Black Holes and Moving Mirrors

Black Holes and Moving Mirrors Niels Tuning Abstract In this rep ort I treat the sub ject of black hole radiation Until the early seventies black holes were considered to b e eternal ob jects in our universe but the striking analogy with the laws of thermo dynamics and the explicit calculation of the black hole radiation due to quantum eects by Hawking showed a dierent scenario Hawking recognized already the problems concerning quantum purity due to thermal radiation o the black hole and the o ccurence of divergent energies due to the gravitational blue shift Backreaction eects were neglected by Hawking but could lead to corrections of Hawkings result and resolve the information problem Furthermore including backre action ensures energy conservation I included backreaction eects in the simple moving mirror mo del and found that the temp erature of the mirror is reduced by a factor relative to the standard result However the WKB approximation I used breaks down in the case of dilaton gravity and therefore I could not extend the result to real black holes Contents Introduction Black Hole Thermo dynamics Black Hole Thermo dynamics Information Problem Remnants Nice Slices Complementarity SMatrix Ansatz The Unruh Eect Particle Detector The Hawking Eect Stress Tensor Moving Mirrors Fixed Trajectory Saddlep oint Approximation Uniform Acceleration Backreaction on the Mirror Dilaton Gravity Discussion Conclusion A WKB Approximation B Thermal Sp ectrum Preface This thesis is the conclusion of a year work under sup ervision of profdr Herman Verlinde on the Institute of Theoretical Physics of the University of Amsterdam The thesis is the nal pro ject of my study of theoretical physics I tried to write this thesis on the level of graduate students At the b eginning of my research I encountered some problems and questions some of them trivial without nding explicit answers I tried to answer these questions and hop e that it might help other students who are trying to understand the dicult sub ject of evaporating black holes Amsterdam August Chapter Introduction A great deal of eort has b een made the last few decades to construct a unied theory of the forces of nature In Glashow prop osed the gauge symmetry SUU as the unication of the weak interactions and the theory of quantum electro dynamics also known as QED which has gauge symmetry U The gauge theory with sym metry SU quantum chromodynamics QCD b ecame in a strong candidate for the strong interactions The standard electroweak theory and QCD together form the standard mo del A p ossible candidate for a grand unied theory GUT ie a gauge theory with a single coupling constant is a theory with gauge symmetry SU The last few years there has also b een a revival of string theory which incorp orates gravity and is considered as another serious candidate for the Theory of Everything Until now it has not b een p ossible to include the fourth force namely gravity Prob lems arise while constructing a quantum theory of gravity b ecause it app ears that the coupling constant is not dimensionless making p erturbation expansion imp ossible Another problem is that gravity couples to everything b ecause it inuences the metric A graviton is thus b oth a source of gravitation and sub ject to gravitation Therefore one considers the gravitational eld as a classical background obtaining a semiclassical theory Hawking considered quantum pro cesses in a classical background geometry and found that vacuum uctuations near the black hole horizon results in thermal radiation o the hole Because of those quantum pro cesses the black hole lo oses mass and can eventually disapp ear while black holes were considered classically as eternal ob jects Hawkings result op ened a new area in physics and is derived in dierent ways by several authors One of the strongest arguments to supp ort the Hawking eect is the deep connection b etween the laws of black hole physics and the laws of thermo dynamics This is also the ma jor reason why Hawkings discovery is now widely b elieved to b e real The pap er is organized as follows I will start more or less historically by discussing the sub ject of black hole thermo dynamics developed in the late sixties anticipating Hawking by suggesting that black holes b ehave as black b o dies with a temp erature CHAPTER INTRODUCTION In chapter I discuss briey the problems which arise assuming black holes having a temp erature and emitting thermal radiation The dierent p oints of view in litera ture are given Instead of treating subsequently the Hawking eect I will discuss the Unruh eect in chapter The Unruh eect is in some sense more general b ecause it states that particle pro duction o ccurs wherever a bifurcate killing horizon app ears Even an accelerating observer in Minkowski spacetime detects thermal radiation The breakdown of Poincare invariance as we have in curved spacetime is already manifest in noninertial systems in at spacetime In chapter the Hawking eect is explic itly calculated giving a strong supp ort to the theory of black hole thermo dynamics of chapter In Hawkings calculation pure states can evolve into mixed states hereby violating the basic principle of unitary evolution in quantum mechanics In chapter it b ecame clear that backreaction eects are imp ortant and may lead to corrections of Hawkings result Furthermore due to the gravitational redshift the energy near the horizon diverges In order to investigate back reaction eects and to implement energy conservation I treat in chapter a mo del which mimics many of the features of gravitational collapse With this moving mirror mo del I showed that including back reaction eects and thus energy conservation leeds to a correction of the temp erature by a factor half relative to the standard result Chapter Black Hole Thermo dynamics Black Hole Before we start talking ab out black holes I will give a short review of what a black hole is I will use units where h c In Einstein found his classical eld equations of general relativity G T If one imp oses the metric to b e static and spherically symmetric one nds an exact solution for the vacuum eld equations G namely the Schwarzschild solution which has line element m dt ds dr r d sin d m r r This solution is physically interesting b ecause the ob jects in our universe are mostly spherically symmetric and the surrounding space can b e considered empty We see that the line element is singular at the co ordinates r and r m The coordinate singularity at r m is however removable by p erforming a co ordinate transforma tion The other singularity at r is an intrinsic singularity and is not removable by co ordinate transformations Although r m is no real singularity it remains an future singularity singularity r=const. t=const. I- horizon r=0 r=2m past singularity Figure Two dierent gures of a black hole a Schwarzschild solution b Compactied Kruskal solution CHAPTER BLACK HOLE THERMODYNAMICS interesting p oint or sphere called the Schwarzschild radius If one writes the solu tion in EddingtonFinkelstein co ordinates one sees the lightcones p ointing towards the center of the hole and it b ecomes clear that nothing can come back after passing the Schwarzschild radius r m is therefore called an event horizon It is imp ossible to b ecome aware of events that take place b ehind the horizon Therefore this ob ject is referred to as a black hole By p erforming a suitable co ordinate transformation one removes the co ordinate singularity at r m obtaining the EddingtonFinkelstein co ordinates The Kruskal solution is the maximal analytic extension of the geometry ie every geo desic ends in innity or in a singularity and can b e compactied by mapping the innite interval to a nite interval In this way one obtains the p enrose diagram of gure b We can imagine dierent types of black holes Beside the collapsing star we can imagine huge black holes formed after a collapsing star cluster On the other hand we can imagine so called primordial black holes with masses of kg One often hears the question if elementary particles could b e black holes The answer is no b ecause the Compton wavelength of a particle is much bigger then its Schwarzschild radius Those p two length scales are equal up to a factor at the Planck mass Gm h m k g P l c mc By combining the fundamental constants c h and G one nds units of length time and mass the so called Planck units s ch m k g P l G s Gh l m P l c s Gh t s P l c E m c GeV P l P l The Planck values mark the frontier where a full theory of quantum gravity is indis p ensable As we will see in the following chapters quantum eects alter the course of gravita tional collapse As seen by an asymptotic observer collapsing matter evaporates b efore forming a real curvature singularity see gure However after stellar collapse the evaporation of a large heavy ob ject pro ceeds very slowly so we refer to this ob ject as a black hole keeping in mind that it is a rather classical notion THERMODYNAMICS Thermo dynamics In the late sixties and b egin seventies

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