We provide a fuzzball picture for Schwarzshild black holes, in which matters and energy consisting the hole are not positioned on the central point exclusively but oscillate around there in a serial of eigen-modes, each of which features a special level of binding degrees and are quantum mechanically possible to be measured outside the horizon. By listing these modes explicitly for holes as large as $6M_\mathrm{pl}$, we find that their number increases exponentially with the area. Basing on this picture, we present a simple but explicitly unitary derivation of hawking radiations.
These lecture notes present an introduction to the fuzzball proposal and emission from the D1-D5 system which is geared to an audience of graduate students and others with little background in the subject. The presentation begins with a discussion of the Penrose process and Hawking radiation. The fuzzball proposal is then introduced, and the two- and three-charge systems are reviewed. In the three-charge case details are not discussed. A detailed discussion of emission calculations for D1-D5-P black holes and for certain non-extremal fuzzballs from both the gravity and CFT perspectives is incl…
We investigate the recent black hole firewall argument. For a black hole in a typical state we argue that unitarity requires every quantum of radiation leaving the black hole to carry information about the initial state. An information-free horizon is thus inconsistent with unitary at every step of the evaporation process (in particular both before and after Page time). The required horizon-scale structure is manifest in the fuzzball proposal which provides a mechanism for holding up the structure. In this context we want to address the experience of an infalling observer and discuss the recen…
We construct a model which illustrates the conjecture of fuzzball complementarity. In the fuzzball paradigm, the black hole microstates have no interior, and radiate unitarily from their surface through quanta of energy $E\sim T$. But quanta with $E\gg T$ impinging on the fuzzball create large collective excitations of the fuzzball surface. The dynamics of such excitations must be studied as an evolution in superspace, the space of all fuzzball solution $|F_i\rangle$. The states in this superspace are arranged in a hierarchy of `complexity'. We argue that evolution towards higher complexity ma…
The black hole information paradox has caused enormous confusion over four decades. But in recent years, the theorem of quantum strong-subaddditivity has sorted out the possible resolutions into three sharp categories: (A) No new physics at $r\gg l_p$; this necessarily implies remnants/information loss. A realization of remnants is given by a baby Universe attached near $r\sim 0$. (B) Violation of the `no-hair' theorem by nontrivial effects at the horizon $r\sim M$. This possibility is realized by fuzzballs in string theory, and gives unitary evaporation. (C) Having the vacuum at the horizon, …
The fuzzball proposal says that the information of the black hole state is distributed throughout the interior of the horizon in a `quantum fuzz'. There are special microstates where in the dual CFT we have `many excitations in the same state'; these are described by regular classical geometries without horizons. Jejjala et.al constructed non-extremal regular geometries of this type. Cardoso et. al then found that these geometries had a classical instability. In this paper we show that the energy radiated through the unstable modes is exactly the Hawking radiation for these microstates. We do …
The fuzzball proposal states that associated with a black hole of entropy S there are exp S horizon-free non-singular solutions that asymptotically look like the black hole but generically differ from the black hole up to the horizon scale. These solutions, the fuzzballs, are considered to be the black hole microstates while the original black hole represents the average description of the system. The purpose of this report is to review current evidence for the fuzzball proposal, emphasizing the use of AdS/CFT methods in developing and testing the proposal. In particular, we discuss the status…
It is possible to construct a special family of nonextremal black hole microstates. These microstates are unstable, and emit radiation at a rate which is found to exactly equal the Hawking radiation rate predicted for them by the dual CFT. In this paper we analyze in more detail the nature of the radiation created by these unstable modes. The energy and angular momentum of the mode is found to be localized in two regions: one near infinity corresponding to the emitted quanta, and the other in the ergoregion which is deep inside the interior of the geometry. The energy and angular momenta are e…
It has been conjectured that the dual of the eternal black hole in AdS is two entangled but disconnected CFTs. We show that the entanglement created by the process of Hawking radiation creates several challenges for this conjecture. The nature of fuzzball states suggests a different picture, where the dual to two entangled CFTs is two entangled but disconnected spacetimes. We argue for a process of `quick tunneling' where the Einstein-Rosen bridge of the eternal hole tunnels rapidly into fuzzball states, preventing the existence of the eternal hole as a semiclassical spacetime. The regions beh…
