Spherically symmetric trapping horizons, the Misner-Sharp mass and black hole evaporation
aa r X i v : . [ g r- q c ] A p r Spherically symmetric trapping horizons, the Misner-Sharp mass and black holeevaporation
Alex B. Nielsen and Dong-han Yeom Center for Theoretical Physics and School of Physics,College of Natural Sciences Seoul National University, Seoul 151-742, Korea ∗ and Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea † Understood in terms of pure states evolving into mixed states, the possibility of information lossin black holes is closely related to the global causal structure of spacetime, as is the existence ofevent horizons. However, black holes need not be defined by event horizons, and in fact we arguethat in order to have a fully unitary evolution for black holes, they should be defined in terms ofsomething else, such as a trapping horizon. The Misner-Sharp mass in spherical symmetry showsvery simply how trapping horizons can give rise to black hole thermodynamics, Hawking radiationand singularities. We show how the Misner-Sharp mass can also be used to give insights into theprocess of collapse and evaporation of locally defined black holes.
PACS numbers: 04.70.-s, 04.70.Bw, 04.70.DyKeywords: black holes, information loss, Hawking radiation, trapping horizons
I. INTRODUCTION
The possibility of information loss in black hole evap-oration has been with us for many years. There havebeen many attempts over the years to resolve the issueone way or another, but it is probably fair to say thatthere is still no consensus on what the correct picture ofa black hole and its evolution should be.Indeed, the possibility that the evolution of a blackhole may be a non-unitary process has many disturbingconsequences. We will not attempt here to provide adefinitive answer to the question of whether black holeevaporation is unitary or not. We believe that this isa question that can only be answered in the context ofa theory that we do not yet possess. Instead what wewant to do here is to highlight some of the semi-classicalissues involved in the problem and the implicit assump-tions that are often under-appreciated in the analysis.Specifically we wish to examine the role that the eventhorizon plays and its connection to the global causalstructure of spacetime. It is precisely the non-trivialcausal structure related to event horizons that leads toproblems with unitarity. Because we can ”throw away”entangled particles in an absolute sense, we can havepure states evolving into mixed states. To this end wewish to argue that there are relatively few options fordoing away with information loss other than doing awaywith event horizons. However, even if the spacetime doesnot contain event horizons it is possible to have a physi-cally meaningful notion of a black hole. To replace eventhorizons, we propose a formalism for black holes thatis already well established in the literature, based onmarginally trapped surfaces.To free us from this event horizon constraint we will ∗ Electronic address: [email protected] † Electronic address: [email protected] use a quasi-local definition of the black hole, in terms oftrapping horizons. At least at the semi-classical level,trapping horizons will be useful whatever the ultimateresolution of the issue of black hole evolution is.To clarify what one might expect from black hole evap-oration and give as clear a picture as possible, with-out solving the full back-reacting semi-classical Einsteinequations with renormalised stress-energy tensor, wepresent here a simple, spherically symmetric picture mak-ing use of the Misner-Sharp mass function. In sphericalsymmetry the Misner-Sharp mass function can be inter-preted as measuring the amount of energy within a sphereof areal radius r at a time τ . Therefore the Misner-Sharpmass function gives a reasonable, quasi-local definitionto the concept of the curvature-producing energy con-tained within a black hole. We can use the Misner-Sharpmass to locate black hole horizons. It can give us insightinto black hole thermodynamics and clarify our pictureof what is happening when the black hole evaporates.This paper is organised as follows. In section II wereview the standard arguments for how black holes can beused to turn pure states into mixed states. We emphasizethe role played by the event horizon and by extension thenon-trivial causal structure related to it, usually in termsof a spacelike singularity. In section III we give a briefreview of trapping horizons in spherical symmetry andhow they give rise to black hole thermodynamics andHawking radiation without any event horizon necessary.In section IV we describe how the Misner-Sharp mass canbe used in spherical symmetry to follow the evolution ofthe black hole spacetime and how the various possibleoutcomes can be explained in terms of it. II. INFORMATION LOSS IN BLACK HOLES
