ARTIFICIAL
BLACK HOLES
Mario Novello Matt Visser Grigori Volovik
World Scientific
ARTIFICIAL
BLACK HOLES
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ARTIFICIAL
BLACK HOLES Editors
Mario Novello Centra Brasileiro de Pesquisas Fisicas
Matt Visser Victoria University of Wellington
Grigori Volovik Helsinki University of Technology Et Landau Institute for Theoretical Physics
, © World Scientific m
New Jersey • London • Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Cover illustration by Enrique Arilla ARTIFICIAL BLACK HOLES Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4807-5
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Preface This is a book about artificial black holes. Specifically, it is based on the premise that working with gravitational black holes in the laboratory appears to be impossibly difficult. Because of this, considerable attention is now turning to the possibility of simulating black holes in the laboratory using various "analogue models", typically based on condensed matter physics. The connection with gravity is that the propagation of photons and quasiparticles in a medium can often be phrased in terms of an "effective metric", using the same mathematical tools as general relativity (Einstein gravity). Condensed matter systems can therefore provide analogues of quantum field effects in curved spacetime backgrounds, and they also suggest possibilities for spacetime structure at short distances that might emerge from the currently unknown quantum theory of gravity. In the other direction, geometric methods originally developed for gravity can be applied to studying some condensed matter systems. The book contains 14 chapters, written by experts from general relativity, particle physics, and condensed matter physics, that explore various aspects of this two-way street. Considerable care has gone into the presentation — you will not need to be an expert in general relativity to understand key elements of the physics. The book is also aimed at condensed matter physicists and particle physicists, both theorists and experimentalists. For those unfamiliar with the machinery of general relativity, the introductory chapter gives a few simple examples of what we mean by an "effective metric", and the appendix gives a brief two-page summary of the essentials of Einstein's theory. In summary, we hope that this book will serve to provoke debate and ignite interest in the collection of ideas collected under the rubrics of "analogue models", "effective metrics", and "analogue black holes". Since this is cutting-edge research, the reader may note that not all of the authors agree v
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Artificial Black Holes
with each other on all issues. Though this is a book written by theorists, we strongly want to encourage experimental participation in these ideas — not all of these approaches will necessarily pan out in feasible laboratory experiments but we feel that there is already enough known to strongly suggest that it is only a matter of time before a laboratory-scale "analogue black hole" is actually built. The sooner the better. Experimental feedback on these subtle issues will be extremely useful. One of the most exciting aspects of these developments is the possibility of experimentally detecting analogue Hawking radiation. The Hawking radiation effect is a subtle quantum mechanical effect that destabilizes black holes in general relativity. Crudely speaking, Hawking radiation permits a small amount of quantum mechanical leakage back up across the event horizon. Once quantum physics is taken into account black holes are not completely black. Prospects for observationally detecting this effect in astrophysical black holes is bleak; but detecting the analogue of this effect for the artificial black holes considered in this book looks much more promising. This is not to say that detecting analogue Hawking radiation would be easy, doing so will certainly require extensive research and new experimental techniques, but order-of-magnitude estimates indicate that the situation is promising. Mario Novello Matt Visser Grigori Volovik May 2002
Contributors • Brandon Carter Departement d'Astrophysique Relativiste et de Cosmologie Centre National de la Recherche Scientifique Observatoire de Paris, 92195 Meudon France E-mail:
[email protected] • George Chapline Physics and Advanced Technology Division Lawrence Livermore National Laboratory Livermore, California 94550 USA E-mail:
[email protected] • Luis Garay Instituto de Matematicas y Fisica Fundamental CSIC Madrid Spain E-mail:
[email protected] • Ted Jacobson University of Maryland College Park, Maryland USA E-mail:
[email protected] vn
viii
Artificial Black Holes • Tatsuhiko Koike Keio University Hiyoshi, Kohoku Yokohama 223-8522 Japan E-mail:
