems of reference 3. Propagation tensor g 4. Lorentz transformation 5. Standard form of linear coordinate transformations B. Vectors and tensors 1. Two-dimensional tensors a. Invariant products b. Pseudo scalar
253 255 255 256 256 257 258 258 260 260 261 262 264 264 265 266 267 270 270 271 272 272 273 276 276 276 278 279 280 282 282 282 283 283 284 285 286 287 288 289
C. Fields 1. The 3Í operator a. The Grad operator b. Further operations
289 291 291 292
Appendix II. TENSORS I N INHOMOGENEOUS REGIONS A. More-dimensional measures 1. Jfc-dimensional measures 2. Multiplication of more-dimensional quantities B. Permutation operators 1. Cyclic permutations 2. The transposition of a mátrix 3. The n, operators C. The 91 operator in curved representations 1. Tensor of several dimensions 2. Symmetry properties
293 293 293 294 295 297 298 299 299 300 301
(4)
3. The antisymmetric tensor e D. Tensor fields 1. The Christoffel bracket symbols 2. The covariant differentiation E. Criteria for homogeneous regions 1. Almost straight representations 2. Tensor character of the Riemann-Christoffel tensor
302 303 305 306 308 310 312
(4)
3. Symmetries of the R tensor
314
(4)
4. The reduced form of the R tensor
316
INTRODUCTION
1. In this book I attempt to give a re-evaluation of the facts which led to establish the special and generál theories of relativity. As the result of this analysis I obtain a mathematical formalism which is strictly equivalent to the known formalism of relativity. Nevertheless my approach to phenomena differs from the usual ones; it resembles sometimes that which is found in the classical literature of relativity rather than the treatments given in modern textbooks. I have always treated with great respect Laue's book* on relativity and many of the ideas elaborated in this volume have their origin in remarks I found there. 2. At first sight it appears a dangerous attempt to reformulate a wellestablished theory in particular — as it is in the case of relativity •— if the old theory gives mathematically a correct description of phenomena. I am, however, of the opinion, that the task of a theory is not merely to give mathematical formuláé which happen to describe correctly certain physical phenomena — a theory must give alsó a physical insight into the particular laws of nature it deals with. The theory of relativity in its originál formulation is certainly not a mere attempt to describe phenomena by suitable mathematical expressions — the theory is a far reaching attempt to give a theory of space and time. Our criticism of the theory is just connected with this latter feature. We think that the theory reflects correctly certain generál physical laws, but these laws — in our opinion — have nothing to do with the "structure of space and time". Therefore our attempt is to give a physical interpretation of the relativistic formuláé, which is different from the old one. 3. We want to give a number of personal experiences which led us to our attempts of reformulation of the theory of relativity; we think that possibly somé of our colleagues had similar experiences. I got acquainted with the theory of relativity at a comparatively early age—I read the famous popular book written by Einstein.** Reading the * M. v. Laue: Die Relativitatstheorie. Fr. Vieweg und Sohn, Braunschweig, 1923. ** A. Einstein: Über die spezielle und die allgemeine Relativitatstheorie. Fr. Vieweg und Sohn, Braunschweig, 1921.
latter I had difficulties with somé of Einstein's concepts; however, having been young and enthusiastic, I convinced myself in the end that I could understand those concepts — to prove this I tried to explain the theory to everybody who was interested. In the course of such attempts I learned the "language of relativity" and gradually I "got used" to the theory. A certain bad feeling never ceased altogether. Many years later I reád several years in succession a course of physics at the university of Manchester. My course contained alsó the special theory of relativity. As the years went on I developed a technique of presenting the subject so that in the end I could convince my students that they really understood the theory. However, as my technique of presenting the theory improved my own belief in the adequateness of the concepts vanished. In the end I became convinced that from the philosophical point of view the concepts had to be changed. Since about 1950 I have struggled with the problem of the reformulation of the theory and the results of my deliberations are found in this volume. 4. Reading Einstein's book on relativity I still can remember the thrill I felt when I met the suggestion that performing a coordinate transformation, it appears to be proper not to transform the space coordinates alone, but it seems necessary to transform alsó the time! Analysing, however, the above suggestion — which started the revolutionary ideas about space and time — I have arrived to a more sober view. We may describe events @ by four coordinates, i.e. three space coordinates giving a position vector r and for the fourth coordinate we can take the measure of time / at which the particular event happens. We may write x = r, t for the four-coordinate of an event. Changing from one system of reference to another we can introduce transformed coordinates x' = f(x),
(1)
where f(x) is somé reversible four-function of its variable x. If the coordinates x are suitable for describing events, then the transformed coordinates are alsó suitable. Introducing particular measures x or x' for events we give merely somé kind of names to events with the help of which we can recognize them — it is a more or less trivial thing whether we take x or x' as the measure of an event. The fact that a transformation of the type (1) mixes the measures of the time and space coordinates does not seem to be of particular importance and it does not imply any properties of time and space. An objective physical process develops according to its own laws and it can be described in terms of arbitrary measures. In a correct description of a phenomenon the role of the representation must vanish, i.e. it must be immaterial whether we use for the measures of events occurring in a process x or x'. 5. Sometimes it is stated that the special relativity gives the laws of nature in a form invariant with respect to linear transformations, while
the generál theory gives invariant formulation with respect to arbitrary coordinate transformations. We do not agree with the above statement. In our opinion — and this is elaborated in detail in this book — the laws of the special theory just as those of the generál theory can be formulated independent of the representation, i.e. in terms of arbitrary coordinate measures and alsó arbitrary time measures. The real difference between the theories is that the special relativity refers to phenomena in an approximation in which gravitational effects can be neglected, while the generál theory is an attempt to give the laws of physics valid in regions where gravitational effects cannot be neglected. This point of view is held among others by A. V. Fock; we think that somé of our arguments resemble those of Fock. Linear transformations play an important role in regions without gravitation — this is so because in a region free of gravitation a closed physical system can be made to move with a constant velocity and (using a suitable representation) this process can be described with the help of linear transformations. In a region where there are gravitational effects a closed system which is made to move changes in the course of its motion its gravitational surroundings and thus suffers deformations which cannot be expressed by linear transformations whatever representation we choose. 6. In our approach of physics in generál and of the theory of relativity in particular we think it very important always to remember that we are dealing with objective physical quantities and that we attempt to describe the latter in terms of measures. A physical quantity can be described by measures in a very large number of ways. To make clear the distinction between a physical quantity and its measures, we use Gothic letters for the quantities themselves and Latin letters for their measures. We write e.g. R(á) = a,
R\a) =
a',...
where we take a to be somé physical quantity and a, a',. .. the measures we find for a using particular representations R, R',. . . Looking in this way at the transformation of time, as has been mentioned above, we can say that a coordinate transformation involves the changes of the measures of the components of the coordinate vector and alsó the change of the measure of time. An event (5 is thus represented in terms of a coordinate measure and a time measure so tbat = r, t,
K'm =
r', t',. . .
the measures t and /' are just parts of the representation of the event 6 and thus / and t' do not give "the time" but two measures of the time of an event 6. 7. We can define measures a, a',. . . for a quantity a; it is obvious that these measures are not all equally useful. We shall see how one can choose from the various possible measures of a quantity such measures that reflect
clearly certain physical properties of the quantity. Such distinguished measures play a great role in the building up of the theory. The Lorentz transformation produces in Einstein's terminology the "transformation of time"; in our concept it gives the connection between equally distinguished measures of time. 8. In our presentation we feel that the ether appears as a médium which is the carrier of electromagnetic waves and alsó of other processes, e.g. it appears as the carrier of matter waves. This point of view does not essentially contradict somé of the views of Einstein. Wequote in the text (p. 49) a passage from an article by Einstein where he expresses views very similar to our own. The question of physical reality appears in our view in a form quite different from the way it is treated in many textbooks. To make this point clearer we quote from Laue's book* a sentence with which we do not agree: "Indem die allgemeine Relativitátstheorie von derartigen Normalkoordinaten vollstandig absieht, entkleidet sie nach einem Ausspruch Einsteins: 'den Raum (wie auch die Zeit) des letzten Restes physikalischer Gegenstándlichkeit'." "As the generál theory of relativity ceases to use any such normál coordinates, as Einstein pointed out, it takes away 'the last remainder of physical reality of space (and alsó of time)'." (My own translation.) It must be pointed out, however, that Laue has left out the above remark in the newer edition of his book and one finds in the newer editions ideas which resemble to somé extent our own ideas given in this book. In our view from the fact that the measures of space and time coordinates can be chosen in an arbitrary manner it does not follow that these coordinates do not express realities. In fact any physical quantity, e.g. temperature, can be expressed with the help of arbitrary scales and thus we can obtain very different measures for a temperature — nevertheless we do not doubt that temperature is an objectively existing quantity. 9. What we have to be careful about is that between measures taken on arbitrary scales there exist connections which connections remain whatever measures we introduce and the latter connections express physical realities. Somé physical quantities are expressed by tensors like the energy-momentum tensor 2 of the electromagnetic field, the energy-momentum tensor £ of a mechanical system, and so on. Among these tensors the tensor g plays a great role which is hormally supposed to give the "metric of the fourdimensional space-time continuum". We introduce the tensor as propagation tensor of electromagnetic waves — it appears at the beginning of our deliberations as a tensor with the help of which the mode of propagation of light can be described. It turns out that the tensor g appears alsó ( e / )
( m )
* D r . M. von Laue: Die Relativitátstheorie. Band II, p. 25. Verlag Friedr. Vieweg und Sohn, Braunschweig, 1923.
in the equations of motion of a partiele and alsó in other physical laws. It seems natural to connect g with the energy-momentum tensor of the carrier of the various phenomena, i.e. with the energy-momentum tensor of the ether. Laue in the newer editions of his book speaks about g as giving a "Führungsfeld", thus he is alsó of the opinion that g deseribes a kind of field. 10. The tensors % , $ , g,. . . have elements the numerical values of which depend on the representation we choose. Nevertheless, there are mathematical connections between these tensors which are independent of the representation and these connections express physical laws connecting matter, field and gravitation. The role of g in the latter relations is analogous to those of the other tensors and, therefore, we think that g represents something physical just as the other tensors do. These considerations are given in more detail in the last chapter of the book. We think the above philosophical considerations to be essential. Nevertheless we wish to point out that all our statements found in this book could alsó be translated into what may be called an orthodox language. Since our mathematical formulations — although strictly equivalent to the usual ones — are presented in a form noticeably different from the usuai formulation, we think that the book may be of interest alsó to readers who do not accept our philosophical point of view. I want to thank many colleagues for interesting and useful discussions on the subject of the book while in preparation. In particular to Mr P. Király wo took active part in clearing somé particular questions. I wish to thank to Mr A. Werner who gave very effective help in editing the manuseript. - I am greatly indebted to my stepfather G. Lukács with whom I had very many fruitful exchanges of point of views in connection with philosophical problems. íel)
( m )
J
2 Relativity
FIZIKUS TAN5Z£KcK)PORT «* KÖNYVTARA ^ ,
r
CHAPTER I
A SURVEY OF SOMÉ EXPERIMENTÁL RESULTS RELATING TO THE PROPAGATION OF LIGHT A N D CONNECTED PHENOMENA
11. In any attempt of dealing with the problems of the theory of relativity the question of the mode of propagation of light plays an important role. Maxwell's electromagnetic theory of light gives a concise description of the phenomena of light. We shall describe somé experiments which support Maxwell's theory. In particular according to Maxwell's theory the velocity c of propagation of light can be determined from measurement on the interaction of charges and magnets without using observations which are concerned directly with the propagation of light.
A. THE CRITICAL VELOCITY 12. Maxwell's equations representing the laws of the electromagnetic field contain a constant c which has the dimension of a velocity. This constant which can be obtained from the measurement of the forces acting between charges and currents is usually called the critical velocity. Its numerical value was determined experimentally with great precision and was found to be equal to the value of the velocity of propagation of light in vacuum. 13. In order to see more clearly the significance of the critical velocity we note the following. Electric charges act upon each other according to the Coulomb law; in particular in the case of two point charges e and e we have for the force F with which e acts upon c 1
12
x
2
2
where r is the vector pointing from e to e . Similarly, the force with which a magnetic pole m acts upon another magnetic pole m can be written x
2
x
2
F&> = ^
.
(2)
However, as single magnetic poles do not exist in nature, the above equation can be verified only indirectly, e.g. by using long raagnetized rods, which behave as if they possessed opposite magnetic charges at their ends. 14. An electric charge does not exert a force upon a magnetic pole if the charge and the pole are both at rest. However, if, say the pole is at rest and the charge is moving with a velocity v, we find that the force acting upon the magnetic pole is given by
^)
=
_ ^LL, a
( 3 )
r where r is the vector pointing from m to e and a is a constant. Comparing (1), (2) ánd (3) we see that the dimension of a is that of an inverse velocity. The íatter velocity can be determined numerically by measuring the forces acting between moving charges and magnetic poles. The velocity c' = l/a is called the critical velocity — or sometimes the Kohlrausch-Weber constant. From the most recent measurements of the interaction between moving charges and magnets it was found that c' = (299 790 ± 30) km/sec*
(4)
while the best value for the velocity of light is given as c = (299 792.50 + 0.10) km/sec** 15. The fact that a moving charge acts upon a magnet was first shown by Rowland. Exact experiments for the determination of the critical velocity appearing in (4) were carried out by Kohlrausch and Weber. In the latter experiment the magnetic action of the current arising when a condenser is discharged is compared with the action of the same charge when distributed electrostatically on the plates of the condenser. 16. From the empirically established fact that c = c' Maxwell concluded that light is an electromagnetic phenomenon. As it will be seen further below (in chapt. VIII) Maxwell's results apply to two phenomena.
* F. Kohlrausch: Lehrbuch der praktischen Physik. Verlag und Druck von B. G. Teubner, Leipzig und Berlin, 1930. p. 684. ** E. R. Cohen and J. W. M. DuMond: Rep. to the Com. on Nuclidic Masses and Related Atomic Constants of UIPAP. June 1963.
