Astrophysical Black Hole horizons in a cosmological context: Nature and possibleconsequences on Hawking Radiation

George F R Ellis,1, ∗ Rituparno Goswami,2, † Aymen I. M. Hamid,2, ‡ and Sunil D. Maharaj2, §

1Mathematics Department and ACGC, University of Cape Town, Rondebosch, 7701 South Africa2Astrophysics and Cosmology Research Unit, School of Mathematics,

Statistics and Computer Science, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, South Africa.

This paper considers the nature of apparent horizons for astrophysical black holes situated in arealistic cosmological context. Using semi-tetrad covariant methods we study the local evolutionof the boundaries of the trapped region in the spacetime. For a collapsing massive star immersedin a cosmology with Cosmic Background Radiation (CBR), we show that the initial 2-dimensionalmarginally trapped surface bifurcates into inner and outer horizons. The inner horizon is timelikewhile the continuous CBR influx into the black hole makes the outer horizon spacelike. We discussthe possible consequences of these features for Hawking radiation in realistic astrophysical contexts.

PACS numbers: 04.20.Cv , 04.20.Dw

I. INTRODUCTION

This paper is about the location of apparent horizonswhen astrophysical black holes form from the collapse ofa single astrophysical object in a cosmological context. Acurrent view with many proponents is that when blackhole formation takes place, a singularity forms and ishidden by an event horizon, but then the blackhole com-pletely evaporates away within a finite time due to Hawk-ing radiation [7, 33, 37].The outcome however has re-cently been challenged by various people proposing thatthe black hole will not evaporate away [1, 21, 29, 62],while others propose, using different arguments, that aglobal event horizon will never form [3, 46].

A key issue in this debate is the location of local appar-ent horizons, characterised as marginally outer trapped3-surfaces (MOTS), which are arguably crucial to Hawk-ing radiation emission. MOTS are important featuresof the geometry of black holes, inter alia leading to theprediction of the existence of singularities in the classicalblack hole case [36, 56]. They are distinct from the globalevent horizon in realistic dynamical cases, when a pair ofMOTS surfaces form (an inner one - the IMOTS and anouter - the OMOTS) as the radius of the collapsing ob-ject falls in past the critical radius r = 2M (see Fig.1and Fig. 2 and their descriptions). They will be affectedby infalling cosmic background radiation (CBR), as wellas by backreaction from ingoing and outgoing Hawkingradiation; and these effects must be taken into accountin realistic models of black hole radiation.

In this paper we will show under what conditions each

∗Electronic address: George.Ellis@uct.ac.za†Electronic address: Goswami@ukzn.ac.za‡Electronic address: aymanimh@gmail.comPermanent Address: Physics Department, University of Khartoum,Sudan.§Electronic address: Maharaj@ukzn.ac.za

of these surfaces is timelike or spacelike, and where theyare located. We find that the effect of the CBR on the na-ture of the outer MOTS surface may be a crucial featurein determining the context of Hawking radiation. Also,in realistic astrophysical contexts, the inner surface doesnot lie where Bardeen places it in his recent paper onblack hole existence [3]; this may significantly affect theoutcomes he claims.

A further paper will explore the implications for emis-sion of Hawking radiation and black hole evaporation.This paper only considers spherically symmetric collapse;the situation will be much more complex in the case ofrotating black holes, but the spherical case indicates thekind of outcomes we may expect in that case also.

A. The context

This section sets the general context within which theissue of the location of the event horizon arises. We areconcerned with the case of astrophysical black holes sit-uated in a realistic cosmological context. There are thenthree forks in the possibilities that will determine theoutcome of Hawking radiation emission processes. Thefirst two possibilities motivate the further studies in thispaper.

1. The cosmic context

Considering black hole formation in a cosmologicalcontext, two features need to be taken into account,based on our present understanding of the universe.

Cosmic Background Radiation. Since the time ofdecoupling, the universe is everywhere pervaded by cos-mic background radiation (CBR), initially at a temper-ature of 4000K but then cooling down to the present

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2.73K, when the radiation is microwave background ra-diation (CMB) [22, 57].

Consequently any black hole formation will be subjectto this infalling radiation [67]. This will affect the natureof the OMOTS surface, as discussed below.

Cosmic constant. The late universe is dominated bydark energy that is consistent with existence of a smallpositive cosmological constant. As the cosmological con-stant does not decay away, the late time evolution ofthe universe is de Sitter, and future infinity is spacelike[36, 55].

This means that, unlike the asymptotically flat case,there is no single event horizon for any observer outside ablack hole. Different cosmological event horizons (whichare in the far future) occur for different observer motionsthat end up at different places pi on future infinity (seeFigure 4 below), because the event horizon for an ob-server ending up at pi is, by definition, the past lightcone of pi [36, 55].

2. Key question 1: Local of global?

The first key question is whether the location of Hawk-ing radiation emission is locally or globally determined.Is it just outside a globally determined event horizon, asargued strongly by Don Page (private communication),or just outside a local horizon (a MOTS surface) that islocally determined, as argued strongly by Visser [66] andsupported by others [11, 48, 52]. In fact the source ofthe Hawking radiation is still a grey area and we need toperform very detailed and deep semiclassical investiga-tions on the global aspects of physically realistic gravita-tional collapse to pinpoint such a source. The attemptsso far, to introduce a timelike emitting surface outsidethe event horizon (for example [3, 38, 39, 63]) are ad-hocin the sense that the existence of such a surface lacks aproper geometrical interpretation. In the last two worksmentioned above, this emitting surface has zero energydensity but non-zero surface stress which is unphysical.It is hard to motivate why such a surface should existin an absolutely regular and future asymptotically sim-ple spacetime outside the black hole. On the other handtrapped regions (which have proper geometrical interpre-tation in terms of null congruences) do play an importantrole in the “particle pair-creation picture” of Hawkingradiation. Hence for a local analysis it is quite natu-ral to assume that the boundary of the trapped regionis the source of the radiation. In the case of an unper-turbed Schwarzschild black hole, since this boundary ex-actly coincides with the future event horizon, the eventhorizon and it’s vicinity can be considered as a globalsource. However the problem arises when this degener-acy is broken by infalling matter in the black hole. Theviewpoint underlying the basis of this paper is that theHawking radiation emission must be locally, rather thanglobally, determined. This is for two reasons. Firstly,

this seems to be the only way the concept of a blackholelocally emitting radiation makes sense (it seems absurdthat we have to wait until the end of the universe beforewe know what happens locally [66]). Secondly, because ofthe point just made: in a realistic context, future infinityoutside the black hole is spacelike rather than null, so dif-ferent observers who are initially near the black hole willexperience different event horizons. Which one shouldwe associate with the radiation? The prescription is notwell defined.

3. Key question 2: Timelike or spacelike?

Assuming that Hawking radiation emission is associ-ated with a MOTS surface, the question then is, doesit matter whether this surface is timelike, spacelike, ornull? If the calculation is based on the idea of tunnelling(e.g. [52]) then it will be applicable only if the MOTSsurface is timelike or null (when “inside” and “outside”can be defined). If it is spacelike, then a tunnellingviewpoint is simply not applicable (spacetime regionswill be “before” or “after” the surface, but not inside oroutside it). The same applies to heuristic explanationsbased on one of a pair of virtual particles being trappedbehind the horizon (they cannot be trapped behind aspacelike surface) or using S-matrix ideas (scatteringtakes place off a timelike world tube, not a spacelikesurface). These mechanisms can only work if the horizonis timelike.

Now the particle picture is considered suspect by manyworkers. As emphasized by Paul Davies and MalcolmPerry (private communication) what is needed to makethis conclusive is to calculate the stress energy tensorassociated with local horizons (see e.g. [7, 11]), and seeif this confirms what is suggested by the particle pictureor not. That will be the subject of a separate paper.

