BLACK HOLE EVAPORATION: VALIDITY OF QUASI-STATIC APPROXIMATION

Transcription

1 BLACK HOLE EVAPORATION: VALIDITY OF QUASI-STATIC APPROXIMATION By KARTHIK SHANKAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2007 Karthik Shankar 2

3 To the entire physics community 3

4 ACKNOWLEDGMENTS I would like to thank my advisor Prof. Bernard Whiting for his valuable guidance and insightful comments that helped me complete this work. I would like to thank all my teachers, in particular all the professors at UF physics who contributed to my physics education. In this respect, I would specifically like to thank Prof. John Klauder, Prof. James Fry and Prof. Pierre Ramond. I am extremely grateful to Prof. Steve Detweiler and Prof. Richard Woodard for their time and availability to help me clarify most of the fundamental concepts in physics. I would also like to acknowledge my friends and colleagues Dr. Aparna Baskaran, Mr. Anand Balaraman, Mr. Ian Vega and Dr. Kyongchul Kong, for the time they spent with me in stimulating physics discussions. Most of all, I would like to thank my parents for being supportive of my education throughout my life. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Classical Black Holes Gravitational Collapse Theorems Black Hole Dynamics Hawking Radiation Particle Creation Back Scattering Quantum Stress Tensor Back Reaction Quasi-static (Qs) Approximation Motivation For The Problem: Is The Qs Approximation Valid? Violation Of WEC Method Of Approach MODEL Classical Metric Schwarzschild Exterior Jump conditions on the Null shell Gauge choice General Exterior Quantum Field T µν In Two Dimensions D-Model Stress Tensor In Schwarzschild Exterior QUASI-STATIC APPROXIMATION Constructing The Metric Computing The Energy Flux Analysis The Complete Quasi-Static Geometry

6 4 NUMERICAL EVOLUTION Algorithm Initial Data Constraint Equations Algorithm Critical Radius Adaptive Meshing Testing The Code Accumulated Error Constraint Violation Correspondence Check With Schwarzschild Geometry Validity Of Gauge Choice At The Surface S Retrieving Output Trace A Constant r Surface Apparent Horizon Negative Energy Density Energy Flux RESULTS AND DISCUSSION Position Of Apparent Horizon Violation Of WEC Comparison Of Mass Loss Apparent Horizon mass SUMMARY AND CONCLUSION Future Directions APPENDIX A INTERPOLATION B DERIVATIVE TERMS B.1 Nonlinear Equations B.2 Linearized Equations B.3 Schwarzschild Derivatives REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 5-1 Position of the Bulge Position of markers P (α) and S(α)

8 Figure LIST OF FIGURES page 1-1 Gravitational collapse Penrose diagram for a collapse geometry Geometric optics approximation Evaporation geometry Quasistatic geometry Collapse and Evaporation without singularity Quasi-static geometry Quasi-static geometry Algorithm for evolution Adaptive Mesh Constant r curves of Schwarzschild Constant r curves of evaporating BH AH and Tvv Mass loss α= Mass loss α= Mass loss α= Mass loss α= Mass loss α= Mass loss α= P (α) plot S(α) plot Bondi mass

9 Chair: Bernard F. Whiting Major: Physics Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BLACK HOLE EVAPORATION: VALIDITY OF QUASI-STATIC APPROXIMATION By Karthik Shankar May 2007 Hawking s discovery that a black hole quantum mechanically radiates energy like a black body suggests that its mass should decrease, leading to a process known as black hole evaporation. Solving for the evaporating black hole geometry (that is, its metric) exactly doesn t seem possible because it involves many complications. The most serious complication is that we do not have an analytic functional form for the quantum stress energy tensor in terms of the unknown metric. One approach to solving this problem is to use a Quasi-static approximation, which assumes that the evaporating black hole at every instant can be approximated by a stationary black hole. Effectively, it assumes that the luminosity of the black hole at any instant goes as 1/M 2, where M is the mass of the black hole at that instant. In this dissertation, the validity of this approximation is examined in the context of a simple model where exact numerical calculations can be performed. In this model, we assume an analytic form for the quantum stress energy tensor in terms of the unknown metric. This model is a four dimensional extension of the two dimensional black hole geometry originally investigated by Unruh, Fulling and Davies. We explicitly compare the results obtained from the quasi-static approximation and the exact numerical calculation performed in this model. We observe that there is a significant difference between the quasi-static approximation and the exact calculation whenever the quantum effects are large. When the quantum effects are very small, as in astrophysical black holes, the quasi-static approximation matches the exact calculations very closely. 9