[email protected] • Robert Laughlin Stanford University Stanford, California USA E-mail:
[email protected] • Ulf Leonhardt University of Saint Andrews North Haugh, Saint Andrews, KY16 9SS Scotland E-mail:
[email protected] • Mario Novello Centro Brasileiro de Pesquisas Ffsicas Rio de Janeiro Brazil E-mail:
[email protected] • Renaud Parentani Laboratoire de Mathematiques et Physique Theorique CNRS UMR 6083, Universite de Tours 37200 Tours France E-mail:
[email protected] • Haret Rosu Instituto Potosino de Investigacion Cientifica y Tecnologica Apartado Postal 3-74 Tangamanga, San Luis Potosi Mexico E-mail:
[email protected] Mario Novello/Matt
Visser/Grigori Volovik
• David Santiago Stanford University Stanford, California USA E-mail:
[email protected] • Michael Stone University of Illinois Urbana,IL 61801 USA E-mail:
[email protected] • William U n r u h University of British Columbia Vancouver Canada E-mail:
[email protected] • M a t t Visser Victoria University School of Mathematics and Computer Science Wellington New Zealand E-mail:
[email protected] • Grigori Volovik Helsinki University of Technology Finland and Landau Institute for Theoretical Physics Moscow Russia E-mail:
[email protected] ix
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Plan of the Book The main body of the book consists of 14 chapters which explore different aspects of the concept of analogue models. The authors are a diverse group of physicists spanning five continents and, depending on whether you count citizenship or residency, over 10 countries and 15 research institutions. Most, but not all, of the authors were present at the workshop on Analogue Models of/for General Relativity, held in Rio de Janeiro in October 2000. Additional background material can be found at the workshop website which for the foreseeable future will be maintained in Brazil: http://www.cbpf.br/~bscg/analog Mirror sites will be maintained in the USA http://www.physics.wustl.edu/~visser/Analog and in New Zealand http://www.mcs.vuw.ac.nz/~visser/Analog Turning to the contents of the book itself, we present a brief outline below. • The introductory chapter, by Matt Visser, gives background information and sets the scene. Then there are four chapters specifically on the most physically plausible ways of experimentally building laboratorybased "analogue black holes". - The chapter by Luis Garay discusses the notion of using acoustics in Bose-Einstein condensates as a way of modelling black holes and Hawking radiation. These are the so-called "dumb holes" or "acoustic black holes". - Ulf Leonhardt considers the use of the optical properties of BoseEinstein condensates, in particular the notion of "slow light" engendered by the process of EIT (electromagnetically induced transXI
Xll
Artificial Black Holes parency). This "slow light" is a key ingredient in possibly developing "optical black holes". — Condensed matter analogies in general, and "quasiparticle black holes" based on Helium-3 in particular, are then discussed by Ted Jacobson and Tatsuhiko Koike. They address the information problem, black hole entropy, and the analogue of "baby universe" production in condensed matter models. — Bill Unruh then discusses some of the problems and observational difficulties to be expected when attempting to measure Hawking radiation from such a system. Issues of this nature will be faced in any attempt at rendering analogue Hawking radiation experimentally observable. These four chapters use condensed matter physics to model parts of Einstein's theory of gravity; without claiming that gravity is identical to condensed matter physics. These are models of gravity. • The next two chapters are perhaps bolder; these two chapters specifically address the notion of using ideas gleaned from condensed matter physics to replace Einstein's theory of gravity. These are models for gravity. — Grigori Volovik explores an extension of Sakharov's idea of "induced gravity". Specifically he considers the idea of using the vacuum states of some condensed matter system (qualitatively similar to liquid He) as the "fabric of space", a "quantum aether", with all the ordinary objects of particle physics being viewed as quasiparticle excitations of this quantum vacuum state. — Robert Laughlin, George Chapline, and David Santiago consider the idea of describing black hole event horizons via some "phase transition" in the quantum vacuum. In this case the interior of the black hole is radically different from the standard interpretation in terms of Einstein's theory of gravity, though Einstein's equations continue to hold both inside and outside the event horizon, suffering a discontinuity at the event horizon. • The preceding six chapters all deal with various versions of "pseudogravity" , in contrast, in the next chapter:
Mario Novello/Matt
Visser/Grigori
Volovik
xm
- Brandon Carter discusses the way in which the currently popular "braneworld cosmologies" can lead to quasi-gravity forces, which while they are not strictly speaking gravitational in nature, nevertheless are easily confused with true physical gravity. • The next five chapters provide a wide range of examples of the cross fertilization of ideas: either because tools of condensed matter physics are used to provide insight into general relativistic problems, or because tools of general relativity are used to provide insight into condensed matter or quantum field theory problems. - To confront the trans-Planckian problem which exists in black hole evaporation, Renaud Parentani analyses the gravitational interactions which are neglected in the semi-classical scenario. Once these are taken into account, the propagation of Hawking photons in the vacuum exhibits close analogies with that of phonons in the acoustic geometry of a random medium. - Brandon Carter then applies the notion of the acoustic metric to help understand aspects of the internal structure of neutron stars. This is a situation in which real physical gravity is certainly strong, and in addition sound itself is important and relativistic. Because the speed of sound is then comparable to the speed of light the "acoustic metric" has to be dealt with in a fully relativistic manner. - In his contribution, Mario Novello addresses the general issue of effective geometries arising from nonlinear field theories. In particular one-loop quantum effects will modify the Lagrangian of electrodynamics so that it is no longer just the standard Maxwell action; it is replaced by the nonlinear Euler-Heisenberg Lagrangian. This Lagrangian modifies the characteristics for the photon, and the presence of a background electromagnetic field can then lead to an "effective metric" (not equal to the physical spacetime metric) and to birefringence. - Haret Rosu next discusses the way in which non-inertial motion of a detector can influence the measured properties of the ordinary (Minkowski) quantum vacuum. This chapter presents a generalization of the notion of Unruh radiation detected by a uniformly
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Artificial Black Holes accelerated observer. Certain special paths through spacetime lead to stationary time-independent spectra of detected particles, and it has been argued (with mixed success) that synchrotron radiation may be interpreted in this manner. - In the penultimate chapter, Mike Stone takes the techniques of general relativity and applies them to standard condensed matter physics of fluids — in particular to the long-standing confusion surrounding the notion of "acoustic momentum". • Finally the last chapter, a brief "coda", wraps up the book with a summary of what has been achieved and some thoughts for the future.
Contents Preface
v
List of contributors
vii
Plan of the book 1
Introduction and survey Matt 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10
2
xi 1
Visser The notion of curved space Adding a dimension: curved spacetime Event horizons and ergoregions Physical models Kinematics versus dynamics Wave equation in the acoustic analogy Examples Hawking radiation? Horizon entropy? Summary
Acoustic black holes in dilute B o s e - E i n s t e i n condensates Luis Garay 2.1 Introduction 2.2 Sonic black holes in condensates 2.3 Black/white holes in a ring 2.4 Sink-generated black holes 2.5 Quasiparticle pair creation 2.6 Conclusions xv
3 6 9 11 13 15 19 26 27 28 35 36 38 40 47 53 55
Artificial Black Holes
XVI
3
4
5
6
Slow light Ulf Leonhardt 3.1 Motivation 3.2 Light-matter interaction 3.3 Ordinary media 3.4 Electromagnetically-Induced Transparency 3.5 Dark-state dynamics 3.6 Slow-light pulses 3.7 Effective field theory 3.8 Moving media 3.9 Summary
61
Black hole and baby universe in a thin film of 3 H e - A Ted Jacobson and Tatsuhiko Koike 4.1 Introduction and motivation 4.2 Black hole analogues using 3 He 4.3 Effective spacetime from a moving domain wall 4.4 Hawking effect in the thin-film domain-wall model 4.5 Conclusion
87
Measurability of dumb hole radiation? William Unruh 5.1 Introduction 5.2 Hypersonic flow 5.3 Roton creation 5.4 Vorticity 5.5 Density changes 5.6 Slow light 5.7 Conclusion
61 62 65 69 73 77 78 80 82
88 91 94 99 105 109 110 113 117 119 120 121 122
Effective gravity and quantum vacuum in superfluids 127 Grigori Volovik 6.1 Introduction 128 6.2 Einstein gravity and cosmological constant problem 130 6.3 Microscopic 'Theory of Everything' in quantum liquids . . . . 135 6.4 Weakly interacting Bose gas 138 6.5 Quantum liquid 148 6.6 Vacuum energy and cosmological constant 152
Mario Novello/Matt 6.7 6.8
Visser/Grigori Volovik
xvii
Effects of discrete number N of particles in the vacuum . . . . 163 Conclusion 172