Firstly, a wave packet, e.g. a short signal emitted by a source moves as a whole with the velocity c. Secondly, in a monochromatic beam of light the surfaces of constant phase alsó move with this velocity. We presently describe very briefly a few experimentál methods with the help of which these results of Maxwell's theory were verified experimentally. B. MEASUREMENT
OF THE VELOCITY OF OF LIGHT SIGNALS
PROPAGATION
1. A GENERAL REMARK
17. In principle the velocity of propagation of light might be measured in the following way. A short signal is emitted from a source near a point A. The arrival of the signál is detected in a point B at a distance / from A. If the time of departure of the signal is t , that of its arrival t , we have for the velocity of the signal x
c=
/
2
.
Such a measurement cannot, however, be carried out in a straightforward way, because in order to measure the times t and t two clocks, say P and P one near A and the other near B have to run synchronously with sufficient an accuracy to render the difference t — t significant. The first problem — the solution of which is not obvious — is thus how to synchronize clocks. 18. The simplest way of doing this seems to be by direct comparison. We might e.g. observe from the point B by means of a telescope the clock P near A. However, when comparing the image P' of the clock P — as seen through a telescope situated near B — with the clock P , one has to remember that light takes a certain time to travel from A to B and therefore the image P' must be expected to be delayed in phase relatíve to P . This means that for the synchronization of P and P in this way it is already necessary to know the velocity c of propagation of light. The above difficulty may be overcome by observing both P and P through a telescope near a third point C situated at equal distances from A and B. The clocks P and P have to be regulated until the images P" and P" of P and P as seen in C appear to be in phase. The synchronization thus described supposes that light is propagated with the same velocity along AC and BC. This method, although feasible, has not in fact been used in any real experiment. x
2
A
B
2
x
A
A
A
B
A
A
A
B
A
A
B
A
B
B
B
A
2. THE SUGGESTION OF GALILEI
19. Galilei* supposed that light is propagated with a finite velocity and for its determination he suggested the following experiment. Two covered lanterns are situated at a certain distance from each other and near each there stands an observer. The first lantern is suddenly uncovered by the first observer. The second observer near the other lantern uncovers his own lantern as soon as he notices the light from the first. Now the first observer must be expected to see the lighting up of the distant lantern with a delay equal to the time the light takes to travel to and fro between the two lanterns. When the experiment was actually carried out no delay could be observed in this way when the lanterns were placed a few miles apart, and Galilei came correctly to conclude that the velocity of propagation of light is very large. He suggested as a possible method for the determination of this velocity with the help of the observations of the Satellites of Jupiter.
3. ASTRONOMICAL METHODS a. OBSERVATIONS OF RÖMER
20. The difficulty of synchronizing distant clocks might alsó be overcome in the following way. The clocks P and P are placed at first close to each other and are thus synchronized. The synchronized clocks are then transported to their positions near A and B respectively. If the transport takes place with sufücient care, we may hope that the clocks remain synchronized during their travel to their final positions and the experiment described in 17 can now be carried out with these clocks. In his investigations which led to the first numerical determination of the velocity of light (in 1676) the astronomer Olaf Römer used the same principle. The Satellites of Jupiter circling periodically around Jupiter provide a clock in space which can be observed from the Earth. While the Earth moves along its orbit the distance between Jupiter and Earth changes and the Jupiter-clock appears to be slow when the Earth is moving away from Jupiter and to be fast when it is approaching Jupiter. The actual method is as follows: Suppose that one of the Satellites of Jupiter actually completes its revolutions at times A
í = v
t + vT, 0
B
v = 1, 2 , . . . ,
k
(5)
where Tis the time taken for one revolution and t is the time when the first observed revolution has started. We observe the completion of the revo0
* Discorsi e demostrazione mat., Elzevir, 1638. p. 43; see alsó R. J. Seeger: Galileo Galilei, his life and his works. Pergamon Press, Oxford, 1966. pp. 1 8 5 - 1 8 7 .
lutions at times t' = t + IJc, v
(6)
P
where l is the distance Earth-Jupiter at the time t' * From (5) and (6) we find for the times observed for the first k revolutions v
v
t' -t' k
0
= kT+
(l - l )/c, k
0
and for the subsequent k revolutions 4 -t'
k
= kT+
(/
2fc
-
l )lc. k
The difference between the times observed for the first and second sequence of k revolutions is thus 4 - 2/* + t' = (l 0
2k
- 24 + l )/c, 0
thus _ hk — 2/fc + /p tzk — 2íjt + t'
0
Römer did not observe complete revolutions of the Satellites, but observed the times of their eclipses; for the sake of simplicity we have described here the method somewhat simplified in terms of full revolutions. Römer observing a few revolutions öf the Satellites found c = 220 000 km/sec.**
(Römer)
The above value is rather inaccurate, nevertheless the fact that the correct order of magnitude was obtained at all must be regarded as a remarkable achievement. The exact instant of an eclipse cannot be determined with an uncertainty less than somé fraction of one minute. The procedure — even using modern methods of astronomical observation — can therefore not be improved sufficiently to give a precise determination of c. b. THE ABERRATION OF LIGHT
21. The first reasonably accurate determination of the velocity of light was carried out by Bradley*** (1727) who predicted and alsó observed the effect of the aberration of light. We shall discuss this effect further below * It may be supposed that the velocity of the Earth relatíve to that of light is so small that it makes no noticeable difference whether we consider / to be the distance Earth-Jupiter at í or at t' . ** R. Römer: Mem. Acad. des Sciences,Paris, 1675; see alsó C. Ramsauer:Grundversuche der Physik in historischer Darstellung. I., Springer Veri., 1953. p. 63. *** J. Bradley: Phil. Trans., London, 35, 637, 1728. v
v
v
(chapt. IX). It will be seen that this procedure yields a determination of the critical velocity c' rather than that of the velocity of propagation c. 4. LABORATORY METHODS a. METHODS OF FIZEAU AND OF FOUCAULT
22. The difficulty in the measurement of the velocity of light mentioned in 17 arising from the need to synchronize distant clocks, can be avoided by making use of return signals. This circumstance was already appreciated by Galilei. Sending light signals from a fixed point A to a point B at a distance / from A and reflecting the light signal by means of a suitable mirror back towards A we may measure the time of departure í of the signal and the time of its arrival t back in A by one and the same clock P . We may thus write x
3
A
21 '3
'l
When using (7) we suppose that t , the time of arrival of the signal in B, is given by 2
i.e. we suppose that the velocity of propagation c of the signal from A to B is equal to its velocity of propagation c_ from B to A. Instead of registering the return times the signals take to move from A to B and back, we may use a clock P near A and view its mirror image obtained on a mirror situated near B. Viewing the mirror image P' in a telescope near A we find the phase shift At = 21)'c between the phase of the clock P and that of its mirror image P' . The velocity of light is thus obtained +
A
A
A
A
c = 21/At. b. THE EXPERIMENTS OF FIZEAU AND FOUCAULT
23. Experiments using the principle explained in the previous paragraph were carried out (in 1849) by Fizeau* and led to a precise determination of the velocity of propagation of light. In the experiment of Fizeau the clock is replaced by a fast moving cogwheel. The light of a small source is made to fali on the rim of the wheel * H. Fizeau: Compt. Rend. Hebd., 29, 90, 1849; Ann. d. Phys., 79, 167, 1850.
and is passing between the cogs to a mirror placed at a distance /. The beam refiected from there is projected by means of a suitable optical arrangement back on the cog-wheel. The optical arrangement, which is shown in a little more detail in Fig. 1, is so adjusted that when the cog-wheel is at rest the returning beam falls on the same part of the rim of the wheel through which it has passed originally and in this way the light source can be seen through the gap between the cogs. If, however, the cog-wheel is set in motion and has turnéd by, say,
F
Fig. 1. Determination of the velocity of light (method of Fizeau)
half a cog during the to and fro passage of the light beam, this can no longer pass through the gap and the image of the light source is extinguished. When the wheel moves faster the returning beam may just fali on the gap following the one through which the originál beam has passed and the image of the source becomes again visible. In the experiment the relation between the angular velocity of the cogwheel and the re-appearances of the image were registered. In the actual experiment of Fizeau, the distance was taken to be /=8633 m and a wheel with n = 720 cogs was chosen. The rate of revolutions at which the image appeared were therefore the integrál multiples of c N = —— = 22.6 revolutions/sec. 2nl
24. In a further experiment Fizeau and Foucault* used an improved method suggested by Arago.** In the latter arrangement a rotating mirror is made use of. A beam of light falling on a rotating mirror placed near A is reflected in such a way that it falls on a stationary mirror placed near B at a distance / from A (see Fig. 2). The mirror in B reflects the beam back to the mirror in A but during the time At = 2l/c which the light takes to travel from A to B and back the mirror turns by an angle ö=a>At where co is the angular velocity of the mirror. Measuring the angle fi the value of At can be obtained as the value of co is known and thus c can be determined. The results of the most accurate determination of c were quoted in 14.
Fig. 2. Determination of the velocity of light
5. THE PROPAGATION OF LIGHT I N REFRACTIVE MEDIA a. THE REFRACTIVE INDEX
25. It follows from Maxwell's theory that in a refractive médium light is propagated with a velocity different from c. We shall discuss this aspect of the theory further below (in chapt. IX). We note here the following: 1) The velocity of propagation of light in a médium with refractive index n is obtained from the theory as V = cjn,
-
(8)
where in a first approximation n = ^/ey. (e is the dielectric constant, /z the magnetic permeability of the substance). * L . Foucault: Compt. Rend. Hebd., 30, 551, 1850; ibid. 55, 501, 1862. * * F . Arago: Compt. Rend. Hebd., 7, 954, 1838; 30, 489, 1850; 55, 792, 1862.
A more detailed analysis shows that (8) is valid for monochromatic light only the propagation of planes of constant phase of a monochromatic beam having velocities V=cln(y), (9) where «(v) is the refractive index for light of frequency v. 2) V gives not only the velocity of propagation of phase planes but alsó that of wave packets, the width of which much exceeds the wave length. Relation (9) can be checked experimentally by measuring directly the velocity V of propagation of planes of constant phases of light of frequency v or by measuring the velocity of wave packets in a transparent médium. The refractive index n(v) may be determined from the observed angles of refraction of a beam of light of frequency v. Thus V and n(v) may be determined by independent experiments and the relation (9) thus checked. The actual experimentál findings described below support the validity of relation (9).
b. THE DETERMINATION OF THE VELOCITY OF LIGHT IN A REFRACTING MÉDIUM
26. The experiment of Foucault using a rotating mirror can be carried out when the distance / is comparatively short. The arrangement could therefore be used alsó to measure the velocity of light in a refracting médium. For this purpose Foucault used in his arrangement a container fiiled with water, which was placed between the rotating and the stationary mirror (Fig. 3). In this way he could ascertain that the velocity of propagation of
Fig. 3. Determination of the velocity of light in a refractive médium
ight through water was indeed equal to V = c/n, where n is the refractive index of the water. So as to avoid the necessity of measuring the angular velocity of the rotating mirror Fizeau improved this experiment by comparing directly the times light needs to traverse a layer of water of depth / = l/n with the time of travel taken for a distance / in air. He found the times to be equal and thus verified the relation (9). x
C. INTERFEROMETRIC METHODS FOR THE MEASUREMENT OF THE VELOCITY OF LIGHT 27. The methods described so far were used to determine the velocity of signals of light. Indeed, in the experiment of Fizeau the rotating cogwheel cuts the incoming beam into packets (which may be regarded as signals) and it is the velocity of these packets that is determined. Similarly in the experiment of Foucault the rotating mirror produces short flashes and the speed of propagation of these flashes is measured. We shall discuss presently interferometric methods of the measurement of the velocity of light. These methods make use of beams of monochromatic light and sérve for the observation of the velocities of planes of constant phase. 1. THE MICHELSON INTERFEROMETER
28. The most suitable arrangement for such interferometric measurements is the interferometer of Michelson.* We give a short description of the arrangement. Schematically the Michelson interferometer is shown in Fig. 4 and can be described as follows. A beam of light starting from a source S is falling with an incidence of 45° on a semi-transparent mirror SM. The beam is split by the mirror into two components. The components fali on mirrors M and M and are reflected back into SM. The returning beams are split again and we obtain one component of each returning beam which passes into the telescope T. In the telescope we obtain a system of interference fringes with the help of which it is possible to determine the difference of the lengths of the return paths 1
2
2/j = SM -* M - SM x
and
2/ = 2
SM -* M -» SM 2
* A. A. Michelson: Sill. Journ., 15, 394, 1878; 18, 390, 1879; Nature 21, 94 and 120, 1880; Naut. Alm. p. 235, 1885; Astrophys. J., 60, 256, 1924; 65, 1, 1927.
of the components of the incident beam. More precisely the system of fringes permits to determine the difference AT = T — T of the times of return travels of the two components between SM and M respectively M . 29. For the better understanding of what happens in the Michelson interferometer it is useful to give a somewhat fuller description. As shown in Fig. 4, light coming from the source S passes through a lens L and an approximately parallel beam is obtained; this beam appears to come from a source 5 ' far to the left from S. 2
x
x
s,f s
2
2
Obstacle
-s' s SM'
Fig. 4. The Michelson intirferometer
The parallel beam is now split by the semi-transparent mirror, the components are reflected back and produce components passing into the telescope. If we put an obstacle between SM and M and thus prevent the latter component to fali into the telescope, then we see in the telescope an image Sí of 5 which appears to come from a source S situated at somé distance behind the mirror M . This image is produced by the reflections of the incident beam first on Af and then on SM. Similarly, when we stop the beam between SM and M by somé obstacle, we see in the telescope an image S of S which image appears alsó to come from a source S situated alsó somewhere behind M . The images Sí and S are coherent and therefore if both paths SM -> M and SM -> M are free we see in the telescope the interference pattern which arises from the superposition of the two images — the pattern is such as if we observed through the telescope the images of two coherent sources S[ and S . If the interferometer is built up symmetrically, i.e. if the mirror M is placed exactly in the pláne into which falls the mirror image of M produced by SM, then the virtual sources S and S coincide and no interferences are 2
x
2
x
±
2
2
2
2
x
2
2
2
x
±
2
"1
i
produced and we observe a bright picture of the originál source 5 in the telescope. We denote this bright picture of the source alsó as the zero interference pattem. If we shift, say, the mirror My parallel to itself by a small amount Al, then the virtual image S[ moves by an amount 2 Al behind and thus we observe in the telescope circular fringes corresponding to two coherent sources Sy and S placed behind each other at the distance 2 Al. Knowing the wavelength X of the light of the source, we can determine from the distribution of the interference rings the numerical value of Al. 2
Fig. 5. Radius of interference fringes
30. Indeed, suppose the distance between the virtual sources Sy and S to be a = nk, n = n + e,
2
0
where s is the non-integral part of n. Bright interference rings will then appear in the field of view when the &-th ring having the radius r = Ftg& k
k
with
n —k cos# = — , A
k = 0, 1 , 2 , . . .
and F being the effective focal length of the telescope. If the & are small one finds, comparing Fig. 5, in good approximation k
íi«M \ n +e 0
*=
0,1,2,...