10 CHAPTER 1 INTRODUCTION 1.1 Classical Black Holes General relativity views gravity as a manifestation of curvature in the geometry. The spacetime is not flat. There exists a metric on the spacetime manifold and test particles move on geodesics with respect to this metric. Gravity, which is now defined by the metric on the manifold is in turn produced by the matter. In Newtonian gravity, the sources were just masses (single component). In General relativity, the source is a 10 component object called the stress-energy Tensor. The equation which relates the stress energy tensor to the curvature of the metric is called Einstein s field equation. G µν = 8πG c T µν. 2 Here, G µν is the Einstein tensor which can be calculated from the metric and T µν is the Stress energy tensor. In this entire thesis, we will work in a system of units called geometrized units, where G = c = 1. With respect to this system, length, mass and time, all have the same dimensions. The Einstein equations are then G µν = 8πT µν. There are some very interesting solutions to Einstein s field equations known as black holes. There is a variety of such solutions. But all these solutions carry a common property, namely, a section of the spacetime manifold (say BH) is causally disconnected from the rest of the manifold (say Ext), that is, no information from the section BH can leak out to the other section Ext. In other words, no timelike or null curve can originate inside BH and enter Ext, but there exist timelike and null curves that originate in the Ext and enter BH. The boundary between Ext and BH is commonly known as the Event horizon. Another common feature of these black hole solutions is the existence of singularities. A point in the spacetime is supposed to be singular if there exists a geodesic which cannot be continued through that point. In most situations, this happens at a point where the curvature tensor diverges. 10

11 The mathematical existence of such black hole solutions does not tell us how these black holes are physically formed. To understand that, in section 1.1.1, we will briefly go over the key points involved in a gravitational collapse. In this respect, we will also introduce the notion of Penrose diagrams. Representing the causal structure of the spacetime geometry using Penrose diagrams is very useful in interpreting many of the physical properties of the geometry, and they will be frequently used in this work. To better understand the singularities, black holes and the collapse process that gives rise to black holes, there is a number of theorems we need to be familiar with. We will spend Section and Section doing that. In Section 1.1.2, we will briefly discuss the relevant theorems which categorized the various black hole solutions, and in Section 1.1.3, we will go over the classical laws of black hole mechanics which resemble the laws of thermodynamics Gravitational Collapse The physical picture of a star collapsing under its own gravity to form a black hole is fairly simple. If the star is so massive that the matter pressure in it is not strong enough to overcome gravity, the star will collapse until it becomes a single point with infinite density. Let us see what General relativity has to say about such a collapse. To keep things simple, let us restrict ourselves to spherical collapse. There is a simple theorem by Birkhoff which is very useful in the context of collapse. It states that the metric on the manifold is static and uniquely determined if we assume that the spacetime is spherically symmetric and that there is no matter (T µν = 0). In particular, when a spherical star collapses symmetrically, its exterior geometry has to be static (independent of time) and the metric outside the surface of the star is given by the Schwarschild solution. ds 2 = (1 2M r )dt2 + 1 (1 2M r )dr2 + r 2 dω 2 11

12 t r=0 singularity AH EH r Figure 1-1. Gravitational collapse of spherical star which results in formation of a black hole. A thorough description of the Schwarzschild geometry and the various coordinate systems generally used to describe it can be found in any standard general relativity text book (for eg. [23], [34]). Hence let us not discuss these details. Nevertheless, we would like to explicitly mention one of the important properties of Schwarzschild geometry. The r =constant surfaces are timelike surfaces outside the event horizon (r = 2M) and they are spacelike surfaces inside the event horizon. This implies that any timelike curve inside the horizon should reach the singularity at r = 0. Hence any r =constant surface inside the horizon is a Trapped surface, basically because even light cannot move out in the direction of increasing r. In a general geometry, a Trapped surface is defined as a closed spacelike 2-surface on which all null geodesic congruences have a positive convergence. Coming back to the collapse of a spherical star, Birkhoff s theorem tells us that the geometry outside the surface of the star is the Schwarzschild geometry. This means, once the surface of the star shrinks to a size less than its gravitational radius (r = 2M), an event horizon forms and outgoing light rays from the surface of the star will move in the direction of decreasing r. In Fig. 1-1, we trace the outgoing light rays from the point r = 0 12

13 r = o i+ H I+ r = 2m i0 R(t) I- i- Figure 1-2. Penrose diagram representing a gravitational collapse which results in formation of a black hole. at various stages of collapse. The x-axis is the r coordinate and the y-axis corresponds to some time coordinate such that each of the t =constant surfaces are spacelike surfaces. We can clearly see that the rays that come out of r = 0 after a particular time, never get out to infinity. They reach a maximum value of r, then they turn back and move in the direction of decreasing r. The null ray which stays at a constant value of r after leaving the surface of the star is the Event horizon. The formation of a singularity is shown in the figure by thickening the r = 0 line after a particular time. Let us now look at the behavior of r = constant surfaces in the collapse geometry. Let us first draw the conformal diagram (Penrose diagram) for the spherically symmetric collapse geometry. Look at Fig At every point, the null cones are 45 o lines. The surface of the star R(t) is shown in red. The various r =constant curves are shown in the diagram. The boundary of the diagram is the future and past null infinities, I + and I respectively. The point where r=0 line becomes horizontal is the beginning of the singularity. The event horizon, denoted by H, coincides with the curve r = 2M in the exterior of the star surface. The radius of the star s surface decreases from infinity and crosses the event horizon. Observe that the r=constant curves outside the surface of 13