In the traditional picture of black holes Hawking radiation is created by pair creation from the vacuum at the horizon. In the fuzzball proposal, individual microstates do not have a horizon with the `vacuum' state in its vicinity. For a special family of non-extremal microstates it was recently found that emission occurs due to pair creation in an ergoregion, rather than at a horizon. In this paper we extend this result to a slightly larger class of microstates, again finding exact agreement between the emission in the gravity picture and the CFT dual. We write down an expression for emission…
In the LLM bubbling geometries, we compute the entropies of black holes and estimate their "horizon" sizes from the fuzzball conjecture, based on coarse-graining on the gravity side. The differences of black hole microstates cannot be seen by classical observations. Conversely, by counting the possible deformations of the geometry which are not classically detectable, we can calculate the entropy. We carry out this method on the black holes of the LLM bubbling geometries, such as the superstar, and obtain the same result as was derived by coarse-graining directly on the CFT (fermion) side. Sec…
It is sometimes believed that small quantum gravity effects can encode information as `delicate correlations' in Hawking radiation, thus saving unitarity while maintaining a semiclassical horizon. A recently derived inequality showed that this belief is incorrect: one must have order unity corrections to low energy evolution at the horizon (i.e. fuzzballs) to remove entanglement between radiation and the hole. In this paper we take several models of `small corrections' and compute the entanglement entropy numerically; in each case this entanglement is seen to monotonically grow, in agreement w…
In this paper, we successfully derive the Bekenstein-Hawking entropy for Schwarzschild black holes in various dimensions by using a non-trivial phase space. It is appealing to notice that the thermodynamics of a Schwarzschild black hole actually behaves like that of a $1$-dimensional quantum mechanical system. Our result suggests that black hole should be viewed as a system with the equation of state $P=ρ$, and it also suggests that a holographic stage should exist in the early universe.
We investigate the effect of quantum correction on Hawking radiation for Schwarzschild black holes. We consider Hawking temperature and entropy to order G^2 and find that the area law of black holes should be modified. We think of Hawking radiation as tunneling and find that at certain frequency ω the radiation spectrum of black holes can be pure blackbody spectrum. Therefore, it is possible to break down information conservation. In order to ensure information conservation at any energy scales, we suggest that black holes cannot release all information they have. We briefly discuss the bound …
Hawking flux from the Schwarzschild black hole with a global monopole is obtained by using Robinson and Wilczek's method. Adopting a dimension reduction technique, the effective quantum field in the (3+1)--dimensional global monopole background can be described by an infinite collection of the (1+1)--dimensional massless fields if neglecting the ingoing modes near the horizon, where the gravitational anomaly can be cancelled by the (1+1)--dimensional black body radiation at the Hawking temperature.
The idea of holography in gravity arose from the fact that the entropy of black holes is given by their surface area. The holography encountered in gauge/gravity duality has no such relation however; the boundary surface can be placed at an arbitrary location in AdS space and its area does not give the entropy of the bulk. The essential issues are also different between the two cases: in black holes we get Hawking radiation from the `holographic surface' which leads to the information issue, while in gauge/gravity duality there is no such radiation from the boundary surface. To resolve the inf…
We study the thermodynamic consequences of a recently proposed description for a Schwarzschild black hole based on Euclidean (D3,D3)+(\bar{D3},\bar{D3}) brane pairs described in terms of chain-like excitations. A discrete mass-spectrum of Bekenstein-type is inferred and upon identification of the black hole mass with the chain's energy the leading corrections to both Hawking-temperature and specific heat of the black hole are obtained. The results indicate that for small black holes the evaporation process will be considerably altered.
This paper has been withdrawn by the authors due to an incorrect statement on the viability of the Robinson-Wilczek method when applied to the case when the metric determinant vanishes at the horizon. This paper has been replaced by arXiv:0803.1338 [hep-th].
