Quantum gravitational dust collapse does not result in a black hole
aa r X i v : . [ g r- q c ] D ec Quantum gravitational dust collapse does not result in a black hole
Cenalo Vaz
Department of Physics, University of Cincinnati, Cincinnati, OH 45221-0011. ∗ Quantum gravity suggests that the paradox recently put forward by Almheiri et. al.(AMPS) can be resolved if matter does not undergo continuous collapse to a singularity butcondenses on the apparent horizon. One can then expect a quasi-static object to form evenafter the gravitational field has overcome any degeneracy pressure of the matter fields. Weconsider dust collapse. If the collapse terminates on the apparent horizon, the Misner-Sharpmass function of the dust ball is predicted and we construct static solutions with no tangentialpressure that would represent such a compact object. The collapse wave functions indicatethat there will be processes by which energy extraction from the center occurs. These leavebehind a negative point mass at the center which contributes to the total energy of thesystem but has no effect on the the energy density of the dust ball. The solutions describea compact object whose boundary lies outside its Schwarzschild radius and which is hardlydistinguishable from a neutron star.
PACS numbers: 04.60.-m, 04.70.Dy, 97.60.Lf, 97.60.Jd
I. INTRODUCTION
The final fate of gravitational collapse has long been a mystery. Classical collapse modelssuggest that a star that is massive enough to overcome all degeneracy pressures will undergocollapse beyond the apparent horizon [1, 2] eventually forming a naked or covered singularity ofspacetime, depending on the initial conditions. But there is something deeply unsatisfying aboutthis picture since it does not take into account quantum gravity, which is expected to play asignificant role in the final stages of collapse.If the collapse begins with initial data that lead to the formation of a naked singularity [3] thena semi-classical treatment of the radiation (assuming the validity of effective field theory) from thesingularity suggests that the final stages will be catastrophic [4, 5]. It is not known what the finalfate of such a collapse is: either the collapsing star will dissipate entirely or a remnant will attemptto form a covered singularity. However, if the initial data are such as to lead to the formation of acovered singularity and an event horizon forms then, as Hawking pointed out [6], the semi-classicaltheory would yield thermal radiation from the point of view of the observer who remains outsidethe black hole provided that the freely falling observer detects nothing unusual while crossing thehorizon. The semi-classical analysis would seem to suggest that information is lost if the black holeevaporates completely, since what is left is a density matrix and not a wave function. But if thequantum theory is unitary then either (a) the evaporation is not in fact thermal and the Hawkingradiation is pure or (b) the thermal evaporation process should, by an as yet unknown mechanism,leave behind a stable remnant that contains all the information that fell into the hole. The secondoption is difficult to imagine since a relatively small object would be required to possess a hugedegeneracy while remaining stable. Moreover, it is ruled out if we assume that quantum gravity is ∗ Electronic address:
CPT invariant.This leaves just the first option, that the Hawking radiation is pure. In 1993, Susskind et. al [7], building on the work of ’t Hooft [8] and Preskill [9], proposed that the unitarity of the Hawkingradiation could be preserved if information is both emitted at the horizon and passes through it, soan observer outside would see it in the Hawking radiation and an observer who falls into the blackhole would see it inside but no single observer would be able to confirm both pictures. Althoughthere is no precise mechanism by which it is can be said to occur, thought experiments that appearto support this picture of “Black Hole Complementarity” rely on three fundamental assumptions, viz., (a) the unitarity of the Hawking radiation, (b) the validity of effective field theory outside a“stretched” horizon and (c) the equivalence principle. Recently, however, Almheiri et. al. (AMPS)have argued that the three assumptions are logically inconsistent and would lead to a violation