Information loss in black holes is related to the possi-bility of pure states evolving into fundamentally mixedstates (for example see [1, 2]). A fundamentally mixedstate is one that cannot be turned into a pure state byfiner-graining or further measurements. In a black holecontext, the mutation of pure states into mixed statescomes about most strikingly in the context of Hawkingradiation. Essentially virtual pair-production can lead toHawking radiation if one of the virtual particles falls intothe black hole, allowing the other particle to escape toinfinity with real positive energy. The two particles areinitially entangled with one another and are described byan entangled state | ψ i = Ce γb † c † | i , (1)where C is a normalisation factor and b † is a creationoperator for the particles that escape to infinity and c † isa creation operator for particles that fall into the blackhole. For simplicity we can write a similar state as | ψ i √ | i b ⊗ | i c + | i b ⊗ | i c ) . (2)This state is similar to (1) if we just ignore all two andmore particle states in the expansion of e γb † c † . This stateis entangled because we cannot write it as a factored stateof the form | ψ i ⊗| ψ i . While we still have both particleswe can write the density operator for the system as a purestate ρ pure = | ψ ih ψ | = 12 | i b ⊗ | i c h | b ⊗ h | c + 12 | i b ⊗ | i c h | b ⊗ h | c + 12 | i b ⊗ | i c h | b ⊗ h | c + 12 | i b ⊗ | i c h | b ⊗ h | c . (3)However, if we ‘ignore’ the particle that falls into theblack hole, then we must sum or partial trace over theunknown state and we obtain a mixed state ρ mixed = 12 | i b h | b + 12 | i b h | b . (4)This state is mixed because it cannot be written as singlestate | φ i in the form | φ ih φ | . In ordinary quantum systemsthis procedure of taking the partial trace over unmea-sured subsystems is common and leads to ‘statistically’mixed states. The subsystem we focus on is described bya mixed state but this is only because we choose to focuson a part of the entire system. This subsystem is en-tangled with some other subsystem and the entire state,including the subsystem we choose not to measure, is stilla pure state. If we chose to measure the full system wewould recover the full pure quantum state.However, when one of the subsystems has fallen over anevent horizon or perhaps more importantly, has reacheda boundary of spacetime and terminated, then there is noway, even in principle, that we could measure the stateof the entire system, because one of the subsystems will never return to our past light cone. For all intents andpurposes it is gone and we cannot choose to measure itsometime in the future. Therefore we must take a traceover the lost subsystem and the remaining state that isaccessible to us will be a fundamentally mixed state.It is this tracing over the unknowable degrees of free-dom that leads to pure states turning into fundamen-tally mixed states. While both particles still exist onecan claim that the full quantum state is pure. One couldeven claim that a particle that had fallen over the eventhorizon was still in principle measurable. However, ifthere is a spacetime boundary inside the black hole, suchas a singularity, then the particle will reach the end of itsworldline in finite proper time. After this it will cease toexist. If the black hole evaporates completely then theoutgoing Hawking radiation particles will be entangledwith nothing and they must represent a mixed state.Whether one wants to associate the tracing over theunknowable states with them falling over the event hori-zon or with them falling into a singular spacetime bound-ary and ending is largely a matter of taste. Particlesthat fall over the event horizon cannot, by definition, re-turn to the exterior asymptotic region, unless they moveacausally. In the classical picture of a Schwarzschildblack hole, particles that fall into black holes reach thesingularity in a finite proper time and at the singularitythey meet a spacetime boundary where their worldlinescannot be continued. In a more realistic, perhaps