For the present paper, the issue is that whether aMOTS surface is timelike or spacelike is not only geo-metrically important, because it determines the relationof the MOTS surface to global event horizons, but it mayalso play a key role in the Hawking radiation emission.

4. Key question 3: vacuum or fluid?

The above two questions assume that the emission ofradiation due to quantum processes in a collapsing blackhole context is due to the properties of the vacuum do-main near the event horizon or trapping surface. Howeverthere is another effect at work: the interior of the collaps-ing fluid is a time-varying gravitational field due to thecollapse process, and this can potentially lead to particlecreation in an evolving fluid filled spacetime, as notedlong ago by Leonard Parker [53]. This is the mechanismof particle creation by a fluid collapsing as it forms a blackhole that is discussed by Hawking ([34]: pp 207-208) and

3

Birrell and Davies ([7]: pp 250-262). In this case the roleof the horizon is not creation of the radiation, rather itis modulation of the propagation of the radiation thathas already been emitted in the time-dependent gravita-tional field of the fluid; the result is that the outcome isindependent of the details of the fluid collapse [7, 34].

This is also the mechanism considered by Mersini-Houghton in her recent paper [46], based on the Hartle-Hawking vacuum, and the assumption that the energymomentum tensor of the ingoing Hawking radiation isthat of radiation in thermal equilibrium. However theHartle-Hawking vacuum is based on the unphysical caseof a white hole being in thermal equilibrium with theblack hole; also the ingoing radiation may have a formdifferent from thermal equilibrium. Thus it is still anopen question as to how significant this mechanism iscompared to the horizon based mechanisms usually asso-ciated with Hawking radiation [34].

What we need to do to convincingly determine the out-come of this fluid based mechanism is to study the evolu-tion of a collapsing body, and then ask what difference doquantum fields make to this dynamical situation. It willbe significant to see how this relates to the nature andlocation of global and local horizons that are discussedin this paper.

B. The dynamical horizons

There is a marginal outer trapped 3-surface (theOMOTS) that lies outside the outgoing initial null sur-face generated by the initial marginally trapped 2-surface(the SMOTS) that marks the onset of black hole dy-namics. The OMOTS surface is locally determined; itmoves outwards with time because of incoming Cos-mic Microwave Background radiation, so that the asso-ciated mass mout(u) increases with time, and the surfacer = 2Mout(u) is spacelike. Because the OMOTS boundsthe trapped domain in spacetime (the outward directednull geodesics are converging inside this surface), it de-termines the outside edges of any future singularity thatmay occur. Thus it non-locally determines the locationof the event horizon. OMOTS is a spacelike dynamicalhorizon whose properties characterise the exterior massmout and angular momentum Jout of the body. How-ever its nature could possibly be changed to timelike atlate times by the back reaction of Hawking radiation onits geometry. That is one of the important issue to beinvestigated.

We make the case below that there is an inner timelikemarginally outer trapped 3-surface (the IMOTS), whichis a dynamical horizon [2]. This surface lies inside boththe OMOTS surface and the event horizon and it’s lo-cation is locally determined. It is timelike and also em-anates from the SMOTS 2-surface, which is therefore abifurcation surface originating both MOTS surfaces. Itis potentially possible that backreaction from Hawkingradiation causes them to merge again in the future at a

final FMOTS 2-surface, if the OMOTS surface eventuallybecomes timelike. Whether that happens or not dependson detailed balance between local dynamics of the fluidand the incoming Hawking radiation in those cases wherethe OMOTS is timelike at late times.

C. This paper

The paper considers these issues in the case of a singleblack hole with spherical symmetry, where the relevantexterior solution is the exterior Schwarzschild solutionsurrounding the single collapsing mass. The result isprobably stable in the case of more general geometriessuch as rotating black holes, perturbed black holes, andif there are later infalling shells of matter. The paperdoes not consider multiple black holes or black hole col-lisions. Since we consider only astrophysically relevantsituations, it also does not consider charged black holeseither.

As we confine our attention to spherically symmetricblack holes produced by the gravitational collapse of amassive star, the Schwarzschild solution is the basic rele-vant exterior metric but the interior will be modelled byspherical fluid body: it might be a Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric, a Lemaıtre-Tolman-Bondi (LTB) metric, or a spherical metric with pressure[22]. However in addition there may be incoming or out-going radiation in both the exterior and interior regions;we therefore need to consider generic spherically sym-metric metrics. For technical reasons it is convenient toconsider a class of spacetimes which are a generalisationof spherically symmetric metrics: namely Locally Rota-tionally Symmetric (LRS) class II spacetimes [18, 61, 65].These are evolving and vorticity free spacetimes with a1-dimensional isotropy group of spatial rotations at ev-ery point. Except for few higher symmetry cases, thesespacetimes have locally (at each point) a unique preferredspatial direction that is covariantly defined.

To describe this class of spacetimes in terms of metriccomponents, we use the most general line element forLRS-II can be written as [61]

ds2 = −A2(t, χ) dt2 +B2(t, χ) dχ2

+C2(t, χ) [ dy2 +D2(y, k) dz2 ] , (1)

where t and χ are parameters along the integral curvesof the timelike vector field ua = A−1δa0 and the pre-ferred spacelike vector field ea = B−1δaν . The func-tion D(y, k) = sin y, y, sinh y for k = (1, 0,−1) respec-tively. The 2-metric dy2 + D2(y, k) dz2 describes spher-ical, flat, or open homogeneous and isotropic 2-surfacesfor k = (1, 0,−1). Spherically symmetric spacetimes arethe k = 1 subclass of LRS-II spacetimes.

We can easily see that the physically interest-ing spherically symmetric spacetimes (for example theSchwarzschild, FLRW, LTB, and Vaidya solutions) fallin the class LRS-II. Hence both the interior and the ex-terior of a collapsing star can be described by this class,

4

which can be characterised covariantly through the prop-erties of the vector fields ua and ea [65]. A parallel paperis being developed by Hellaby to study the same problemusing a metric based formalism. The basic results pre-sented in this paper are supported by studies using thatdifferent formalism.

Unless otherwise specified, we use natural units (c =8πG = 1) throughout this paper, Latin indices run from0 to 3. The symbol ∇ represents the usual covariantderivative and ∂ corresponds to partial differentiation.We use the (−,+,+,+) signature.

II. COVARIANT DESCRIPTION OF LRS-IISPACETIMES

In this section we give a brief summary of the semi-tetrad 1+1+2 covariant description of LRS-II spacetimes.For more details we refer to [6, 9, 10, 24, 65].

As the first step of the covariant description [19], wedefine a time-like congruence with a unit tangent vectorua. One natural and obvious choice of this vector maybe the tangent to the matter flow lines. Any 4-vectorXa in the manifold can then be projected onto the 3-space by the projection tensor hab = gab+uaub as Xa =Xua +X〈a〉, where X is the scalar along ua and X〈a〉 isthe projected 3-vector. A natural definition for two kindsof derivatives are then:

• The covariant time derivative along the observers’worldlines (denoted by a dot). Hence for any tensor

Sa..bc..d we have Sa..bc..d ≡ ue∇eSa..bc..d.

• Fully orthogonally projected covariant derivativeD with the tensor hab, with total projection onall the free indices. Hence we have DeS

a..bc..d ≡

hafhpc...h

bghqdhre∇rSf..gp..q.

