We argue that a process where a fuzzy space splits in two others can be used to explain the origin of the black hole entropy, and why a "generalized second law of thermodynamics" appears to hold in the presence of black holes. We reach the Bekenstein-Hawking formula from the count of the microstates of a black hole modeled by a fuzzy space. In this approach, a discrete area spectrum for the black hole, which becomes increasingly spaced as the black hole approaches the Planck scale, is obtained. We show that, as a consequence of this, the black hole radiation becomes less and less entropic as the black hole evaporates, in a way that some information about its initial state could be recovered.
We extend the recently proposed Black Hole Entropy Distance Conjecture to the case of charged black holes in de Sitter space. By systematically studying distances in the space of black hole geometries with multiple horizons, we find that the distance is generically related to the logarithm of the entropy. From the infinite distance conjecture this predicts the appearance of a massless tower of modes in the limit of infinite entropy. Further, we study the evaporation of these black holes and relate it to the geometric distance. We find that the corresponding distance to the final stage of evapo…
We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS$_2$), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS$_2$ with a vanishing dilaton, i.e. a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation, as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS$_2$ gravity. We…
We analyze the Generalized Uncertainty Principle (GUP) impact onto the nonextensive black hole thermodynamics by using Rényi entropy. We show that when introducing GUP effects, both Rényi entropy and temperature associated to black holes have finite values at the end of the evaporation process. We also study the sparsity of the radiation, associated with Rényi temperature, and compare it with the sparsity of Hawking radiation. Finally, we investigate GUP modifications to the sparsity of the radiation when GUP effects are introduced into Rényi temperature.
Based on the discretized horizon picture, we introduce a macroscopic effective model of the horizon area quanta that encapsulates the features necessary for black holes to evaporate consistently. The price to pay is the introduction of a "hidden sector" that represents our lack of knowledge about the final destination of the black hole entropy. We focus on the peculiar form of the interaction between this hidden sector and the black hole enforced by the self-consistency. Despite the expressive power of the model, we arrive at several qualitative statements. Furthermore, we identify these state…
Non-thermal correction to the emission probability of particles from black holes can be obtained if the backreaction or self-gravitational effects of the emitted particles on the black hole spacetime are taken into consideration. These non-thermally emitted particles conserve the entropy of the black hole, i.e, the entropy of the system of radiated particles after complete evaporation of the black hole matches the initial entropy of the black hole. Using the non-thermal emission probability, we have determined the probability for a black hole of mass $M$ to be completely evaporated by a given …
In a recent article, we have shown how quantum fluctuations of the background geometry modify Hawking's density matrix for black hole (BH) radiation. Hawking's diagonal matrix picks up small off-diagonal elements whose influence becomes larger with the number of emitted particles. We have calculated the "time-of-first-bit", when the first bit of information comes out of the BH, and the "transparency time", when the rate of information release becomes order unity. We have found that the transparency time is equal to the "Page time", when the BH has lost half of its initial entropy to the radiat…
We study the dynamics of vacuum entanglement in the process of gravitational collapse and subsequent black hole evaporation. In the first part of the paper, we introduce a covariant regularization of entanglement entropy tailored to curved spacetimes; this regularization allows us to propose precise definitions for the concepts of black hole "exterior entropy" and "radiation entropy." For a Vaidya model of collapse we find results consistent with the standard thermodynamic properties of Hawking radiation. In the second part of the paper, we compute the vacuum entanglement entropy of various sp…
We argue that a unitary description of the formation and evaporation of a black hole implies that the Bekenstein-Hawking entropy is the "entropy of a vacuum": the logarithm of the number of possible independent ways in which quantum field theory on a fixed classical spacetime background can emerge in a full quantum theory of gravity. In many cases, the covariant entropy counts this entropy--the degeneracy of emergent quantum field theories in full quantum gravity--with the entropy of particle excitations in each quantum field theory giving only a tiny perturbation. In the Rindler description o…
We investigate the evaporation process of a Kerr-de Sitter black hole with the Unruh-Hawking-like vacuum state, which is a realistic vacuum state modelling the evaporation process of a black hole originating from gravitational collapse. We also compute the greybody factors for gravitons, photons, and conformal-coupling massless scalar particles by using the analytic solutions of the Teukolsky equation in the Kerr-de Sitter background. It turns out that the cosmological constant quenches the amplification factor and it approaches to zero towards the critical point where the Nariai and extremal …
We propose a model with multiple qubits that reproduces the thermal properties of 4-dimensional (4-dim) Schwarzschild black holes (BHs) by simultaneously taking account of the emission of Hawking particles and the zero-energy soft hair evaporation at horizon. The results verify that the entanglement entropy between a qubit and other subsystems, including emitted radiation, is much larger than the BH entropy analogue of the qubit, as opposed to the Page curve prediction. Our result suggests that early Hawking radiation is entangled with soft hair, and that late Hawking radiation can be highly e…