;
The concentric rings appearing in the field of view show decreasing spacing when going outwards (see Fig. 6). From the pattern both n and £ can be determined. Denoting the effective distances SM -» M by l and SM -* M by l (i.e. travelling time multiplied by c) we can determine with great precision the difference Ál = l — l = nk, 0
x
2
x
2
2
x
if A is known. In particular, we can shift one of the mirrors M or M parallel to itself until the interference fringes disappear and we obtain the zero interference pattern, i.e. we see a bright image of the light source in the telescope. We have then n = 0 and therefore
2
x
Ál = 0.
Fig. 6. Scheme of interference fringes
If we tilt, say, M by a small angle instead of shifting it from its symmetrical position, then the virtual images Sí and S will bé placed side by side and we obtain a system of parallel fringes instead of a system of rings; for practical consideration often the interferometer is adjusted so as to give parallel fringes instead of rings — the exact way of adjustment has no great significance, therefore we discuss further below the applications of the interferometer supposing the mirrors M and M to be adjusted so as [to'give a system of interference rings. x
2
x
2
2. MEASUREMENT OF THE PHASE VELOCITY OF LIGHT
31. The Michelson interferometer described above allows the measurement not so much of the difference ál of the lengths of its arms, but rather gives the difference AT in the times of travel of the two wave fronts produced by the mirror SM from the moment of their separation until they unité again.
If the wave fronts travel along the arms l and l with different velocities, say, with velocities C] and c , then from the interference pattern we do not determine Al = l - h but rather t
2
2
2
The fact that we are not measuring the difference of lengths of light paths but rather the difference of times of fiight was made use of for to measure the velocity of propagation of wave fronts in a refracting médium. Inserting a refracting médium in one of the arms of an interferometer and adjusting the interferometer so as to give AT = 0, we find from (10) i '•
c
2
c
— h '• h-
Thus, if e.g. Cj = c, c — c/n, we have 2
n =
ljl . 2
The above experiment extends the result of that described in 26. Considering both types of experiments we find that wave fronts like wave packets both propagate with velocities c/«(v). .• 3. THE EXPERIMENT OF FIZEAU
32. Similar methods led to the measurement of the velocity of phase planes in a moving médium. The arrangement for such a measurement is shown schematically in Fig. 7.
Fig. 7. Determination of the velocity of light in a moving médium (Fizeau)
The incident beam is split into two components and these are made to pass through columns of water flowing with a velocity v. The one front moves into the direction of flow, the other in a direction opposite to it. As will be seen further below (in chapt. IX), the phase velocities of light, passing through the moving médium, are expected theoretically to be equal to
Thus re-uniting the beams a shift of interference fringes depending on the velocity v of flow is to be expected. The measurements of Fizeau* confirmed at least qualitatively the relation (11). A good agreement between experiment and theory was obtained by Zeeman** when the values observed directly were corrected for dispersion. Evaluating the experimentál results one has to put in place of (11) more precisely c { 1 1
K . - in- r í l - : '
1
l
»(v ) J' 2
2
where n(vj) and n(v ) are the values of the refractive index for frequencies and 2
v =v(l 2
+
|J,
where v is the originál frequency of the source and \\ and v respectively are the frequencies shifted by the Doppler effect, i.e. the frequencies with which the light acts upon the atoms of the moving liquid. We shall come back to the theoretical interpretation of this effect in chapt IX. 2
D. THE DOPPLER EFFECT*** 33. It follows from Maxwell's theory that the velocity of propagation of light is independent of the velocity of the source at the instant of emission. Sometimes it causes confusion that the frequency of the light emitted by a moving atom is nevertheless affected by the state of motion of the atom. The latter effect predicted by Doppler is, however, quite compatible with * H. Fizeau: Compt. Rend., 33, 349, 1851. ** P. Zeeman: Amst. Akad. Vrl. p. 245, 1914; and p. 18, 1915; see alsó A. Sommerteid: Vorlesungen über theor. Physik. Optik, p. 64. *** Ch. Doppler: Abh. d. K. Böhmischen Ges. d. Wiss., 2, 465, 1892.
3 Relativity
the fact that the velocity of propagation of light is independent of the state of motion of the source. Indeed, the frequency of light we observe is determined by the rhythm in which consecutive wave fronts reach us. This rhythm may be affected by the motion of the source but the wave fronts themselves will always proceed with the velocity c independent of the motion of the source. 34. The emission of waves by a moving atom can be treated with the help of Maxwell's theory. The exact treatment shows that an atom emits spherical waves, the surfaces of equal phase expanding isotropically with a velocity c. That this treatment leads to correct results can be seen most clearly if we express the field of the emitting atom in terms of retarded potentials (see chapt. VIII). Consider thus an atom which is at rest in the point r = 0 and which emits radiation of constant frequency v. Fronts of constant phase are then emitted at intervals T =- , v
(12)
and the fronts may be supposed to start at times t = t + kT, k
0
k = 0, 1, 2 , . . .
At any time t the fronts are distributed as a system of concentric spheres around the point r = 0. 1. MOVING SOURCE
35. If the emitting atom moves with a constant velocity v the position of the atom at the time t is given by r(í) = yt and the wave fronts will be eccentric spheres with centres r = vt . k
k
If the fronts are observed from a point A towards which the atom moves (see Fig. 8), they will appear to be crowded together and to arrive at intervals shorter than T while in a point B from which the atom moves away the wave fronts will arrive at intervals longer than T. Of course the velocity of the individual fronts is equal to c both when passing through A and alsó when passing through B. The points of the fc-th front at the time t > t obey the relation k
(r - yt f k
= c\t - t f k
for
t > t. k
The k-th front arrives at the time t' in the point A with coordinate vector x = R so that (K-yt f = c\t' -t f, thus = + | R - yt l/c. k
A
k
k
k
k
Supposing the distance R to be much longer than vt we may write — neglecting terms (vtJR) as compared to unity k
2
_
,
Wtcos^i
„r
|R-tM=JI
V " ) '
1
Fig. 8. Wave fronts in the Doppler effect
where cos = Rv/Rv is the cosine of the angle between R and v. Thus we have in this approximation t' = Rlc+ | l - -^-cos^J t . k
k
(13)
Writing t' i-t k+
k
= T'
and
v' = ^ L ,
(14)
we find from (12), (13) and (14)
• —i 1
• c
cos &
(15)
We see thus that the observed frequency v' varies with the angle & between v and the direction of observation. We note that formula (15) is asymptotically correct for large distances R. However, in practice we observe the radiation of atoms always at such great distances that the deviation from expression (15) which arises from having neglected terms of the order of vt /R can always be neglected. The deviation can be neglected evén if v ~ c. k
2. OBSERVATION OF THE DOPPLER EFFECT
36. The Doppler effect was first observed by Galitzin and Belepolsky (1895);* they compared the frequency of the light emitted by a monochromatic light source with the frequency which the mirror image of the
Fig. 9. Scheme of Doppler effect produced by a moving mirror
source obtained on a moving mirror seems to have. The above procedure was chosen as it seemed difficult at that time to move a real source of light with a sufliciently large velocity. Apart from the fact that it is easier to move a mirror than a whole light source the procedure had alsó the advantage that the mirror image appears to move with a velocity 2v, twice the velocity v of the mirror. The frequency observed from the moving mirror image was found to be ,
v 1 — 2v cos é/c
in accordance with (15). That the observed frequency of the moving image changes just as the observed frequency of a moving source can be understood readily (see Fig. 9). If a mirror moves towards the source with a velocity v perpendicular to the wave fronts, then subsequent wave fronts are reflected from diminishing * Quoted from Grirasehl: Lehrbuch d. Phys., Band III. p. 290.
distances. The times of travel of the wave fronts — from the source to the mirror, and alsó from the mirror to the observer — decrease for subsequent wave fronts and therefore the fronts arrive near the observer crowded together in the same fashion as if they had started originally from the moving image S of S. The Doppler effect was observed later by many others. Stark observed the Doppler effect of the radiation emitted by atoms moving in a beam. He alsó confirmed relation (15).* The Doppler effect can now be considered to be a well-established effect and it may be made use of for the determination of the velocity of moving sources. The velocity of fast moving atoms can conveniently be determined by measuring the frequencies of the light emitted by them. 37. In astronomy the velocity of motion of the components of double stars can be obtained with the help of the Doppler effect. From the observation of double stars evidence was obtained for the fact that the velocity of light is independent of the motion of the source. If we were to suppose that the velocity of light depended on the motion of the source, then we had to expect that a component of a double star emitted light which is propagated towards us with a velocity changing with the state of motion of the star. Because of the large distance the light has to travel we would observe the motion of the star in a distorted fashion. By analysing the observational data of the double star /?-Auriga, it was found that these are incompatible with the assumption that the velocity of light depends on that of the source. x
E. THE PERPENDICULAR DOPPLER EFFECT 1. EXPERIMENTÁL OBSERVATIONS
38. More recent experiments by íves and Stillwell** in 1938 and later by Otting*** in 1939 showed that the frequencies of radiations emitted by very fast atoms are given by
v-wW -^', 1
1
c
(16)
cosi?
the observed frequencies v" being a little smaller than the frequencies predicted from simple geometrical considerations and given by (15). It fol* J. Stark: Phys. Zs., 6, 893, 1905; Ann. d. Phys., 21, 401, 1906. ** H. E. íves and G. R. Stillwell: J. Opt. S o c , 28, 215, 1938. *** G. Otting: Phys. Zs„ 40, 681, 1939.
ows in particular from (16) that for •b = nj2 we find = v / l -v lc* = v*. 2
N
(17)
Relation (17) describes the so-called perpendicular Doppler effect. The frequency of the radiation emitted at right angles to the direction of motion of an atom is found to be v* — less than the frequency v of the radiation emitted by the same atom when at rest. Equation (16) can be written in accordance with (17) as v'(#) =
£ 1
•
(18)
cos#
c Relation (18) gives the frequencies of the radiation we expect to receive from an atom at various directions when the atom oscillates with a frequency v* =
/l-v lc . 2
2
Vy
Therefore (18) can be interpreted phenomenologically by supposing that the frequency of a radiating atom is slowed down from v to v* if the atom is made to move with a velocity v. 39. Since the ordinary Doppler effect described by (15) can be observed on moving mirror images the question arises whether relation (15) has to be replaced by (16) in the case of observing a source through a moving mirror. Thus the question may be raised whether a moving mirror image shows a perpendicular Doppler effect? The latter question can be answered in the negatíve as the result of the following consideration. On viewing a distant star through a mirror and tilting the mirror slightly, the image of the distant star will move with an enormous velocity. If relations (16) and (17) were to apply to a moving image, then the colour of a distant star as viewed through such a moving mirror ought to change radically. (It can easily happen that such an image moves with a velocity considerably exceeding that of light and then the relations (16) and (17) lose their meaning.) Such effects obviously do not occur and therefore we see that formuláé (16) and (17) cannot be applied to moving mirror images. 2. PHYSICAL INTERPRETATION OF THE PERPENDICULAR DOPPLER EFFECT
40. Regarding the physical interpretation of the perpendicular Doppler effect we have to conclude that an atom when accelerated so as to move with a velocity v, then its inner rhythm is reduced by a factor y/l — v jc . 2
2
As there exist misunderstandings in connection with this question we discuss the above statement still further. We emphasize that (15), i.e. the relation
1 ——cos# c is derived as the result of very simple kinematic considerations and that these considerations must be accepted under any circumstances. Observing thus the frequencies of radiation v"(#) received from a moving source we expect from our geometrical considerations that v"(#) (l — - c o s # = independent of d l I = frequency of moving atom = v(v). c
(19)
Observing v"(é) the numerical values of the left hand expression of relation (19) can be determined for various values of é by experiment. If the latter expression is found to be independent of % indeed, then we have proved by experiment our theory of the Doppler effect to be correct. We have to identify the empirically found value v(v) with the frequency v* emitted by the moving atom. Carrying out the experiment for various values of v we may establish empirically the relation / ^ v(f) = v ( 0 ) / l - - = v*. x
?
41. That the above interpretation of the Doppler effect is correct indeed — and that the factor y/l — v /c is not caused by somé "geometrical" phenomenon which we have overlooked — is further supported by the fact that a moving mirror image does not show the perpendicular Doppler effect. Indeed, the difference between a mirror image and a real atom is that the former has no inner mechanism, therefore its oscillations reflect simply those of the light source. A real atom — having an inner structure — may be affected physically when set to move. The perpendicular Doppler effect observed on moving atoms indicates therefore that the process of acceleration slows down indeed the inner frequencies of atoms. The perpendicular Doppler effect shown by a moving atom is thus caused by the slowing down of its inner rhythm. The lack of this effect when observing the radiation of a mirror image is caused by the fact that a mirror image is not a physical system and it does not reduce its inner motion when put to move. 2
2
We shall see further below that taking together a number of phenomena there is good reason to believe that the inner forces inside an atom and alsó inside other closed physical systems are always such that a change of velocity of the system as a whole produces a reduction of the inner rhythms.