This 1+3 splitting defines the 3-volume element naturallyas εabc = −

√|g|δ0

[a δ1b δ

2cδ

3d]u

d. For the LRS-II class of

spacetimes, the covariant derivative of the timelike vectorua can be irreducibly split in the following way

∇aub = −Aaub +1

3habΘ + σab, (2)

where Aa = ua is the acceleration, Θ = Daua is the ex-

pansion, σab = D〈aub〉 (where the angle brackets denotethe projected symmetric trace-free part) is the shear ten-sor. Also for these spacetimes all the independent compo-nents of the Weyl tensor can be expressed in terms of theprojected symmetric trace free gravito-electric tensor de-fined as Eab = Cacbdu

cud = E〈ab〉. The gravito-magneticpart of Weyl tensor [45] is identically zero. Using thistimelike vector ua, the energy momentum tensor for ageneral matter field is decomposed as follows

Tab = ρuaub + qaub + qbua + phab + πab , (3)

where ρ = Tabuaub is the energy density, p = (1/3)habTab

is the isotropic pressure, qa = q〈a〉 = −hcaTcdud is the3-vector defining the heat flux and πab = π〈ab〉 is theanisotropic stress.

The symmetry of the LRS-II spacetimes, having a pre-ferred spatial direction, points towards an extension ofthe 1+3 splitting mentioned above, by further splittingthe 3-space by the preferred spatial vector ea. This iscommonly known as 1+1+2 splitting. This allows usto derive a set of covariant scalar variables which aremore advantageous to treat these systems. This spatialvector is orthogonal to ua such that it satisfies eaea =1, uaea = 0. The 1+3 projection tensor h b

a ≡ g ba +uau

b

combined with ea defines a new projection tensor de-fined as N b

a ≡ g ba + uau

b − eaeb which projects vectorsorthogonal to ea and ua (eaNab = 0 = uaNab) onto a 2-surface which is defined as the sheet (N a

a = 2). The vol-ume element of this 2-surface is the Levi-Civita 2-tensorεab ≡ εabcec = udηdabce

c. The preferred spatial vector ea

introduces two new derivatives as a natural result of thenew splitting of the 3-space, for any 3-tensor ψa...b

c...d :

• The hat derivative is the spatial derivative along the

vector ea. Thus we have ψa..bc..d ≡ efDfψa..b

c..d.

• The delta derivative is the projected spatial deriva-tive on the 2-sheet by the projection tensor N b

a andprojected on all the free indices. Hence δfψa..b

c..d ≡Na

f ...NbgNh

c..NidNf

jDjψf..gi..j .

The set of 1+1+2 covariant scalars that fully describeLRS-II spacetimes are {A,Θ, φ,Σ, E , ρ, p,Π, Q}, whichare defined as follows

ua = Aea , φ = δaea , (4)

σab = Σ(eaeb − 1

2Nab), Eab = E

(eaeb − 1

2Nab),(5)

πab = Π(eaeb − 1

2Nab), qa = Qea . (6)

And the full covariant derivative of ea and ua is nowwritten as

∇aeb = −Auaub +(Σ + 1

3Θ)eaub + 1

2φNab, (7)

∇aub = −Auaeb + eaeb(

13Θ + Σ

)+Nab

(13Θ− 1

2Σ).(8)

We also write the useful relation

ua =(

13Θ + Σ

)ea . (9)

We now derive the equations governing the evolutionand propagation of the covariant scalars that fully de-scribes the LRS-II spacetimes. Using the Ricci identi-ties for the vectors ua and ea and the doubly contractedBianchi identities, we obtain the following:

5

• Propagation:

φ = − 12φ

2 +(

13Θ + Σ

) (23Θ− Σ

)− 2

3 (ρ+ Λ)− E − 12Π, (10)

Σ− 23 Θ = − 3

2φΣ−Q , (11)

E − 13 ρ+ 1

2 Π = − 32φ(E + 1

2Π)

+(

12Σ− 1

3Θ)Q. (12)

• Evolution:

φ = −(Σ− 2

3Θ) (A− 1

2φ)

+Q , (13)

Σ− 23 Θ = −Aφ+ 2

(13Θ− 1

2Σ)2

+ 13 (ρ+ 3p− 2Λ)− E + 1

2Π , (14)

E − 13 ρ+ 1

2 Π = +(

32Σ−Θ

)E + 1

4

(Σ− 2

3Θ)

Π + 12φQ−

12 (ρ+ p)

(Σ− 2

3Θ). (15)

• Propagation/evolution:

A − Θ = − (A+ φ)A+ 13Θ2 + 3

2Σ2 + 12 (ρ+ 3p− 2Λ) , (16)

ρ+ Q = −Θ (ρ+ p)− (φ+ 2A)Q− 32ΣΠ, (17)

Q+ p+ Π = −(

32φ+A

)Π−

(43Θ + Σ

)Q− (ρ+ p)A . (18)

The 3-Ricci scalar of the spacelike 3-space orthogonal toua can be expressed as

3R = −2[φ+ 3

4φ2 −K

], (19)

where K is the Gaussian curvature of the 2-sheet definedby 2Rab = KNab. In terms of the covariant scalars wecan write the Gaussian curvature K as

K = 13 (ρ+ Λ)− E − 1

2Π + 14φ

2 −(

13Θ− 1

2Σ)2

. (20)

Finally the evolution and propagation equations for theGaussian curvature K are

K = −(

23Θ− Σ

)K , K = −φK . (21)

III. NULL GEODESICS IN LRS-II SPACETIMES

On any spacetime (M, g), null geodesics (light rays)are characterised by the curves xa(ν), where ν is anaffine parameter along the geodesics. The tangent to

these curves is defined by ka = dxa(ν)dν , where ka is a null

vector obeying kaka = 0. Also, since the tangent vectorto the geodesic is parallely propagated to itself, we canwrite

kb∇bka =δka

δν= 0 , (22)

where δδν = kb∇b as the derivative along the ray with

respect to the affine parameter.On LRS-II spacetimes if light rays moves along the pre-

ferred spatial direction (for the special case of sphericalsymmetry this is equivalent to radial null rays), then thesheet components of these null curves are zero. Also at

this point we may define the notion of locally outgoingand incoming null geodesics with respect to the preferredspatial direction. Consider any open subset S of (M, g)and let xa(ν) be a null geodesic in S. Let ka be the tan-gent to this geodesic. If eaka > 0 in S then the geodesicis considered to be outgoing with respect to the preferreddirection in S. Similarly eaka < 0 denotes an incominggeodesic. Therefore for LRS-II spacetimes, the equationof the tangent to the outgoing null geodesics along thepreferred spatial direction can be written as

ka =E√

2(ua + ea) , (23)

where E : [u, e]→ R denotes the energy of the light ray.The propagation of E along the null geodesic is given bythe geodesic equation (22) as (using (7) and (8))

E′ ≡ δE

δν= ∓E2A− E2

(Σ + 1

3Θ). (24)

We can easily see that the hypersurface orthogonal tonull vector ka, contains ka. Therefore we need a dif-ferent construction to define a locally orthogonal spacewith respect to the outgoing null geodesics. Let us nowdefine the projection tensor hab, which projects tensorsand vectors into the 2-D screen space orthogonal to ka,as [16]

hab ≡ gab + 2k(alb), haa = 2, hach

cb = hab, habk

b = 0,(25)

where la is the null ingoing geodesic that obeys

lala = 0, kala = −1 andδla

δν= kb∇bla = 0. (26)

Using these definitions, the general form of la can be

6

written as

la =1√2E

(ua − ea) , (27)

and substituting (27) into (25) the screen-space projec-tion tensor is obtained as

hab = gab + uaub − eaeb = Nab (28)

Hence we note that the 2-D screen projection tensor isabsolutely same as the 2-sheet projection tensor Nab. Forthe LRS-II spacetimes the 1+3 decomposition of the co-variant derivative of the null vector ka is given as [16]

∇bka = 12 habΘout + σab + Xakb + Ybka + λkakb, (29)

where

Xa =1

Eed∇dka, Ya =

1

Eed∇akd, λ = − 1

E2eced∇dkc,

(30)

and Θout and σab represent the expansion and shear ofof the outgoing null congruence respectively. From theabove definitions and the geometry of LRS-II spacetimeswe can easily calculate that

Θout ≡ hab∇bka =E√

2Nab∇a (ub + eb) . (31)