ofthe strong subadditivity of the entanglement entropy [10–12]. To resolve the paradox the authorssuggested giving up the third assumption, i.e., the equivalence principle.But Hawking has proposed an intriguing alternative, suggesting that no event horizon wouldform in the first place if somehow the collapse did not continue beyond the apparent horizon [13]. Ifindeed no event horizon is formed, the entire discussion about information loss becomes moot. Yet,one is left with the question of how the system evolves after the formation of the apparent horizon.There appears to be ample experimental evidence supporting the existence of very massive, quasi-stable, compact objects located in galactic centers that are consistent with black holes, althoughit is not known for certain if these supermassive configurations are indeed black holes with eventhorizons. In this paper, we will examine Hawking’s conjecture as it relates to dust collapse, byre-examining some results of an exact quantization [14, 15] of the LeMaˆıtre-Tolman-Bondi familyof solutions [16].In previous work [17] we have shown that two kinds of functional solutions (analogous to planewaves) of the Wheeler-DeWitt equation for dust collapse may be given. In one, dust shells coalesceonto the apparent horizon on both sides of it. Exterior, infalling waves representing the collapsingshells of dust are accompanied by interior, outgoing waves, which are produced with a relativeprobability given by the Boltzmann factor at the Hawking temperature of the shells. These interiorwaves, which are of quantum origin, represent an interior Unruh radiation. In the other solution,dust moves away from the apparent horizon on both sides of it. Interior, infalling waves representingthe continued collapse of the dust shells across the apparent horizon are accompanied by exterior,outgoing waves, which are produced with a relative probability again given by the Boltzmannfactor. at the Hawking temperature. These latter ougoing waves represent the exterior Unruhradiation.Continued collapse across the apparent horizon from an initial diffuse state and to a centralsingularity can be achieved by combining the two solutions and requiring the net flux to vanishat the apparent horizon as in [17]. The net effect is that the collapse is accompanied by Unruhradiation in the exterior, as is well known [18], but ends in a central singularity. However, if thecollapse does not continue past the apparent horizon, there will be no exterior radiation during thecollapse. Furthermore, as the shells coalesce on the apparent horizon, no event horizon will formand the AMPS paradox is resolved. This picture is captured by the first of the exact solutions ofthe Wheeler-DeWitt equation discussed in the previous paragraph [19].In this paper, we will examine the consequences of taking seriously the possibility that continuedcollapse does not occur, i.e., that quantum collapse is described by the first solution described above.The collapsing matter is then accompanied by Unruh radiation in the interior of the apparenthorizon. In this case, we expect to end up with a spherically symmetric, quasi-static configurationof finite extension and with a specific mass function as the end state of the collapse. Even thoughno classical , static, extended dust configuration can exist, we will show that the interior Unruhradiation that accompanies the infalling dust shells during the collapse will generate the conditionsappropriate for a quasi-static configuration to exist. In effect it creates a negative mass pointsource at the center of the star, which is enveloped by the collapsed matter. We allow for radialbut no tangential pressure. This is in keeping with the midi-superspace quantization that informsour construction [14, 15]. With the inclusion of a constant negative vacuum energy and radialpressure, unique static solutions exist. There are no horizons and the matter itself extends to twicethe Schwarzschild radius.In section II we briefly summarize our previous work on the wave functionals describing thecollapse. In section III we construct the static, spherically symmetric solutions described aboveand analyze the solutions. In section IV we estimate the size of the central negative mass. Weconclude with a discussion on the our results and possible implications for future observations insection IV. We take ~ = c = 1 in what follows. II. QUANTUM DUST COLLAPSE
