8/20/2019 1509.09010

1/12

 a r X i v : 1 5 0 9

 . 0 9 0 1 0 v 1

 [ g r - q c ]

 3 0 S e p 2 0 1 5

Holography, mass area relation and discrete quantum spectrum

of black holes

Kinjalk Lochan   ∗ and Sumanta Chakraborty   †

IUCAA, Post Bag 4, Ganeshkhind,

Pune University Campus, Pune 411 007, India 

October 1, 2015

Abstract

The quantum genesis of Hawking radiation is a long-standing puzzle in black hole physics.Semi-classically one can argue that the spectrum of radiation emitted by a black hole look verymuch sparse unlike what is expected from a thermal object. It was demonstrated through a simplequantum model that a quantum black hole will retain a discrete profile, at least in the weak energyregime. However, it was suggested that this discreteness might be an artifact of the simplicity of eigenspectrum of the model considered. Different quantum theories can, in principle, give rise todifferent complicated spectra and make the radiation from black hole dense enough in transitionlines, to make them look continuous in profile. We show that such a hope from a geometry-quantizedblack hole is not realized as long as large enough black holes are dubbed with holographic relationwhich tells that the entropy of the black hole can be obtained from the area of the horizon and theyhave a classical mass area relation. We show that the smallest frequency of emission from black

hole in any quantum description, is bounded from below, to be of the order of its inverse mass.That leaves the emission with only two possibilities. It can either be non-thermal, or it can bethermal only with the temperature being much larger than 1/M. In order to populate the emissionprofile with the holographic relation preserved, one must change the area mass relation of the holesor vice versa.

1 Introduction

Black holes are very fascinating ob jects appearing as solutions of classical general relativity. Eventhough they are supposed to be perfect trapping systems at the classical level, there are enough evi-dences to point out that black holes seem to behave much like as a thermal object   [1, 2]. This belief was strengthened once Hawking showed that black holes seem to be radiating at a characteristic tem-perature inversely proportional to their masses [3]. Therefore, similar to a thermal body a black holeis also prescribed to have some entropy which turns out to be proportional to their area of horizon(This thermodynamic interpretation of black hole horizons can be extended to arbitrary null surfacesas well, where also these results continue to hold showing a deep interrelationship between gravitational

∗kinjalk@iucaa.in†sumanta@iucaa.in; sumantac.physics@gmail.com

1

http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1http://arxiv.org/abs/1509.09010v1

8/20/2019 1509.09010

2/12

dynamics and thermodynamics [4, 5]) in the Einstein theory. One way to look for this intriguing areadependence is to calculate the entanglement entropy for the black hole which scales as area  [6–9]. As

and when a correct theory of quantum gravity is achieved, a natural onus on it will be to show the cor-responding microstates in a black hole making up for the entropy. Not only this, a consistent quantumtheory of gravity is also expected to shed light on the thermal behavior of black holes from microscopicpoint of view.

All these results follow from semi-classical arguments and are studied extensively for observers inthe regions outside black hole event horizon. Recently in [10,11] it has been shown that energy densityof a quantum field dominates over the classical background, near the singularity and have the potentialto alter the singularity structure. Also in  [12] the semi-classical approximation is shown to receivecorrection leading to some non-trivial implications for information content in the Hawking radiation(see also [13–16]). All these results motivate us to study the semi-classical thermal Hawking spectrumin a full quantum picture, if possible. One way to do that will be to treat the black hole as a highlyexcited quantum object1 or a macroscopic construct of a consistent microscopic theory of gravity. Alarge black hole can then be thought of as either a highly excited state of the fundamental quantumdescription or a macroscopic limit of the underlying quantum description (i.e.   N  → ∞ where N   is thetotal number of quanta constituting the hole).

Although there are semi-classical strong arguments suggesting a thermal behavior of Hawking ra-diation, there could be various factors which can distort the thermal profile of the radiation. Thereare various classical phenomenon, e.g., grey-body factor, non-adiabaticity which can produce distor-tions in the Infra-red regime of the Hawking spectra  [20]. More particularly, it has been arguedpreviously   [17–19] that macroscopic black holes should have large elapse time resulting in a sparsespectrum. One may expect from a potential quantum theory of black holes, that some of these prop-erties might find some backing from quantum point of view. One can also expect that the essentialclassical characteristic features of black holes get quantized too, in a quantum theory of gravity. Ithas been argued by Bekenstein et. al. [21] that if one takes the black hole as a highly excited state of the quantum description, the emission profile of the Hawking radiation remains very sparse such that

in the dominant part of the thermal spectra only a few lines contribute and most of the continuumfeature is present towards the tail. So   the Hawking radiation is quantum mechanically silent in the region where bulk of the thermal radiation comes from.  This effect was taken as a stumbling block forsome earlier quantization models [22–25] which predicted the mass of the hole to be quantized in integersteps. However, it was still to be seen whether such a phenomenon is general enough to accommodatedifferent quantum spectra arising from different quantum gravity theories.