Now using (7) and (8) in (31) we obtain,

Θout =E√

2

(23Θ− Σ + φ

). (32)

A similar 1+3 decomposition of the covariant derivativeof the ingoing null geodesic la can be performed, giving

Θin =1√2E

(23Θ− Σ− φ

). (33)

IV. MARGINALLY OUTER TRAPPEDSURFACES

As described in [36], let us consider a spherical emitterof light, surrounding a massive body, emitting a flash oflight. In a normal situation, (like Minkowski spacetime orSchwarzschild spacetime outside the horizon) the volumeexpansion of the outgoing null congruence orthogonal tothe sphere is always positive (Θout > 0) while that of the

incoming congruence is always negative (Θin < 0). How-ever, if a sufficiently large amount of matter is presentwithin the emitting sphere, the volume expansion of theoutgoing null congruence orthogonal to the sphere be-comes negative. This is the case where both outgoingand ingoing wavefronts collapse towards the centre ofthe emitting sphere. In this case, the emitting sphereis then called a Closed trapped 2-surface. The collec-tion of all closed trapped 2-surfaces in a four dimensionalmanifold constitutes a four dimensional trapped region.

Marginally outer trapped surfaces (MOTS) are the 3 di-mensional boundary of the trapped region. We will nowderive the equations for a MOTS surface and it’s evolu-tion in the [e, u] plane. Similar derivations were done in[8] using a different formalism.

A. Equation for MOTS is LRS-II spacetimes

To define MOTS we generalise the definition of dynam-ical horizons by Ashtekar and Krishnan [2] by removingthe restriction that it is a spacelike surface. Thus,

Definition: Marginally Outer TrappedSurface (MOTS): A smooth, three-dimensional sub-manifold H in a spacetime(M, g) is said to be a Marginally outertrapped surface if it is foliated by a preferredfamily of 2-spheres such that, on each leaf S,the expansion Θout of the outgoing null nor-mal ka vanishes and the expansion Θin of theother ingoing null normal la is strictly nega-tive.

Consequently a MOTS is a 3-manifold which is foliatedby marginally trapped 2-spheres. On this definition, such3-surfaces can be timelike, spacelike, or null, with ratherdifferent properties. The essential point is that they arelocally defined, and therefore are able respond to localdynamic change. We don’t need to know what is hap-pening at infinity in order to determine a local physicaleffect.

At this point let us consider spacetimes that have apositive sheet expansion, φ ≡ δaea > 0. In the geometri-cal context this implies that at any given epoch, the localGaussian curvature of the two sheets are monotonicallydecreasing functions of the affine parameter along thepreferred spatial direction. Schwarzschild, FLRW, andLTB spacetimes have this property. Furthermore, by thesymmetries of LRS-II spacetimes, we can easily see thatit suffices to study the one dimensional MOTS curve inthe local [u, e] plane to determine it’s local properties.Now using equations (32,33) it is evident that the equa-tion of the MOTS curve in the local [u, e] plane is givenby

Ψ ≡(

23Θ− Σ + φ

)= 0 . (34)

Thus, if φ > 0, on the MOTS curve we have Θout = 0 andΘin < 0, as it should be by the above definition. Here wewould like to show another important geometrical prop-erty of MOTS. Let us calculate the quantity ∇aK∇aKfor a spherically symmetric spacetime (where K 6= 0):

∇aK∇aK = −K2 + K2. (35)

Now using (21) in (35) we get

∇aK∇aK =(

23Θ− Σ + φ

) (23Θ− Σ− φ

)K2 . (36)

7

Hence for any 2-sheet on the MOTS, the gradient of it’slocal Gaussian curvature is null. This property allows usto easily locate the MOTS in a given spacetime.

As an example let us consider the spherically symmet-ric vacuum spacetime. Then by Birkhoff’s theorem [36]we know that the spacetime has an extra symmetry, it iseither static or spatially homogeneous. Let us considerthe static exterior part. Existence of a timelike Killingvector implies that Θ = Σ = 0 [31]. Thus the horizon isdescribed by the curve φ = 0. This is the event horizonof the Schwarzschild spacetime. Indeed if we calculate φfrom the LRS-II field equations in Schwarzschild coordi-nates, we get

φ =2

r

√1− 2m

r, (37)

and φ = 0 corresponds to the event horizon at r = 2m.

B. Evolution of MOTS

Let us now describe how the MOTS evolve geometri-cally. What are the local conditions in terms of geome-try or the matter energy momentum tensor that deter-mine whether MOTS will be locally timelike, spacelikeor null? As we stated earlier, due to the symmetries ofLRS-II spacetimes, it suffices to study the behaviour ofthe MOTS curve in the local [u, e] plane. Let the vectorΨa = αua+βea be the tangent to the curve Ψ = 0 in thelocal [u, e] plane. Then we must have Ψa∇aΨ = 0. Since

we know that ∇aΨ = −Ψua + Ψea, we can immediatelysee that the slope of the tangent to the MOTS on the

local [u, e] plane is given by αβ = − Ψ

Ψ. Now using this

decomposition with the field equations (10) to (18), weobtain

∇aΨ =(

13 (ρ+ Λ) + (p− Λ)− E + 1

2Π−Q)ua

+(− 2

3 (ρ+ Λ)− 12Π− E +Q

)ea, (38)

and hence

α

β=

23 (ρ+ Λ) + 1

2Π + E −Q− 1

3 (ρ+ Λ)− (p− Λ) + E − 12Π +Q

. (39)

We now define the notion of Future Outgoing (Ingoing)in the following way. If α

β > 0 then the MOTS is said

to be future outgoing and if αβ < 0 then the MOTS is

said to be future ingoing. The nature of the MOTS interms of it being timelike, spacelike or null can also bedetermined by the ratio α

β . We can easily see that

ΨaΨa = β2(1− α2

β2 ). (40)

Therefore α2

β2 > 1(< 1) denotes the MOTS to be locally

timelike (spacelike). If α2

β2 = 1 (as in the case of vac-

uum spacetime, Schwarzschild or Schwarzschild-de Sit-ter, where all the thermodynamical terms vanish), theMOTS are null.

It is interesting to see how the local matter thermody-namical quantities, along with the Weyl curvature en-tirely determine the nature of the dynamical horizon.In special cases, where the matter is a perfect fluid(Q = Π = 0), then the condition for the MOTS beingtimelike is given by (assuming Λ = 0)(

13ρ− p+ 2E

)(ρ+ p) < 0 . (41)

Thus if the weak energy condition is satisfied so thatρ + p > 0, then ρ > (3p − 6E) would ensure a timelikeMOTS.

V. MISNER-SHARP MASS FOR LRS-IISPACETIMES

In this section, we derive the Misner-Sharp [47] massequation for LRS-II spacetimes in terms of the 1+1+2kinematical quantities. We know, for the metric (1), wecan define the mass function as [60]:

M(r, t) =C

2(k −∇aC∇aC) , (42)

where C represents the physical radius and k = +1, 0,−1represents closed, flat or open 3-space geometry. Here weare concentrating on spherically symmetric spacetimesand hence we will only consider the case k = 1. Then wecan write C = 1√

K, where K is the Gaussian curvature

of the spherical 2-sheets. Then we obtain the expressionof the mass as

M =1

2√K

(1− 1

4K3∇aK∇aK

). (43)

Geometrically, the above expression gives the amount ofmass enclosed within the spherical shell at a given valueof affine parameter of the integral curves of ea at a giveninstant of time. Using the 1+1+2 decomposition of thecovariant derivative for LRS-II together with (20, 21),the Misner-Sharp mass takes the form

M =1

2K3/2

(1

3(ρ+ Λ)− E − 1

2Π

). (44)

We can easily see from this equation, even in the case ofvacuum spacetimes the mass does not vanish due to theelectric part of Weyl Curvature E , which plays the roleof the mass source.