Dust collapse in any dimension, with or without a cosmological constant, is described bythe LeMaˆıtre-Tolman-Bondi family of solutions [16]. In comoving and synchronous coordinates,( t, ρ, θ, φ ), one has ds = dτ − R ′ ( τ, ρ ) f ( ρ ) dρ − R ( τ, ρ ) d Ω , (1)where the area radius, R ( τ, ρ ) obeys the Einstein equation˙ R ( τ, ρ ) = s f ( ρ ) + 2 GF ( ρ ) R ( τ, ρ ) + 13 Λ R ( τ, ρ ) (2)and the energy density is given by ǫ ( τ, ρ ) = F ′ ( ρ ) R ( τ, ρ ) R ′ ( τ, ρ ) . (3)Λ is the cosmological constant. There are two integration functions, F ( ρ ) and f ( ρ ), that areinterpreted as the twice the gravitational (Misner-Sharp) mass contained within a shell locatedat ρ and the total energy contained within the same shell respectively. They are the “mass” and“energy” functions of the collapse [3, 20].By considering the expansion of an outgoing, radial null congruence,Θ = 2 R ′ ( τ, ρ ) R ( t, ρ ) " − s GF ( ρ ) R ( τ, ρ ) + 13 Λ R ( τ, ρ ) , (4)one sees that the condition for trapping is met when2 GF ( ρ ) R ( τ, ρ ) + 13 Λ R ( τ, ρ ) = 1 , (5)which can be used to determine the time of formation, τ ah ( ρ ), of the apparent horizon once asolution of (2) is determined.The canonical dynamics of the collapsing dust shells is described by embedding the sphericallysymmetric ADM metric in the LTB spacetime of (1). After a series of canonical transformations[14, 21, 22], they are described in a phase space consisting of the dust proper time, τ ( t, r ), the arearadius, R ( t, r ), the mass density, Γ( r ) = F ′ ( r ), and their conjugate momenta, P τ ( t, r ), P R ( t, r ) and P Γ ( t, r ) respectively by two constraints, H r = τ ′ P τ + R ′ P R − Γ P ′ Γ ≈ H = P τ + F P R − Γ F ≈ , (6)where the prime denotes a derivative with respect to the ADM radial label coordinate, r , and F def = 1 − GFR −
13 Λ R . The apparent horizon occurs when F = 0. In the absence of a cosmological constant, this saysthat on the apparent horizon the physical radius of each shell is given by R ( τ ah , ρ ) = 2 GF ( ρ ) . (7)Dirac’s quantization procedure may be employed to turn the classical constraints in (6) intoquantum constraints, which act on wave functionals. The Hamiltonian constraint then yields a for-mal Wheeler-DeWitt equation and the momentum constraint imposes diffeomorphism invariance.We begin with an ansatz for the wave functional [14],Ψ[ τ, R, Γ] = exp (cid:20) − i Z dr Γ( r ) W ( τ ( r ) , R ( r ) , Γ( r )) (cid:21) , (8)which automatically satisfies the momentum constraint if W has no explicit dependence on r . TheWheeler-DeWitt equation must be regularized before solutions can be obtained. This regularizationwas performed on a one dimensional lattice [23, 24] given by a discrete set of points, r i , representingdust shells and separated by a spacing σ . One then finds that the ansatz in (8) yields a product ofwhat may be thought of as shell wave functions,Ψ = lim σ → Y i ψ i ( τ i , R i , Γ i ) = lim σ → Y i e ω i b i × exp − iω i a i τ i ± Z R i dR i q − a i F i F i , (9)with a well defined continuum limit ( σ → a i = 1 / √ f i is related to the energy functionand ω i = σ Γ i /
2. Diffeomorphism invariance also requires that both a i and b i depend on r via themass function, i.e., a i = a i ( F i ) and b i = b i ( F i ).These solutions are defined everywhere except at the apparent horizon. Thus there are “exte-rior” wave functions that must be matched to “interior” wave functions across the horizon. As canbe seen, however, the phases of the interior and exterior wave functions diverge there. A standardtechnique used in such cases is to analytically continue the solutions into the complex plane. Thistechnique was used to derive the Hawking radiation as a tunneling process in [25]. Thus, analyti-cally continuing into the complex R plane, taking F i = ǫ exp[ iϕ ], with ǫ >
0, and comparing themat ϕ = π/