3. CONSIDERATIONS I N CONNECTION WITH THE DOPPLER EFFECT a. THE QUESTION OF THE MOVING OBSERVER
42. Consider an atom A emitting a radiation of frequency v and an observer B both moving on one straight line such that the atom moves to the
X
Fig. 10. x — t diagram of comparison of rates of moving clocks
left with a velocity v and the observer to the right with a velocity w (see Fig. 10). The coordinates of the atom and observer at a time t can thus be written = a-vt,
XA( ) 1
x (t) = b + wt. B
(20)
The atom emits phase surfaces at times t = t + kT, k
T = —,
Q
k = 0,1,2,...
(21)
v
from points with coordinates x (t ); the latter arrive to the observer at times t' so that xA*k) + c(t' - t ) = x (tí). (22) A
k
k
k
k
B
It follows from (20) and (22) (c + v)t =
(c-w)t' -l,
k
k
where / = b — a, and therefore with the help of (21)
writing 4
+ 1
— t' = T = 1/v' we have alsó k
v' = ^ v ,
(23)
C+V
where Y is the frequency with which the phase planes arrive to the observer in B. b. ACOUSTICAL AND ELECTROMAGNETIC DOPPLER EFFECTS
43. The formula (23) is remarkable because it is unsymmetric with respect to v and w thus replacing v -» w and w -* v, i.e. interchanging the roles of the source and the observer we obtain a changed value of v'. In particular if we put v = 0, w = V, i.e. if we consider the case where the source is at rest and the observer moves, then we find
while in the case when the source moves and the observer is at rest, i.e. if we put v = V, w = 0 we have
It is a well-known acoustical phenomenon that the Doppler shift appearing in the case of the source moving with a velocity v towards the observer differs from the shift which appears if the observer moves with the same velocity v towards the source. It is not always realized that the relation (23) expressing this asymmetry applies to electromagnetic waves also. Indeed the formalism making use of retarded potentials leads to (23) in a straightforward way — and the reason for the asymmetry thus obtained is analogous to that in the case of sound waves. 44. Considering, however, real observations the asymmetry of relation (23) disappears for the following reason. A real experiment in a somewhat simplified manner can be described as follows. Consider two atoms A and B moving in opposite directions on a straight line. The velocities of A and B may be v and w related to a fixed point of the line.
The frequencies of the atoms thus moving are expected to be "A
v
0
V1
- v /c\
v = v
2
B
0
71
- v?\c\
(24)
where v„ is the common frequency of A and B at rest. The atom A emits radiation with a frequency v , the phase planes thus emitted reach the moving observer B, they arrive with a frequency v' and according to (23) we expect c—w v' = . (25) c+v A
A
AA
A
v
;
Dividing both sides of (25) with v we obtain a quantity which we denote Q and find thus with the help of (24) and (25) B
= ^ = í ^ . . ^ ] ' . v [c + v c + w) 1
e
2
(26)
B
We note that the quantity Q is the quantity we can determine by experiment: it is the ratio of the frequency v' falling upon B and the frequency v of B itself. The latter expression is symmetric with respect to v and w. Thus in the case of electromagnetic waves — unlike the case of sound — a real experiment does not permit us to distinguish whether the source or the observer is in motion. Interchanging the roles of A and B we find with the help of (26) A
B
Úlv = v lv B
B
A
= Q.
(27)
As already pointed out the ratio Q is a quantity which can be measured directly by forming the ratio of two frequencies. It follows from (27) that the numerical value of this ratio is expected to have the same value no matter whether we compare the two frequencies at the position of A or of B. However, the ratios v' jv and v' /v are determined by two different experiments. Thus whether or not (27) stands can be established in principle by experiment. If (27) is found experimentally to be correct, then the latter experiment can be taken to support the assumption to the fact that the frequencies of the atoms change indeed with velocity according to (24). Real experiments which effectively support relation (27) were carried out by Isaak, Champeney and Khan.* We shall come back to this question further below. 45. It is remarkable that with the help of the perpendicular Doppler effect relations (24) can be checked without knowing the numerical value A
B
B
A
* G. R. Isaak et al., Phys. Letters 7, 241, 1963.
of v and w explicitly. The quantity which can be measured directly is ^
c+v
c+w
v
'
So as to see the significance of the latter quantity more clearly, we may introduce a quantity V with the dimension of a velocity such that c- V
^-V-
e =
(29) c+V Comparing (28) and (29) one finds as the result of a simple calculation 2
v v
=
+
w
vw 1
+—
c The velocity V is, apart from small terms, equal to the sum of the velocities v and w. From the point of view of the Doppler effect the latter quantity characterizes the relatíve motion of observer and source. Provided relations (24) hold the Doppler effect is characterized by the quantity V which may be called the "relativistic relatíve velocity". We shall come back to this question in greater detail. l
F. SOMÉ RELATIVISTIC EFFECTS 46. We discuss now two effects which are not connected directly with the propagation of light, the results of which will be, however, of importance further below. 1. VARIATION OF DECAY TIME WITH VELOCITY*
47. As has been seen in 38, we are led to conclude that the frequency of oscillations of an atom is slowed down if the atom is made to move with a velocity v. A similar effect was observed with unstable elementary particles. Direct observation of /í-mesons in cosmic radiation showed that the mean life of a |t-meson when it is brought to rest in an absorber is equal to T = 2.1983 + 0.0008 //sec.** 0
* See for details e.g. L. Jánossy: Cosmic Rays. Clarendon Press, Oxford, 1950. 2nd ed. ** From Review of Partiele Properties UCRL-8030.
The cosmic ray /i-mesons are förmed in the higher layers of the atmosphere by primary cosmic rays. Supposing that the mesons move with velocities approximately equal to that of light, the mean rangé of them would be of the order I « CT « 650 m. 0
0
The experiments show that ^-mesons travel distances of many kilometers, i.e. distances much exceeding the above value. The behaviour of the /imesons can be explained readily by the fact that the mean life of a /x-meson travelling with a velocity v is given by T = Wl
- t> /c 2
2
(30)
Experiments of Rossi* and others have shown this relation to be correct. It is interesting that the cosmic ray experiments give evidence concerning (30) for velocities v x c, where the change of the decay time is considerable and the ratio T/T is of the order 10-30. It must be noted, however, that in the latter experiments v could not be measured directly. The observed results were analysed using the relation between velocity and momentum. We shall discuss this question further below. Relation (30) was first established for the /x-mesons of cosmic rays and has since been proved to be correct for a number of other unstable elementary particles. Somé of the evidence was obtained making use of observations of fast particles accelerated artificially. 0
2. CHANGE OF MASS WITH VELOCITY
48. Another effect which will be found relevant to our analysis is the change of the mass of a partiele with velocity. From theoretical arguments Ábrahám** came to conclude that the mass of an electron should increase, when the electron is accelerated; the increase of mass was derived considering the action of an accelerated charge upon itself. The theory of Ábrahám was modified by Lorentz*** who obtained an expression somewhat different from that of Ábrahám. The experiments of Kaufmann**** (1901) proved that the mass of the electrons changes indeed with velocity, his measurements were, however, * B. Rossi and D. B. Hall: Phys. Rev., 59, 223, 1941; B. Rossi and others: Phys. Rev., 61, 675, 1942. ** M. Ábrahám: Ann. d. Phys., 10, 105, 1903; see alsó A. Sommerfeld: Atombau u. Spektrallinien, 1950. p. 313. *** H. A. Lorentz: The Theory of Electrons. Leipzig, 1909. **** W. Kaufmann: Ann. d. Phys., 19, 487, 1906.
not sufficiently accurate to decidé between the theoretical formuláé of Lorentz and those of Ábrahám. 49. From theoretical considerations it was supposed that the formuláé of Lorentz and not those of Ábrahám are correct and a large number of experiments were carried out later with the object of trying to proveLorentz's formuláé to be correct. The latter experiments claimed to support the Lorentz formuláé, however, as a detailed analysis showed on the whole they did not have sufficient accuracy to decidé the question experimentally (see the analysis given by Faragó and Jánossy*). The first experiment which proved convincingly that Lorentz's formula and not that of Ábrahám described the change of mass of the electron correctly is that carried out by Rogers, Reynolds and Rogers.** The experiment was later improved by Staub*** who attained still greater accuracy. 50. Experiments to determine the change of thé mass of protons with velocity were carried out by Zreliov, Tapkin and Faragó.**** In the latter investigation the velocity of the protons which was 83 per cent of that of light was determined directly with the help of Cerenkov radiation. The Lorentz formula at that velocity was found to be correct inside the margin of experimentál error which was estimated not to exceed 0.1 per cent. Taken all the evidence together it supports the validity of the Lorentz formula for electrons and for protons. It seems reasonable to suppose that the formula has generál validity. 3. REMARK ON THE METHODS OF MEASURING THE MASS VELOCITY RELATION
51. The experiments for determining the change of mass with velocity are based on the observation of the orbits of charged particles in electric and magnetic fields. The analysis of an orbit may give information about velocity and acceleration of the partiele — so as to obtain information about the mass of the partiele dynamical considerations have to be introduced. For the analysis of the experiments it is supposed that the force F acting upon a partiele of charge e and velocity v is given by the Lorentz formula. F = e |E + — (v x B) , l I c
* P. Faragó and L. Jánossy: Nuovo Cim., 5, 1411, 1957. ** Rogers, Reynolds and Rogers: Phys. Rev., 57, 379, 1940. *** Staub and others, Helv. Phys. Acta, 36, 981, 1963. **** V. P. Zreliov, A. A. Tapkin, P. Faragó: Soviet Physics, JETP 7, 384, 1958,
where E and B are the electric and magnetic field strengths. Secondly it is supposed that the force produces a change of momentum, i.e. f =
Supposing that
F
-
p = m(v)y,
with m(v) =
° yi-^/c ' m
2
we can write in place of (31) my
F=
\ _
0
J
dt j l
- v \c ) 2
mv
m„(vV)v/c
:
0
2
/ l - v jc 2
v
2
(1 -
W
2
We may alsó write
where _ m
l
_
fn
0
2 ~ ~T\ ..2f.2\3/2 ' (l-v2JcY' '
m, ' —
w (1 - «; /c ) 0
2
2
1/2
and = (vv)v/i; , 2
V l
v = v - Vi 2
are the components of the acceleration which are parallel respectively perpendicular to the velocity. We see thus that if we define the mass as the ratio of force and acceleration then the mass thus defined depends on the angle between velocity and acceleration. The extrémé cases are ni] the mass which appears when the partiele is accelerated in the direction of velocity and iw, the mass which appears when the partiele is accelerated in a direction perpendicular to v. 52. The real experiments can be divided into two types. In the experiment of Rogers and co-workers and alsó of Staub and co-workers particles are made to move along a circular orbit. So as to maintain a partiele of velocity c o n a circular orbit with radius R a radial force F = mv lR 2
is required. In the radial electric field F—Ee, while in a homogeneous magnetic field B perpendicular to the orbit we have F = eBv/c.
We may adjust E and B until we arrive at such a state that our beam of charged particles moves along the same orbit whether it is under the influence of the electric field E or alternatively under the influence of the magnetic field B. From the equations of motion we find for the fields thus adjusted eE = eBv/c = mv /R.
(32)
2
With the help of (32) we can determine both m and v. We find v = cE/B,
m = eRB^c
2
E.
Comparing the values of m obtained for particles with different velocities, the change of mass with velocity can be obtained. In the experiment of Zreliov et al. the velocity v of the proton beam was determined directly with the help of Cerenkov radiation and thus only the magnetic deflection had to be measured for to determine m. 53. We note that the above experiments give only information about the ratio ejm. Interpreting the various experiments it is always supposed that the charge e is independent of the velocity. We come back to this question in 280. While the assumption that the measure e of a charge remains unchanged if the charge is set to move is partly a definition, it contains alsó an element which can be checked experimentally. Without going into details we mention that in the Stern-Gerlach experiment neutral atoms possessing magnetic moments are deflected by a strongly inhomogeneous magnetic field; the field strength of the deflecting field is alsó considerable. The atoms thus deflected must be very exactly neutral as otherwise they would suffer very noticeable deflections by the magnetic field. From the fact that in the Stern-Gerlach experiment an atomic beam passing through a strongly inhomogeneous magnetic field can be satisfactorily focused, one concludes that the charge of the nucleus compensates exactly that of the electrons. Furthermore, because of the motion of the electric charge insides the atoms, it must be concluded. that the totál electric charge is not affected by the internál motion of the electrons.
CHAPTER II
INVESTIGATIONS CONCERNING THE CARRIER OF ELECTROMAGNETIC WAVES
A. THE QUESTION OF THE ETHER 54. From Maxwell's theory it follows that light in particular, and all electromagnetic action in generál, is propagated with a velocity c = c' where c' is the critical velocity. We shall analyse Maxwell's equations in more detail in chapL VIII and show there that the above conclusion is indeed an integrál part of Maxwell's theory. The question cannot be avoided relatíve to what are electromagnetic waves propagated with the velocity cl A simple answer to this question could be obtained claiming that light is propagated with the velocity c relative to its source. The latter assumption contradicts, however, the well-estabüshed theory of Maxwell and seems alsó to be contradicted directly by experiment. The latter assumption, sometimes referred to alsó as the "ballistic theory of light", must therefore be rejected. An electromagnetic perturbation once it has left its source is propagated thus with a velocity c independent of how the perturbation has come about. The only reasonable interpretation of this is to assume that the perturbation moves with a velocity c relative to its carrier. The carrier may be denoted, using Maxwell's terminology, the ether. We shall in accord with the ideas of Maxwell alsó assume that light is propagated with a velocity c relative to the ether. 55. So as to avoid misconceptions we wish to emphasize that we regard the ether merely as the carrier of electromagnetic waves and possibly of the waves associated with other fields and of elementary particles. In the last century a number of mechanical models were proposed so as to explain the properties of the ether by rather artificial mechanisms. Such models are meaningless as there is no reason why the ether should have properties like e.g. solids consisting of atoms and molecules. On the contrary one might suppose that the properties of atoms, molecules and solids are ultimately determined by the properties of the ether. We think of the ether more or less in a manner which was discussed by Einstein in a not very known article.* * A. Einstein: Über den Áther. Verh. d. Schweizer. Nat. Ges., 105, Teil II. 8 5 - 9 3 , 1924.