As an example lets us consider the Schwarzschild staticspacetime again. Since it is a vacuum spacetime the ther-modynamical quantities vanish and in Schwarzschild co-ordinates we have

E = −2m

r3. (45)

In this case we see that the Misner-Sharp mass is thesame as the Schwarzschild mass. As discussed in the pre-vious section, on the MOTS ∇aK∇aK = 0 [32]. There-fore the mass enclosed within the MOTS at a given in-stant of time (which can be regarded as the local mass of

8

the black hole) is given by

MBH =1

2√KBH

, (46)

where KBH is the Gaussian curvature of the black holesurface at any given instant of time.

VI. OPPENHEIMER-SNYDER-DATTCOLLAPSE

This is the simplest model of black hole formation fromthe dynamical collapse of a massive star [13, 50]. The in-terior of the spherically symmetric star is considered tobe dustlike and is described by a flat FLRW metric. Theexterior of the star is spherically symmetric vacuum andhence, by Birkhoff’s theorem, is a static Schwarzschildspacetime. These two spacetimes are matched usingIsrael-Darmois matching conditions (the first and secondfundamental forms) at the boundary of the collapsingstar, which is described by a comoving radius rB in theinterior spacetime. When the star collapses to the singu-larity at the centre the spacetime becomes Schwarzschild.Though this model is a simplified toy model, it is widelyaccepted that any realistic massive star would collapse toa black hole in a similar fashion [56].

FIG. 1: The spacetime Diagram showing the formation ofEvent Horizon and Apparent Horizon (MOTS) for a spheri-cally symmetric dust collapse.

From the Einstein’s field equations one can explicitlyderive the dynamics of the trapped region in the space-

time, which are as follows (see Figure 1) (see [41–43] fordiscussions on the global evolutions of trapped regions):

• In the interior spacetime the boundary shell of thestar gets trapped first. This is the instant of timewhen the radius of the star becomes equal to theSchwarzschild radius. The interior of the star iscausally cut off from the exterior spacetime fromthis instant. As described in the previous section,at this instant the gradient of the Gaussian curva-ture of the boundary shell (∇aKB ) becomes null.Thus a 2 dimensional initial MOTS is formed whichwe will denote as the SMOTS . This initial SMOTS

is described by a point in the [u, e] plane, which isa bifurcation point. From this point two separatebranches of 3-MOTS develop. The one which is inthe exterior spacetime is called the Outer MOTSor OMOTS while the one which is in the interior ofthe star is called the Inner MOTS or IMOTS.

To understand geometrically why the two MOTS sur-faces exist, consider the domains where Θout is positiveand negative (Figure 2). The boundary between them is

where Θout = 0. It is essentially a geometrical necessitythat one cannot have a single starting trapped 2-sphereSMOTS with only one MOTS surface emanating from it.A single surface where Θout = 0 can’t bound a domainwhere Θout < 0: there have to be two surfaces to makegeometrical sense.

To understand physically why Θout = 0 on two sur-faces, consider a sphere of radius r round the centre ofthe collapsing fluid containing a mass of fluid m(r). Closeenough to the centre of the collapsing object, for smallr, there is insufficient mass inside to cause refocusing:m(r) < r/2. That is why there is no refocussing close

enough to the centre ( Θout > 0). The possibility of localtrapping 2-spheres existing arises when r is large enoughthat there is sufficient mass inside a sphere to cause re-focusing: m(r) > r/2. The inner horizon IMOTS surface

where Θout = 0 thus exists at any time t at which thereexists a first radius which includes sufficient matter tocause refocussing at that time (its like the refocussingof the past light cone in a FLRW model that occursat z = 1.25 in the Einstein de-Sitter case). The outerOMOTS horizon exists because outside the star there isvacuum, and the mass is then constant as the radius in-creases. So going to a sufficient radius one will eventuallyagain find r > 2m, and there is no refocussing outside thebounding surface where this first occurs.

• Since the exterior spacetime is vacuum, all thematter thermodynamic terms are identically zero.Hence we have α

β = 1. Consequently, the OMOTS

is future outgoing and null. This exactly coincideswith the event horizon of Schwarzschild spacetime.

• The interior spacetime has flat FLRW geometrywith dust like matter. Hence apart from the en-ergy density ρ all the other thermodynamic terms

9

vanish and also the Weyl scalar is zero. Hence weget α

β = −2. Thus the IMOTS is a future ingoing

timelike 3-surface. This surface reaches the centreof the collapsing star at the singularity.

The trapping region III and its boundaries separating itfrom (non-trapped) regions I and II are show in Figure 2.It is clear from this characterisation of the domains whereθ+ is greater and less that zero that the MOTS surfaceshave to be created as a pair at the SMOTS 2-sphere.

A Ricci tensor singularity will occur inside the fluid asthe fluid collapses and KB(τ)→∞ within a finite propertime. This may or may not be accompanied by a Weylsingularity inside the fluid; such a singularity will notoccur for example if the fluid is a Friedmann-Lemaıtremodel, as the Weyl tensor is then zero inside the fluid.

The Ricci tensor is zero outside the collapsing star;it is the Weyl tensor that diverges at the spacelikeSchwarzschild singularity in the vacuum domain. Specif-ically, the Kretschman scalar is

K = CabcdCabcd = α

M2

r6, (47)

where α is a constant, so this diverges as r → 0.

It is the spatial inhomogeneity of the matter distribu-tion (non-zero density inside the fluid, zero outside) thatgenerates this singularity in the conformal structure ofspacetime.

As seen from the outside, the mass of the star neveralters; it is always equal to the initial value M0:

MBH =M0 = const. (48)

This will of course not be the same if matter falls intothe black hole, thereby increasing it’s mass; then the hori-zon is a dynamic horizon [2] and the laws of black holethermodynamics [4] come into play to characterise theresulting changes. We also note that the above pictureof the black hole formation changes drastically if insteadof a homogeneous dust ball we consider an inhomoge-neous one with the energy density decreasing monoton-ically from the centre to the surface. In that case, theIMOTS at the central singularity is future outgoing andnull [41, 42] and hence the central singularity is locallynaked. In such cases the spacetime will cease to be fu-ture asymptotically simple [36] and proofs of a numberof results of black hole thermodynamics will break down.However, in our analysis here, we only concentrate onfuture asymptotically simple spacetimes.

VII. AN ASTROPHYSICAL BLACK HOLE: THEIDEALISED PICTURE CHANGES

Let us now consider a gravitational collapse scenariowhich is embedded in a expanding cosmology. Thereemerge a few crucial differences to the idealised scenariodescribed in the previous sections. They are as follows:

• The universe is permeated by cosmic black bodyradiation (CBR) radiation emitted from the hot

10

big bang era in the early universe [57], [22]. Thatblackbody background radiation was emitted bythe last scattering surface at the end of the HotBig Bang era in the early universe. It then propa-gates through the universe with a temperature

TCBR = (1 + z)TCBR|0 , (49)

where z is the redshift characterising the cosmolog-ical time when the radiation temperature is mea-sured. It now pervades the universe as microwavebackground radiation (CMB) at a temperature

TCBR|0 = 2.7K. (50)

In effect, this radiation reaches local systems at thepresent time from an effective “Finite Infinity”([20],[23]) at about one light year’s distance that emitsblack body radiation at a temperature TCBR|0.This is the effective sky for every local systemat present. It was hotter in the past, as shownby (49); at very early times TCBR ' 4000K andρCBR ' (1 + z)4ρCBR|0 ' 1012ρCBR|0. The CBRradiation has the stress tensor of a perfect fluid withpositive energy density and pressure:

pCBR = ρCBR/3, ρCBR = a T 4CBR. (51)

It has an entropy density given by

SCBR =4

3aT 3

CBR + b (52)

where b is a constant independent of the energy Eand volume V .