2. One then finds two sets of matched solutions, with support everywhere; the first isgiven by [17] ψ (1) i, col ( τ i , R i , F i ) = e ω i b i × exp (cid:26) − iω i (cid:20) a i τ i + R R i dR i √ − a i F i F i (cid:21)(cid:27) F i > e − πωigi,h × e ω i b i × exp (cid:26) − iω i (cid:20) a i τ i + R R i dR i √ − a i F i F i (cid:21)(cid:27) F i < ψ (2) i, col ( τ i , R i , F i ) = e − πωigi,h × e ω i b i × exp (cid:26) − iω i (cid:20) a i τ i − R R i dR i √ − a i F i F i (cid:21)(cid:27) F i > e ω i b i × exp (cid:26) − iω i (cid:20) a i τ i − R R i dR i √ − a i F i F i (cid:21)(cid:27) F i < , (11)where g i,h = ∂ R F ( R ) | R i,h / i th shell at the apparent horizon.These are the shell wave functions we described in the introduction. The first (in (10)) repre-sents a flow toward the apparent horizon on both sides of it: an infalling shell in the exterior isaccompanied by an interior, outgoing shell, produced with a relative probability determined by theBoltzmann factor at the Hawking temperature of the shell. The second (in (11)) describes a flowaway from the apparent horizon: an infalling shell in the interior, which represents its continuedcollapse past the apparent horizon and to a central singularity is accompanied by an exterior,outgoing shell, with a relative probability also determined by the Boltzmann factor. It representsthe thermal radiation in the exterior.One might in principle be interested in constructing wave packets that represent an evolutionfrom a configuration in which the dust cloud begins far from the apparent horizon. Such a wavepacket would serve to clarify the semi-classical description of the collapsing ball and would beconstructed by superposing the solutions given above with different energies, a i ( F i ). This difficultproblem, which is currently under investigation, does not seem feasable at present as both the factorordering and the diffeomorphism invariance depend on the energy function [24]. Nevertheless, someuseful conclusions can be drawn from the “plane wave” solutions we have presented above, as onedoes, for example, in ordinary quantum scattering theory. We notice that if we take the wavefunctions in (10) to form the basis for the quantum collapse of dust then there will be thermalUnruh radiation inside the apparent horizon but no thermal radiation outside, accompanying thecollapse. There will also be no continued collapse to a central singularity; the collapse wouldterminate at the apparent horizon ( F i = 0), which agrees with Hawking’s proposal [13, 19]. Theseconclusions would hold true even if one could find a way to construct diffeomorphism invariantwave packets from (10) representing the collapse. III. A QUASI-CLASSICAL CONFIGURATION
As there is good experimental evidence for the existence of very masive, quasi-stable compactobjects, we look for static, spherically symmetric solutions of Einstein’s equations satisfying thefollowing criteria: • the collapsed dust ball should occupy a finite region and posesses an energy density that ischaracteristic of a dust cloud that has condensed onto its apparent horizon, i.e., given by(7), • the solutions should incorporate the effect of the internal Unruh radiation that has occurredduring the collapse phase and • they must match smoothly to the Schwarzschild vacuum at the boundary.Within the dust ball the metric will be of the form ds = e A dt − e B dr − r d Ω , (12)where A = A ( r ), B = B ( r ) and r represents the physical radius. In this coordinate system, if wetake the components of the stess-energy to be T µν = diag( − ε ( r ) , p r ( r ) , p θ ( r ) , p θ ( r )) but impose noequations of state, the field equations are1 − e B − rB ′ = − πGr e B ε − e B + 2 rA ′ = 8 πGr e B p r rA ′ − B ′ + A ′ (1 − rB ′ ) + rA ′′ = 8 πGre B p θ , (13)where a prime indicates a derivative with respect to the radius, r . The conservation of energy-momentum gives a constraint, εA ′ + p ′ r + p r (cid:20) r + A ′ (cid:21) − r p θ = 0 , (14)which represents the condition for static equilibrium. Two of the stress-energy components maybe chosen arbitrarily and then the third is determined by either Einstein’s equations or by theconservation law. Below we will choose the energy density and set the tangential pressure to zero.The first equation in (13) may be re-expressed as[ r (1 − e − B )] ′ = 8 πGr ε, (15)which is straightforwardly integrated to give r (1 − e − B ) = 8 πG Z r dr ′ r ′ ε ( r ′ ) − r , (16)where r is an integration constant. This is usually set to zero in stellar models to avoid a centralsingularity, but we will not do so here for reasons that will become clear in the following. TheMisner-Sharp mass function of the dust is to be identified with the integral on the right, F ( r ) = 4 π Z r dr ′ r ′ ε ( r ) . (17)Now, according to (7), the mass function that may be expected of a dust ball whose collapse hasterminated at the apparent horizon is F ( r ) = r G , (18)for a total gravitational mass of M ms = F ( r b ) = r b / G , where r b denotes its boundary. It corre-sponds to an energy density of ε ( r ) = 18 πGr (19)and (16) gives e B = r/r . (20)We see that the constant r > B ( r ) with the desired mass function, even if pressure is included. Strictly itdescribes a negative mass point source the center. Such a negative mass source is actually predictedby the wave functions in (10) to form during the collapse as energy is extracted from the centerby the interior, outgoing Unruh radiation that accompanies the exterior, collapsing shells. Thisprocess of energy extraction from the center continues until the collapse terminates. In the nextsection we will estimate its size.With B ( r ) given in (20) and no tangential pressure, we solve the Riccati equation in (13) for A ( r ) and find ds = r (cid:16) γr / (cid:17) dt − rr dr − r d Ω , (21)where γ is another integration constant. There are curvature singularities at r = 0 and at r =( − γ ) / . To avoid the singularity at r = ( − γ ) / , either γ must be positive or ( − γ ) / must lieoutside the outer boundary of the collapsed star, where the solution no longer applies. We willsoon show that the second condition cannot be met.We determine the pressure directly from the second equation in (13) p r ( r ) = − πGr (cid:20) − r r (cid:16) γr / (cid:17) − (cid:21) , (22)so with γ ≥ r b denotes the outer boundary of the collapsed star, we want to match the interior geometryto an external vacuum, described by the Schwarzschild metric ds = f ( R ) dT − f − ( R ) dR − R d Ω , (23)where f ( R ) = (cid:18) − GM s R (cid:19) and M s is the Schwarzschild mass of the dust ball. The junction conditions require that R b = r b , T b = e A ( r b ) p f ( r b ) te − B ( r b ) = p f ( R b ) , A ′ ( r b ) = (ln f ) ′ | R b (24)and therefore r = r b − r s , γ = 2 r / b (cid:18) − r s r b (cid:19) , (25)where we have let r s = 2 GM s be the Schwarzschild radius.The first condition says that the physical radius of the boundary must lie outside itsSchwarzschild radius. Therefore, as expected, the Schwarzschild mass of the star is less thanthe Misner-Sharp mass of the dust, M s = r s G = r b − r G = M ms − M , (26)by precisely the negative central mass, − M . If γ ≥
0, the second condition requires that r b ≥ r s / γ < (cid:18) r s r b − (cid:19) > , (27)but this would imply that r s > r b . As this is not possible, the star will be singularity free (exceptat the center) only if r b ≥ r s /
2. This implies that r b ≤ r and r s ≤ r . IV. ESTIMATING r We can provide a simple estimate of the radius, r , as follows. The energy extraction fromthe center occurs during the collapse because every collapsing shell is accompanied by an interior,outgoing wave, which will extract energy from the center. We want to estimate how much energyis extracted in this process. For the given mass function, the energy density of the dust is constant,Γ = 1 / G . If σ represents the shell thickness, the average energy, ω i , of each shell will also beconstant, ω i = ω = σ Γ / σ/ G .The collapse of the i th shell will have been accompanied by the emission in the interior of anoutgoing wave of the same frequency, with a probability that is given by the Boltzmann factor, e − β i ω , at the Hawking temperature, β i = 2 πr i , of the shell. It follows that the average energy ofthe outgoing shell is h ω i = ωe − β i ω and, to get the total energy extracted, we must sum over allcollapsed shells, M = 1 σ Z r b drωe − πωr = 12 πσ (cid:2) − e − πωr b (cid:3) . (28)Replacing ω by σ/ G and taking the limit as the shell spacing approaches zero then gives M = r b G = 12 M ms , (29)which implies that r = r b /
2. By the matching conditions, it follows that r = r s , therefore theregion of negative energy occupies the Schwarzschild radius of the star. Although it extends to halfthe boundary radius of the collapsed dust ball and is necessarily surrounded by a cloud of ordinarymatter, this is a surprisingly large length scale over which quantum gravitational effects shouldpredominate. There is no event horizon. A photon, emitted near the boundary of this cloud, wouldexperience a relatively tame redshift of z = r r b r − √ − ≈ . , (30)which is compatible with the gravitational redshift of neutron stars of low core densities [26],suggesting that, in a collapse of realistic matter, quantum gravity could “kick in” much beforepreviously imagined, very near the time at which nuclear densities are achieved. This is consistentwith the idea that in extreme conditions quantum gravity may be relevant on distance scales muchlarger than previously anticipated. V. DISCUSSION
