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Vacuum Effects in Gravitational Fields: Theory and Detectability QUANTUM VACUUM EFFECTS IN GRAVITATIONAL FIELDS: THEORY AND DETECTABILITY Thesis submitted for the degree of “Doctor Philosophiæ” Stefano Liberati International School for Advanced Studies Via Beirut 2-4, 34014 Trieste, Italy. E-mail: [email protected] arXiv:gr-qc/0009050v1 14 Sep 2000 June 2000 QUANTUM VACUUM EFFECTS IN GRAVITATIONAL FIELDS: THEORY AND DETECTABILITY Stefano Liberati — Ph.D. Thesis Supervisor: D.W. Sciama Co-Supervisors: M. Visser and J.C. Miller Abstract This thesis is devoted to the study of quantum vacuum effects in the presence of strong gravitational fields. We shall see how the quantum vacuum interacts with black hole geometries and how it can play an important role in the interpretation of the gravitational entropy. In this respect particular attention will be given to the peculiar role of the extremal black hole solutions. From this branch of our research we shall try to collect some important hints about the relation between quantum gravity theories and the semiclassical results. After these investigations we shall move our attention toward possible experimental tests of particle creation from the quantum vacuum which is an indirect confirmation of the Hawking effect. This aim will lead us to study acoustic geometries and their way of “simulating” General Relativity structures, such as horizons and black holes. We shall study the stability of these structures and the problems related to setting up experimental detection of phonon Hawking flux from acoustic horizons. This research will naturally lead us to propose a new model for explaining the emission of light in the phenomenon of Sonoluminescence, based on the dynamical Casimir effect. Possible experimental tests of this proposal will be discussed. In this way we shall set up one of the few available models of quantum vacuum radiation amenable to observational test within the next few years. After this journey in the condensed matter world we shall move to the other arena where our efforts to test the effects of the quantum vacuum in gravitational fields can find a positive solution in the future: the high energy phenomena in the early universe. We shall concentrate our attention on inflation and its possible alternatives for solving the cosmological puzzles. This will lead us to propose a new way to reheat the universe after inflation via pure gravitational effects. We shall finally show how some known phenomena related to the vacuum polarization in the Casimir effects, can naturally suggest new ways to replace (or at least improve) the inflationary scenario. Contents Introduction 1 1 Quantum vacuum and Gravitation 4 1.1 Thenatureofthequantumvacuum . ........ 5 1.1.1 Canonicalquantization . ....... 6 1.1.2 Canonical quantization in external fields . ............. 7 1.1.3 Thequantumvacuum .............................. ... 10 1.2 Vacuumeffectsinstaticexternalfields . ............. 12 1.2.1 TheCasimireffect ............................... .... 12 1.2.2 Othercasesofvacuumpolarization. .......... 18 1.2.3 Effectivevacuumtemperature. ........ 19 1.3 Vacuumeffects in dynamicalexternalfields . ............. 21 1.3.1 Particle production in a time-varying external field . ................ 22 1.3.2 Movingmirrors................................. .... 26 1.3.3 SqueezedStatesandThermality . ........ 28 1.4 Vacuumeffectsingravitationalfields . ............. 30 1.4.1 Renormalization of the stress energy tensor . ............. 32 1.4.2 Particle interpretation and creation in gravitationalfields ............. 35 1.4.3 Hawking radiation for general spacetimes with Killinghorizons .......... 37 2 Quantum Black Holes 45 2.1 BlackHoleThermodynamics . ........ 46 2.2 Alternative derivations of the black hole entropy . ................. 50 2.2.1 Euclidean derivation of black hole Thermodynamics . .............. 50 2.2.2 Derivation of black hole entropy as a Noether Charge . ............. 52 2.3 ExplanationsofBlackHoleentropy. ............ 54 2.3.1 InformationbasedApproaches . ........ 55 2.3.2 QuantumVacuumBasedApproaches . ...... 55 2.3.3 A Casimir approach to black hole thermodynamics . ............ 59 2.3.4 Symmetrybasedapproaches. ....... 64 2.4 Gravitational entropy and global structure . ................ 70 i 2.4.1 Euler characteristic and manifold structure . .............. 71 2.4.2 Entropy for manifolds with a boundary . ......... 72 2.4.3 Gravitational entropy and Euler characteristic for spherically symmetric metrics 73 2.4.4 Kerrmetric.................................... ... 79 2.4.5 Discussion.................................... .... 81 2.5 Incipientextremalblackholes . ............ 82 2.5.1 Kruskal-like coordinates for the extremal RN solution ............... 82 2.5.2 Asymptoticworldlines . ....... 84 2.5.3 Bogoliubovcoefficients. ....... 