In this letter, we argue that the sparseness of Hawking radiation survives the modification of quan-tum eigenspectrum of geometric observables. So, the sparseness of Hawking spectra has nothing todo with precise spectral description of the underlying quantum theory2. We show that the following(semi)-classical inputs — (1)  holography  (by which we mean entropy of the hole is directly proportionalto its area), (2)   classical mass area   relation and (3)   an effectively continuous Hawking spectra — areincompatible with each other in a full quantum theory. In section 2 we will first argue for the generalcase where a black hole gets quantized by some underlying quantum theory (which we are not interested

1In fact there are prescriptions like   fuzzballs   [16] that suggest there should not be any consistent fully classicaldescription of black holes and they are indeed purely quantum mechanical objects.

2It was argued   [26] in the framework of Loop Quantum Gravity (henceforth referred to as LQG), that the areaspectrum remains no more equidistant and gets quantized like an angular momentum operator. This supposedly filled inthe sparse region in the radiation spectrum in the equidistant case, with additional states, introducing more number of transition lines thus populating the discrete profile, making it look more continuous.

2

8/20/2019 1509.09010

3/12

in, presently). Thereafter, we will demonstrate the cases where the quantization scheme is adopted forsome particular geometric quantities of the black hole in section 3. We conclude with a discussion on

our results.

2 Quantized Black hole

Let us consider a black hole which is thought to be quantized by some efficient quantum theory of gravity giving rise to a description where one of the parameter describing the hole gets quantized as

Z n = α̃nβ,   (1)

where Z  is a geometric parameter of black hole in the characteristic theory (for instance such as in [ 27]),α is a numerical constant and  n  is an integer. We will consider the case where in the macroscopic limitthe area of the event horizon gets related to this characterizing geometric parameter as

A ∝ Z γ ⇒ A =  αγ nβγ .   (2)

For large enough black hole, the characterizing geometric variable should be determined in termsof its mass (through a mass-area relation), in order to respect the  no-hair   conjecture. Further, theBekenstein-Mukhanov case can be obtained from  Eq. (2)  by choosing   β   = 1 =   γ . Later on we willgeneralize the following arguments for an arbitrary quantum spectra. Now, if we assume in the spiritof [21], that there is a  holographic  description of black holes, i.e., its entropy and area are related by,

S    =  A

4  (3)

=  αγ nβγ 

4  = log g(n),   (4)

where g(n) is effective number of microstates giving rise to the black hole configuration. This effectivenumber comes from thinking the black hole either as a microscopic limiting case of the quantum

description or the degeneracy of the higher excited state if that be the case. Now,

g(n) = exp [S ] = exp

αγ nβγ 

4

.   (5)

The condition that g(n) should be an integer puts defining criterion on the parameters α, β  and  γ . Theparameter g(n) is supposed to be integer for all integers  n (at least after some large enough macroscopiclimit  n). Therefore,

αγ  = 4 log K,   (6)

for some integer  K   and  βγ  should be a positive integer. At this stage we can assume that massiveblack holes admit the classical area-mass relation3

M n   ∝   A1/2n   (7)

∝   α

γ/2

n

βγ/2

.   (8)3In fact in LQG, this logic is adopted to argue for new mass states by LQG scheme of quantization   [28], where the

presumed number of degenerate area states are argued to be exponentially large. However, in general, a quantizationscheme can give rise to multiple number of degenerate states, it will not shift the mass separation gap if such a mass-arearelation is adopted. We show that modification to this relation offers very little help for filling the emission profile of theblack hole. Moreover the Hardy Ramanujan relation for the degeneracy of mass states is not applicable once the partsare assumed to carry large spin values.

3

8/20/2019 1509.09010

4/12

If the black hole is considered as a highly excited state of such a theory, then the emission profile of the black holes will be obtained from the transitions to lower mass eigenstates which will be obtained

as the quasinormal frequency [23, 24] associated with the black hole geometry. The minimum of suchfrequency gap is

ω = ∆M 

   =

 M n −M n−1 

  ∝ nβγ/2 − (n− 1)βγ/2

   .   (9)

Clearly for a macroscopic hole,  n ≫ 1,  resulting in

ω = M n −M n−1

   ∝

 nβγ/2 − (n− 1)βγ/2

   ∝

 n(βγ/2)−1

   .   (10)

Now, as we learnt from Eq. (6), the exponent βγ  is constrained to be a positive integer, the minimumenergy gap scales at the smallest as

ω ∝   1M 

 , constant, M 1/3...   corresponding to   βγ  = 1, 2, 3...   (11)

Clearly, for macroscopic black holes the spectra line separation is bounded from below by 1/M   in thisscheme. Thus such a kind of non-integer separation in mass eigenstates does not help. Admittedly wehave used, for this demonstration some classical inputs, like mass-area relation, apart from consideringthe system as a single excited state of the fundamental theory, rather than a collection of large numberof quantized microscopic systems.

The first of these issues can be trivially handled by choosing an arbitrary relation between theaverage area (expectation value) and the macroscopic mass which approaches the standard classicalrelation in large  N   limit, and repeating the above calculation. This scheme will just introduce an-other parameter in the description without disturbing the salient features discussed above unless theparameters in the theory lie in a particular range, which we will see later.