Former part of this article is the proceedings for my talk on 2004.07474, which is a report on the issue in the title of this article. Later part is the detailed description of 2004.07474.
The massive particles tunneling method has been used to investigate the Hawking non-thermal and purely thermal radiations of Schwarzschild Anti-de Sitter (SAdS) black hole. Considering the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles the imaginary part of the action has been derived from Hamilton-Jacobi equation. Using the conservation laws of energy and angular momentum we have showed that the non-thermal and purely thermal tunneling rates are related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviate…
Motivated by the universality of Hawking radiation and that of the anomaly cancellation technique as well as that of the effective action method, we investigate the Hawking radiation of a Schwarzschild black hole in the isotropic coordinates via the cancellation of gravitational anomaly. After performing a dimensional reduction from the four-dimensional isotropic Schwarzschild metric, we show that this reduction procedure will, in general, result in two classes of two-dimensional effective metrics: the conformal equivalent and the inequivalent ones. For the physically equivalent class, the two…
It has recently been shown that the Hawking evaporation process of $(3+1)$-dimensional Schwarzschild black holes is characterized by the dimensionless ratio $η\equivτ_{\text{gap}}/τ_{\text{emission}}\gg1$, where $τ_{\text{gap}}$ is the characteristic time gap between the emissions of successive Hawking quanta and $τ_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the Schwarzschild black hole. This strong inequality implies that the Hawking cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is extremely {\it …
By globally embedding curved spaces into higher dimensional flat ones, we show that Hawking thermal properties map into their Unruh equivalents: The relevant curved space detectors become Rindler ones, whose temperature and entropy reproduce the originals. Specific illustrations include Schwarzschild, Schwarzschild-(anti)deSitter, Reissner-Nordstrom and BTZ spaces.
A sufficiently large Schwarzschild AdS black hole with the Hawking temperature has a positive heat capacity and undergoes the Hawking-Page phase transition. Even though an arbitrary isothermal surface of a cavity is introduced, the essential features about the stability and phase transition of the Schwarzschild AdS black hole remain the same. In this paper, we study the stability and the Hawking-Page phase transition of the Schwarzschild AdS black hole by employing an alternative local temperature satisfying the Hartle-Hawking vacuum condition which states that the ingoing and outgoing fluxes …
Several approaches to Hawking radiation on Schwarzschild spacetime rely in some way or another on the fact that the Kruskal manifold has two causally disconnected exterior regions. We investigate the Hawking(-Unruh) effect for a real scalar field on the $\RPthree$ geon: an inextendible, globally hyperbolic, space and time orientable eternal black hole spacetime that is locally isometric to Kruskal but contains only one exterior region. The Hartle-Hawking-like vacuum~$\hhvacgeon$, which can be characterized alternatively by the positive frequency properties along the horizons or by the complex …
It has been suggested by Giddings that the origin of Hawking radiation in black holes is a quantum atmosphere of near-horizon quantum region by investigating both the total emission rate and the stress tensor of Hawking radiation. Revisiting this issue in the exactly soluble model of a dimensionally reduced Schwarzschild black hole, we shall confirm that the dominant Hawking radiation in the Unruh vacuum indeed occurs at the quantum atmosphere, not just at the horizon by exactly calculating the out-temperature responsible for outgoing Hawking particle excitations. Consequently we show that the…
Following the recent observation that the Ricci flow and the infinite distance swampland conjecture are closely related to each other, we will investigate in this paper geometric flow equations for AdS space-time geometries. First, we consider the so called Yamabe and Ricci-Bourguignon flows and we show that these two flows - in contrast to the Ricci flow - lead to infinite distance fixed points for product spaces like $AdS_d\times S^p$, where $AdS_d$ denotes d-dimensional AdS space and $S^p$ corresponds to a p-dimensional sphere. Second, we consider black hole geometries in AdS space time geo…
Analytic semiclassical backreaction of a Schwarzschild black hole in a finite cavity: horizon shift, temperature renormalization, and canonical stability in the Hartle-Hawking State
Centre for Theoretical Physics, The British University（英国大学理论物理中心）；Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊马姆穆罕默德·本·沙特伊斯兰大学数学与统计系）；Faculty of Physics, Babeş-Bolyai University（巴比什-博雅依大学物理学院）；Astronomical Observatory（天文台）
The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any ass…
Some authors have recently found that the tunneling approach gives a different Hawking temperature for a Schwarzschild black hole in a different coordinate system. In this paper, we find that to work out the Hawking temperature in a different coordinate system by the tunneling approach, we must use the correct definition of the energy of the radiating particles. By using a new definition of the particle energy, we obtain the correct Hawking temperature for a Schwarzschild black hole in two dynamic coordinate systems, the Kruskal-Szekers and dynamic Lemaitre coordinate systems.