7
Emergent relativity and the physics of black hole horizons 179 George Chapline/Robert Laughlin/David Santiago 7.1 Introduction 180 7.2 Horizons in Bose fluids 183 7.3 Horizons in quantum magnets 187 7.4 Quantum criticality 190 7.5 Discussion 193
8
Quasi-gravity in branes Brandon Carter 8.1 Introduction 8.2 Equation of motion of brane worldsheet 8.3 Perturbed worldsheet configuration 8.4 Quasi-gravitational metric perturbations 8.5 Jordan-Brans-Dicke type theories 8.6 Linearised local scalar tensor field configurations 8.7 Conclusion
9
Towards a collective treatment of quantum gravitational interactions Renaud Parentani 9.1 Outline 9.2 Introduction 9.3 The model 9.4 The gravitational interactions between _ and <j)+ 9.5 Non-vacuum gravitational effects 9.6 Modified Green function 9.7 Conclusions 9.8 Appendix: The large N limit
10 Role of sonic metric in relativistic superfiuid Brandon Carter 10.1 Introduction 10.2 Single constituent perfect fluid models 10.3 Single constituent superfiuid models
199 200 201 202 203 204 206 208
213 214 216 221 224 227 230 236 237 245 246 247 253
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Artificial Black Holes
10.4 Landau-type two-constituent superfluid models 11 Effective geometry in nonlinear field theory (Electrodynamics and Gravity) Mario Novello 11.1 Introduction 11.2 Nonlinear electrodynamics 11.3 Nonlinear dielectric media 11.4 Moving dielectrics 11.5 Non-trivial quantum vacua 11.6 Preliminary synthesis 11.7 The case of spin 2 (gravity) 11.8 Conclusions
257
267 268 269 279 281 289 296 298 304
12 Non-inertial quantum mechanical fluctuations Haret Rosu 12.1 Introduction 12.2 Vacuum Field Noise — VFN 12.3 Circular electromagnetic vacuum noise 12.4 Unruh effect versus anomalous Doppler effect 12.5 Summary
307
13 P h o n o n s and forces: M o m e n t u m versus p s e u d o m o m e n t u m Michael Stone 13.1 Introduction 13.2 Momentum and pseudomomentum 13.3 Radiation pressure 13.4 Mass flow and the Stokes drift 13.5 The Unruh wave equation 13.6 The acoustic metric 13.7 Second-order quantities 13.8 Conservation laws 13.9 Phonons and conservation of wave action 13.10 Summary
335
14 Coda Matt Visser 14.1 Introduction
365
308 308 319 326 328
336 337 341 345 349 350 352 353 358 361
366
Mario Novello/Matt Visser/Grigori Volovik 14.2 14.3 14.4 14.5
Effective metric techniques Analogue models for Einstein gravity Approximate Lorentz symmetry Last words
xix 366 373 374 375
Appendix: Elements of general relativity Matt Visser
383
Index
384
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List of Figures 1.1 1.2 1.3 1.4 1.5
Spherical pulses Laval nozzle Pair of Laval nozzles Vortex geometry Cavitating bubble
8 20 20 22 24
2.1 2.2 2.3 2.4 2.5 2.6
Ring-like configuration for an acoustic black hole 41 Stability diagram for acoustic black holes 44 Stability diagram for extremal acoustic black holes 45 Simulation of the creation of an acoustic black/white hole. . . 46 Sink configuration for an acoustic black hole 47 Speed of sound profile for a "sinkhole" 48
3.1 3.2 3.3
Spectral susceptibility: ordinary dielectric medium Electromagnetically Induced Transparency Spectral susceptibility: EIT
4.1 4.2 4.3 4.4 4.5 4.6
Thin-film domain-wall texture Quasiparticles in a moving domain-wall texture Carter-Penrose diagrams Static wall Causal diagrams of moving domain-wall textures Temporary one-way window
94 96 100 101 103 105
5.1 5.2 5.3
Positive and negative norm states Sketch of a slot type de Laval nozzle Dispersion curve for superfluid 4 He
113 114 118
7.1 7.2
Phenomenological equation of state Critical surface with zero sound speed
184 186
xxi
68 69 74
Artificial Black Holes
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7.3 7.4 7.5 7.6 7.7 7.8
Spin wave dispersion relations Lowest-order scattering processes in the critical region Reflectivity near the critical surface Differential cross-section for inelastic scattering Thermal energy per unit volume for a critical surface Time dilation factor in the vicinity of an event horizon
189 190 191 192 193 194
12.1 Normal and anomalous Doppler effects
327
13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8
339 341 342 343 344 344 345 348
High-speed train Force on overhead wire Vibrating string Radiation pressure Rayleigh radiation pressure Langevin radiation pressure Typical experimental situation Momentum flux without mass
flow
Chapter 1 Introduction and survey Matt Visser Victoria University School of Mathematics and Computer Science Wellington New Zealand E-mail:
[email protected] Abstract: General relativity can be partially simulated by using condensedmatter analogues. Some of these analogue models seem amenable to laboratory implementation on a 5 to 10 year timescale. This might be enough to give us an experimental handle on Hawking radiation.