56. Einstein's polemic against the ether concerned mainly the assumption that the ether is at "absolute rest". Thus Einstein denied the existence of a system of reference K which is at "absolute rest". We think that the assumption that electromagnetic waves possess a carrier has nothing to do with the question of absolute rest. The concept of "absolute rest" is a metaphysical concept which must be rejected. However, the concept of the ether as the carrier of electromagnetic and other phenomena is quite a different one. 57. Maxwell thus supposed the carrier of electromagnetic waves to be the ether. He supposed electromagnetic waves to be perturbations of the ether propagated in a way somewhat similar to that in which sound waves are propagated in air. Using a terminology similar to that introduced by Einstein we may denote by K a system of coordinates which we suppose to be at rest relatíve to the ether, i.e. at rest relatíve to the carrier of the electromagnetic waves. As clearly explained above we do not suppose the system K to be that of "absolute rest". Whether or not the ether, i.e. the carrier of electromagnetic waves, is at rest or even at "absolute rest" is a question which does not arise here and certainly has no significance in relation to our problems. 0
0
0
"Der Áther der allgemeinen Relativitatstheorie unterscheidet sich alsó von demjenigen der klassischen Mechanik bezw. der speziellen Relativitatstheorie dadurch, dass er nicht 'absolut', sondern in seinen örtlich variablen Eigenschaften durch die ponderable Materié bestimmt i s t . . . Dass es in der allgemeinen Relativitatstheorie keine bevorzugten, mit der Metrik eindeutig verknüpften raumzeitlichen Koordinaten gibt, ist mehr für die mathematische Form dieser Theorie als für ihren physikalischen Gehalt charakteristisch." "Aber selbst wenn diese Möglichkeiten zu wirklichen Theorien heranreifen, werden wir des Áthers, d. h. des mit physikalischen Eigenschaften ausgestatteten Kontinuums, in der theoretischen Physik nicht entbehren können; denn die allgemeine Relativitatstheorie, an derén grundsátzlichen Gesichtspunkten die Physiker wohl stets festhalten werden, schliesst eine unvermittelte Fernwirkung aus: jede NahewirkungsTheorie aber setzt kontinuierliche Felder voraus, alsó auch die Existenz eines 'Áthers'." The translation of this remarkable statement is as follows: "The ether of the generál theory of relativity diífers from that of classical mechanics or from that of the special theory of relativity in so far as it is not 'absolute' but its spatial distribution is determined by that of matter. The fact that, in the framework of the generál theory of relativity, there are no distinguished space-time representations connected in an unambiguous manner with the metric — is rather a characteristic of the mathematical methods of the theory than a characteristic of its physical contents." "However, even if these possibilities developed into a real theory we shall not be able to dispense, in the field of theoretical physics, with the ether, i.e. a médium which possesses physical properties; indeed, the generál theory of relativity — to the principles of which physicists will probably always adhere — excludes any direct distant action. Every theory based on close action supposes the existence of continuous fields thus they suppose alsó the existence of an 'ether'." (My own translation.)
4 Relativity
Furthermore, for our considerations it is alsó immaterial whether or not various parts of the ether move relative to each other. It seems quite plausible that considered on. a cosmic scale distant parts of the ether are streaming with various velocities and thus the system K , we consider, has only local significance. K is supppsed to float together with the ether. We may assume that in a certain vicinity of the origin of K the ether has negligible velocity relative to K . 0
0
0
0
B. EXPERIMENTÁL INVESTIGATIONS 58. The question arises whether it is possible to determine experimentally the state of motion of the ether in somé definite region. In particular the question arises whether it is possible to ascertain the state of motion of the Earth relative to the surrounding ether. To answer the question, we have to consider separately the effects of the rotation of the Earth and its translational motion relative to the ether. 1. ROTATION OF THE EARTH
59. The fact that the Earth is rotating around its axis can be seen from the apparent motion of the stars in the sky. The rotation can alsó be observed by mechanical experiments carried out on the surface of the Earth, i.e. with the help of Foucault's pendulum, or by observing the motion of a fast rotating gyroscope. It is interesting that the rotation of the Earth can alsó be observed by optical experiments. The experiment in question is the extended form of the experiment of Sagnac (1913), i.e. the experiment of Michelson and Gale (1925). We give a description of the experiments. We note that if we speak of the rotation of the Earth, we imply thus rotation relative to the carrier of electromagnetic waves, i.e. rotation relative to the ether. 2. THE S A G N A C EXPERIMENT*
60. Before describing the real experiment we describe a schematical version of it which will elucidate its essential features. Consider a disc of radius R rotating with an angular velocity co around its axis. Suppose a large number of mirrors arranged on its periphery in such a way that a light signal starting, say, from a point A of the periphery is guided along a path very nearly coinciding with the edge of the disc. * G. Sagnac, Compt. Rend., 157, 708, 1913; J. de Phys., 5, 177, 1914.
If the disc is at rest a signal starting at the time t = 0 from a point A on the periphery arrives back into A at a time T = InRIc. [f, however, the disc is rotating with an angular velocity co and the light signal is moving in the direction of rotation it will reach at the time T = = 2izR/c a point A located in the place which A had left at t = 0. The signal has to catch up the point A which is moving away and it will be reached by the signal at a later time T so that 0
+
cT
+
therefore
= 2nR + RcoT , +
2%R
=
=— > c — Rco T could alsó have been obtained by supposing that the light signal moves relatíve to the edge of the disc with a velocity c+ = c - Rco. (1) T
T
+
We emphasize, however, that we have made no assumption about the velocity of the beam relatíve to the disc but have simply calculated the time the signal starting from A needs to catch up again the point A moving away. If the light signal moves in the opposite direction it reaches A sooner than at / = T as the point A moves then towards the signal. In this case we find for the time at which the signal reaches A T~ =
2nR — < T.
C + RÍO
The latter result could be obtained directly if we were to assume that the signal moves with the velocity c~ = c + Rco,
(2)
relatíve to the edge of the disc. Here again we note that (2) üke (1) is not an assumption but follows from the consideration of the time of flight of a signal. The difference in the times needed to circle around the disc in opposite directions is thus r + - T~ = 2nR
L-L
L_] = ***V „ 4Sa>/c>
\c-Rco c + Rcoj where S = nR is the area of the disc circled 61. Sagnac carried out his experiment with schematically in Fig. 11. A light source, mirrors 2
c - R co round by the beams. an arrangement reproduced and a telescope are mounted 2
2
2
1
on a disc which can be made to rotate. The beam leaving the source S is split into two coherent components by the semitransparent mirror SM. The components are led around a square path with the help of the mirrors Mi, M and M . After having circled round in opposite directions the beams meet on SM, and split again. The components of the two returning beams which are moving back towards the source are led with the help of a second semitransparent mirror SM' into the telescope T, where an interference pattern is produced. Just üke in the case of the Michelson interferometer 2
3
M
2
Fig. 11. Scheme of the Sagnac experiment
(see 31) the interference pattern can be used to determine the difference of the times of füght of the beams circling round SM, M M , M in opposite directions. In the originál experiment carried out by Sagnac and that repeated later more carefully by Pogány* (1926) the phase difference between the two beams was observed when the arrangement was at rest. The arrangement including light source and telescope was made to rotate with an angular velocity co and a shift of fringes of magnitude 1;
2
3
AX = 4Sco/c was found. S is the area which the beams are made to circle round. This shift is exactly what is expected from the calculations. * B . Pogány: Ann. d. Phys., 80, 217, 1926; 85, 244, 1928; Naturwiss., 15, 177, 1927.
3. THE EXPERIMENT OF MICHELSON A N D GALE*
62. The experiment of Sagnac was developed further by Michelson and Gale who succeeded in observing the effect of the rotation of the Earth with the help of an interferometer of the Sagnac type. The angular velocity of the Earth is about 0.8 • 10~ s e c ; the distance between the mirrors were of the order of 1 km, thus the effect to be expected was 4
-1
JA = 10 000 Á and in the experiment a shift of about / fringe was found. It must be noted that the experiment is carried out on the rotating Earth; it is impossible to "stop" the Earth while adjusting the arrangement. The required adjustment is, however, rendered possible by the fact that the effect of rotation increases with increasing area S. In particular for a return beam the encircled area is zero and the effect of rotation is not felt in this case. One might in principle adjust the lengths of the sides of the square path with the help of return signals and measure the fringe shift obtained with beams circling round the square path, the sides of which have thus been adjusted. In the actual experiment Michelson and Gale compared interferences obtained with beams circling round a smaller rectangular and a larger quadratic area. From the difference of the observed fringe shifts they could determine the angular velocity of the system as a whole. The Michelson-Gale experiment can be repeated by making use of lasers. The fringe shift thus obtained can be observed more easily because the line width of the laser beam is extremely small. 1
i
C. TRANSLATIONAL MOTION RELATÍVE TO THE ETHER 63. Unlike the rotation of the Earth its translational motion relatíve to the ether cannot be observed by mechanical experiments. The fact that the translational motion of a system cannot be observed was already recognized by Galilei.** We quote an interesting passage: "For a final indication of the nullity of the experiments brought forth, this seems to me the place to show you a way to test them all very easily. Shut yourself up with somé friend in the main eabin below decks on somé * A. A. Michelson and H. G. Gale: Astrophys. J., 61, 140, 1925. ** R. J. Seeger, Galileo Galilei, his life and his works. Pergamon Press, Oxford, 1966, pp. 2 3 6 - 3 7 .
large ship, and have with you there somé flies, butterflies, and other small flying animals. Have a large bowl of water with somé fish in it; hang up a bottle that empties drop by drop into a wide vessél beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fali into the vessél beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you teli from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow, even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fali as before into the vessél beneath without dropping toward the stern, although while the drops are in the air the ship runs mony spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning somé incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air alsó." At first sight it may appear as if the translational motion of the Earth could be observed by optical experiments. Although in reality this cannot be done it is necessary to analyse the question in greater detail. The experiments of Fizeau and Foucault determined the velocity of light relative to the solid Earth. The question arises whether or not the motion of the Earth relative to the ether effects the results of the experiments? The question is not a trivial one as can be seen from the following argument. 64. Consider a signal of light to be emitted at the time t = 0 from a point A which we suppose to be at rest relative to the ether; the signal will reach 0
a point B which is alsó at rest and is situated at a distance / from A at a time T = l/c. 0
0
Considering further two points A and B both moving with a velocity v in the direction A -> B such that at the time t = 0 A coincides with A and B with B , then the time T which is needed for the signal emitted from A to reach B differs from T. Indeed supposing the points A and B to move along the x-axis of the system of reference their respective coordinates can be written as 0
0
0
+
0
XAQ)
=
>
VT
B(()
X
= vt + I,
v>0.
(3)
The signal emitted at / = 0 from A will have at the time t a coordinate x {t) = ct.
(4)
s
When writing down (4) we remember that the propagation of light is independent of the motion of the source, therefore (4) gives the motion of the signal no matter whether it was emitted from A or from A . (We may e.g. suppose that when A approaches A$ a spark takes place between A and A and the light of the spark gives the signal.) The signal reaches at the time t = T = l/c the point B but it does not reach B since B has moved away from B during the time T. The signal reaches B at a time t = T > T, T satisfies the condition 0
0
0
0
+
+
*s(T ) = x (T ). +
+
B
(5)
Thus introducing into (5) the expressions (3) and (4) we find r+ = //(c
-1-).
(6)
If we consider a signal moving from B towards A, then we find with the help of a similar argument that the time of travel T~ from B to A is equal to T~ = U(c + v).
(7)
We see thus that T > T > T~ therefore the time of travel between the points A and B is affected by the motion of these points relatíve to the ether. 65. Relations (6) and (7) can alsó be interpreted as follows. Consider a number of points C, D, . . . which all like A and B move with the same constant velocity v relatíve to K. The points A, B, C, D,.. . define a system of reference K' relatíve to which all the above points are at rest. We can define as the velocity of light relatíve to K' in the direction from +
A -* B respectively B -» A as c+ = //r+,
c- = //r-.
(8)
Thus we find from (6), (7) and (8) c
+
= c—
V,
C~ = C + V.
Relations (6) and (7) are based on a purely phenomenological consideration. The argument leading to (6) is the same as if we were asked the following question: a dog runs with the velocity c after a hare which runs with a velocity v < c, how long will it take that the dog reaches the hare, if the initial distance is /? When discussing the latter problem it is quite immaterial what runs after what? Whether the dog catches up with the hare or whether a light signal catching up with the moving Earth. Furthermore it is evident that if the hare loses its nerve and starts to run towards the dog instead of running away, the dog will catch the hare more quickly than if it was running away from him, thus T > T~ in any case. The above argument is supported — if support is at all needed — by the Sagnac experiment where the delay is directly observed with which the light going round a closed path reaches the semitransparent mirror moving away from it. +
1. PROPAGATION OF A SPHERICAL SIGNAL
66. Formulating the problem a little more generally we may state that a light flash emitted at a time t = 0 will expand and it will be found at a time t distributed on a sphere with points
r (0 = ch\
(9)
2
5
Considering a point B moving with a velocity v the coordinate vector of which is given as = 1 + vf,
x (i) B
(10)
we find that the flash emitted at t = 0 from a point A the coordinate vector of which at t = 0 is given by r (0) = 0, A
will reach B at the time T
+
such that r (T+Y = r (r+) s
B
2
or with the help of (9) and (10) (1 + v r + ) - c T 2
2
+2
= 0.