quan-tum description, whether their worldines actually cometo an end at some finite parameter time depends on thefull causal structure and the details of what one believeshappens in the singular regions of general relativity.A similar thing can happen if we allow naked time-like singularities. One of the entangled particles can fallinto the singularity and disappear while the other par-ticle escapes to infinity and becomes a fundamentallymixed state. This would appear to provide a mecha-nism for turning pure states into mixed states withoutevent horizons, although the existence of a naked singu-larity would cause other problems as well. In some senseblack holes are most commonly associated with informa-tion loss, only because the horizon readily creates the en-tangled particles and the spacetime singularity/boundaryallows one of the particles to be erased.The possibility of pure states turning into mixed statesis therefore related to the possibility of the worldlines ofparticles coming to an end. This suggests that in or-der to understand whether information can be lost inblack holes one must look at the possible endings of par-ticle worldlines. There are in fact several possibilities forworldlines. Worldlines will either end at finite parametertime, in which case we would say they have reached aspacetime boundary, or they can be continued to infiniteparameter time, in which case we would say that theyend at infinity. If continued to infinite parameter timethey can either end up in the same infinity of externalobservers or some other infinity, as occurs for example,for certain worldlines in the Reissner-Nordstr¨om space-time.The are other possibilities for worldlines that areslightly less clearcut. One can consider the possibilitythat worldlines become undefined and it is meaningless totalk about their continuation. This may occur if space-time because somehow ‘fuzzy’ because of quantum ef-fects. One can also consider the possibility that somenew physics conspires to induce a breakdown of locality,or allows particles and observers to move along space-like curves, so that causal Penrose diagrams are largelymeaningless. This might occur for example if the inte-rior boundary is defended by some membrane-like struc-ture that prevents all freely-falling particles from passingthrough itself, along their geometric geodesic paths [3].In most situations a spacelike boundary will give riseto an event horizon. This is the case with the spacelikesingularity in the Schwarzschild solution. The spacelikeboundary casts a shadow on light rays continued backfrom future infinity. Of course, spacelike singularitiesare not a necessary condition for event horizons. TheReissner-Nordstr¨om spacetime has no spacelike bound-aries but does have event horizons. The singularities inthe Reissner-Nordstr¨om spacetime are timelike and occurbecause of the disjoint null infinity structure.If we want to avoid pure states evolving into mixedstates we must not throw away the ingoing Hawking par-ticles. If the ingoing particles reach a boundary in finiteparameter time there will be hypersurfaces in the space-time that their worldlines do not intersect and it is hardto argue that they should not be ‘thrown away’. Thereare several possibilities to avoid this outcome.The first, perhaps most severe approach, is to deny thephysical existence of black holes. This can be done, forexample, by arguing that some non-trivial physics comesinto play when matter attempts to collapse to form ablack hole, as in for example [4–7].The second approach is to deny the existence of aspacetime boundary within the black hole that entangledparticles can fall over. This is for example the purely clas-sical proposal of Dymnikova [8] and Hayward [9] and thequantum geometrical proposal of [10]. The possibilitythat one could have horizons without singularities seemsto run foul of the singularity theorems but these can ei-ther be circumvented with non-global hyperbolicity [11]or energy condition violation [9] or a breakdown of thedifferentiable manifold picture [10]. Violation of the en-ergy conditions is perhaps not as drastic as it may sound,because Hawking radiation is known to violate nearly allof the energy conditions [12]. III. TRAPPING HORIZONS