14 star and inside the event horizon are spacelike. So, once the surface of the star crosses its gravitational radius, it has no choice but to hit the singularity. So, classical general relativity predicts the fate of a sufficiently massive star to be a singularity Theorems To have a reasonably complete discussion, it is necessary to mention some of the milestone theorems proved with respect to collapse and classical black holes. A thorough description of these theorems can be found in [18]. Israel(Uniqueness theorem): A static black hole in vacuum spacetime necessarily has to be spherically symmetric, that is, it must correspond to the Schwarzschild solution. Hawking: A Stationary black hole has to be either Static or Axially symmetric. Carter: A stationary and axially symmetric black hole in vacuum has a unique form given by the Kerr Solution. No hair theorem: A stationary black hole in vacuum is uniquely determined by its mass, charge and angular momentum. An external observer cannot find anything about the black hole other than these three parameters. That is, an observer at future infinity has no way of finding whether the black hole was formed out of collapse of Television sets or collapse of cars. Hence, with respect to an external observer, there is a huge information loss when black holes are formed. Price: During a realistic gravitational collapse, all modes of perturbations are radiated away as gravitational waves and the spacetime will eventually settle down to a stationary geometry. All the above theorems lead to a wonderful implication: The final result of any gravitational collapse is a stationary black hole characterized by its mass, charge and angular momentum. In particular, if the initial angular momentum of the star is zero, and if the star is uncharged, then the final black hole geometry is just the Schwarzschild geometry. 14

15 Weak energy condition (WEC) is the requirement that the local energy density measured by any observer must be positive. Mathematically this amounts to asking for T µν W µ W ν 0 for all possible timelike and null W µ vectors. Singularity theorems : There are a few of these theorems. The first of these is given by Penrose [28]. It states that a singularity will necessarily form if the following three conditions are satisfied, (i) Weak Energy condition is satisfied, (ii) There exists a non-compact Cauchy surface in the space-time manifold, (iii) there exists a trapped surface in the manifold. This is a very powerful theorem in the context of collapse, for it states that the moment the first trapped surface forms within a collapsing star, its fate is determined to be singular. Hawking: Two-Surface area of the event horizon never decreases, if Weak Energy condition is satisfied. This theorem is very general and it does not require the black hole to be stationary, nor does it require it to be a vacuum solution Black Hole Dynamics Before we discuss the consequences of quantum effects on black holes, let us first briefly go over some important properties of classical black holes. Four laws were derived according to which the black holes behave. These had an excellent mathematical analogy with the laws of thermodynamics. A good description of these laws can be found in [29]. They are briefly summarized below. 1. The surface gravity κ of a stationary black hole 1 is uniform over the event horizon. {κ 2 = 1 2 ξα;β ξ α;β. For Schwarzschild black hole, κ = 1/4M}. The analogous statement in thermodynamics is that a system in equilibrium has uniform temperature. Thus, we make a connection between surface gravity and temperature. 1 The geometry should have a killing vector ξ α, which is timelike outside the event horizon. 15

16 2. If the charge Q, mass M, angular momentum J, of a stationary black hole changes infinitesimally due to some process of external interference, then the area of the event horizon changes by δm = κ 8π δa + Ω HδJ + ΦδQ. This equation is analogous to the first law of thermodynamics wherein M is proportional to energy, A is proportional to entropy, κ is proportional to temperature, and the last two terms of the equation correspond to the work done in the quasi-static limit. 3. If null energy condition is satisfied, then the surface area of the black hole never decreases δa 0. This is just Hawking s Area theorem. This is analogous to the second law of thermodynamics, where a system s entropy always increases if it absorbs positive energy from the surroundings. It can also be shown that the total entropy of the black hole and the external universe together should always increase. This is called the Generalized Second Law (GSL). 4. The κ of the black hole cannot be reduced to zero within a finite advanced time. The thermodynamic analogue of this statement is that no system can be brought down to zero temperature in finite time. Bekenstein [3] showed that these laws bear more than just mathematical resemblance with thermodynamics. He argued that information content is thermodynamically nothing but entropy, and the information content in a black hole can be evaluated by considering the bits of information lost into it during collapse. With this argument, he finds that the entropy of a black hole is 1 (ln 2)A/. 2 The analogy between black holes and thermodynamics is further strengthened when Hawking showed that black holes would radiate as though it were a black body with a temperature of κ. If we assume that this quantity is the correct expression for the 2π temperature of the black hole, then by law 2, the correct expression for entropy should be A/4, which is very close to the result of Bekenstein s argument. So, it appears that there should be some deeper connection between thermodynamics and black holes. With these four powerful laws we can ask the question : How does the event horizon behave after the collapse? Is it possible to extract energy from it? 16