These lecture notes are an elementary and pedagogical introduction to the black hole evaporation, based on a lecture given by the author at the Ninth Modave Summer School in Mathematical Physics and are intended for PhD students. First, quantum field theory in curved spacetime is studied and tools needed for the remaining of the course are introduced. Then, quantum field theory in Rindler spacetime in 1+1 dimensions and in the spacetime of a spherically collapsing star are considered, leading to Unruh and Hawking effects, respectively. Finally, some consequences such as thermodynamics of black…
We consider the black hole information problem in an explicitly defined spacetime modelling black hole evaporation. Using this context we review basic aspects of the problem, with a particular effort to be unambiguous about subtle topics, for instance precisely what is meant by entropy in various circumstances. We then focus on questions of unitarity, and argue that commonly invoked semiclassical statements of long term, evaporation time, and Page time "unitarity" may all be violated even if physics is fundamentally unitary. This suggests that there is no horizon firewall. We discuss how the p…
On Radiative Fluxes and Coulombic Charges in the Balance Law for Black Hole Evaporation
The Pennsylvania State University（宾夕法尼亚州立大学）；Department of Physics, The Pennsylvania State University（物理学系，宾夕法尼亚州立大学）；Applied Research Lab, The Pennsylvania State University（应用研究实验室，宾夕法尼亚州立大学）
Corrected Hawking Temperature and Final State of Black Hole Evaporation Under GEVAG Framework
Center for the Cross-disciplinary Research of Space Science and Quantum-technologies (CROSS-Q), College of Physics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学物理学院空间科学与量子技术交叉研究中心）
In this review, we describe recent progress on the black hole information problem that involves a new understanding of how to calculate the entropy of Hawking radiation. We show how the method for computing gravitational fine-grained entropy, developed over the past 15 years, can be extended to capture the entropy of Hawking radiation. This technique reveals large corrections needed for the entropy to be consistent with unitary black hole evaporation.
A new simple qubit model of black hole evaporation is proposed. The model operates on four qubits and is defined in terms of quantum gates and a quantum circuit. The chief features of the model include explicit unitarity and (most notably) causality which is understood as the impossibility of the transfer of information from the interior of the black hole through its horizon. The corresponding von Neumann entanglement entropy yields a crude (four-qubit) approximation of the Page curve.
A reformulation of the calculation of the semi-classical energy-momentum tensor on a Schwarzschild background, the Bousso covariant entropy bound, and the ER=EPR conjecture of Maldacena and Susskind taken together suggest a scenario for the evaporation of a large spherically symmetric black hole formed in gravitational collapse in which 1) the classical r = 0 singularity is replaced by an initially small non-singular core inside an inner apparent horizon, 2) the radius of the core grows with time due to the increasing entanglement between Hawking radiation quanta outside the black hole and the…
The existence of a minimal length, predicted by different theories of quantum gravity, can be phenomenologically described in terms of a generalized uncertainty principle. We consider the impact of this quantum gravity motivated effect onto the information budget of a black hole and the sparsity of Hawking radiation during the black hole evaporation process. We show that the information is not transmitted at the same rate during the final stages of the evaporation and that the Hawking radiation is not sparse anymore when the black hole approaches the Planck mass.
We focus on the entropy relations of black holes in three, four and higher dimensions. These entropy relations include entropy product, "part" entropy product and entropy sum. We also discuss their differences and similarities, in order to make a further study on understanding the origin of black hole entropy at the microscopic level.
Since Hawking's 1974 discovery, we expect that a black hole formed by collapse will emit radiation and eventually disappear. Closely related to the information loss puzzle is the challenge to define an objective notion of physical entropy which increases throughout this process in a way consistent with unitarity. In recent years, this has been addressed with certain notions of coarse grained entropy. We have suggested instead that physical entropy should be identified with matter-gravity entanglement entropy and that this may offer an explanation of entropy increase both for the black hole col…
We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black body radiation, its loss of entropy is always 3/4 of the entropy of the emitted radiation. This proposition enables us to relate the entropy of an evaporating black hole to the entropy of its Hawking radiation. Explicitly, by calculating the entropy of the Hawking radiation emitted in the full period of evaporation of the black hole, we find the Bekenstein-Hawking entropy of the i…
An alternative derivation of generalized gravitational entropy associated to co-dimension 2 'entangling' hypersurfaces is given. The approach is similar to the Jacobson-Myers 'Hamiltonian' method and it does not require computations on manifolds with conical singularities. It is demonstrated that the entangling surfaces should be extrema of the entropy functional. When our approach is applied to Lovelock theories of gravity the generalized entropy formula coincides with results derived by other methods.