If one follows the approach of denying the existence ofspacetime boundaries within black holes, but one doesnot want to deny the existence of a black hole, one is leftwith the question of what one means by the black holeregion. Traditionally, one defines the black hole using an event horizon. If there is no internal boundary withinthe black hole and no wormholes that connect the inte-rior region to a different disjoint asymptotic region, sincethe worldlines of particles have to end somewhere, theonly place left for them to end is at asymptotic future in-finity. This means that technically there will be no eventhorizon.If there is no event horizon, then how should one de-fine the black hole? Here we suggest that one simplyadopt the proposal of using some quasi-local definitionof the black hole in terms of for example, a trappinghorizon. Trapping horizons are defined quasi-locally interms of the geometrical structure and are independentof the global causal structure. To see how trapping hori-zons provide an acceptable definition of black holes andeven give rise to black hole thermodynamics and Hawk-ing radiation, we give here a brief summary of trappinghorizons in spherical symmetry. Further details can befound in the literature, see [13] and references therein.While spherical symmetry is not a reasonable assump-tion for astrophysical black holes it simplifies the dis-cussion sufficiently to justify its adoption for ‘conceptualclarity’. Spherical symmetry is a reasonable assumptionto study the formation and evaporation of black holes ata toy model level and it makes the analysis much simplerand clearer. Furthermore, in spherical symmetry, we canuse the Misner-Sharp mass as a definition of quasi-localmass, which as we will see, also provides conceptual clar-ity to the issue of black hole evaporation.A trapping horizon, more properly a future outer trap-ping horizon, is defined by Hayward [14] as the closureof a three-surface which is foliated by marginal surfaces,for which θ l = 0, and which, in addition, satisfies i. θ n < ii. n a ∇ a θ l < s = − e − t,r ) (cid:18) − m ( t, r ) r (cid:19) d t +d r (cid:16) − m ( t,r ) r (cid:17) + r dΩ , (5)in so-called Schwarzschild coordinates, where m ( t, r ) isimmediately recognizable as the Misner-Sharp mass func-tion. The Misner-Sharp mass function can be interpretedas the quasi-local mass contained within a sphere of ra-dius r at a time t . Unfortunately, this coordinate systemis undefined at r = 2 m ( t, r ) and cannot be continuedthrough this hypersurface and cannot be relied upon forcalculating properties at r = 2 m . A more appropriatecoordinate system is perhaps the Painlev´e-Gullstrand co-ordinate systemd s = − e − τ,r ) (cid:18) − m ( τ, r ) r (cid:19) d τ +2 e − Φ( τ,r ) r m ( τ, r ) r d τ d r + d r + r dΩ . (6)This coordinate system is regular at future horizons. Theradial null geodesics for this metric can be easily foundby setting d s = dΩ = 0. For this we findd r d τ = − e − Φ( τ,r ) ± r m ( τ, r ) r ! , (7)where the plus sign denotes the ingoing geodesics. Thuswe can find outgoing geodesics l a and ingoing geodesics n a with components l a = e Φ( τ,r ) , − r m ( τ, r ) r , , ! , (8) n a = 12 e Φ( τ,r ) , − − r m ( τ, r ) r , , ! . (9)The factor of two ensures that the cross normalisation isthe conventional n a l a = −
1. Then we can compute θ l = 2 r − r m ( τ, r ) r ! , (10) θ n = − r r m ( τ, r ) r ! . (11)We see that the expansion of n a is always negative andthat at r = 2 m ( τ, r ) the expansion of l a is zero. We canalso compute the value of n a ∇ a θ l at r = 2 m ( n a ∇ a θ l ) H = − (1 − m ′ H ) r H (cid:18) r H e − Φ H (cid:19) , (12)where a subscript H denotes functions to be evaluatedat the horizon and we have used a dash to denote partialderivative with respect to r and a dot to denote the par-tial derivative with respect to the time τ (here, since r H isonly a function of τ it is actually an ordinary derivative).For the horizon to be an outer horizon in a spacetimewith a regular asymptotic region, we require 2 m ′ H < m ( τ, r ) must be less than r for large r . In addition,we can see from (7) for the ingoing null geodesic n a that˙ r = − e − Φ H . Thus we see that we have a trapping hori-zon at r = 2 m if the horizon is outer and not movinginwards faster than ingoing null geodesics.The normal N a to the surface r = 2 m has norm N a N a = − m H e H − m H e Φ H (1 − m ′ H ) . (13) If ˙ m H = 0 the trapping horizon will be a null hypersur-face, and, assuming 1 − m ′ >
0, it will be a spacelikehypersurface if ˙ m >