17 By analyzing the above laws, one can come to the conclusion that energy can be extracted from a charged rotating black hole until the black hole stops rotating, Ω = 0 and the black hole becomes neutral, Q = 0. The processes which leave the entropy (Area) unchanged are called reversible processes. These processes can of course change the mass, charge and angular momentum, but with a constraint that M 2 = (M ir + Q2 ) + J 2, 4M ir 4Mir 2 where M ir = ( A 16π )1/2. This constraint is derived from the second law. By definition M ir is invariant under a reversible process. An irreversible process is one which increases M ir. It is clear that when the maximum possible energy is extracted from the black hole, M ir would be its final mass. Hence, as a consequence of the second law, no energy can be removed from a Schwarzschild black hole. Let us now take a small digression to emphasize that there exist at least two well known processes by which energy can be extracted out of a Kerr black hole until it looses all its angular momentum and becomes a Schwarzschild black hole. For details, refer to [23]. Penrose process : Let a particle get into the ergosphere of a rotating black hole and break into two pieces (say a bomb explodes). Let one piece have a positive energy with respect to the observer at infinity and let the other piece have a negative energy with respect to infinity. The important point to note is that, inside the ergosphere, there can exist a timelike four velocity for which the energy is negative (which is not possible outside the ergosphere). Now, when the positive energy particle comes out to infinity, by conservation of energy, the particle has more energy than the initial particle. So, we have succeeded in extracting energy from a black hole which has an ergosphere (Ω 0). Note that, there is no such ergosphere in a Schwarzschild black hole. Superradiance (Misner effect) : Consider a classical wave incident on a rotating black hole. It can been shown that, if the wave has the same direction of angular 17

18 momentum as the black hole, then it get scattered off with an increased amplitude and in the process reducing the angular momentum of the black hole. Thus, again we have retrieved energy from a rotating black hole. The above discussions were all about classical processes. One main assumption that goes into our conclusion that no energy can be extracted from a Schwarzschild black hole is that the WEC is satisfied. All known classical matter satisfy this assumption. But quantum fields violate the WEC. Hence our conclusion is not justified when quantum fields are present in the black hole geometry. In fact, in 1975, Hawking showed that even Schwarzschild black holes would radiate energy quantum mechanically [16]. The result of this process is referred to as Hawking radiation Particle Creation 1.2 Hawking Radiation Hawking s original derivation of black hole radiation [16] was followed by mathematically more precise derivations [12] that confirmed his predictions on black hole radiation. In this Section, we shall go over the key steps involved in Hawking s original derivation. Let us start with a brief discussion on the process of particle creation in curved spacetime. In flat spacetime, the concept of particles arise from second quantization of fields. We shall first generalize the concept of particles to curved spacetime. A detailed description of field theory in curved space, can be found in [35], [13] and [4]. To keep things simple, we will restrict ourselves to the simplest of the field theories; a real, massless, scalar field. 1. We first solve the classical field equations µ µ Φ(x) = 0. We then define the Klein-Gordon inner product of two solutions ϕ(x), ψ(x) as (ϕ, ψ) i dσ µ {ϕ(x) µ ψ (x) ψ (x) µ ϕ(x)}. Σ The field equations guarantee that the above integral when evaluated over any Cauchy surface Σ gives the same result. We define H to be the Hilbert space of smooth solutions to the field equations which vanish sufficiently rapidly at spatial infinity and have a finite Klein-Gordon norm. We then obtain a orthogonal basis to H. Let us call the individual elements of the basis set mode functions. These mode functions are split into two sets, positive frequency modes {u i } and negative 18

19 frequency modes {u i }, such that (u i, u j ) = δ ij, (u i, u j) = δ ij and (u i, u j) = 0. In flat spacetime, these modes actually correspond to plane waves exp(i k. x iωt). The modes with ω > 0 are the positive frequency modes and the modes with ω < 0 are the negative frequency modes. Moreover, in flat spacetime, the splitting into positive and negative frequency modes is natural, in the sense that, with respect to the global Minkowski time coordinate t, the positive frequency solutions have the property that t u j(t, x) = iω j u j (t, x), ω j > 0. In an arbitrary space time, which does not have a natural time axis, there is no such natural splitting. 2. We then expand the field operator Φ(x) with respect to the mode functions in terms of creation and annihilation operators as Φ(x) = Σ i (a i u i (x) + a i u i (x)). 3. Now, the canonical quantization procedure leads to the following commutation relations between the creation and annihilation operators. [a i, a j ] = δ ij and [a i, a j ] = [a i, a j ] = 0. Note that, the quantization procedure very much depends upon our choice of the mode functions. 4. Let us now construct the Fock space, which is defined as the space of all possible states of the quantum field. The vacuum state vac or the zero particle state is the one which is annihilated by all the annihilation operators, that is, a vac = 0. If we index the set of mode functions by the momentum associated with that mode, then the single particle state, characterized by its momentum p, is defined as the state created by the creation operator a p when it acts on the vacuum state vac. Extending this definition, the multi particle states are defined by a set of creation operators acting on the vacuum state. For example, a three particle state with momenta p 1, p 2, p 3 is defined as a (p 1 a p 2 a p 3 ) vac.(since these particles are bosons, we symmetrize with respect to p 1, p 2, p 3 ). The set of all possible multi particle states forms the basis of the Fock space. 2 2 The basis states can be normalized by a multiplicative factor. 19