The crucial outcome is that the OMOTS surface becomesspacelike [21] (see Figure 3). The infalling (positive den-sity) CBR radiation (51) increases the interior mass andmakes the OMOTS surface spacelike, therefore enlargingthe trapping region to include domain VI and extendingthe spacelike future singularity further out from P1 toP2. This moves the globally defined event horizon out-wards so that the spacelike OMOTS surface lies insidethe event horizon. In fact most astrophysical black holeshave an accretion disk and the rate of infalling of mat-ter in general is much larger than the CBR. However theCBR influx rate can be considered as a lower limit to thenet accretion.

• The late universe is dominated by dark energy thatis consistent with existence of a small positive cos-mological constant Λ. As the cosmological constantdoes not decay away, the late time evolution of theuniverse is de Sitter, and future infinity is spacelike[36, 55] (see Figure 4).

This means that, unlike the asymptotically flat case,there is no single event horizon for observers outside ablack hole. Different cosmological event horizons (whichare in the far future) occur for different observer motions

that end up at different places pi on future infinity,because the event horizon for an observer ending up atpi is, by definition, the past light cone of pi [36, 55].The first event horizon is the past light cone of P2 (thepast light cone of P1 is no longer an event horizon as itis trapped), but it is irrelevant to most observers, suchas those that end up at P3.

Since these cosmological effects are much smaller thanthe thermodynamical and gravitational effects of a mas-sive collapsing star locally, we can treat these effects asa perturbations of the idealised scenario. Let us con-sider that the thermodynamical variables of the CBR are(ρCBR, pCBR, QCBR,ΠCBR) and these are much smallerthan the local gravitational tidal effects (E) outside a col-lapsing star. In this perturbed case the ratio α

β for the

OMOTS outside the collapsing star is given as

α

β=

1 + 23ρCBR+ΛE + 1

2ΠCBR

E − QCBR

E

1− 13ρCBR+ΛE − pCBR−Λ

E − 12

ΠCBR

E + QCBR

E. (53)

Here we note that the CBR is falling “into” the blackhole and hence we have Q < 0. Furthermore in the back-ground Schwarzschild spacetime E < 0. Using this, we

find the condition for a spacelike OMOTS (α2

β2 < 1) is

given by

(ρCBR + pCBR + 2|QCBR|+ ΠCBR) > 0 . (54)

The above condition is generally satisfied for CBR andhence we can easily conclude that in the presence of CBR

11

falling in the black hole, the OMOTS is locally spacelike(see Fig. 3 for the compact description).

Since the OMOTS is spacelike, there always exist anopen set U in the vicinity of the OMOTS from which notimelike or null trajectories can escape to infinity. Toillustrate this by an example, let the instant of time forthe SMOTS to appear in the spacetime be denoted byτ0. Let us now consider the open set U in the [u, e] planewith the following closure: the curves Ψ = 0, t = τ0 andthe outgoing null curve ν from t = τ0 intersecting thecurve Ψ = 0 at some later time t = τ1 (this is possibleas Ψ = 0 is spacelike between τ0 and τ1). One can easilysee that any non-spacelike trajectory from U cannotescape to infinity, the trajectories will fall back to thesingularity. By construction this open set U extends tospacelike future infinity for a LCDM universe.

Now let us consider a realistic astrophysical Black Holeimmersed in an environment containing matter. In thatcase there will be continuous in-falling matter into theblack hole and the black hole mass will vary continuously.Now the variation of Misner-Sharp mass along ua and thepreferred spatial direction ea can be calculated using thefield equations together with (44) as

M =1

4K3/2

[φ(ρCBR + Λ)−

(Σ− 2

3Θ)QCBR

],(55)

M =−1

4K3/2

[(23Θ− Σ

)(pCBR − Λ) +QCBRφ

].(56)

Using the above equations along with the equation ofhorizon, we can find the evolution equation governingthe mass of a black hole enclosed within the dynamicalhorizon as

MBH =1

4K3/2BH

φBH [(pCBR − Λ) + |QCBR|] , (57)

where KBH and φBH are evaluated on the MOTS lo-cally. For in-falling CBR we have pCBR > 0. Also asalready stated earlier, in the interior and exterior space-time we have φ ≥ 0 . Hence we see as the CBR falls inthe black hole, its mass MBH increases monotonically.As the CBR becomes weaker and weaker in far future thethermodynamic terms will be less than the cosmologicalconstant Λ. When Λ dominates, the solution tends to aSchwarzschild de Sitter solution where we have α

β = 1 and

the OMOTS continuously tends to a null surface from aspacelike one, with φBH → 0 till the future spacelike in-finity. Hence during this phase near the future spacelikeinfinity MBH → 0 and the black hole mass will asymp-totically reach a constant value.

VIII. DISCUSSIONS: CONSEQUENCES FORHAWKING RADIATION

The results obtained in the previous section for lo-cation of horizons of realistic astrophysical black holes

immersed in a LCDM cosmology with CBR, may haveimportant consequences on the Hawking radiations fromthese black holes. We return now to the key questionsraised in the introductory section.

1. Local or Global:

As Fig 4 depicts, we would like to emphasise that thereare no global event horizons in an asymptotically de-Sitter Universe. The event horizon for each particle isdifferent, being the past null cone of it’s world line at fu-ture spacelike infinity.Even the particle P1, whose world-line ends at a singularity, has every right to refer to it’spast lightcone also as a future event horizon. In the con-text of string theory these null boundaries are sometimesreferred as observer horizons. Hence there exist an infi-nite number of world lines that will never see Hawkingradiation from a black hole, as this radiation (if any) willlie outside their cosmological event horizon. Thereforeit makes no sense to talk about Hawking radiation froma global event horizon in this context. Although thereis a notion of this happening in the Gibbons-Hawkingvacuum, each observer sees radiation coming from theirown personal event horizon. But the relation of such ra-diation to a black hole horizon (which may lie entirelyoutside the observer’s event horizon) is yet to be deci-phered properly.

We can argue that there is an inmost event horizonassociated with the black hole, so one might proposeradiation is associated with this non-locally determinedsurface. We would like to clarify here that this inmosthorizon is the part of the event horizon that forms beforethe incoming CBR shell crosses the horizon. Howeverfrom Fig. 1, it can be seen that this event horizon formseven before parts of the spacetime become trapped. Itseems unlikely that the event horizon in such a regularand untrapped epoch will start radiating as there is nolocal occurrence there that could lead to this happening.Hence it is probable the radiation must come from thevicinity of the boundary of the trapped region, that is,from near an apparent horizons (a MOTS).

2. Timelike or spacelike:

If the radiation is locally emitted [66], it will be asso-ciated with either the IMOTS surface or the OMOTssurface. As was suggested in the introduction, if thecalculation for Hawking radiation is based on the ideaof tunnelling [52], then it will only be applicable if theMOTS surface is timelike or null. If it is spacelike, thena tunnelling viewpoint is simply not applicable. Hencein the scenario discussed in this paper only the timelikeIMOTS could radiate; and any such radiation emittednear the IMOTS will necessarily fall into the singularityand so be shielded from outside observers [21].

12

However that conclusion depends on the particle pic-ture, which can be called into question. Consider thenpossible emission near the OMOTS surface. As weproved in the previous sections, the CBR makes theOMOTS spacelike, which ensures the existence of anopen set in the vicinity of the OMOTS from which notimelike or null trajectories can escape to infinity. Henceif any Hawking radiation is generated from this open setit is bound to fall back into the black hole rather thanescaping to infinity. This suggests that there may be aremnant mass of any black hole at future infinity, and theblack hole will not evaporate entirely away. QuantumField Theory calculations suggest radiation is emittedfrom the vicinity of the OMOTS surface, rather than justfrom the surface itself, so on this picture a small amountof radiation that is emitted from outside the open set de-scribed above can in principle escape to infinity. Howeverexistence of such open set would necessarily imply a partof the radiation would fall back to the black hole andtherefore the entire black hole will not totally evaporateaway.