2.1 Black hole as a macroscopic system

Now we study the second scheme of description where a classical black hole is thought to be obtainedin a thermodynamic limit of the quantized theory. Thus the expectation value of the parameter willbe obtained as

Z  = α̃j

nj jβ ,   (12)

where njs are the number of quanta occupying  j−th quantum level. A macroscopic black hole will bemarked by condition

j nj  ≫   1. The previous case can be captured in this scheme with occupancy

profile ni =  δ ij, with j  ≫ 1 for a large mass black hole. Now appealing to  Eq. (2), we can write

A ∝ (α̃j

nj jβ)γ  = (αj

nj jβ)γ .   (13)

Again, repeating the arguments about the number of microstates, we see that

g(n) = exp S  =  eA4 = e

αγ

4  (

j njjβ)γ .   (14)

4

8/20/2019 1509.09010

5/12

Clearly, for the right hand of side of  Eq. (14) to be able to churn out an integer, we require  αγ /4 = log K 

for some integer  K . But additionally, we have another constraint that j n

j jβγ  has to be integer

for all possible  {nj}  (at least) for which

j nj  ≫ 1 otherwise, the integer criterion, i.e.,  Eq. (14) willnot work out for arbitrary high occupancies. Now again assuming a mass-area relation like Eq. (7), wehave

∆M 

   ∝

j nj j

βγ/2

−

j n′

j jβγ/2

   .   (15)

Now, we know that the quantity

j nj jβγ 

has to be integer for all possible   {nj}  which satisfyj nj  ≫ 1 for a stable system. Thus mass difference turns out to be difference of square roots of two

such integers constructed from two different set of occupancies  {nj}  and  {n′j}. Clearly, the smallest,the integers could change would be by unity and hence the smallest frequency of radiation for a massive

black hole will be

ω =  ∆M 

   ∝

  I 1/2 − (I − 1)1/2

   ∝

  1

 I 1/2,   (16)

with   I   =

j nj jβγ 

. Therefore, the leading order frequency dependence of the emitted radiation

goes as ω ∝ 1/M . We note here, the precise spectral form of the geometric variable was never requiredfor this demonstration. Thus, exactly the same result can be obtained for the spectral profile madetotally arbitrary

Z  = α̃j

njf ( j),   (17)

with an arbitrary   f ( j). In this case, following the lines of discussion, we obtain the criterion thatj njf ( j)

γ = I  should be an integer. Of course, this criterion will not be satisfied by all arbitrary

f ( j) and γ , but again this criterion is mandatory for a quantum gravity theory which will explain theholographic character of the event horizon of the black hole. It can be an interesting number theoryexercise to obtain possible pairs of (f ( j), γ ) leading to a consistent holographic description. Secondly,it is not clear which kind of geometric spectra will provide a thermal envelope to the emission profile.We will not comment further on these issues and pursue them elsewhere. However, for the moment,adopting such a spectra and the constraint, rest of the arguments follow and with the classical massarea relation we again obtain the same dependence of smallest quasi-normal modes being 1/M . Sowe show that irrespective of details of quantization, the sparseness of Hawking spectrum emerges as arobust feature which can be applied to any geometric quantization scheme and which yields a classicalholographic   and mass area relations. We will see that it is necessary to give up one of these twocharacterizing features of the macroscopic black hole, if denseness of radiation is to be obtained.

As pointed out in  [29–31] the mass dependence of the minimum frequency quanta radiated by theblack hole can also be related to the “cavity size”4 of the black hole. However in the case of anycovariant gravity theory the cavity size of a black hole can depend on the slicing of the spacetime

4We note that even in standard black body radiation there will be an apparent cutoff when the wavelength of theblack body radiation  ∼  the size of the cavity.

5

8/20/2019 1509.09010

6/12

manifold (in particular the region interior to the black hole) [32]. Even though the cavity size maydiffer depending on the slicing, but to an exterior observer receiving the Hawking radiation the area

radius plays the pivotal role, showing the holographic character of black holes.By altering the classical mass-area relation it is possible to generate a dense spectrum. The best

way to see this is to start with a mass-area relation of the form  M n  =  Aχ/2, where  χ  = 1 yields the

standard classical mass-area relation. In this case, by straightforward algebra it turns out that lowestfrequency corresponds to  ω  =  M 1−(2/χ). Thus for χ  = 1 we have the  ω  ∼  (1/M ) result as obtainedbefore. However if the mass-area relation were different and χ

8/20/2019 1509.09010

7/12

Then if there is a transition between (n + 1) and  nth level then the mass change is given by (withn ≫ 1),

∆M  = M n+1 −M n =  α

(n + 1)β − nβ

 =  α

nβ

1 + 1

n

β− nβ

,

= αβnβ−1.   (19)

Hence the quasi-normal mode frequency is given by,

ω̄ = αβ 

   nβ−1 =

 αβ 

 

M n

α

β−1β

.   (20)

The area turns out to be,

A = 4π(2M )2

= 16πM 2

= 16πα2

n2β

.   (21)