20 To calculate the number of particles in a given state, let us define the number operator as N = Σa i a i, so that in any given state ψ, the expectation value of the total number of particles can be obtained as ψ N ψ. Let us now choose another set of mode functions, say {v j, v j } and call the creation and annihilation operators with respect to these mode functions as b j and b j, so that the field expansion is, Φ(x) = Σ j (b j v j (x) + b j v j (x)). Quantization with respect to any set of mode functions would ultimately give rise to the same Fock space, but the basis of the Fock space constructed in terms of multi particle states would differ depending on our choice of mode functions. In other words, the definition of Particle depends on our choice of mode functions. The vacuum state vac with respect to the old set of mode functions {u i, u i }, is no longer a zero particle state with respect to these new mode functions {v j, v j }. To calculate the expectation value of the number of particles, we make use of the Bogolubov transformations [11], [4]. It turns out that, vac Σ j b j b j vac = Σ ij β ij 2, where β ij is defined as the inner product of the two sets of mode functions, β ij = (v i, u j). This is essentially the concept of particle creation. Given a background geometry and two sets of mode functions, we now know the procedure to calculate the particle creation. At the first glance, the concept of particle creation seems to be a purely mathematical artifact, because it ultimately just depends on the choice of the mode functions, and there is no natural choice of mode functions except in flat spacetime. But that is not true, there are some situations where the concept of particle creation does have a physical meaning. To understand this, we should first note an important property of the mode functions: if the mode functions are fixed on any one Cauchy surface, the field equations determine them everywhere in spacetime. Let us now ask how we should choose the mode functions on a surface so that corresponding states 20

21 have the correct physical meaning with respect to the relevant observers? If this surface is from a region in spacetime which is almost flat, then we know the answer to that question: We make the natural choice of mode functions as in flat spacetime. Otherwise, we do not know the answer. Let us now consider a spacetime which is almost flat at early times (t = ) and late times (t = ). Consider two spacelike Cauchy surfaces at these two times Σ and Σ respectively. The natural choice of mode functions on these two surfaces may not match, because the spacetime is not flat everywhere. That is, when one set of mode functions is propagated through time from one Cauchy surface to the other, they need not match the other set of mode functions. This might lead to a nonzero Bugolubov coefficient β ij. Hence, the vacuum state with respect to an observer at early times on the surface Σ would be seen as a nonzero particle state by an observer at late times on the surface Σ. A variety of examples of particle production has been worked out. Particle creation effects on different cosmological backgrounds have been calculated [4]. Unruh calculated the particle production in flat spacetime with respect to an accelerated observer [32]. It turns out that the accelerated observer would see a thermal bath of particles. Hawking applied the concept of particle creation to the gravitational collapse geometry, and he found that an observer at late times would observe particles in accordance to a thermal spectrum [16], [17]. Let us now briefly discuss Hawking s result. In a real collapse geometry as in Fig. 1-2, we have two asymptotically flat surfaces {I } and {I + }, and hence two sets of inertial flat space observers. {I } is a Cauchy surface by itself, and {I +, H} together form a Cauchy surface. The observer at I will choose his natural set of mode functions f ωlm such that f ωlm e iωv Y lm (θ, φ) on I. r The observer at I + will choose a natural set of mode functions p ωlm such that p ωlm e iωu Y lm (θ, φ) on I + r 21