This conclusion would however be wrong if the backre-action from ingoing negative density Hawking radiationovercame the CBR effect and made the OMOTS surfacetimelike. Can this occur? If we consider the usual par-ticle picture of Hawking radiation from an open set Udescribed above between two closely spaced interval τ1and τ2, we can easily see that both the positive energyparticle and the negative energy particle will enter sub-set of the trapped region between τ1 and τ2. Hence therewill be no net change in the average energy (mass) ofthe black hole due to Hawking radiation from this openset. This can be better understood in the context of lo-cal black hole mass. As we have seen, in-falling CMBRincreases the black hole mass monotonically, and thatimplies the temperature of the horizon decreasing mono-tonically. We know at the present epoch the temperatureof CBR is much greater than the Hawking temperature ofa solar mass black hole. From equation (57) we can easilysee that the mass of the black hole will go on increasing(and hence the Hawking temperature will go on decreas-ing) till pCMB and |QCMB | becomes smaller than the cos-mological constant. From this point in far future to thefuture spacelike infinity the black hole mass asymptoti-cally tend to a constant value with the OMOTS asymp-totically becoming null from a spacelike surface but neveractually being null: it is always spacelike. Hence muchof the emitted Hawking radiation will be trapped, as theOMOTS surface will remain inside the black hole eventhorizon. In the thermodynamic context, black holes notradiating to a hotter background makes perfect sense.

This argument is suggestive but not conclusive: to ob-tain a definitive result, we must calculate the expectationvalue of the stress tensor in the present context (work un-der way). However insofar as the particle picture is valid,the above conclusion seems inevitable.

We would like to clarify here that our conclusion is dif-ferent from that in Bardeen’s paper [3] inter alia becausehis picture of the horizon geometry is different from ours.

In that paper he used as a classical background a some-what artificial construction of a singularity free black hole[27, 38] leading to the existence of a particle creation sur-face “outside” the black hole with zero energy density butnon-zero surface stress. The existence of such a surfaceis unphysical, and we argue that it cannot develop for arealistic astrophysical black hole. It is the use of this un-physical equation of state that leads to an understandingof the inner horizon (see Fig 2 in [3]) different than thatpresented here.

3. Vacuum or fluid:

It is quite interesting that most of the existing litera-ture on Hawking radiation from a black hole talks aboutvacuum excitations near the event horizon. Very fewworks discuss the effect of a time varying collapsing mat-ter field on the particle creation; however there may wellbe particle emission there too [7, 34, 53].

In [7] it is argued that if particles created by vacuumexcitation in the exterior of an collapsing star travelsthrough the star, then the difference in blueshift of aparticle falling in from the redshift of the same parti-cle coming out becomes important in the vicinity of thehorizon. This difference in particle energy manifests ina diminishing energy of the collapsing star, which mightstop the formation of singularities and trapped surface[46]. In that case all the above argument is called intoquestion.

Whether such effects can still occur in the cosmologicalcontext discussed here, where a spacelike OMOTS sur-face occurs inside the event horizon, is an open question,and is worthy of future investigation.

Acknowledgments

GFRE thanks Paul Davies, Don Page, Tim Clifton,and particularly Malcolm Perry for helpful interactionsthat have shaped much of this work, and Hadi and MahdiGodazgar and Laura Mersini Houghton for useful discus-sions on the topic at Great Brampton House in Marchthis year. We thank Charles Hellaby, Alton Kesselly andJavad Firouzjaee for helpful discussions in Cape Town,and Tim Clifton, Malcolm Perry, and Naresh Dadich forcomments on an earlier draft that have helped improvethis paper.

AH and RG are supported by National Research Foun-dation (NRF), South Africa, and GFRE receives supportfrom the NRF and from the UCT research committee.SDM acknowledges that this work is based on researchsupported by the South African Research Chair Initia-tive of the Department of Science and Technology andthe National Research Foundation.

We thank the anonymous referee for his valued com-ments on this paper.

13

[1] Y. Aharonov, A. Casher, and S. Nussinov (1987) “Theunitarity puzzle and Planck mass stable particles”, Phys.Lett. 191B: 51-55.

[2] A Ashtekar and B Krishnan (2002) “Dynamical Horizons:Energy, Angular Momentum, Fluxes and Balance Laws”Phys.Rev.Lett. 89:261101 [arXiv:gr-qc/0207080].

[3] J M. Bardeen(2014) “Black hole evaporation without anevent horizon ”arXiv:1406.4098

[4] J. M. Bardeen, B. Carter and S. W. Hawking, “The Fourlaws of black hole mechanics,” Commun. Math. Phys. 31,161 (1973).

[5] L. Bell, Compt. Rend. Acad. Sci. Colon., (1958), 246,3015; (1958), 247, 1094; (1958), 247, 2096; (1959), 248,1297; (1959), 248, 2561; Cahiers Phys., (1962), 16, 59.

[6] G. Betschart and C. A. Clarkson, Scalar and electromag-netic perturbations on LRS class II space-times, Class.Quant. Grav. 21, 5587 (2004) [gr-qc/0404116].

[7] N D Birrell and P C W Davies (1984) Quantum Fields inCurved Space (Cambridge: Cambridge University Press)

[8] I Booth, L Brits, J A Gonzalez, and C Van Den Broeck(2005) “Marginally trapped tubes and dynamical hori-zons ” Class.Quant.Grav. 23 (2006) 413-440 [arXiv:gr-qc/0506119]

[9] C. Clarkson, “ A Covariant approach for perturbationsof rotationally symmetric spacetimes” Phys. Rev. D 76,104034 (2007) [arXiv:0708.1398 [gr-qc]].

[10] C. A. Clarkson and R. K. Barrett, “Covariant perturba-tions of Schwarzschild black holes”, Class. Quant. Grav.20, 3855 (2003) [gr-qc/0209051].

[11] T Clifton (2008) “Properties of Black Hole Radia-tion From Tunnelling” Class.Quant.Grav. 25:175022[arXiv:0804.2635]

[12] T. Clifton, G. F. R. Ellis and R. Tavakol, “A Grav-itational Entropy Proposal,” Class. Quant. Grav. 30,125009 (2013) [arXiv:1303.5612 [gr-qc]].

[13] B. Datt (1938) “Uber eine Klasse von Losungen der Grav-itationsgleichungen der Relativitat” Z. Phys. 108:314.Reprinted as a Golden Oldie, General Relativity andGravitation, 31, (1999), 1615.

[14] P. C. W. Davies (1976) “On the Origin of Black HoleEvaporation Radiation” Proc Roy Soc (London) 351:129-139.

[15] P C W Davies, S A Fulling, and W G Unruh (1976)“Energy-momentum tensor near an evaporating blackhole” Phys Rev D13: 2720-2723.

[16] B. de Swardt, P. K. S. Dunsby and C. Clarkson, “ Grav-itational Lensing in Spherically Symmetric Spacetimes”,arXiv:1002.2041 [gr-qc].

[17] O Dreyer, B Krishnan, E Schnetter, and D Shoemaker(2003) “Introduction to Isolated Horizons in NumericalRelativity” Phys.Rev.D67:024018 [arXiv:gr-qc/0206008]

[18] G. F. R. Ellis (1967) “ The dynamics of pressure-freematter in general relativity” Journ Math Phys 8, 1171 –1194.

[19] G F R Ellis (1971) “Relativistic Cosmology” In: Pro-ceedings of the International School of Physics ”EnricoFermi”, Course 47: General relativity and cosmology,Ed. R.K. Sachs (Academic Press): 104-182. Reprinted:Gen. Rel. Grav. 41:581 (2009).