0. For − (1 − m ′ ) e Φ < ˙ m < m < − (1 − m ′ ) e Φ the horizon is spacelike, but evap-orating ‘faster than the speed of light’ and so all timelikecurves from a region just inside the horizon must moveto the outside [13].The surface r = 2 m ( r, t ) does not however, define thelocation of the event horizon in a dynamical spacetime.The event horizon is always a null surface and so thespherically symmetric trapping horizon at r = 2 m canonly be an event horizon if ˙ m = 0 (note however thatthis is necessary but not sufficient). To find the eventhorizon, firstly one would need an explicit form for m ( r, t )and then one would look for radial null vectors that arenot able to reach infinity by propagating them outwardsfrom the centre of the spacetime.It has been well-known for a long time that locallydefined horizons can give rise to equations of black holethermodynamics [15]. At the horizon we have ∂m∂τ = 18 π (1 − m ′ )2 r d A d τ , (14)where m ′ = ∂m∂r and τ is a parameter labeling ‘time-slicings’ of the horizon. This equation has the same formas the first law of black hole thermodynamics δm = π κ δA with a surface gravity that agrees with otherdefinitions of surface gravity [16]. In order to obtain aversion of the second law we can just compute G ab l a l b ,where G ab is the Einstein tensor. This gives G ab l a l b = 2 e Φ r ∂m∂τ r mr − r ∂ Φ ∂r − r mr ! . (15)Rearranging, and imposing (14) at r = 2 m ( τ, r ) we find ∂A∂τ = 16 πr e − Φ − m ′ G ab l a l b . (16)Thus we see that the area of the horizon A is increasing if G ab l a l b >
0. By the Einstein equations we can write thiscondition as T ab l a l b >
0, which is just the null energycondition (NEC). The area of the horizon is increasing ifthe NEC is satisfied and can decrease only if the NEC isviolated.Furthermore, using the Parikh-Wilczek tunneling ap-proach to Hawking radiation we may expect to haveHawking radiation produced in the vicinity of the trap-ping horizon [17–19]. This is perhaps not surprising,since the effect is just a result of quantum field theoryon a curved background and quantum field theory is alocal theory that is unlikely to depend on global proper-ties like an event horizon [20].Consider the equation for a massless scalar field on acurved background¯ h √− g ∂ a (cid:0) g ab √− g∂ b (cid:1) φ = 0 . (17)We look for solutions of the form φ = exp ( − iS ( τ, r ) / ¯ h ).Taking this limit as ¯ h →
0, to lowest order this equationgives the Hamilton-Jacobi equation g ab ∂ a S∂ b S = 0 . (18)Invoking the geometrical optics approximation S ( τ, r ) = ωt − Z k ( r )d r, (19)this equation gives ω + 2 e − Φ r mr ωk − e − (cid:18) − mr (cid:19) k = 0 . (20)Rearranging gives k = ± ωe Φ ∓ q mr . (21)The plus sign denotes the outgoing modes and the nega-tive sign denotes the ingoing modes. The outgoing modescontain a pole at the horizon r = 2 m . S = ωt + 2 r H ωe Φ H (1 − m ′ H ) Z d r ( r − r H ) . (22)The integral can be performed by deforming the contourinto the lower half of the complex plane, which gives acomplex part to S Im S = 4 πr H ωe Φ H (1 − m ′ H ) . (23)At this level of approximation this corresponds to ther-mal radiation with a temperature T = 12 π e − Φ H r H (1 − m ′ H ) , (24)which agrees with the calculations in [16]. IV. BLACK HOLE EVAPORATION WITH THEMISNER-SHARP MASS
The Misner-Sharp mass [22] can be very useful for in-vestigating the behaviour of black holes [23]. For a spher-ically symmetric metric, using the Einstein equations, wehave T = m ′ πr , (25) FIG. 1: The radial mass profile as it might appear for a starof radius r star and mass M star . where m ′ denotes partial differentiation with respect tothe coordinate r . This will correspond to the local energydensity ρ as measured by an observer with constant coor-dinate r . Hypersurfaces of constant Painlev´e-Gullstrandtime τ are spacelike since the normal to the surface ∇ a τ satisfies ∇ a τ ∇ a τ = − e τ,r ) . (26)On a τ = constant hypersurface σ we can write a quasilocal Misner-Sharp mass as M = Z Σ ρ p − m/r d V, (27)where d V is the volume element on the hypersurface.In the words of Misner and Sharp [22], the function m includes contributions from the kinetic energy and grav-itational potential energy. Since the volume element isjust r sin θ/ p − m/r we obtain M = Z m ′ d r = m. (28)The Misner-Sharp mass reduces to the Arnowitt-Deser-Misner (ADM) mass at spacelike infinity and reduces tothe Bondi-Sachs mass at null infinity. In the Newtonianlimit is also gives the Newtonian mass to first order [23]. FIG. 2: As the star collapses, a horizon forms, initially witha small radius.