Thus, entropy and hence the multiplicity of level can be given by,

S  = 4πα2n2β ;   g(n) = exp(S ) = exp

4πα2n2β

.   (22)

Now we will choose  α  to be,

α2 =  1

4π ln K ;   g(n) =  K n

2β

.   (23)

Hence 2β  has to be integer which leads to,

β  = 1

2, 1,

 3

2, 2, . . .   (24)

Therefore, quasinormal modes turn out to have the following frequencies,

ω̄ ∼ M −1n   (β  = 1

2); ω̄ ∼ constant(β  = 1),   ω̄ =  M 1/3n   (β  =

 3

2). . . .   (25)

Thus, we see that that in this scheme the smallest frequency emitted by the hole scales inversely as itsmass. All subsequent frequencies have larger values so the minimum separation remains of the orderof the thermal maxima making the spectra discrete. We shall see below that such a feature is borneout by other geometric quantities like area and volume.

3.2 Area Quantization

Under this scheme of quantization, the area becomes one of the fundamental variable which becomes

quantized. such a quantization appears readily in the literature on quantum black holes5 [34–36].

5In particular, in LQG at the kinematic level, quantization of area element of a spatial slice in spacetime in termsof   spin quantum numbers   is known to indicate ultraviolet finiteness of the theory. We show that even this scheme of quantization reveals a sparse emission profile for black hole.

7

8/20/2019 1509.09010

8/12

Again we consider a simple enough quantization of the area of black hole horizon. Most generalquantization results can be argued again on the lines discussed in section 2 (see Eq. (17) in particular).

Therefore, we consider that area of the black hole is quantized such that,

An =  αnβ ,   (26)

where n  is an integer depicting the quantum level  6. First we consider the black hole as a single highlyexcited state of such a theory. A macroscopic description can follow on previous lines. Thus, if theblack hole made a transition between (n + 1)th and nth area level the area change is given by,

A = 16πM 2; ∆A = 32πM ∆M  = 32πM ω̄;

∆A =  An+1 −An =  αβnβ−1.   (27)

This immediately leads to the following expression for quasi normal frequency,

ω̄ =  αβ  nβ−1

32πM    =

 αβ 32π

16πα 

β−1β

M β−2

β .   (28)

Now, the entropy and multiplicity of nth level can be given by,

S  = α

4nβ;   g(n) = exp(S ) = exp

α4

nβ

.   (29)

This requires α  = 4ln K  and β  to be integer in order to make  g(n) integral. Thus quasinormal modeshave the following scaling relations,

ω̄ ∼ M −1(β  = 1),   ∼ constant(β  = 2),   ∼ M 1/3(β  = 3). . . .   (30)

Thus, clearly such a scheme does not make the spectrum dense. As before the minimum frequency is

bounded from below by ω̄ ∝ 1/M .

3.3 Volume Quantization

Lastly, on a 3-dimensional surface let us consider the model where the volume form gets quantized.Classically, the volume of interior of black hole is a foliation dependent quantity [ 32]. However for eachof the foliations, the volume remains proportional to  M 3. In some co-ordinates, the interior becomesdynamic so the interior geometry evolves in time. However, at each constant time hypersurface thevolume remains proportional to  M 3. The constant of proportionality does not play much of a role,so we make a simple choice. Also, as before we only demonstrate the case for a highly excited statesuch that the above mentioned classical notions are applicable approximately. Thus, the volume of then−th level can be quantized as,

V n =  αnβ = 32π3

  M 3.   (31)

6In fact, in order to argue for the denseness of Hawking spectra in LQG, it is suggested  [34–36] that the so called spinnetwork punctures the horizon with high spin quantum numbers. In that case the area spectrum approximately followsa quantization like that given in  Eq. (26)   with  β   = 1. We argue that such an approximation is of a little help as far asdensitization of the emission profile is concerned.

8

8/20/2019 1509.09010

9/12

If the black hole makes a transition between (n +1)th and nth level then the associated volume changeis given by,

∆V   = αβnβ−1 = 32πM 2∆M.   (32)

Again the frequency of emission is obtained by the relation ∆M  =   ω̄, the quasi normal mode frequencyturns out to be,

ω̄ =  αβ 

32π 

32π

3α

β−1β

M β−3β .   (33)

The area and hence the entropy leading to the multiplicity turns out to be,

A = 16π 3α

32π2

3

n2β3 ;   g(n) = exp(S ) = exp

4π

 3α

32π2

3

n2β3

.   (34)

The multiplicity would be integer only if 2β/3 is integral, i.e.,   β   = 3/2, 3, 9/2, . . .  and   α   is chosenappropriately. Then the quasinormal modes behave as,

ω̄ ∼ M −1(β  = 3/2),   ∼ constant(β  = 3),   ∼ M 1/3(β  = 9/2), . . .   (35)

Thus, we see that the quasi normal mode frequencies in all the three cases follow identical behavior,i.e., it has the minimal separation of the order of 1 /M . It is a trivial extension of the arguments insection 2, to see the sparseness persists even if the occupancy across the quantum levels is of the kindnj  = δ ij   with arbitrary spectral distribution function.