The correction of Hawking temperature of Schwarzschild-de Sitter (SdS) black hole is investigated using the generalized Klein-Gordon equation and the generalized Dirac equation by taking the quantum gravity effects into account. We derive the corrected Hawking temperatures for scalar particles and fermions crossing the event horizon. The quantum gravity effects prevent the rise of temperature in the SdS black hole. Besides correction of Hawking temperature, the Hawking radiation of SdS black hole is also investigated using massive particles tunneling method. By considering self gravitation eff…
In this paper, we have extended the previous study of the thermodynamics and phase transition of the Schwarzschild black hole in the rainbow gravity to the Schwarzschild-AdS black hole where metric depends on the energy of a probe. Making use of the Heisenberg uncertainty principle and the modified dispersion relation, we have obtained the modified local Hawking temperature and thermodynamic quantities in an isothermal cavity. Moreover, we carry out the analysis of constant temperature slices of a black hole. As a result, we have shown that there also exists another Hawking-Page-like phase tra…
We study, by separately calculating the contributions of vacuum fluctuations and radiation reaction to the atomic energy level shift, the Lamb shift of a static two-level atom interacting with real massless scalar fields in the Boulware, Unruh and Hartle-Hawking vacuums outside a Schwarzschild black hole. We find that in the Boulware vacuum, the Lamb shift gets a correction arising as a result of the backscattering of vacuum field modes off the space-time curvature, which is reminiscent of the correction to the Lamb shift induced by the presence of cavities. However, when the Unruh and Hartle-…
The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string.
We study whether the Hawking-Page phase transition may occur in topological de Sitter spaces (TdS) and Schwarzschild-de Sitter black hole (SdS). We show that at the critical temperature $T=T_1$, TdS with hyperbolic cosmological horizon can make the Hawking-Page transition from the zero mass de Sitter space to TdS. It is also shown that there is no Hawking-Page transition for TdS with Ricci-flat and spherical horizons, when the zero mass de Sitter space is taken as the thermal background. Also we find that the SdS undergoes a different phase transition at T=0 which the Nariai black hole is form…
Using a graphical analysis, we show that for the horizon radius $r_h\gtrsim 4.8\sqrtθ$, the standard semiclassical Bekenstein-Hawking area law for noncommutative Schwarzschild black hole exactly holds for all orders of $θ$. We also give the corrections to the area law to get the exact nature of the Bekenstein-Hawking entropy when $r_h<4.8\sqrtθ$ till the extremal point $r_h=3.0\sqrtθ$.
Recently a method has been developed for relating four dimensional Schwarzschild black holes in M-theory to near-extremal black holes in string theory with four charges, using suitably defined ``boosts'' and T-dualities. We show that this method can be extended to obtain the emission rate of low energy massless scalars for the four dimensional Schwarzschild hole from the microscopic picture of radiation from the near extremal hole.
Quantum gravity has exciting peculiarities on the Planck scale.The effect of generalized uncertainty principle (GUP) to the entangled scalar/fermion particles' tunneling from a Schwarzschild black hole immersed in an electromagnetic Universe is investigated by the help of semi-classical tunneling method. The quantum corrected Hawking temperature of this black hole with an external parameter "a" modifies the Hawking temperature for the entangled particles.