22 and p ωlm = 0 on H +. For brevity, we will suppress the angular dependence in the modes, but we should keep in mind that all of the following arguments apply to all the (l, m) modes. We see that f ω forms a complete basis, but p ω does not form a complete basis by itself. To specify the basis completely, we have to specify some in falling modes q ω with nonzero behavior on the horizon. The form of q ω is not needed to compute the particle creation at I +. We should just make sure that whatever be the q ω we choose, it should have zero data at I +. Our aim is to find the number of particles observed at I + as a function of the retarded time u. Since the mode functions p ω do not have a compact support at I +, when we calculate the particle creation, the particles will be delocalized everywhere on I +. To calculate the number of particles observed at I + as a function of time, the mode functions p ω should not be used. We have to construct a complete set of wave packets out of these mode functions. Let us use P u 0 ω 0 to denote a wave packet of characteristic frequency ω 0 and peaked around u 0 with a width of u 0. We will not get into construction of these wave packets, a detailed description can be found in [11]. Let us consider the field to be in a vacuum state with respect to the observer at I. To compute the number of particles emitted at I + in a time interval u 0 around the time u 0, with frequency ω 0, we need to calculate the Bugolubov coefficient β u 0 ω 0 ω = (P u 0 ω 0, f w), which involves evaluating the inner product integral on a Cauchy surface. We will evaluate the inner product on I. For this, we first need to know the behavior of the modes P u 0 ω at I. The way to obtain the functional form of P u 0 ω at I is to back propagate it from I + using the classical field equations. When we back propagate the modes from late times u 0, these modes would travel very close to the the event horizon r 2M, and are the highly blue-shifted. We now invoke Geometric optics approximation [23], [34], which says that, if the effective wavelength of the wave is very short compared to the curvature scale of the geometry, the surfaces of constant phase of a wave can be approximated by null rays. 22

23 0 e -iwu U= 0 u 0 H λ γ e -iwv v0 Figure 1-3. Diagram representing Geometric optics approximation by which modes at I + are traced back to I. Hence, we can back propagate the P u 0 ω modes using null rays. This is pictorially shown in the Fig Back Scattering The classical Klein-Gordon field equation in Schwarzschild geometry can be viewed as a free field equation (as in flat spacetime) with a potential term. The potential term peaks at r = 3M and vanishes at r = 2M and r =. It depends on the l value of the wave mode. The potential is larger for a higher value of l. If we neglect the potential, then we can freely propagate the wave through spacetime to get the functional form of P u 0 ω at I (that is, we do not even need the help of Geometric optics approximation). By evaluating the inner product integral at I, we can compute the Bugolubov coefficient β u 0 ω 0 ω, and hence the number of particles emitted. It turns out that at sufficiently late times (large u 0 ), the flux of particles (number of particles emitted per unit time) at I + is a constant (independent of the time u 0 ). n ω = 1 e 8πMω 1 23

24 The flux is similar to a thermal flux, in the sense that it obeys Planck distribution with a temperature T H = 8πk B M. Since we neglected the effects of the potential term which depends on the l value of the modes, we will find that every l mode contributes to the energy flux equally. The energy flux or the luminosity of the black hole in each l mode is L l = 2l + 1 2π 0 ω e 8πωM 1 dω = (2l + 1) 768πM 2. If we sum over all the l modes, we see that the total luminosity of the black hole is infinite. This happens because we neglected the effects of the potential. Let us now include the effects of the potential for each l mode. Since, the potential term is larger for a higher value of l, the higher l modes will be back scattered more. It turns out that including the effect of back scattering modifies the results of particle flux and luminosity by just a multiplicative factor T ωl. n ω = T ωl e 8πMω 1, L l = (2l + 1) 768πM 2 T ωl. This can be physically understood in the following way. A fraction R ωl of the back propagated mode from I + is reflected by the potential back to I with the same frequency. This piece of back propagated mode does not contribute to particle creation when we calculate the Bugolubov coefficient integral at I. The other fraction(t ωl ) propagates along null rays (geometric optics approximation) close to the horizon and gets reflected at r = 0 back to I. It is this piece of the back propagated mode that contributes to particle creation when we calculate the Bugolubov coefficient integral at I. Hence, it is understandable how the transmission coefficient (T ωl ) enters the results. For each mode ω, l, we can numerically compute the transmission coefficient by solving the Klien-Gordon wave equations with appropriate boundary conditions. For a given value of ω, T ωl 0 as l becomes large. If we compute the total luminosity of the 24

25 black hole by summing over all the l modes, it now turns out to be finite [25]. Explicit numerical computations have been performed [10] and it turns out that the luminosity of the l = 0 mode is L l=0 = and the total luminosity of the black hole is L = πM 2, πM 2 It is interesting to note that more than 90 percent of the black hole luminosity is from the s-wave modes (l = 0). Another point to note is the importance of backscattering. Without back scattering, the s-wave luminosity would be around five to ten times bigger than what it should be. Yet another point we would like to note is that, even though the magnitude of luminosity is strongly affected if we neglect back scattering, the functional form of luminosity (L /M 2 ) is nevertheless preserved. So, in a sense, neglecting back scattering is acceptable if we are interested in studying just the qualitative aspects of black hole radiation Quantum Stress Tensor So far, we used the concept of particle creation to discuss black hole radiation. But as we discussed in Section 1.2.1, the concept of particle has a physical meaning only with respect to inertial observers at I and I +. Let us now discuss a more rigorous way to approach the problem of black hole radiation. In General relativity, the covariant stress energy tensor of the matter field at any point in space time contains all the physically important information about the energy fluxes and momentum fluxes. The quantum field which apparently contributes to an energy flux at I + (as seen in Section 1.2.2), should have a corresponding stress energy tensor. From the Lagrangian of the Klein-Gordon field, we see that classically, it has 25