In this paper we have speculated on the consequences of a simple quantum model of dustcollape. We have argued that the AMPS paradox is avoided if continued collapse does not occurand all dust shells coalesce onto the apparent horizon. We showed that the collapse process isthen accompanied by Unruh radiation within the apparent horizon. We argued that the effect ofthe interior Unruh radiation is energy extraction from the center of the cloud, leaving behind anegative mass singularity as the cloud settles into a quasi-stable equilibrium.Stable classical solutions, with the given mass function and including pressure were determined.The solutions are governed by two parameters, the Schwarzschild radius, r s , of the dust ball,equivalently its mass as measured by a distant observer, and the boundary radius, r b . The differencebetween the two is the radius, r , of a region inside of which the total energy is negative. Thereare strong constraints on the parameters r , r s and r b if the interior geometry is required to bewell behaved everywhere (except at the center). We have shown that the r should extend to morethan one half the Schwarzschild radius and more than one third the radius of the entire star, so itwill occupy a significant fraction of the star. A more detailed analysis of the Unruh radiation fromthe center during the collapse indicated that r = r s = r b / r b /r s & ∼ M ⊙ [31, 32] appear to rule out exotic matter EOSs, which tendto become soft at high densities. Under the same assumption, this leaves “ordinary” (nucleonic)matter EOSs with comparatively large radii >
11 km for a 2 M ⊙ CS [31]. Quantum black holeswould be both more massive and possess smaller radii than neutron stars, but larger radii thanclassical black holes of the same mass. Therefore it is necessary to measure the radius of CSs in aprecise and model independent way to provide this information. While this has proved difficult sofar, the proposed Large Observatory for X-ray timing (LOFT) has claimed to be able to measurethe radii of some CSs with a precision of up to 1 km [33].Another difference between them will be their luminosities in the presence of accretion flowssuch as would occur in X-ray binaries or in galactic centers where supermassive black holes arethought to exist. One may expect accretion onto the surface of an ordinary neutron star to lead tohigher luminosities than accretion onto the surface of a quantum black hole because an accretingshell of matter encounters a hard surface as it collapses onto an ordinary neutron star, but thequantum theory dictates that it should slow down and coalesce onto the apparent horizon as itapproaches the “surface” of a quantum black hole. Accreting quantum black holes will thereforelook fainter than accreting neutron stars. The reason for the darkness of the quantum black holeis quantum mechanics and not the absence of a surface, but the outcome agrees qualitatively withthe predictions of [34, 35].Very large compact objects, such as the supermassive black holes that are thought to exist atthe centers of galaxies make excellent candidates for verifying or falsifying the existence of quantumblack holes, if their radius can be determined accurately. In the near future, observations of thesupermassive black hole Sgr A* by the Event Horizon Telescope (EHT) are expected to be sensitiveto distance scales of better than a horizon length in the 1 mm range and direct measuremnts ofSgr A*’s size are expected to become possible [36, 37].Finally, we also mention that a recent study of the periodic modulation in the intensity vs.1frequency spectrum of galactic centers seems to support the similarity between behaviors of certainpulsars and supermassive black holes [38]. These issues are under investigation, as is also theproblem of constructing wave packets representing a collapsing dust ball.
Acknowledgement
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