FIG. 3: As the collapse proceeds the area of the horizon grows.FIG. 4: Hawking radiation occurs in the vicinity of the hori-zon. This means that positive energy flows outwards and acorresponding negative energy flows inwards. The size of thiseffect for a macroscopically sized black hole is greatly exag-gerated here.
We can also see from (25) that if the local energy den-sity is negative then m ′ will also be negative. A negativeenergy density corresponds to a violation of the WeakEnergy Condition (WEC). Using the Page approxima-tion [24] for the renormalised energy-momentum tensorin the Hartle-Hawking vacuum of a Schwarzschild blackhole we know that Hawking radiation is expected to vi-olate the WEC and NEC [12]. Thus we might expect,in a fully backreacting spacetime, that in the region nearthe black hole horizon m ′ should be negative. In fact,we know that the null energy condition must be violatedin the vicinity of an evaporating, area decreasing horizon(16). This also means that the surface gravity and there-fore temperature of the black hole can be greater thanthe corresponding temperature of a Schwarzschild blackhole with the same mass [16].The Misner-Sharp mass allows us to construct the fol-lowing heuristic picture of black hole formation and evap-oration. By taking equal time “snapshots” of the Misner-Sharp mass function on various constant τ coordinateslices we can imagine the evolution of a star as it under-goes gravitational collapse to form a black hole and howHawking radiation evaporates away the mass-energy ofthe star. FIG. 5: As the Hawking evaporation proceeds, the horizonarea shrinks. The flux of positive energy moves outwards togreater and greater radius, and the flux of negative energymoves inwards to the centre.FIG. 6: Eventually, sufficient energy has evaporated awayfrom the black hole to leave the intermediate spacetime ef-fectively flat. An observer in this region would feel almostno gravitational field from the collapsed star. Whether theinterior region is flat as well depends on the physics at small r An important feature of this picture is the r = 2 m con-dition that allows us to see the location of the trappinghorizon boundary of the black hole. This is indicatedin the plots by a red line. Initially, Fig.1, the star hassome stellar mass profile, without necessarily any preju-dice as to whether the density goes like ρ r − as depictedhere, and vacuum outside. At this time, nowhere is thecondition r = 2 m reached. As the star collapses, a hori-zon forms, initially with a small radius, Fig.2. As thecollapse proceeds the area of the horizon grows, Fig.3.Hawking radiation is expected to occur in the vicinity ofthe horizon, Fig.4. This means that positive energy flowsoutwards and a corresponding negative energy flows in-wards. The size of this effect for a macroscopically sizedblack hole is greatly exaggerated here. As the Hawk-ing evaporation proceeds, the horizon area shrinks, Fig.5.The flux of positive energy moves outwards to greater andgreater radius, and the flux of negative energy moves in-wards to the centre. Eventually, sufficient energy hasevaporated away from the black hole to leave the inter-mediate spacetime effectively flat, Fig.6. An observer in FIG. 7: How the ingoing flux of negative energy interactswith the interior mass of the star depends on the details ofthe calculation at small radii. This is the region where thematter of the originally collapsed star is compressed to greatdensities and the region where one might expect quantumgravitational effects to be important. If the negative energyand original star matter annihilate completely, one will endup with a spacetime that is effectively flat Minkowski space. this region would feel almost no gravitational field fromthe collapsed star.How the ingoing flux of negative energy interacts withthe interior mass of the star depends on the details ofthe calculation at small radii. This is the region wherethe matter of the originally collapsed star is compressedto great densities and the region where one might ex-pect quantum gravitational effects to be important. Ifthe negative energy and original star matter annihilatecompletely, one will end up with a spacetime that is ef-fectively flat Minkowskian space, Fig.7.If we want to view the collapse purely classically thenthe boundary term at r = 0 is crucial. In the aboveplots we have imposed m ( τ, r = 0) = 0 as was done, forexample, in [9]. To investigate the behaviour at r = 0we can compute the Kretschmann scalar for the simplestatic