We examine the Hawking effect by studying the asymptotic entanglement of two mutually independent two-level atoms placed at a fixed radial distance outside a Schwarzschild black hole in the framework of open quantum systems. We treat the two-atom system as an open quantum system in a bath of fluctuating quantized massless scalar fields in vacuum and calculate the concurrence, a measurement of entanglement, of the equilibrium state of the system at large times, for the Unruh, Hartle-Hawking and Boulware vacua respectively. We find, for all three vacuum cases, that the atoms turn out to be entan…
We develop a Hamiltonian formalism which can be used to discuss the physics of a massless scalar field in a gravitational background of a Schwarzschild black hole. Using this formalism we show that the time evolution of the system is unitary and yet all known results such as the existence of Hawking radiation can be readily understood. We then point out that the Hamiltonian formalism leads to interesting observations about black hole entropy and the information paradox.
We propose a model for the geometry of a dynamical spherical shell in which the metric is asymptotically Schwarzschild, but deviates from Ricci-flatness in a finite neighbourhood of the shell. Hence, the geometry corresponds to a `hairy' black hole, with the hair originating on the shell. The metric is regular for an infalling shell, but it bifurcates, leading to two disconnected Schwarzschild-like spacetime geometries. The shell is interpreted as either collapsing matter or as Hawking radiation, depending on whether or not the shell is infalling or outgoing. In this model, the Hawking radiati…
The spontaneous excitation of a two-level atom held static outside a four dimensional Schwarzschild black hole and in interaction with a massless scalar field in the Boulware, Unruh and Hartle-Hawking vacuum is investigated and the contributions of the vacuum fluctuations and radiation reaction to the rate of change of the mean atomic energy are calculated separately. We find that for the Boulware vacuum, the spontaneous excitation does not occur and the ground state atoms are stable, while the spontaneous emission rate for excited atoms in the Boulware vacuum, which is well-behaved at the eve…
This paper considers the effect of Hawking radiation on an eternal black hole - that is. a maximally extended Schwarzschild solution. Symmetry considerations that hold independent of the details of the emission mechanism show there is an inconsistency in the claim that such a blackhole evaporates away in a finite time. In essence: because the external domain is static, there is an infinite time available for the process to take place, so whenever the evaporation process is claimed to come to completion, it should have happened earlier. The problem is identified to lie in the claim that the loc…
We study the Hawking effect in terms of the geometric phase acquired by a two-level atom as a result of coupling to vacuum fluctuations outside a Schwarzschild black hole in a gedanken experiment. We treat the atom in interaction with a bath of fluctuating quantized massless scalar fields as an open quantum system, whose dynamics is governed by a master equation obtained by tracing over the field degrees of freedom. The nonunitary effects of this system are examined by analyzing the geometric phase for the Boulware, Unruh and Hartle-Hawking vacua respectively. We find, for all the three cases,…
We investigate whether the equality found for the response of static scalar sources interacting (i) with {\em Hawking radiation in Schwarzschild spacetime} and (ii) with the Fulling-Davies-Unruh thermal bath in the Rindler wedge is maintained in the case of electric charges. We find a finite result in the Schwarzschild case, which is computed exactly, in contrast with the divergent result associated with the infrared catastrophe in the Rindler case, i.e. in the case of uniformly accelerated charges in Minkowski spacetime. Thus, the equality found for scalar sources does not hold for electric c…
In this paper we study through tunneling formalism, the effect of noncommutativity to Hawking radiation and the entropy of the noncommutative Schwarzschild black hole. In our model we have considered the noncommutativity implemented via the Lorentzian distribution. We obtain non-commutative corrections to the Hawking temperature using the Hamilton-Jacobi method and the Wentzel-Kramers-Brillouin (WKB) approximation. In addition, we found corrections of the logarithmic and other types due to noncommutativity and quantum corrections from the generalized uncertainty principle (GUP) for the entropy…
We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semi-classical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation. This also shows that there is …