(11)
From (11) it follows /,
V
c
•2
2
V Q
sir # c
1
+ — COS 1 2
where we have written vl = vl cos thus # is the angle between I ard v. The time of travel T~ of a signal from B to A is obtained by replacing •& by n — thus we find ,1
/ V
r- =
d
2
sin # = °-
We note that Rif)l{t' — f) ~ c, thus neglecting small terms of the order (D/C) we find from the above relation 2
dt' , vR(/') — « 1 +— dt c 2
We expect therefore the radiation emitted with the constant frequency v from A to arrive in B with a varying frequency Y(f) so that .
vR(í') c
V
)
(29)
2
If the emitter ^4 is moving with the Earth the frequency of radiation received in B should fluctuate periodically in accord with (29). In the experi-
ment of Isaak and co-workers a y-emitter was placed in the centre of a disc and an absorber B was placed on the edge of the disc and resonance absorption was observed with the help of the Mössbauer effect. If the frequency of the radiation received in B would fiuctuate indeed according to (29), then the resonance would take place only during part of the motion and only the measure of the absorption would be expected to vary with the position of B relative to A. 86. In the actual experiment no effect on the absorption was found, when the disc supporting the absorber was made to rotate. The experiment was sufficiently accurate that a small fraction of the effect expected according to (29) could have been observed. v+w
Y
r v-w
\
Fig. 13. The orbit of the source used in the experiment of Isaak and co-workers
The negatíve result of the above experiment can be understood supposing that the frequency of the absorber B changes with the velocity of B in a manner as it suggests itself from the perpendicular Doppler effect. Indeed, if the inner frequency of the absorber depends on its velocity w(í) = v + R(í),
(30)
so that v*(0 = v ^ 1 - " ^ . o
(31)
then we find inserting (30) into (31) and neglecting small terms v*(í) = v 1 0
vR(0) c
2
and we find thus
r
v*(í') = v'(0, .e. the frequency v*(í') of B at any time t' will be equal to the frequency v'(í') falling upon B. Summarizing the above considerations we can thus state the following. If the emitter A moves with a constant velocity v relative to the ether, then the absorber B circling round A moves along a cycloid path (see Fig. 13).
The phase surfaces emitted by A in equidistant times will reach B in a sequence corresponding to a periodically varying frequency v'(f). However, the velocity w(í) = v + R(í) of the absorber along the cycloid alsó changes periodically and if the inner frequency \*{f) of B adjusts itself at any instant t' to the velocity of motion of w(í) according to (31), then the radiation of A falling on B with periodically changing frequency v ' ( 0 will be exactly in resonance with the changing inner frequency v*(/') of B. Thus we find just as in the cases of the Michelson-Morley and TroutonNoble experiments that alsó in the experiment of Isaak and co-workers the effects which occurred on purely geometrical grounds are compensated by other effects and therefore the two types of effects caused by the translational motion of the system compensate each other and no observable effect remains. E. GENERAL REMARKS CONCERNING THE SERIES OF NEGATÍVE RESULTS 87. The series of failures to find effects suitable for to determine the translational velocity v of the Earth relatíve to the ether might appear at first sight as a series of "accidents". In this view the Michelson-Morley experiment might have been suitable for to determine v, but for the "accidental" fact that the arms of the interferometer deform. Similarly the Trouton-Noble experiment is rendered inconclusive by the apparently accidental fact that mechanical torques appear to compensate the electric ones. The fact that the mechanical torque should change exactly to the same amount as the torque inside the suspended condenser must be regarded as a rather remarkable fact which, however, is proved directly by the experiment itself. In the Isaak experiment we are prevented from determining v by the apparently accidental fact that the inner frequencies of the absorber vary with the velocity w(í) of the absorber relatíve to the ether. Similar apparent accidents prevent the determination of v by a number of other methods. In a scientific analysis a series of apparent accidents cannot be accepted as real accidents. 88. So as to remind of precedents in the history of science, we note that all the attempts to build perpetuum mobile seemed at that time to be prevented by a series of "accidents". Somehow the proposed mechanisms did not work because the one or the other disturbing circumstance was underestimated. The fact that this series of accidents were not real accidents but they reflected somé generál law of nature, was first clearly pronounced when the
French Academy of Sciences declared that it will not be concerned in future with perpetuum mobiles. The above declaration is only a negatíve result. It was realized that the series of failures encountered cannot be accidental but that there must be a generál law of nature which makes it impossible to construct such a machine. The second step was to find and to formulate the law underlying the negatíve experiences. This law was the law of conservation of energy. It must be emphasized that the law of conservation of energy is a law which (as far as we know) is valid for all phenomena. This law gives, however, only a framework for many particular laws of physics without determining these laws precisely. 89. It must be noted that should one find somé new phenomena where the conservation of energy is not satisfied, then this phenomenon would restrict the validity of the conservation law without making the law itself invalid. Indeed, the law of conservation of energy has proved its validity upon a very large field of phenomena and thus it is clear that this principle does indeed refíect correctly an aspect of laws of nature. Its success would not be undone even if we were to find somé particular phenomena outside its validity. The above remark can be supported if we remember the activities of the alchemists. Trying unsuccessfully to produce gold, the alchemists found a large number of laws of chemistry. Eventually it was realized that there exists a law of nature which had made the efforts of the alchemists futile. The law itself is that the chemical elements consist of indivisible atoms. Today we can produce gold from suitable elements by nuclear chemical methods. Nevertheless the laws discovered by the alchemists remain the foundation of chemistry. The fact that we know now the limitations of a law does not affect its importance inside its region of validity. 90. It is often pronounced as a principle of science that a hypothesis can only be maintained until no single fact is known which contradicts it and it is claimed as soon as there is one fact which contradicts a given hypothesis that the hypothesis must be dropped. It is a fortunate thing for the development of science that this principle is only pronounced in textbooks but is not taken seriously by scientists. Indeed, if a hypothesis proved its worth by explaining a series of facts then this is an indication that this hypothesis reflects correctly at least part of reality and therefore such a hypothesis has a content of reality. If new phenomena seem to contradict the hypothesis a careful analysis is necessary to the effect of whether and how the new phenomena affect the validity of the hypothesis. It is very important for the stability of scientific progress that hypotheses which have proved their worth are not thrown away when the first difficulty arises.
1. GENERALIZATION OF NEGATÍVE EXPERIENCES
91. We have brought a number of generál remarks upon methodological questions so as to make more clear our procedure. From the series of negatíve results like that obtained by Michelson and Morley it seems reasonable to suppose that it is not accidental that various experiments trying to determine v, the velocity of the Earth relatíve to the ether, remained unsuccessful. And we may accept the view that there exists a generál law of nature which prevents us to determine v by laboratory experiments. As a second step we have to formulate this law in a positive form. The usual formulation of this law is the principle of relativity by Einstein. We shall, however, give an alternative formulation of the generál law which we shall denote the Lorentz principle. We shall give the formulation of this principle further below.
CHAPTER III
THE PROBLEM OF MEASUREMENT
92. In 75 we have claimed that the Michelson-Morley experiment amounts to a measurement of the Lorentz contraction. So as to support this statement more deeply and alsó for to formulate the Lorentz principle, it is necessary to analyse the problem of measurement in somé detail.
A. THE PROBLEM OF MEASURES 1. REPRESENTATIONS
93. Presently we give a generál analysis of measures and quantities. In these considerations we want to distinguish clearly between the real quantities* which exist objectively independent of whether or not we attempt to measure them and the measures of quantities, or shorter measures which describe the quantities by means of numbers (or sets of numbers). It is quite clear that real quantities can be described in very many ways by numbers; we have merely to try to choose for the description of particular quantities numbers which reflect adequately certain physical properties. 94. To make the distinction between quantities and measures quite clear we shall use Gothic symbols for the real quantities and Latin letters for their representations. Thus we may write, e.g. e,
%
G, • • •
(i)
for certain real quantities and write
m) = £, xm = p>
*(&) = e . . .
for certain representations. E,P,Q . .. are thus numbers or sets of numbers describing the quantities (1) in a representation R. The quantities (1) can * We use the term "real quantity" because we cannot find a better expression. We understand as explained in the text the quantity as it exists objectively as distinct from its measure.
(2)
alsó be described in a different representation, say R', and we have thus
R'(&) = E', R'{%) = P\ R'(£i) = Q'... The E', P', Q are of course functions of the E, P,Q; the latter functions give the transformation between the representations R and K. 95. In particular if © stands for an event then it can be represented relatíve to a system of coordinates K by K{ 3 charges e e , . . ., e„ then we can determine the measure e of e in more than one manner. We find thus with the help of (7) 1;
k
2
k
'
1
'
lm
i
l' ' m
where k, l, m and k, /', m' refer to groups of three charges with the help of which we attempt to determine e . From (8) we see that e can be determined in a number of different ways. We must expect that the various methods for determining e lead to the same numerical value. In other words relations (8) give a check of consistency of the Coulomb formuláé (7). If we carry out measurements with one fixed distance r only, then the checks support the assumption that the force acting between two charges e and e, is indeed proportional to the measures e and e, of the charges. 98. Sometimes it is claimed that relation (7) gives simply the definition of the measures of charges. This claim cannot, however, be maintained altogether. — Indeed we have seen that relation (8) provides checks of k
k
k
k
k
relation (7); if e.g. having four charges and we were to find from measurement that ^12^13 TI ••23
, ^12^14 '
^24
then we are led to a result contradicting the law (7) in a particular instance. Thus (7) is not merely a definition; nevertheless it contains a definition also. Indeed, if a set of checks of the form (8) is satisfied, we cannot conclude that Coulomb's law has necessarily the form (7). If we were to assume instead of (7) a law of the form *u =
p
,
(9)
where q>(e') is a monotonous function of its argument, then starting from (9) in place of (7) we find in place of (8)
where
0.
The r' are the lengths expressed in a new scale. We may also introduce a new scale using a factor a < 0. Doing so we reverse our convention and ascribe smaller measures to larger rods. Such a convention is unusual but nevertheless is internally consistent. kl
B. SYSTEMS OF SPACE COORDINATES 1. DETERMINATION OF COORDINATE VECTORS
110. We may now analyse the method of how to construct three-dimensional systems of coordinates. Consider for this purpose a set of fixed points . .., ^S„. The distances between the pairs of points can be measured with the help of rods and thus we can obtain (additive) measures K(*ki) = r„
k,l = 0, 1, 2 , . . . N
of the distances between various pairs ?$ and % of points. k
We expect the measures thus obtained to obey the relations +r
ri k
lm
r
=
k m
(18)
,
where the equality sign stands if the points ty , ty, and ty lie along a straight line. If the relations (18) are found to hold for the measures of distances taken for any group of three points then this can be taken as a qualitative check of consistency of our method. The above result supports in particular the assumption that our measuring rods behave like solids. 111. For to obtain measures of the coordinates of the points ty we may introduce relative position vectors x so that certain vector quantities k
m
k
kI
= *« define (in a representation K) the position of % relative to % . From experience we can take it that the position of a point relative to another is given by a three component quantity, therefore we suppose t
OT/,2' kl,3
kl — kl,l>
T
k
r
r
thus r i = 1, 2, 3 are the components of r . We may try to look for such a representation K in which the relative position vectors are additive, i.e. such that kIi
w
r« + f/m = r*m-
(19)
The relation (19) stands for three relations, i.e. for one relation for each of the three components of the vectors. The relations (19) are automatically satisfied if we suppose r « = i-/ -
r
k = 0, 1,. .., n
/,
fc
(20)
where r
/
=
r
k
0/»
T
0k •
=
T
We can take the r as the coordinate vectors of the points ?$ k = 0, 1, 2 , . . . the point ty with coordinate vector r = 0 being taken as the origin of K. 112. So as to obtain a statement which can be checked experimentally we make an assumption as to the connection between the position vectors r and the measures r . The distances r are supposed to be measured with solid rods, therefore we suppose the law of Pythagoras to be valid, thus fc
k
0
kl
0
kt
r£, =
kl
for
rf,
k, 1 = 0, 1 , . . .,
where we have written short 3 kl
T
=
X kl,i • I=L r
n
(21)
Relation (21) is valid in an orthogonal system of reference. In terms of skew coordinates we can suppose in the place of (21) rf, = T Gr kl
(22)
kl
where G is a symmetric and positive definite mátrix. In particular supposing G = 1, (22) reduces to (21). The following considerations will be carried through in representations where we do not specify whether they are orthogonal or skew. In this way we can considerably simplify the actual calculations and the results can be applied in orthogonal representations if this is desirable. For the moment we take thus (22) as a hypothetical connection between coordinate measures r and measures r of distances. The precise meaning of "orthogonal" respectively "skew" representations will be elucidated further below. 113. So as to check (22) experimentally we can write in place of (22) making use of (20) also w
kl
(r.-rOGfo-r*)^,.
(23)
In the case of N + 1 points ty , ^ . . . , ty the relations (23) give N(N + + l)/2 equations for the 3N components of the vectors r k = l,2,...,N. For sufficiently large values of N the system (23) becomes overdetermined and from the purely mathematical point of view (23) need not admit of solutions. If nevertheless in a particular case the overdetermined system (23) does admit of solutions, then this circumstance cannot be taken to be accidental. If in a given case relations (23) do not lead to contradiction this circumstance reflects on the properties of the measuring rods used. We may conclude (generalizing the result of 108) that the measuring rods can be taken to behave like ideál solids — if the measures of distances r obtained with them can be expressed in terms of quadratic expressions of the form (23). 0
u
N
k
kl
2. EXPLICIT DETERMINATION OF COORDINATE MEASURES
114. Explicit solution of (23) can be obtained in the following manner. Consider four points % (24) which four points should not lie in one pláne. (How this can be ascertained for four given points will be discussed further below.) We may consider a coordinate system such that ^ fixes its origin and the axes of the system lie in the direction $ ^P/t k = 1, 2, 3. Furthermore choosing the units along the axis suitably we can take the coordinate vectors of these points to be given by 0
0
r = 0,0,0, 0
r
1
=
1,0,0,
r = 0, 1, 0, 2
r = 0, 0, 1. 3
(25)
Writing down (23) for k = 0, / = 1, 2, 3 we find G„ = rf,
/ = 1, 2, 3
(26)
where we have written r, in place of r . Making use of (26) we obtain from (23) for k, l = 1, 2, 3 0/
G = j{ri
+ r?-rl).
u
(27)
Thus (26) and (27) give the elements of G if the coordinates of the four points (24) are to be given by (25). It will be necessary to suppose further below that det G # 0.