4 Conclusions

In this paper we try to obtain quantum support for a dense emission profile for a black hole. It has beenargued previously [17–19]   that the emission spectra of a black hole should look very discrete. Thereis a general hope that including the quantum description of gravity will make black hole a quantumobject. So large black holes may support many new transition states which it can jump into and emita spectral line. Thus, in principle there can be many transition lines and that can make the spectrumrich. However, semi classically, quasi normal modes tell us about the emission profile of the black hole.The frequency of the quasi normal modes can be thought to be the energy extracted by the quantumfield in the background spacetime from the black hole. So quantum mechanically the black hole isexpected to settle down in a new mass state allowed by the quantum theory after emitting one quantaof radiation. So a quantum theory which gives rise to a dense Hawking radiation profile should havemany nearby mass states for black holes, so that the emission frequency can be extremely small. Inother words, the quantum cut-off on frequency in IR regime must be extremely small, so that the empty

region in the IR domain as prescribed in semi-classical treatment gets densely occupied.We consider an arbitrary quantum description of a black hole where one of its characterizing vari-

ables gets quantized. We assume that this characterizing parameter at the macroscopic limit shouldget related to the mass of the hole, owing to the celebrated  no-hair  conjecture. For that purpose wetake two inputs about a large mass black hole which a classical black hole is expected to satisfy. Firstwe assume that the holography of the black hole is recovered for large N  (or large mass) limit. That is

9

8/20/2019 1509.09010

10/12

to say for such black holes, the entropy, at the leading order, can be given as the macroscopic averagearea. Secondly, we assume that the mass-area relation for the classical hole is also obtained in that

limit. We show that these inputs are sufficient to rule out any quantum geometric description of ablack hole if a dense Hawking spectrum is taken as a guiding criterion. Alternatively, one can saythat the semi-classical sparseness of a black hole radiation is also supported by the quantum geometricapproach. This suggests that either the emission profile of the black hole is non-thermal, or, it mayonly be thermal with a temperature much larger than the IR cut-off which is 1 /M . We demonstratethis feature through a few simple quantum models as well, apart from a general framework. Therefore,we argue that a fully quantum mechanical description is unlikely to ascribe the hole with a thermalcharacter with the temperature decided by inverse of its mass. Such a possibility can only arise if themass and area of the black hole are related in a way different than the classical one. Or alternatively,the entropy of the black hole must not be related to its area in a fashion true in Einstein theory.

It is interesting to ask which kind of geometric quantization can resemble a thermal profile for largemass black holes. We also propose a constraint on models of geometric quantization which is obtainedfrom the requirement of holographic description of black hole. These two criteria may allow only afew possibilities for underlying quantum spectrum. Furthermore, with a specified area mass relation,area entropy relation can be tweaked to obtain a dense spectrum. Such a relation may be realized in agravity theory different from Einstein’s theory of relativity. We will pursue and address these aspectsin a subsequent work.

Acknowledgements

Research of S.C. is funded by a SPM fellowship from CSIR, Government of India. The authors thankT. Padmanabhan for illuminating discussions.

References

[1] J. Bekenstein, “Black holes and the second law,” Lett. Nuovo Cimento Soc. Ital. Fis.  4 (1972)737–740.

[2] W. G. Unruh and R. M. Wald, “What happens when an accelerating observer detects a Rindlerparticle,”  Phys. Rev.  D29  (1984) 1047–1056.

[3] S. Hawking, “Particle Creation by Black Holes,”  Commun.Math.Phys.   43 (1975) 199–220.

[4] S. Chakraborty and T. Padmanabhan, “Thermodynamical interpretation of the geometricalvariables associated with null surfaces,”  arXiv:1508.04060 [gr-qc].

[5] T. Padmanabhan, “Emergent Gravity Paradigm: Recent Progress,”Mod. Phys. Lett.  A30 no. 03n04, (2015) 1540007,  arXiv:1410.6285 [gr-qc].

[6] J. Eisert, M. Cramer, and M. B. Plenio, “Area laws for the entanglement entropy - a review,”Rev. Mod. Phys.  82 (2010) 277–306,  arXiv:0808.3773 [quant-ph].

[7] R. Brustein, M. B. Einhorn, and A. Yarom, “Entanglement interpretation of black hole entropyin string theory,”  JHEP  01 (2006) 098,   arXiv:hep-th/0508217 [hep-th].