case with Φ( τ, r ) = 0. If we take m ( r ) ∼ r n toleading order we get R abcd R abcd ∼ (cid:0) n − n + 17 n − n + 12 (cid:1) r n r . (29)This quartic equation in n has no real roots and thereforeif m ( r ) ∼ r n with n ≤ r = 0 in the Kretschmann scalar. Of course, a finiteKretschmann scalar is necessary but not sufficient forthere to be no singularity at r = 0.For the Schwarzschild solution we have the condition m ( τ, r ) = M for all τ and r , Fig.8. This suggests thatthe collapse may proceed in the way depicted in Fig.9.If m > r = 0 then we obtain a spacelike singularityat r = 0. Indeed, it is even possible that we could have m < r = 0 in which case we would have an untrapped(weakly naked) timelike singularity [23].The final thing that it is easy to show using the Misner-Sharp mass is the creation of an inner and outer horizon,as for example considered in [9]. The simple condition FIG. 8: For the Schwarzschild solution, the Misner-Sharpmass function is a constant, M , for all values of τ and r .FIG. 9: We can imagine the possibility that there is a finiteamount of mass contained within a sphere of radius zero. Inthis case there will be a singularity at r = 0. for this to occur is that m ′ > , (30)at the inner horizon where r = 2 m . In Hayward’s sce-nario the trapping horizon forms initially as an extremalhorizon at a finite radius. Then the inner and outer hori-zons separate to form a trapped region (with the regionclose to r = 0 untrapped) and then the inner and outerhorizons join up again forming an instantaneously ex-tremal horizon before disappearing altogether. This isshown in Fig.10. The region between the inner and outerhorizons contains trapped surfaces, while the region in-side the inner horizon does not. V. CONCLUSION
We have seen how the causal structure of spacetimeplays an important role in the semi-classical informa-tion loss argument. In this way, information loss is veryclosely tied up with defining black holes via event hori-zons. One often reasons that in order to have a blackhole one must have an event horizon. In order to have anevent horizon one must have a non-trivial causal struc-
FIG. 10: An inner and outer horizon. The region between theinner and outer horizons contains trapped surfaces, while theregion inside the inner horizon does not. ture. If one has a non-trivial causal structure then onealways has the possibility of information loss.Current notions about what constitutes a black holeare, of course, a result of the history of the developmentof the subject. The early days of Finkelstein, Wheelerand others started out with a purely classical picture ofa black hole based largely on exact solutions, which bytheir very nature, required a large amount of symme-try in order to be discovered as solutions of the Einsteinequations. Penrose, Hawking and others then laid downthe fundamental concepts of what it means to be a blackhole as a region that one can never escape from and devel-oped techniques for dealing with situations beyond exactsolutions. Then the laws of black hole thermodynamicswere discovered, still in the context of black holes thateventually settle down to just being black before finally,Hawking discovered black hole radiation and the possibil-ity that black holes could disappear arose. Had Hawkingradiation been discovered at an earlier stage, the com-munity might not be so insistent on defining black holesin terms of event horizons.We have presented a picture that hopefully clarifies theissue of how event horizons are related to informationloss. It is not based on an exact solution to the semi-classical Einstein equations but gives the properties onemight expect from a full solution. It is also importantto recognise what assumptions have gone into the abovepicture.Firstly, we have assumed spherical symmetry. Thisgreatly simplifies the formalism and allows us to use theMisner-Sharp mass as a quasi-local measure of energy. Itis of course, not a valid model for describing astrophysicalblack holes, but it is a valid assumption to study blackhole evaporation in its purest form. It is not expectedthat deviations from spherical symmetry will qualita-tively change the physics of black hole evaporation.Secondly, we have chosen to define the black hole interms of a trapping horizon. This is not the usual defini-tion of a black hole, but arises from the desire to defineblack holes without appealing to event horizons. There are many issues that remain to be clarified about theuse of trapping horizons and they may ultimately