Numerical results for the semi-classical stress-energy tensor outside the horizon of a Schwarzschild black hole obtained in the 1980's and 1990's are re- examined in order to better understand the origin of Hawking radiation and the implications for the black hole information paradox. Polynomial fits to the numerical results for the 4D transverse stress are obtained for conformally-coupled spin 0 and spin 1 fields in the Hartle-Hawking and Unruh states and. Analysis of the spin 0 Unruh state results clearly shows that the origin of the Hawking radiation is not pair creation or tunneling very c…
Analytic approximations to and numerical results for the semi-classical stress-energy tensor outside the horizon of a Schwarzschild black hole obtained in the 1980's and 1990's are re-examined in order to better understand the origin of Hawking radiation and the implications for the black hole information paradox. Polynomial fits to the numerical results for the tangential stress in 4D are obtained for conformally coupled spin 0 and spin 1 fields in the Hartle-Hawking and Unruh quantum states. What these results show is that the origin of the Hawking radiation is not pair creation or tunneling…
An algorithm for numerical computation of the barrier transparency for the potentials surrounding Schwarzschild black holes is described for massive spinor particles. It is then applied to calculate the total (including electronic neutrino and the contributions of twisted field configurations connected with Dirac monopoles) luminosity for the electron-positron Hawking radiation from a Schwarzschild black hole with mass $M=10^{15}$ g. It is found that the contribution due to monopoles can be of order 12% of the total electron-positron luminosity.
In a Schwarzschild-de Sitter space, we consider an equipotential surface which consists of two holographic screens. Adapting the Bousso-Hawking's reference point of vanishing force, we divide the space into two regions, which are from the reference point to each holographic screen. These two regions can be treated as independent thermodynamical systems, because the Bousso-Hawking reference point with zero temperature behaves like a thermally insulating wall. The entropy obtained in this way agrees with the conventional results; i) when the holographic screens lie at the black hole and cosmolog…
We compute the albedo (or reflectivity) of electromagnetic waves off the electron-positron Hawking plasma that surrounds the horizon of a Quantum Black Hole. We adopt the "modified firewall conjecture" for fuzzballs [arXiv:hep-th/0502050,arXiv:1711.01617], where we consider significant electromagnetic interaction around the horizon. While prior work has treated this problem as an electron-photon scattering process, we find that the incoming quanta interact collectively with the fermionic excitations of the Hawking plasma at low energies. We derive this via two different methods: one using rela…
The semiclassical effects of antievaporating black holes can be discussed in the framework of $f(R)$ gravity. In particular, the Bousso-Hawking-Nojiri-Odinstov antievaporation instability of degenerate Schwarzschild-de Sitter black holes (the so called Nariai space-time) leads to a dynamical increasing of black hole horizon in $f(R)$ gravity. This phenomenon causes the following transition: emitting marginally trapped surfaces become space-like surfaces before the effective Bekenstein-Hawking emission time. As a consequence, Bousso-Hawking thermal radiation cannot be emitted in an antievaporat…
The black hole information paradox is a very poorly understood problem. It is often believed that Hawking's argument is not precisely formulated, and a more careful accounting of naturally occurring quantum corrections will allow the radiation process to become unitary. We show that such is not the case, by proving that small corrections to the leading order Hawking computation cannot remove the entanglement between the radiation and the hole. We formulate Hawking's argument as a `theorem': assuming `traditional' physics at the horizon and usual assumptions of locality we will be forced into m…
We apply the generalization of the Parikh-Wilczek method to the tunneling of massive particles from noncommutative inspired Schwarzschild black holes. By deriving the equation of radial motion of the tunneling particle directly, we calculate the emission rate which is shown to be dependent on the noncommutative parameter besides the energy and mass of the tunneling particle. After equating the emission rate to the Boltzmann factor, we obtain the modified Hawking temperature which relates to the noncommutativity and recovers the standard Hawking temperature in the commutative limit. We also dis…
The total number of Hawking quanta emitted during the evaporation of a Schwarzschild black hole is proportional to the square of the initial mass or, equivalently, to the Bekenstein entropy. This simple, but little appreciated, fact is interpreted in terms of the recent discovery of black hole soft hair.
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penrose's theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metr…