26 following stress tensor. T µν = µ Φ ν Φ 1 2 g µν α Φ α Φ. When we quantize the field, the stress energy tensor becomes an operator. Quantum theory then tells us that the physically measured stress energy tensor is given by the expectation value of this operator T µν. Unfortunately, calculation of the expectation value of the stress energy tensor with respect to any state in curved space time is rather complicated. Even in flat space, when the expectation value of stress tensor operator is calculated, we get an infinite result. This is because when we sum over all the modes, the zero point energy of each of them add up to give infinity. No matter which state we calculate the expectation value of the stress tensor, this infinite zero point energy is always a problem. To rectify the problem, we use the argument that, the absolute energy of a state is really of no consequence, it is the difference in energy between various states that is physically measurable. So, it is justified to ignore the infinite zero point energy. Since in flat space, an inertial observer is supposed to measure zero energy in the vacuum state (ground state with zero particles), the physically measured energy in any other state can be computed by calculating the expectation value of the stress energy tensor in that state and then subtracting out the infinite zero point energy which appears while calculating the expectation value of the stress tensor in the vacuum state. This process of subtracting out infinity is called Renormalization. Mathematically, this renormalization is done by normal ordering the stress tensor, which amounts to rearranging the creation and annihilation operators which appear in the field expansion in such a way that all the creation operators are to the right of all the annihilation operators. We represent the normal ordered stress tensor as : T µν :. The expectation value of the normal ordered stress tensor with respect to any state ψ is then ψ : T µν : ψ. Unfortunately, this process of normal ordering is not a covariant procedure. Hence the resulting expectation value of the normal ordered stress tensor will not be a tensor. A correct procedure for renormalization has to be adapted to obtain a 26

27 covariant stress energy tensor. There are various methods of doing this : dimensional regularization, covariant geodesic point split regularization [8],... A detailed description of all these methods are given [4], [13]. Without getting into the details of these procedures, we shall go over the results relevant to black hole radiation. To calculate T µν, we need to know the state of the quantum field and the background geometry completely. We do not know the collapse geometry completely, all we know is that the exterior geometry is Schwarzschild. It turns out that is possible compute T µν in the exterior Schwarzschild geometry. The expression for T µν contains messy integrals. Using a Gaussian path integral approximation, an approximate analytic expression [24] for it can be derived. But a complete computation can be performed only numerically [5], [6], [21]. Let us now briefly discuss the properties of T µν in the exterior section of Schwarzschild geometry for various states of the quantum field. The region under consideration is bounded by H, I (the past horizon and the past asymptotic null infinity)and H +, I + (the future horizon and the future asymptotic null infinity). The state of the field will be described with respect to inertial observers on these surfaces. Boulware vacuum: This state corresponds to zero particle state with respect to observers at I and I +. When T µν is evaluated with respect to this state, it turns out that it vanishes at I and I + as expected. But, near the horizon, when evaluated in a locally inertial coordinate system, T µν blows up. This tells us that Boulware vacuum state is physically unstable. Hartle-Hawking vacuum: On the horizon H, the Kruskal U coordinate is a locally inertial coordinate and on the horizon H +, the Kruskal V coordinate is a locally inertial coordinate. Hartle-Hawking state is the vacuum state with respect to the standard mode functions exp(iωu) on H and exp(iωv ) on H +. Note that, even though U and V are locally inertial coordinates on the horizon, this state does not physically correspond to a zero particle state at the horizon, because the geometry near the horizon is not flat. 27

28 In fact when T µν is calculated, it turns out to be nonzero and negative near the horizon. At I + and I, the energy density turns out to be a constant. There is an outgoing energy flux at I + matched exactly by an ingoing energy flux at I. Hence, this state represents a black hole in thermal equilibrium. Unruh vacuum: This state corresponds to vacuum state with respect to locally inertial observers at I and H. The stress tensor T µν is regular everywhere on the horizon. Moreover, we find a positive energy flux at I +, and negative energy flux at H +. Note that, this state closely resembles the in vacuum state of the collapse geometry (in a collapse geometry, we do not have a H ). Hence, we expect this state to reproduce the radiation seen in the collapse geometry. In fact, the energy flux at I + has the same form as what we obtained using the concept of particle. This suggests that Hawking s original treatment of the problem is indeed valid. Further discussion on the Quantum stress tensor given in Chapter Back Reaction From Section 1.2, we learn that when we apply quantum field theory on a fixed black hole geometry, we see that the black hole radiates with a constant luminosity at late times. This implies that the total energy radiated out is infinite. This obviously is wrong, because conservation of energy tells us that the energy radiated out of the black hole should somehow be compensated by the decrease in mass of the black hole. Where did we go wrong? As discussed in Section 1.2.3, calculation of the quantum stress energy tensor in the Unruh vacuum state shows a negative energy flux across the horizon. This ingoing negative energy should decrease the mass of the black hole. This process is called Black hole evaporation. A genuine way to treat this problem within general relativity is to consider the back reaction effect of the quantum stress tensor on the geometry. That is, the geometry outside the collapsing star is not Schwarzschild, but is determined by the quantum stress tensor as dictated by the semi-classical Einstein field equations. 28