turnout not to be the optimal choice. Questions about theuniqueness of trapping horizons have not been studiedmuch in the literature and one can certainly ask questionsabout how a spacelike trapping horizon should intersecta given “constant time” spatial hypersurface.Thirdly, we have of course assumed that it is mean-ingful to talk about the matter that falls into the blackhole, and this irrespective of whether it is the originalcollapsing star or matter that is subsequently accreted.This aspect is often ignored in other studies, partly be-cause whatever falls through the event horizon, can, bydefinition, be ignored in terms of its effect of the outsideregion and secondly because there is in some quartersan implicit belief that whatever the state of the interiorblack hole is, it can somehow be read off from physics atthe horizon.Here we argue instead that the collapse of the centralmatter is essential to determining the causal structure ofthe spacetime and this is ultimately responsible for de-termining whether one can turn pure states into mixedstates. Black holes cannot be truly understood until weunderstand what happens to all the neutrons (and pro-tons and other particles) of the original star as they col-lapse under gravity to ever higher densities.It is perhaps tempting to think that since quantumgravity effects are only likely to come into play in regionsof high curvature, that the existence of an event horizon,since it occurs in a region of relatively low curvature, isunlikely to be affected. However, it is important to re-member that the event horizon is not defined in termsof the local geometry, but causally in terms of the globalcausal structure. If the collapse is somehow halted or asingularity avoided then small changes in the microscopiccentral region can have large effects on the global causalstructure, in the same way that adding one electron’sworth of charge to an otherwise supermassive black holecan change its causal structure from Schwarzschild toReissner-Nordstr¨om. This change, while insignificant interms of local physics, corresponds to a radical change ofthe causal structure, turning spacelike singularities intotimelike singularities amongst other things.The picture presented above also downplays severalfeatures commonly associated with information loss inblack holes. Firstly, the uniqueness theorems (oftencalled no-hair theorems) that hold for stationary, asymp-totically flat, electro-vac, Einstein-Maxwell black holespacetimes [25] play little role. Nor does the issue ofwhether the Hawking radiation is exactly thermal or not.All that is essential is that previously entangled parti-cles become entangled with nothing when their entangled“partners” are thrown away. One could imagine a blackhole spacetime with infinitely many independent mea-surable charges at infinity [26], if the spacetime still hasa spacetime boundary inside the black hole, over whichentangled particles can be lost, then there is still thepossibility of information loss. One could also imaginea situation with a single pair of entangled particles. Ifone of them is lost to a spacetime boundary the otherparticle will be in a fundamentally mixed state, and asingle particle can hardly be said to describe a thermalspectrum.Another related feature to uniqueness is the possibilityof retrodicting the original state that collapsed to formthe black hole from measurements of the final state. Thisshould also be conceptually separated from the loss ofunitarity in black holes. Such loss of retrodictablilityoccurs in ordinary particle physics where one knows thatthe S-matrix is perfectly unitary. For example, since bothelectron-positron pairs and muon-anti muon pairs can co-annihilate to produce high energy photons, if one justobserves the high energy photons at late time, one cannotretrodict whether they arose due to electron-positron co-annihilation or a different particle-antiparticle pair.Another implicit assumption that one often meetsis that the evaporating Hawking radiation somehow islinked to the originally collapsed matter. This is becauseenergy is conserved and if the Hawking radiation is carry- ing away energy, then that energy must somehow be theenergy of the original star. What we have shown usingthe Misner-Sharp mass function is that this need not bethe case. One can think of the Hawking radiation as apure quantum field effect on a curved background. It isthe spacetime that it evaporating and not the collapsedstar. 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