10

http://dx.doi.org/10.1103/PhysRevD.29.1047http://dx.doi.org/10.1103/PhysRevD.29.1047http://dx.doi.org/10.1103/PhysRevD.29.1047http://dx.doi.org/10.1103/PhysRevD.29.1047http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://arxiv.org/abs/1508.04060http://arxiv.org/abs/1508.04060http://dx.doi.org/10.1142/S0217732315400076http://dx.doi.org/10.1142/S0217732315400076http://dx.doi.org/10.1142/S0217732315400076http://dx.doi.org/10.1142/S0217732315400076http://arxiv.org/abs/1410.6285http://arxiv.org/abs/1410.6285http://dx.doi.org/10.1103/RevModPhys.82.277http://dx.doi.org/10.1103/RevModPhys.82.277http://dx.doi.org/10.1103/RevModPhys.82.277http://dx.doi.org/10.1103/RevModPhys.82.277http://arxiv.org/abs/0808.3773http://arxiv.org/abs/0808.3773http://dx.doi.org/10.1088/1126-6708/2006/01/098http://dx.doi.org/10.1088/1126-6708/2006/01/098http://dx.doi.org/10.1088/1126-6708/2006/01/098http://arxiv.org/abs/hep-th/0508217http://arxiv.org/abs/hep-th/0508217http://arxiv.org/abs/hep-th/0508217http://dx.doi.org/10.1088/1126-6708/2006/01/098http://arxiv.org/abs/0808.3773http://dx.doi.org/10.1103/RevModPhys.82.277http://arxiv.org/abs/1410.6285http://dx.doi.org/10.1142/S0217732315400076http://arxiv.org/abs/1508.04060http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1103/PhysRevD.29.1047

8/20/2019 1509.09010

11/12

[8] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory: A Non-technicalintroduction,” Int. J. Quant. Inf.  4 (2006) 429,   arXiv:quant-ph/0505193 [quant-ph].

[9] L. Masanes, “An Area law for the entropy of low-energy states,” Phys. Rev.  A80 (2009) 052104,arXiv:0907.4672 [quant-ph].

[10] S. Chakraborty, S. Singh, and T. Padmanabhan, “A quantum peek inside the black hole eventhorizon,”  JHEP  1506 (2015) 192,  arXiv:1503.01774 [gr-qc].

[11] S. Singh and S. Chakraborty, “Black hole kinematics: The in-vacuum energy density and flux fordifferent observers,” Phys.Rev.  D90 no. 2, (2014) 024011,  arXiv:1404.0684 [gr-qc].

[12] K. Lochan and T. Padmanabhan, “Extracting information about the initial state from the blackhole radiation,”  arXiv:1507.06402 [gr-qc].

[13] D. N. Page, “Information in black hole radiation,”  Phys. Rev. Lett.  71 (1993) 3743–3746,

arXiv:hep-th/9306083 [hep-th].[14] S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,”

Phys. Rev.  D14 (1976) 2460–2473.

[15] S. D. Mathur, “The Information paradox: A Pedagogical introduction,”Class. Quant. Grav.  26  (2009) 224001,   arXiv:0909.1038 [hep-th].

[16] S. D. Mathur, “Tunneling into fuzzball states,” Gen. Rel. Grav.  42 (2010) 113–118,arXiv:0805.3716 [hep-th].

[17] F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, “The Hawking cascade from a black hole isextremely sparse,”   arXiv:1506.03975 [gr-qc].

[18] S. Hod, “The Hawking evaporation process of rapidly-rotating black holes: An almost continuous

cascade of gravitons,”  Eur. Phys. J.  C75 no. 7, (2015) 329,  arXiv:1506.05457 [gr-qc].

[19] S. Hod, “Discrete black hole radiation and the information loss paradox,”Phys. Lett.  A299 (2002) 144–148,   arXiv:gr-qc/0012076 [gr-qc].

[20] M. Visser, “Thermality of the Hawking flux,”  JHEP  07 (2015) 009,   arXiv:1409.7754 [gr-qc].

[21] J. D. Bekenstein and V. F. Mukhanov, “Spectroscopy of the quantum black hole,”Phys. Lett.  B360 (1995) 7–12,   arXiv:gr-qc/9505012 [gr-qc].

[22] J. D. Bekenstein, “Black holes and entropy,”  Phys.Rev.  D7 (1973) 2333–2346.

[23] S. Hod, “Gravitation, the quantum, and Bohr’s correspondence principle,”Gen. Rel. Grav.  31 (1999) 1639,   arXiv:gr-qc/0002002 [gr-qc].

[24] S. Hod, “Bohr’s correspondence principle and the area spectrum of quantum black holes,”Phys. Rev. Lett.  81 (1998) 4293,   arXiv:gr-qc/9812002 [gr-qc].

[25] C. Vaz, S. Gutti, C. Kiefer, T. P. Singh, and L. C. R. Wijewardhana, “Mass spectrum andstatistical entropy of the BTZ black hole from canonical quantum gravity,”Phys. Rev.  D77 (2008) 064021,   arXiv:0712.1998 [gr-qc].