29 i + I + singular AH EH i 0 R(t) I - i - Figure 1-4. Penrose diagram representing a gravitational collapse, formation of a black hole and evaporation of the black hole. G µν = 8π T µν Since, as mentioned in Section 1.2.3, we do not have a generic expression for the quantum stress tensor in terms of the metric, we cannot solve the above equation to get a solution to the evaporating black hole geometry. Before considering ways to tackle this problem, let us look at the conformal diagram of the evaporating black hole geometry. We shall assume that the black hole evaporates completely so that its singularity finally disappears and that after all the radiation has escaped, the geometry is flat. The conformal diagram representing such a situation is shown in Fig r =constant curves of this geometry are shown in black. The large r curves are everywhere timelike as expected. For small values of r, including r = 0, the r = constant curves are initially time like, then become spacelike, and once again become timelike. The point where r = 0 curve turns spacelike indicates the formation of singularity. The point where the r = 0 curve turns back to a timelike curve indicates the completion of evaporation. Event horizon (EH) is the outgoing null ray (in red) that starts from the initial timelike piece of the r=0 29

30 curve and ends at the point where it evaporation ends. The region where the r =constant curves are spacelike is the trapped region. That is, any timelike or null curve will have to move in the direction of decreasing r. The boundary of trapped region AH is called the apparent horizon. A crucial point to note is that the apparent horizon is not confined within the event horizon. That is to say that the trapped region extends beyond the event horizon. We shall call this region the Bulge. Note that, in a collapse geometry without evaporation as in Fig. 1-2, there exists no such bulge Quasi-static (Qs) Approximation Let us now get to the problem of solving for the completely back reacting geometry. We already noted that solving this problem completely is not possible since we do not have the form of the quantum stress tensor in an arbitrary metric. The problem with calculating the energy radiated out of a fixed background is that the conservation of energy is violated. A commonly accepted method to fix this problem is the Quasi-static approximation. It says that, since the temperature of macroscopic black holes is very low, the process of evaporation is quasi-static. That is, the radiation from the black hole at any instant can be viewed as thermal radiation from a static black hole of mass M with a temperature T 1/M. Then, the conservation of energy requires that the rate of decrease in mass of the black hole is equal to the luminosity of the black hole. Since the luminosity of the black hole goes as /M 2, we have dm dt M 2. This tells us that the total life time of the black hole τ evap M 3. In terms of physical units, the temperature of a black hole is roughly 10 8 (M 0 /M) o K and the life time is roughly (M/M 0 ) 3 years, where M 0 is the mass of our sun. As the evaporation proceeds, the mass decreases, causing the temperature to increase, and hence the rate of evaporation increases. Since the evaporation is very slow, at any instant, it 30

31 might be possible to find a time scale τ much bigger than the dynamical time scale of the geometry ( τ M), such that (dm/dt) τ M. If there exist such a time scale, then it seems reasonable to neglect the back reaction of the quantum field on the geometry in that period of time τ and approximate the radiation as that emitted by a static black hole. Thus, the quasi-static approximation seems to be justified if there exist such a time scale τ. The two conditions on the time scale (dm/dt) τ M and τ M together imply that dm/dt 1. In the units where G=c=1, we then have /M 2 1. This suggests that the quasi-static approximation should hold good until the late stages of evaporation, that is, until the black hole mass goes down to Planck scale. During the late stages of evaporation, when dm/dt becomes large, the quasi-static approximation does not make sense. During this stage of evaporation, the black hole is very small and very hot and the curvature at the horizon is very high (Planck scale). At this stage there are bigger problems to worry about than validity of quasi-static approximation, the semi classical Einstein equations are themselves not valid, quantum gravity effects must be included Motivation For The Problem: Is The Qs Approximation Valid? Let us now understand the implications of the quasi-static approximation. If we note carefully, the basic assumption is that the fully back reacting evaporating black hole geometry can be considered as a sequence of snapshots of non back reacting black hole geometries. This approximation implies that, at every instant during evaporation, the geometry outside the event horizon(in particular, the region responsible for radiation) can be mapped to a region of Schwarzschild exterior of a given mass. Let us pictorially represent this in Fig We see that, at any instant (a specific snapshot), the apparent horizon and the event horizon coincide outside the surface of the star and the radiation coming out to an observer far away is generated from a Schwarzschild exterior like region. Let us take a look at the fully back reacting geometry. We see that there is a region where the apparent horizon bulges out of the event horizon. Various arguments [33], 31