[20] G F R Ellis (1984) “Relativistic cosmology: its nature,aims and problems”. In General Relativity and Gravita-

tion, Ed B Bertotti et al (Reidel), 215-288.[21] G F R Ellis (2013) “Astrophysical black holes may radi-

ate, but they do not evaporate ”: arXiv:1310.4771[22] G F R Ellis, R Maartens and M A H MacCallum, Rela-

tivistic Cosmology (Cambridge University Press)[23] G F R Ellis and W R Stoeger (2009) “The Evolution

of Our Local Cosmic Domain: Effective Causal Limits”MNRAS 398:c1527-1536 [arXiv:1001.4572]

[24] G. F. R. Ellis & H van Elst, “ Cosmological Models”.Cargese Lectures 1998, in Theoretical and ObservationalCosmology, Ed. M Lachze-Rey, (Dordrecht: Kluwer1999), 1. [arXiv:gr-qc/9812046].

[25] J T Firouzjaee (2011) “The spherical symmetry Blackhole collapse in expanding universe” International Jour-nal of Modern Physics D (IJMPD). [arXiv:1102.1062]

[26] J T Firouzjaee and R Mansouri (2012) “Radiation fromthe LTB black hole ” Europhysics Letter 97: 29002[arXiv:1104.0530]

[27] V P Frolov (2014) “Information loss problemand a ‘black hole’ model with a closed apparenthorizon”arXiv:1402.5446

[28] G. W. Gibbons and S. W. Hawking (1977), “Cosmologi-cal Event Horizons, Thermodynamics, and Particle Cre-ation,” Phys. Rev. D 15, 2738.

[29] S. Giddings (1994) “Comments on information loss andremnants” Phys. Rev. D49: 4078.

[30] E. Gourgoulhon, “3+1 formalism and bases of numericalrelativity,” gr-qc/0703035 [GR-QC].

[31] R. Goswami and G. F. R. Ellis (2011), “ Almost BirkhoffTheorem in General Relativity”, Gen. Rel. Grav. 43,2157 [arXiv:1101.4520 [gr-qc]].

[32] A. I. M. Hamid, R. Goswami and S. D. Maharaj (2014)“Cosmic Censorship Conjecture revisited: Covariantly,”Class. Quant. Grav. 31, 135010 (2014) [arXiv:1402.4355[gr-qc]].

[33] S W Hawking (1974) “Black-hole explosions” Nature248, 30–31.

[34] S W Hawking (1975) “Particle creation by black holes”Communications in Mathematical Physics 43: 199-220.

[35] S W Hawking and G F R Ellis (1968) “The cosmic black-body radiation and the existence of singularities in ourUniverse” The Astrophysical Journal 152: 25-36.

[36] S W Hawking and G F R Ellis (1973) The Large ScaleStructure of Space-Time (Cambridge: Cambridge Uni-versity Press).

[37] S Hawking and R Penrose (1996) The Nature of Spaceand Time (Princeton: Princeton University Press).

[38] S. A. Hayward (2006) “Formation and evaporation ofnon-singular black holes” Phys. Rev. Lett., 96, 031103[arXiv:gr-qc/0506126].

[39] W A Hiscock (1981) “Models of evaporating black holesI” Phys Rev D23:2813-2822

[40] J. L. Jaramillo and E. Gourgoulhon, “Mass and Angu-lar Momentum in General Relativity,” Fundam. Theor.Phys. 162, 87 (2011) [arXiv:1001.5429 [gr-qc]].

[41] P. S. Joshi (1993), Global Aspects in Gravitation and Cos-mology, Oxford University Press.

[42] P. S. Joshi (2007), Gravitational Collapse and SpacetimeSingularities, Cambridge University Press.

[43] P. S. Joshi, J. V. Narlikar Black hole physics in globallyhyperbolic spacetimes Pramana 18 (1982) 385-396.

14

[44] A. Komar (1995) “Covariant conservation laws in generalrelativity,” Phys. Rev. 113, 934.

[45] R. Maartens and B. A. Bassett, “Gravitoelectromag-netism,” Class. Quant. Grav. 15, 705 (1998) [gr-qc/9704059].

[46] L Mersini-Houghton (2014) “Backreaction of HawkingRadiation on a Gravitationally Collapsing Star I: BlackHoles? ”arXiv:1406.1525

[47] C. W. Misner and D. H. Sharp (1964), “Relativistic equa-tions for adiabatic, spherically symmetric gravitationalcollapse,” Phys. Rev. 136, B571.

[48] A B Nielsen (2009) “Black holes and black hole ther-modynamics without event horizons” Gen.Rel.Grav.41:1539-1584 [arXiv:0809.3850]

[49] A B Nielsen and M Visser (2006) “Production and decayof evolving horizons” Class.Quant.Grav. 23 4637-4658[arXiv:gr-qc/0510083].

[50] J. R. Oppenheimer and H. Snyder (1939) “On continuedgravitational contraction” Phys. Rev. 56:455.

[51] A Paranjape and T Padmanabhan (2009) “Radia-tion from collapsing shells, semiclassical backreac-tion and black hole formation” Phys.Rev. D80:044011[arXiv:0906.1768]

[52] M K Parikh and F Wilczek (2000) “Hawking RadiationAs Tunneling” Phys Rev Lett 85: 5042-5045.

[53] L Parker (1969) “Quantized Fields and Particle Creationin Expanding Universes. I ” Phys. Rev. 183, 1057

[54] A Peltola (2009) “Local Approach to Hawking Radia-tion” Class.Quant.Grav.26: 035014 [arXiv:0807.3309]

[55] R Penrose (1964) “Conformal Treatment of Infinity” InRelativity, Groups and Topology, edited by B. de Wittand C. de Witt. (Gordon and Breach, New York - London): 565 - 584. Reprinted Gen. Rel. Grav. 43:901-22 (2011).

[56] R Penrose (1965) “Gravitational collapse and space-time

singularities” Phys. Rev. Lett. 14, 57–59.[57] P Peter and J-P Uzan (2009) Primordial Cosmology (Ox-

ford: Oxford University Press).[58] E Poisson (2004) A Relativist’s Toolkit: The Mathematics

of Black-Hole Mechanics. (Cambridge: Cambridge Uni-versity Press). Section 4.3.5 and Section 5.1.8.

[59] J. M. M. Senovilla, “ Remarks on the Stability Operatorfor MOTS”, Progress in Mathematical Relativity, Grav-itation and Cosmology, Springer Proceedings in Mathe-matics and Statistics Volume 60, 403 (2014).

[60] H. Stephani, E. Herlt, M. MacCullum, C. Hoenselaersand D. Kramer, Exact Solutions of Einstein’s Equations(Cambridge University Press)

[61] J M Stewart and G F R Ellis (1968) “On solutions of ein-stein’s equations for a fluid which exhibit local rotationalsymmetry” J. Math. Phys. 9 1072

[62] A. Strominger and S.P. Trivedi (1993) “Information con-sumption by Reissner-Nordstrom black holes”, Phys.Rev. D 48: 5778.

[63] L. Susskind, L. Thorlacius, J. Uglum (1993) “TheStretched Horizon and Black Hole Complementarity”Phys. Rev. D 48:3743.

[64] W. G. Unruh (1976) “Notes on black hole evaporation,”Phys. Rev. D 14, 870.

[65] H. van Elst and G. F. R. Ellis (1996) “The covariant ap-proach to LRS perfect fluid spacetime geometries” Class.Quantum Grav. 13, 1099 [gr-qc/9510044].

[66] M Visser (2003) “Essential and inessential features ofHawking radiation” Int.J.Mod.Phys. D12 : 649-661[arXiv:hep-th/0106111].

[67] M Wielgus, M A Abramowicz, G F R Ellis, and F H Vin-cent (2014) “Cosmic background radiation in the vicin-ity of a Schwarzschild black hole: no classic firewall”arXiv:1406.6551