11

http://dx.doi.org/10.1142/S021974990600192Xhttp://dx.doi.org/10.1142/S021974990600192Xhttp://dx.doi.org/10.1142/S021974990600192Xhttp://dx.doi.org/10.1142/S021974990600192Xhttp://arxiv.org/abs/quant-ph/0505193http://dx.doi.org/10.1103/PhysRevA.80.052104http://dx.doi.org/10.1103/PhysRevA.80.052104http://dx.doi.org/10.1103/PhysRevA.80.052104http://dx.doi.org/10.1103/PhysRevA.80.052104http://arxiv.org/abs/0907.4672http://dx.doi.org/10.1007/JHEP06(2015)192http://dx.doi.org/10.1007/JHEP06(2015)192http://dx.doi.org/10.1007/JHEP06(2015)192http://dx.doi.org/10.1007/JHEP06(2015)192http://arxiv.org/abs/1503.01774http://arxiv.org/abs/1503.01774http://dx.doi.org/10.1103/PhysRevD.90.024011http://dx.doi.org/10.1103/PhysRevD.90.024011http://dx.doi.org/10.1103/PhysRevD.90.024011http://dx.doi.org/10.1103/PhysRevD.90.024011http://arxiv.org/abs/1404.0684http://arxiv.org/abs/1507.06402http://dx.doi.org/10.1103/PhysRevLett.71.3743http://dx.doi.org/10.1103/PhysRevLett.71.3743http://dx.doi.org/10.1103/PhysRevLett.71.3743http://dx.doi.org/10.1103/PhysRevLett.71.3743http://arxiv.org/abs/hep-th/9306083http://arxiv.org/abs/hep-th/9306083http://dx.doi.org/10.1103/PhysRevD.14.2460http://dx.doi.org/10.1103/PhysRevD.14.2460http://dx.doi.org/10.1103/PhysRevD.14.2460http://dx.doi.org/10.1088/0264-9381/26/22/224001http://dx.doi.org/10.1088/0264-9381/26/22/224001http://dx.doi.org/10.1088/0264-9381/26/22/224001http://arxiv.org/abs/0909.1038http://arxiv.org/abs/0909.1038http://dx.doi.org/10.1007/s10714-009-0837-3http://dx.doi.org/10.1007/s10714-009-0837-3http://dx.doi.org/10.1007/s10714-009-0837-3http://arxiv.org/abs/0805.3716http://arxiv.org/abs/0805.3716http://arxiv.org/abs/1506.03975http://dx.doi.org/10.1140/epjc/s10052-015-3554-yhttp://dx.doi.org/10.1140/epjc/s10052-015-3554-yhttp://dx.doi.org/10.1140/epjc/s10052-015-3554-yhttp://arxiv.org/abs/1506.05457http://arxiv.org/abs/1506.05457http://dx.doi.org/10.1016/S0375-9601(02)00013-0http://dx.doi.org/10.1016/S0375-9601(02)00013-0http://dx.doi.org/10.1016/S0375-9601(02)00013-0http://dx.doi.org/10.1016/S0375-9601(02)00013-0http://arxiv.org/abs/gr-qc/0012076http://dx.doi.org/10.1007/JHEP07(2015)009http://dx.doi.org/10.1007/JHEP07(2015)009http://dx.doi.org/10.1007/JHEP07(2015)009http://arxiv.org/abs/1409.7754http://arxiv.org/abs/1409.7754http://dx.doi.org/10.1016/0370-2693(95)01148-Jhttp://dx.doi.org/10.1016/0370-2693(95)01148-Jhttp://dx.doi.org/10.1016/0370-2693(95)01148-Jhttp://arxiv.org/abs/gr-qc/9505012http://arxiv.org/abs/gr-qc/9505012http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1023/A:1026753914838http://dx.doi.org/10.1023/A:1026753914838http://dx.doi.org/10.1023/A:1026753914838http://arxiv.org/abs/gr-qc/0002002http://arxiv.org/abs/gr-qc/0002002http://dx.doi.org/10.1103/PhysRevLett.81.4293http://dx.doi.org/10.1103/PhysRevLett.81.4293http://dx.doi.org/10.1103/PhysRevLett.81.4293http://dx.doi.org/10.1103/PhysRevLett.81.4293http://arxiv.org/abs/gr-qc/9812002http://dx.doi.org/10.1103/PhysRevD.77.064021http://dx.doi.org/10.1103/PhysRevD.77.064021http://dx.doi.org/10.1103/PhysRevD.77.064021http://arxiv.org/abs/0712.1998http://arxiv.org/abs/0712.1998http://arxiv.org/abs/0712.1998http://dx.doi.org/10.1103/PhysRevD.77.064021http://arxiv.org/abs/gr-qc/9812002http://dx.doi.org/10.1103/PhysRevLett.81.4293http://arxiv.org/abs/gr-qc/0002002http://dx.doi.org/10.1023/A:1026753914838http://dx.doi.org/10.1103/PhysRevD.7.2333http://arxiv.org/abs/gr-qc/9505012http://dx.doi.org/10.1016/0370-2693(95)01148-Jhttp://arxiv.org/abs/1409.7754http://dx.doi.org/10.1007/JHEP07(2015)009http://arxiv.org/abs/gr-qc/0012076http://dx.doi.org/10.1016/S0375-9601(02)00013-0http://arxiv.org/abs/1506.05457http://dx.doi.org/10.1140/epjc/s10052-015-3554-yhttp://arxiv.org/abs/1506.03975http://arxiv.org/abs/0805.3716http://dx.doi.org/10.1007/s10714-009-0837-3http://arxiv.org/abs/0909.1038http://dx.doi.org/10.1088/0264-9381/26/22/224001http://dx.doi.org/10.1103/PhysRevD.14.2460http://arxiv.org/abs/hep-th/9306083http://dx.doi.org/10.1103/PhysRevLett.71.3743http://arxiv.org/abs/1507.06402http://arxiv.org/abs/1404.0684http://dx.doi.org/10.1103/PhysRevD.90.024011http://arxiv.org/abs/1503.01774http://dx.doi.org/10.1007/JHEP06(2015)192http://arxiv.org/abs/0907.4672http://dx.doi.org/10.1103/PhysRevA.80.052104http://arxiv.org/abs/quant-ph/0505193http://dx.doi.org/10.1142/S021974990600192X