86 2.5.4 Preservationofcosmiccensorship . ........... 88 2.5.5 Preservationoftheno-hairtheorems . ........... 89 2.5.6 Detecting radiation from a uniformly accelerated mirror.............. 90 2.5.7 Conclusions................................... .... 93 3 Acoustic horizons 96 3.1 Seeking tests of the dynamical Casimir effect . .............. 97 3.2 AcousticManifolds ............................... ........ 98 3.2.1 Ergoregionsandacoustichorizons . .......... 101 3.2.2 Acousticblackholes .. .. .. .. .. .. .. .. .. .. .. .. ...... 102 3.2.3 PhononHawkingradiation . ...... 105 3.3 Stabilityofacoustichorizons . ............ 107 3.3.1 Regularity conditions at ergo-surfaces . ............. 108 3.3.2 Regularityconditionsathorizons . ........... 109 3.4 Spherically symmetric stationary flow . ............. 110 3.4.1 Constantspeedofsound. ...... 111 3.4.2 Examples ...................................... 112 3.4.3 Viscosity ..................................... ... 116 3.4.4 Discussion.................................... .... 120 3.5 Prospects: condensedmatter blackholes? . .............. 120 4 Sonoluminescence 122 4.1 Sonoluminescence as a dynamical Casimir Effect . .............. 123 4.1.1 Casimir models: Schwinger’s approach . ........... 125 4.1.2 Eberlein’sdynamicalmodelforSL . ......... 128 4.1.3 Timescales: The need for a sudden approximation . ............ 130 4.1.4 BubblesatthevanderWaalshardcore . ........ 132 4.2 Sonoluminescence as a dynamical Casimir effect: Homogeneousmodel . 133 4.2.1 Definingthemodel............................... .... 134 4.2.2 Extensionofthemodel . .. .. .. .. .. .. .. .. .. .. .. ..... 140 4.2.3 Somenumericalestimates . ....... 142 4.3 Sonoluminescence as a dynamical Casimir effect: Finite Volumemodel . 147 4.3.1 Generalizingtheinnerproduct . ......... 148 4.3.2 Fixing the normalization constants . ........... 150 4.3.3 Finitevolumecalculation . ........ 151 4.3.4 Behaviour for finite radius: Numerical analysis . .............. 155 4.3.5 Implementationofthecutoff . ....... 158 4.3.6 Analyticalapproximations. ......... 159 4.3.7 Thespectrum: numericalevaluation . .......... 160 4.4 Experimentalfeatures and possible tests . ............... 163 4.4.1 Squeezed states as a test of Sonoluminescence . ............. 164 4.5 DiscussionandConclusions . .......... 167 5 Quantum vacuum effects in the early universe 169 5.1 The role of the quantum vacuum in modern cosmology . ............. 170 5.2 The topology of the universe and the quantum vacuum . .............. 171 5.3 Inflation........................................ ..... 174 5.3.1 ThecosmologicalPuzzles . ....... 175 5.3.2 Generalframework.............................. ..... 178 5.3.3 Possible implementations of the framework . ............ 179 5.3.4 Reheating..................................... ... 182 5.3.5 BasicPreheating ............................... ..... 183 5.4 Geometricpreheating ............................. ........ 188 5.4.1 Scalarfields................................... .... 190 5.4.2 Vectorfields................................... .... 193 5.4.3 Thegravitoncase ............................... .... 194 5.4.4 Discussion.................................... .... 195 5.5 Alternativestoinflation?. ........... 195 5.5.1 Varying–Speed–of–LightCosmologies. ............ 196 5.6 The χVSLframework ..................................... 199 5.6.1 Stress-energy tensor, equation of state, and equationsofmotion. 202 5.6.2 Cosmological puzzles and Primordial Seeds . ............ 205 5.6.3 Observational tests and the low-redshift χVSLuniverse . 211 5.7 Discussion...................................... ...... 214 Conclusions 216 Bibliography 220 Notation Unless otherwise stated we shall use units for which ¯h = c = 1. We shall make explicit the dependence on the fundamental constants in the most important formulæ. The Boltzmann constant is kB and the gravitational constant will be denoted as GN. The Greek indices take values 0 . 3 while Latin indices denote spatial directions and range over 1 . 3. The wide range of problems treated in this thesis has made it impossible to use the same metric signature through all of the work. Chapters 1 and 4 use the signature common in the literature of quantum field theory (+, , , ) with the Minkowski metric given by η = diag(1, 1, 1, 1). In − − − µν − − − chapters 2, 3 and 5 we use the signature commonly used by general relativists ( , +, +, +) with the − Minkowski metric given by η = diag( 1, +1, +1, +1). µν − These notations are consistent with standard reference books in the subject, which have been used as references for the review parts of this work. For purely general relativistic issues we have mainly used references [1, 2, 3], for quantum field theory in external/gravitational fields, the principal sources are [4, 5, 6]. For the review on black hole thermodynamics
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