(28)
Whether or not the above relation stands, depends on the numerical values of the distances r in terms of which we express the elements of G. We have to require therefore that the points (24) should have distances in terms of which (28) is fulfilled. The latter requirement is equivalent to the requirement that the points (24) should not lie in one pláne. Thus we have to start our procedure with four points (24) with mutual distances satisfying (28). 115. Considering five points M
*o. %,
%
$3,
m>3.
m
(29)
Writing down (23) for the mutual distances we obtain relation 4 - 2(Gr ) + r G r = r | . m
fc
m
m
(30)
m
Supposing for the moment that r Gr = r ,
(31)
2
m
m
we can write in place of (30) G r = DC"> m
where D
( m )
is a vector with components ^ " Y ^ + r i - r í J
* = 1,2,3,
m>4
(32)
thus r m
=
G
-i
D ( m )
The relation (33) gives explicitly the coordinate vector r in terms of measured distances only.
( 3 3 )
m
of a point ty„
3. QUESTION OF CONSISTENCY
116. Introducing (33) into (23) we find that the relations (23) are indeed satisfied for the coordinate vectors of the five points (29) provided (31) is satisfied. Introducing (33) into (31) we find a relation D( >G- D > = r . m
1
(m
(34)
2
m
The latter relation gives a quadratic equation for Solving the above equation, we can express r in terms of the remaining nine distances between the five points (29). [Since (34) is a quadratic equation, we obtain in generál two solutions for r . ] If the measured value of r coincides with one of the solutions of (34), then (25) and (33) give a consistent set of coordinate vectors for the points (29). Relation (34) gives thus a test of consistency; it shows whether or not the distances between the five points (29) can be expressed in a form (23). 117. Adding a sixth point to the five points (29) we can determine the coordinate vectors r„ of *)3„ as m
m
m
r„ =
G-W>
where the elements of D are obtained from an expression of the form (32). The coordinate vectors thus obtained can be taken to be consistent provided we have apart from (34) also (n)
r Gr„ = r\ n
and
r„Gr = i - (r* + r%- r * J . m
Increasing the number of points we can construct expressions which give the values of the coordinate vectors provided a consistent set of coordinate vectors exist at all. In case of N > 4 points, the coordinate vectors obtained have to fulfil
conditions. 4. VARIOUS REPRESENTATIONS
118. It remains to investigate how far our procedure is affected by the arbitrary choice (25) of the coordinate vectors of the four standard points. We note first that equation (33) is only meaningful if (28) is fulfilled, i.e. if G exists. We have thus to choose the points S$ so that the determinant of the mátrix G should not vanish. It can be taken that the latter condition is fulfilled if we choose the four standard points (24) so as not to lie in one pláne. - 1
k
det G = 0
(35)
as the definition for points to lie in one pláne. If det G ^ 0 we find it to b positive definite. 119. If we succeed in determining coordinate vectors r k = 0 , 1 , 2 , . . . , N to N + 1 points such that these vectors satisfy all the relations (23) then we can take linear transforms fc
r' = Sr + s k
k = 0, 1, 2,...
k
N
(36)
alsó as consistent measures of the coordinate vectors. Indeed from (36) it follows that r - r, = S-V/c' - *í) = W - O S " .
(37)
1
fc
Introducing thus (37) into (23) we find (f * ~ ''i)G'(r - r',) = rli
(38)
k
with G' = S - G S - . 1
(39)
1
Thus we see that provided the coordinate vectors r satisfy the overdetermined systems (23), then the transformed coordinate vectors r' satisfy the relations (38). 120. When constructing coordinate measures for the coordinate vectors t of the points ty we assumed in 114 (25) particular values for the measures of the coordinate vectors r of the points ?$ k = 0, 1, 2, 3. If we were to assume in place of (25) that in somé representations K' the coordinate vectors are given by k
k
k
k
k
rí =
r ,
r'
kr2
ktS
k
k = 0,1,2,3
(40)
where the r' are chosen arbitrarily, then we can regard the r' as transforms of the originál r^, the transformation having the form kJ
k
t = Sr + s k
fc
k = 1, 2, 3
with $ik — k,i ~ 'o,t r
r
— ó.i r
k,l=
1,2,3.
(41)
Applying the transformation (41) to any of the coordinate measures of ty in the representation K we obtain its representation in K'. The representation K' could alsó be obtained directly by assuming the values (40) in place of the values (25) for the coordinate measures of the points ty k = 0, 1, 2, 3. The only restriction to be imposed on (40) is that the coordinate measures have to be chosen so that k
k
dct|r^-.ri|5É0 the latter condition is necessary for to make det S # 0 and to make the transformation (41) a reversible one. From the above considerations it follows that providéd consistent coordinate vectors r can be introduced, supposing distances to be given by (23) then consistent coordinate vectors can alsó be found if G is replaced by G' as defined by (39). 121. For given matrices G and G' we can determine S so as to satisfy (39). We may write e.g. S = G'- G . k
1 / 2
1 / 2
We see that if it is possible to construct consistent measures óf coordinate vectors for one assumed value of the mátrix G then it is possible to find consistent representation for any other choice G' in place óf G. We conclude therefore, that we can decidé by measurement whether or not it is possible to represent a set of measured distances r by a positive definite quadraticform of the difference of coordinate vectors. No information can be obtained about the elements of the mátrix G occurring in the quadratic form. In particular we can obtain orthogonal coordinates in which G' = 1 if we put r' = G ' !. u
1 2
Thus the coordinate vector of the point ty„ in the orthogonal representation is obtained as r
'
m
=
G
-l/2 (m)_
(42)
D
122. So as to formulate our results in a more generál form we denote @ the metric tensor. We say that ® is represented in a system of reference K by K(®) = G, where G is always a symmetric positive definite mátrix. The representations G = K(&)
and
G' = K'(®)
of the metric tensor are connected by a relation G' = S ^ G S -
1
provided the coordinate measures relatíve to K and K' are connected by r' = Sr + s. In particular we can obtain an orthogonal representation as described in 121. C. PROBLEMS CONNECTED WITH COORDINATE REPRESENTATIONS 1. REMARK O N " N O N - E U C L I D E A N " GEOMETRY
123. The above statements can also be formulated in another way. If the measured distances r between the points of a set can be expressed by a quadratic form (23), then one might conclude that the space in which the points are situated is "Euclidean". Or if no consistent coordinate measures can be obtained one might conclude that the space involved is "nonEuclidean". We do not think, however, that such a conclusion has any meaning. The fact that the overdetermined system (23) possesses solutions r , k = 0, 1, 2 , . . . , n seems to us to reflect upon the method of measurement of the distances r and in particular upon the physical properties of the measuring rods used. Roughly speaking one may conclude from the consistency of the measures that the measuring rods made use of are behaving like rigid bodies, i.e. if the measuring rods are turnéd or shifted they do not change their length. Of course the procedure described provides only necessary conditions for the rods to behave like rigid rods. 124. A further aspect of the question is as follows. As relation (23) is a (generalized) form of the law of Pythagoras we come therefore to conclude that the latter law can be tested experimentally. This statement appears at first sight paradox as the law of Pythagoras is usually proved with the help of the axioms of geometry. In fact no paradox is involved. The axioms of geometry simply reflect the properties of ideál solids. The experimentál test described above is a test to the effect that our measuring rods behave like ideál solids. u
k
kl
2. COORDINATE TRANSFORMATIONS A N D DEFORMATIONS
125. nr>
x
m
Let us consider a system of points % with coordinate vectors = 1. 2 , . .., n. In a particular representation K we have m
K(tJ
= r . m
We obtain the coordinate vectors in another representation K' with the help of a linear transformation; we may write K\i ) m
= t' = Sr + s. m
(43)
m
A linear transformation of the coordinate vectors may alsó be interpreted in a different manner, by writing in place of (43)
C = Tr
m
+ t
(44)
where detT # 0 and t is a constant vector.
5*
Fig. 14. Scheme of a deformation
We can regard the r* as the coordinate vectors of points ty* m = 1, 2 , . . . , / ! in the representation K. Thus we may suppose In the above sense the transformation (44) produces from a system of points fy *Js , . . . , $ „ another system • • •> The transformation (44) describes thus a deformation of the configuration of a set of points: the deformation being expressed in measures relative to K (see Fig. 14). 126. Let us consider the deformation (44) in measures of coordinate vectors of two systems of reference K and K'. Thus suppose lt
2
K(x) = r,
K\x) = r-
and r and r' are connected by (43). Transforming both sides from (44) according to (43) we find Sr* + s = STr + St + s, m
(45)
expressing r from (43) we have m
r = S - ^ - S - V
(46)
m
Inserting (46) into (45) we find C = Tr'
m
+ t\
T' = S T S -
(47) (48a)
1
t' = (1 - T')s + St.
(48b)
Relation (47) gives the connection between r*' = *'(r*)
and
r' = K'(t ) m
m
i.e. it gives the connection between the coordinate vectors of $ in the representation K'. 127. In a more generál notation we may also write
mj=r ,
m
and ^3*
(49)
m
where % stands for the deformation which shifts the points S$ into points ty*. The representations of % relatíve to K and K' can be written m
K(%) = T, t K'(%) «= T', t'. The relations between the representations T, t and T', t' of £ are given by (48a, b). The points ty m = 1, 2 , . . . n can also be taken to be the points constituting somé physical system m
a = %,
Applying the deformation operation 2 to the points of & we obtain another system q* = ..., «p;, and thus we may also write D* =
£(&)
where £}* is a deformed and displaced version of |Q the operator £ giving the deformation.
3. ORTHOGONAL TRANSFORMATIONS a. DEFINITIONS
128. In particular we may be interested in deformations, which we shall denote by £) which leave the distances i , between pairs of points ty , of JQ unchanged. Writing O in place of T we find from (44) k
k
r* = Or* + t,
(50)
k
and therefore
*ti =
thus
Or , w
r£ = r Ő G O r . 2
w
(51)
w
Provided O satisfies the relation ŐGO = G
(52)
we find from (51) r
r*i = ki
k,l=\,2,
...
Thus the deformation mátrix O produces deformations which leave the measures r unchanged. We denote such matrices orthogonal matrices and transformations (50) orthogonal transformations. In particular in an orthogonal representation K with G = 1 (52) reduces kl
0
to
ŐO = 1. 129. Equation (52) gives the definition of the representation of an ortho gonal mátrix in one particular system of reference. We may write O = X(£>). From (48a) and (39) we find 1
O' = S O S ,
1
1
G' = S G S ,
(53)
where O' = K'(£>)
and
G' = K'{®).
Thus introducing (53) into (52) we find Ő G O ' = G'. We see therefore that reference in which we 130. Relation (52) However, since G is a
the definition (52) is independent of the system of represent © and £). gives nine equations for the nine elements of O. symmetric mátrix only six out of the nine equations
are independent and the matrices obeying (52) form a set depending on three independent parameters. We may denote this by writing O in place of O where p stands for the parameters. We may also write £) for the mátrix giving a particular deformation and p
p
K(£) ) = O
K(ti) = p.
or
p
v
Thus we write p for the parameters defining a particular orthogonal de formation mátrix and write p for its representation in a system K. b. GROUP CHARACTE R OF ORTHOGONAL MATRICE S
131. The orthogonal matrices O , O , . . . form the socalled ortho gonal group. Indeed, taking the determinant of both sides of (52) we find since det G > 0 det O = ± 1 . (54) p
q
We see thus that there exist orthogonal matrices with determinant + 1 and others with determinant —1. From (54) it follows that any orthogonal mátrix O possesses an inverse O p ; thus multiplying (52) from the left with Op and from the right by Op we find l
p
Őp^GOp = G, 1
thus Op" is also an orthogonal mátrix. Furthermore if O orthogonal matrices then 1
ŐpGOp = Ö „ G O , = G
p
and O
q
are (55)
thus the product O, = O O p
q
is also an orthogonal mátrix. Indeed, multiplying (52) from the left by O , and from the right by O remembering O = O Ő we find q
r
q
p
Ő G O = G. r
r
The unit mátrix obeys also (52) and matrices are always associative, thus we see that the matrices obeying (52) fulfil the postulates of a group and thus they form indeed a group. The matrices with det O = + 1 form a subgroup which subgroup may be called the proper orthogonal group. Since the O form a group thus their transforms to another system of reference, i.e. the matrices p
O . = SOpS" , p
1
(56)
form alsó a group. The relation (56) defines alsó the transformation of the representation of the parameters p from one system of reference to another. 132. In particular we may consider coordinate transformations S taken with orthogonal matrices S = O . We write O in place of O, to signify that 0 ( , ) does not represent an operator but is the mátrix of a coordinate transformation. In case of one orthogonal coordinate transformation we may write in place of (56) ( only. As can be seen easily an orthogonal deformation r* = O r„ + t p
can always be regarded as a shift and a turning round through an angle
4. However, in the cases where (4a) admits of solutions, these can be obtained with the help of (33) in 115. In particular using equ. (42) in 121 we may obtain systems of coordinates in an orthogonal representation. Once the r , are determined by return signals and are found to give a consistent set of measures of length we can proceed to adjust the phases of the clocks Indeed, observing E from ty we expect that the image © of &o should show a delay of At = r jc. Thus we have to adjust © so as to show a phase shift At relative to ©o*'. Adjusting thus the phases of the clock ($* k = 1, 2 , . . . to be synchronous to ©o we expect that any pair of clocks and £, so synchronized appear alsó to be synchronous relative to each other. This can be tested by viewing, say, E from ty . If the clocks are consistently synchronized then the image ©J*' of (£ as seen in a telescope near ty should be late by an amount k
k
0
fc)
k
0
k
t
k
;
k
;
At
kl
k
=
r ijc. k
148. Summarizing the procedure we see that using the exchange of light signals between the points 5Js k = 0, 1, 2 , . . . we can synchronize the clocks $ k = 1, 2 , . . . with respect to both rate and phase. Furthermore we can determine the measures of i of the coordinate vectors x of the points The system of coordinate vectors and alsó the modes of synchronization of clocks obtained by the above procedure can be subjected to a number of tests of consistency. If the tests lead to positive results, then it can be taken to support the hypothesis regarding to the mode of propagation of light relative to the system of reference K in which the points 9p are at rest. However, it is important to point out that the fact that there exist consistent coordinate measures in terms of which the light appears to propfc
k
k
k
fc
agate isotropically is only a necessary condition which has to be fulfilled and this condition may be fulfilled even if the mode of propagation of light is not isotropic; we shall return to this question further below.