8/20/2019 1509.09010

12/12

[26] C. Rovelli and L. Smolin, “Discreteness of area and volume in quantum gravity,”Nucl. Phys.  B442 (1995) 593–622,   arXiv:gr-qc/9411005 [gr-qc]. [Erratum: Nucl.

Phys.B456,753(1995)].

[27] C. Vaz and L. C. R. Wijewardhana, “Spectrum and Entropy of AdS Black Holes,”Phys. Rev.  D79 (2009) 084014,   arXiv:0902.1192 [gr-qc].

[28] M. Barreira, M. Carfora, and C. Rovelli, “Physics with nonperturbative quantum gravity:Radiation from a quantum black hole,”  Gen. Rel. Grav.  28 (1996) 1293–1299,arXiv:gr-qc/9603064 [gr-qc].

[29] “Private Communications with T. Padmanabhan,”.

[30] J. D. Bekenstein, “The quantum mass spectrum of the Kerr black hole,”Lett. Nuovo Cim.  11 (1974) 467.

[31] A. V. Kotwal and S. Hofmann, hep-ph/0204117.

[32] B. S. DiNunno and R. A. Matzner, “The Volume Inside a Black Hole,”Gen. Rel. Grav.  42 (2010) 63–76,  arXiv:0801.1734 [gr-qc].

[33] Y. Yoon, arXiv:1210.8355 [gr-qc].

[34] C. Rovelli,  Quantum gravity . Cambridge University Press, 2004.http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521837332.

[35] M. Bojowald,  Canonical Gravity and Applications . Cambridge University Press, 2010.http://www.cambridge.org/mw/academic/subjects/physics/cosmology-relativity-and-gravitation/

[36] A. Perez, “Introduction to loop quantum gravity and spin foams,” in  2nd International Conference on Fundamental Interactions (ICFI 2004) Domingos Martins, Espirito Santo, Brazil,June 6-12, 2004. 2004.   arXiv:gr-qc/0409061 [gr-qc].

12

http://dx.doi.org/10.1016/0550-3213(95)00150-Qhttp://dx.doi.org/10.1016/0550-3213(95)00150-Qhttp://dx.doi.org/10.1016/0550-3213(95)00150-Qhttp://arxiv.org/abs/gr-qc/9411005http://arxiv.org/abs/gr-qc/9411005http://dx.doi.org/10.1103/PhysRevD.79.084014http://dx.doi.org/10.1103/PhysRevD.79.084014http://dx.doi.org/10.1103/PhysRevD.79.084014http://arxiv.org/abs/0902.1192http://arxiv.org/abs/0902.1192http://dx.doi.org/10.1007/BF02109521http://dx.doi.org/10.1007/BF02109521http://dx.doi.org/10.1007/BF02109521http://dx.doi.org/10.1007/BF02109521http://arxiv.org/abs/gr-qc/9603064http://dx.doi.org/10.1007/BF02762768http://dx.doi.org/10.1007/BF02762768http://dx.doi.org/10.1007/BF02762768http://dx.doi.org/10.1007/s10714-009-0814-xhttp://dx.doi.org/10.1007/s10714-009-0814-xhttp://dx.doi.org/10.1007/s10714-009-0814-xhttp://arxiv.org/abs/0801.1734http://arxiv.org/abs/0801.1734http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521837332http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521837332http://www.cambridge.org/mw/academic/subjects/physics/cosmology-relativity-and-gravitation/canonical-gravity-and-applications-cosmology-black-holes-and-quantum-gravity?format=HBhttp://arxiv.org/abs/gr-qc/0409061http://arxiv.org/abs/gr-qc/0409061http://arxiv.org/abs/gr-qc/0409061http://www.cambridge.org/mw/academic/subjects/physics/cosmology-relativity-and-gravitation/canonical-gravity-and-applications-cosmology-black-holes-and-quantum-gravity?format=HBhttp://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521837332http://arxiv.org/abs/0801.1734http://dx.doi.org/10.1007/s10714-009-0814-xhttp://dx.doi.org/10.1007/BF02762768http://arxiv.org/abs/gr-qc/9603064http://dx.doi.org/10.1007/BF02109521http://arxiv.org/abs/0902.1192http://dx.doi.org/10.1103/PhysRevD.79.084014http://arxiv.org/abs/gr-qc/9411005http://dx.doi.org/10.1016/0550-3213(95)00150-Q