We consider rotating Primordial black holes (PBHs) and study the effect of accretion of radiation in the radiation dominated era. The central part of our analysis deals with the role of the angular momentum parameter on the evolution of PBHs. We find that both the accretion rate and evaporation rate decrease with increase in angular momentum parameter, but the rate of evaporation decreases more rapidly than the rate of accretion. This shows that the evaporation time of PBHs get prolonged with increase in angular momentum parameter. We also note that the lifetime of rotating PBHs increase with …
We study a thermofield double type entangled state on two disjoint universes $A$ and $B$, where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy on the universe $A$ through the island formula, and argue that it naturally follows the Pa…
We examine the Kerr/CFT correspondence in Einstein gravity extended with quadratic curvature invariants. We consider two explicit examples in four and five dimensions and compute the central charges of the asymptotic symmetry algebras of the near horizon geometries, using the improved version of the BBC formalism that encompasses the information of the Lagrangian. We find that the resulting Cardy entropy differs from the Wald entropy, caused by the Riemann-squared $R^{μνρσ}R_{μνρσ}$ term.
It was found by Hung, Myers and Smolkin that there is entropy discrepancy for the CFTs in 6-dimensional space-time, between the field theoretical and the holographic analysis. Recently, two different resolutions to this puzzle have been proposed. One of them suggests to utilize the anomaly-like entropy and the generalized Wald entropy to resolve the HMS puzzle, while the other one initiates to use the entanglement entropy which arises from total derivative terms in the Weyl anomaly to explain the HMS mismatch. We investigate these two proposals carefully in this note. By studying the CFTs dual…
In this work, we study the thermodynamic properties on a non--commutative background via gravitational gauge field potentials. This procedure is accomplished after contracting de Sitter (dS) group, $\mathrm{SO}(4,1)$, with the Poincarè group, $\mathrm{ISO}(3,1)$. Particularly, we focus on a static spherically symmetric black hole. In this manner, we calculate the modified Hawking temperature and the other deformed thermal state quantities, namely, entropy, heat capacity, Helmholtz free energy and pressure. Finally, we also investigate the black hole evaporation process in such a context.
We investigate the black hole thermodynamics in a "deformed" relativity framework where the energy-momentum dispersion law is Lorentz-violating and the Schwarzchild-like metric is momentum-dependent with a Planckian cut-off. We obtain net deviations of the basic thermodynamical quantities from the Hawking-Bekenstein predictions: actually, the black hole evaporation is expected to quit at a nonzero critical mass value (of the order of the Planck mass), leaving a zero temperature remnant, and avoiding a spacetime singularity. Quite surprisingly, the present semiclassical corrections to black hol…
For an evaporating black hole which is a radiation-black hole combined system, we express the entanglement entropy and the Page time in terms of the conformal time in the RST model. The entropy change of the black hole is nicely written in terms of Hawking flux. Integrating the first law of thermodynamics, we can obtain the decreasing black-hole entropy and the increasing radiation entropy, and the entanglement entropy for this system based on the Page argument. We also obtain analytically the critical temperature to release black-hole information, which corresponds to the Page time, and discu…
I discuss black hole evaporation in two different coordinate systems and argue that the results of the two are compatible once one takes the holographic principle into account. de Sitter space is then discussed along similar lines. Finally I make some remarks about smooth initial conditions in GR, which evolve to space-times with bifurcate horizons, and emphasize the care one must take in identifying spaces of solutions of General Relativity which belong to the same quantum theory of gravity. No really new material is presented, but the point of view I take on all 3 subjects is not widely appr…
The recently proposed gravitational entropy generalize the usual black hole entropy to Euclidean solutions without U(1) symmetry in the framework of Einstein gravity. The entropy of such smooth configuration is given by the area of minimal surface, therefore explaining the Ryu-Takayanagi formula of holographic entanglement entropy. In this note we investigate the generalized gravitational entropy for general Lovelock gravity in arbitrary dimensions. We use the replica trick and consider the Euclidean bulk spacetime with conical singularity localized at a codimension two surface. We obtain a co…