B. THE LORENTZ TRANSFORMATION AS COORDINATE TRANSFORMATION 149. We may construct a system of reference with the help of signals of light with a method described in the preceding section. The consistency of the measures thus obtained can be ascertained by timing departures and arrivals of light signals. The following relation is expected to hold (r, - r*) - c (t, - ttf = 0 2
t, > t
with
2
(5)
k
for a signal departing at t from x and arriving at t, in r,. Relation (5) can also be written in a different manner. Let us for this purpose describe events by four-vectors; thus we write k
k
Xx, x , x , x
£( = ± 1. q
154. So as to obtain an explicit representation of A containing six independent parameters, we consider two particular types of Lorentz transformations. Let us denote by (,)
O 0
0 1
(a) (13)
with OO = 1
(b)
where O is an orthogonal mátrix of third order. We find that A satisfies (12). Indeed, the transformation (13) corresponds to a turning round of the system of reference leaving the adjustment of the clocks unchanged. A second type of matrices satisfying (12) can be written as* ( t )
( A =
B
0 0
0 1 0 0
0 0 1 0
-vB\ o. 0 B )
(14)
with B=
1-
-1/2
* The transformation (11) with A = AJ") and X = 0 is sometimes referred to as "the Lorentz transformation"; this is of course only a particular case of the Lorentz transformations.
The transformation (14) can be generalized by writing /
V
o
V
l+—-(B-l) \ - yfi/c
-vB B
2
J
Inserting (14) or (15) into (12) one verifies that A is a Lorentz mátrix indeed. 155. Both (13) and (15) represent a three-parameter manyfold. From the group character of the Lorentz matrices it follows that (,)
AA + v(/, -
í ). k
We may thus write in place of (26) (r, + T(Í, - t ) - r )G(r, + v(/, - t ) - i ) - c (í, - t f 2
k
k
k
k
k
= 0.
We may also write (x, - xFC)G(x - x ) = 0 ;
k
(27)
V = Gv
C 2 = c - vV 2
and alsó det g = - c det G. 2
The above consideration leading to the hypothesis (27), (28) is an auxiliary consideration. Independent of the details of the above consideration we shall investigate the law of propagation of light in accord with (27) supposing g to be a symmetric mátrix such that g < 0 and g should possess three positive and one negatíve eigenvalue. u
2. TEST FOR HOMOGENEOUS PROPAGATION OF LIGHT
163. The mode of propagation of light as described by (27) and (28) may be called homogeneous propagation of light. Making an assumption about the elements of g the relation (27) can be tested experimentally in a way which is very similar to the test for isotropy described above. For this purpose we may consider a set of points $ , ^ ••• d clocks (S , Ei, ©2> • • • near the points. Using signals of lights we can synchronize the rates of the clocks just as described in 145. The fact that this synchronization can be carried out consistently is a test of the hypothesis regarding the mode of propagation of light. Once the clocks © k = 0, 1, 2 , . . . are synchronized we can observe the time t = t + t , (29) a n
0
u
0
fc
m
kI
m
a signal takes to move from % to % and back. From (27) and (28) it fol* lows that rGr + 2Vr/< > - C í > = 0 (30) k
1
2
(1
2
where r = r, — r and is the time the signal takes to move from ty to Solving (30) we find (remembering that f > 0) k
k
(1)
_ Vr + Y C ( r G r ) + (Vr) 2
2
C2
The time t^ of the flight back from % to *p is obtained from the above expression replacing r by — r. We find thus }
fc
+ / = 2(rGr) 1 ' 2 /C, 2
(31a)
G = G +
VOV
c
(31b)
2
From (29) and (31a) it follows (r -r,)G(r -r,)--C í| =0.
(32)
2
;
/
/
We may take as the definition of the measures of distances r is given by
kl
that it (33)
Supposing (33) we find from (32) («•; - r )G(r, k
(34)
- r) = r . 2
k
kl
The above relation (34) is analogous to relation (23) of chapt. III, 113. The relation (34) gives a system of n(n— 1 )/2 equations for the 3n unknown coordinates. The right-hand expression can be taken to be the distances measured with return signals. Thus the system is overdetermined. If the overdetermined system (34) can be solved, then it provides us with consistent coordinate vectors r of the points % k = 1 , 2 , . . . and at the same time we obtain support as to our assumptions concerning the mode of propagation of light. 164. Viewing the clock E from a position near E, we observe an image EJjP of E which appears as a clock retarded by an amount k
k
fc
t
í
(i)
(r -r )/C + y í . 2
=
V
/
f c
w
If the coordinate vectors r , r have been determined we can determine also and thus we can adjust the phase of E, so as to be synchronous to E*. In this manner we can adjust the phases of all the clocks E k = 0, 1, 2 , . . .. The synchronization of the phases provides us again with numerous checks upon our assumptions. Carrying out the whole of the procedure we may arrive at consistent values of the coordinate vectors r k = 0, 1, 2 , . . . and to consistent rates and phases of the clocks of E , E If this is the case we may suppose that the assumption (27) about the mode of propagation of light is consistent with the observed behaviour of light signals. k
;
fc
k
0
x
3. CONNECTION BETWEEN VARIOUS REPRESENTATIONS
165. Having found coordinate measures in terms of which the propagation can be described in a consistent way by (27), then we can find other representations in measures of which the propagation of light also appears
homogeneous, but which representations correspond to a mátrix g' different from g. Indeed, we may introduce transformed measures x' = Sx + s,
(35)
where S is a fourth order mátrix with det S # 0. The propagation of light in terms of the new coordinates can be expressed as (*í - x*)g'(*í - 4 ) = o. where (36) g' = S ^ g S " . 1
Relations (36) are mathematically equivalent with (27). Thus provided measures x can be found which are consistent with (27) then the measures x' are automatically consistent with (36). We may write cj for the propagation tensor and thus tf(g) = g
K'(q) = g'.
(37)
We see if there exists a system of reference K where the propagation tensor is represented by g, then there exists alsó a system of reference K' where the propagation tensor is given by g'. a. TRANSFORMATIONS OF THE PROPAGATION TENSOR
166. Presently we show that there exists always a transformation (35) which changes g from a given form to another given form g'. Let us write g in the form (28). We introduce a mátrix a such that &R<x = g;
(38)
we find that a will satisfy (38) provided we put g~ V\
j v2
1I2
G
" ( o
*
)'
P»>
We use c for the velocity of light in the Lorentz system of reference, thus we suppose r _ _ 2 0
1
Further we take
u —
R c
0 •
C 2 = C 2 - Vv.
The inverse of a can alsó be written explicitly; we find «-I = Í
G _ 1 / 2
(O
-G-^O/C] C0/C
Introducing (39) into (38) we obtain an identity.
We note that (38) can be satisfied if we replace a by a
(0)
=
A
(q)
a >
where A is the orthogonal representation of a Lorentz mátrix. 167. In analogy with (38) we can introduce a mátrix a' satisfying (
- 1
(42)
1
In place of S as given by (42) we can also use a transformation mátrix S defined as S«ö = c t ' - ^ a (43) (q)
where A is a Lorentz mátrix. Thus transformations (35) where S is given by (42) or (43) lead from a representation K to one K' satisfying (37). 168. We shall always suppose if we speak of a propagation mátrix (or of a propagation tensor) that it is represented by a symmetric mátrix with three positive and one negatíve eigenvalue such that g < 0. We see therefore that observing the propagation of light in a region 9? we can merely find out whether or not light is propagated homogeneously in that region, i.e. we can find out whether or not the propagation of signals of light can be described by a quadratic expression of the form (27). We cannot, however, determine the elements of the propagation mátrix g itself from the observations of signals of light. Thus the question whether or not light is propagated isotropically in a given region has no particular significance — we can only find out whether or not light is propagated homogeneously in ÍR. (q)
u
D. THE RELATION OF SYSTEMS OF REFERENCES OBTAINED WITH LIGHT SIGNALS AND WITH SOLIDS 169. We have shown in 110 how we can construct, using measuring rods, consistent sets of coordinate vectors. We may denote coordinate vectors obtained with the help of measuring rods by r. The coordinate
vectors of a set of points are thus obtained solving a system of equations (r -r )G(r -i ) l
k
l
= ¥f
k
(44)
k
where the r are the distances between points as measured with the help of rods. Similarly, using signals of light instead of measuring rods we can determine measures of distances r in accord with 163 and we obtain coordinate measures r as solutions of the equations tk
lk
k
VO V \
l
(r, - r )|G + - ^ - j (r, - r ) = r? . k
fc
(45)
k
The question arises whether there exist such representations K where the coordinate measures i satisfying (44) coincide with the measures r satisfying (45) in this turn. So as to answer this question we have first to investigate the relation between the measures r and r of the distances between points ty and As the methods of determining r and r are different there is a priori no reason why these measures should or should not be equal. It is, however, an empirical fact that k
k
kl
kI
k
kl
kl
kfiki — constant,
r
or choosing suitable units r i = hi • k
(46)
170. The experiment proving (46) is the Michelson-Morley experiment and the Kennedy-Thorndike experiment. Indeed, the latter experiments prove the following. Marking two points A, B on a solid, the time return of light T = t + t between the two points remains constant if we turn round the solid, or if we shift it about. Changing the position of a solid, the points A, B are shifted into positions A*, B*. From the deliberations of 134 and 135 we have to suppose that in the case of an ideál solid AB
AB
BA
AB
=
R
thus the measures of the distances between the two points do not change if we shift the solid. From the interferometer experiments we conclude that TAB
—
AB
—
T *B* a
and therefore R
R
A'B*
jf we take the measure determined interferometrically of a distance to be r = CT. Thus if two distances appear equal, i.e. AB — PQ>
r
r
when compared with rods, then they will also appear to be equal when measured with signals of light. If we add to this that the distances of points along straight lines are additive — we see that (46) will be correct for all pairs of points ty and ^5, if it is correct for one particular pair. Thus if we choose the units so that e.g. k
r
12
=
r
12
then we expect (46) to hold for any values of k and /. 171. If we can take it as a result of experiment that (46) is valid, then we can obtain coordinate measures by solving the equation (
VOV)
(T - r,) JG + — ^ - \ (r - r,) = rf,; see 163 equs (31b), (32) and also K
(47a)
k
{i - í,)G(F - F,) = k
(47b)
fc
The two systems admit of solutions r* = í*
k = 0, 1, 2 , . . . , n
provided (46) is valid and G= G+
VOV
.
(48)
We see therefore that provided (1) we can determine in a region 9t consistent coordinates with the help of measuring rods, (2) we can also determine consistent coordinates with the help of light signals and (3) we find empirically that the measures of distances r , are equal to those of r (choosing the units suitably), thus we can introduce a representation K such that k
kl
r* = í*;
(49)
for the purpose we have to take VOV
G = G+ - ^ - ,
(50)
as the connection between the elements of g and the elements of G. 112
172. The requirements (l)-(3) which lead to (49) contain three distinct physical statements. These statements reflect upon: (1) the physical properties óf solid rods, (2) the physical properties of the propagation of light, (3) the connection between the physical properties of solids and those of the propagation of light. We note that the physical properties of solids mentioned under (1) are by no means trivial ones. These include the Lorentz contractions suffered by solids. We cannot, however, determine the elements of the propagation mátrix g itself from the observations of signals of light. Thus the question whether or not light is propagated isotropically in a given region has no particular significance — we can only find out whether or not light is propagated homogeneously in 9í.
CHAPTER V
THE LORENTZ PRINCIPLE
A. THE LORENTZ TRANSFORMATION AS DEFORMATION 173. After the preliminary considerations given above we are now in a position to give the Lorentz principle, i.e. we can formulate the generál law which explains why it is impossible to observe directly the translational motion of a physical system relatíve to the ether. We have seen that in the case of the Michelson-Morley experiment the geometrical effect from which we might hope to determine v is compensated by a deformation of the interferometer which exactly compensates the geometrical effect. Similarly as in the other experiments described in chapter II we have to conclude that experimentál arrangements when moving relatíve to the ether suffer deformations so that these deformations are compensating exactly such effects that otherwise might be suitable for to determine the velocity of the system relatíve to the ether. Presently we analyse in more detail the deformations which are suitable to compensate exactly effects caused by translational motion of a system relatíve to the ether. We shall see that the compensating effects can be described in terms of what we shall denote Lorentz deformations. We give the definition and properties of these deformations below. 1. DEFORMATION OPERATORS
174. Consider a physical system O consisting of particles in points ?$ , k = 1, 2 , . . ., «. The representation Q of D relatíve to K, i.e. k
Q = can be expressed with the help of the four-coordinates of the points ty
k
x
l , 2s . . . , X x
n
.
We do not suppose £t to be at rest and we allow that the points ty of Q may move relatíve to each other. The motion of the points of Q can be expressed in a paraméter representation, i.e. putting k
x* = MP) = kÍP), Kp) T
where i(p) > 0.
* = 1,2,... n
Thus choosing various values of the paraméter p we obtain the positions of tyk various times t(p). Applying a reversible linear transformation we obtain four-coordinates a t
x* (p) = Tx (p) + a. k
k
(1)
The four-coordinate x*(p) defines the orbit of a point in terms of its coordinates relative to K. The operation determined by T and a defines thus the orbit of any point ty to that of another point ^s*.. The points ÍJs*. thus obtained can be taken to form a physical system C* and thus (1) can k
x Fig. 15. Scheme of the Lorentz deformation of an orbit
be taken to define (in the representation K) a physical system O* to a given physical system £}. We can write symbolically &* = £(D). 175. Relation (1) has the mathematical form of a coordinate transformation — it represents, however, not a coordinate transformation, but a deformation. Indeed, the coordinates x*.(p) just as the coordinates x (p) represent the orbits of points relative to one and the same system of reference K (see Fig. 15). One finds that the representations of Q and C* relative to a system K' can be written *k(p) = Sx (/>) + s 1 k
fc
x* '(p) = Sxt(p) k
+ s j
W
and comparing (2) with (1) we find x*'(p) = Tx'(j>) + a', with T' = S T S " a' = (1 - TO s + Sa 1
(3)
We can thus write K(%) = T, A
and
K'