Experiments in general relativity are extraordinarily difficult — simply because generating and manipulating even weak gravitational fields requires one to manipulate planetary-scale masses. This is far beyond the ability of current technology. Additionally, if we ever did manage to generate a black hole in the laboratory, such an undertaking would seem to be rather risky. Faced with these difficulties, some members of the physics community have 1
2
Introduction
and survey
now turned to the possibility of simulating aspects of general relativity by developing "analogue models" of general relativity; these analogue models typically being based on some condensed-matter system. The key questions are: • How much of general relativity can you carry over into these condensedmatter analogues? • How easy is it to actually build these things in the laboratory and do a few experiments? We are still collectively at stage one, trying to see how much we can carry over, and what the best physical model might be. Experiments might reasonably be expected on the 5 to 10 year timescale, and the results are eagerly awaited by the general relativity, particle physics, and condensed matter communities. To start the discussion: The simplest analogue model is that of sound in a flowing fluid. Suppose you consider sound in a body of water that is swirling around, and in particular let some of the water be moving extremely rapidly, indeed supersonically. Then in the region of supersonic flow the sound is completely entrained by the flowing water and cannot fight its way back upstream — this begins to sound a lot like light being trapped by a black hole in general relativity, and the question arises as to whether there is any real connection. Is the similarity more than skin deep? Of course the answer is yes [1], since otherwise we would not be writing this book, we would not have organized the workshop on which this book is loosely based, and physicists would not have published several hundred technical scientific papers discussing various aspects of "analogue gravity" and "effective metric" techniques. The key theme that runs through all of these analogue models is that small oscillations of many continuum systems (e.g., sound in a liquid, phase oscillations in a Bose-Einstein condensate, "slow light" in a dielectric medium) often behave as though they are influenced by an "effective" curved-spacetime. We are not talking here about the physical curved spacetime of general relativity, but instead want to use the notion of an effective curved spacetime to describe focussing or defocussing effects, and in the most extreme cases the trapping of signals behind an "analogue horizon".
Matt Visser
1.1
3
The notion of curved space
To get the ball rolling, I will need to set up some mathematical machinery; I will build it up slowly starting from elementary notions (experts in general relativity can skip the next few pages). Start with Fermat's principle in optics: Take a medium with a position-dependent refractive index n(x), and in that medium consider an arbitrary path 7 described by x(s), with the parameter s being the physical distance along the path. Then as light travels through the medium it is slowed down to a speed c(2) = -%=.
(1.1)
So the total time taken for light to travel along the path is
Fermat's principle then states that the actual path light takes in crossing the medium is that path which minimizes the total transit time. Equivalently we want to minimize the so-called "optical distance" L7=
fdl=
fdsn{x(s)).
(1.3)
Once it is phrased as a minimization problem, it is clear that everything can be reformulated mathematically in terms of the geometrical notion of "geodesies", looking for the "shortest" path through a curved space. The notion of distance in this curved space is described by a 3 x 3 tensor called the metric: 1 gij{x)=n2(x)Sij.
(1.4)
(The amount of tensor algebra required to get the gist of the argument is minimal, and you can find a nice compact survey in the Schaum outline series book "Tensor Algebra" [2].) In this situation you have two distance functions, l Latin indices run from 1 to 3 and typically denote the x, y, and z directions. In considering analogue models we can almost always get away with using what from the point of view of the effective metric are quasi-Cartesian coordinates; they are inherited from the ordinary Cartesian coordinates of the underlying physical flat space.
Introduction
4
and survey
the optical metric and the physical metric, with the optical distance and physical distance between two nearby points being related by 2 d£2 = gij(x)
dx1 dxj = n2{x)
