We study the relative entropy of highly excited quantum states. First, we sample states from the Wishart ensemble and develop a large-N diagrammatic technique for the relative entropy. The solution is exactly expressed in terms of elementary functions. We compare the analytic results to small-N numerics, finding precise agreement. Furthermore, the random matrix theory results accurately match the behavior of chaotic many-body eigenstates, a manifestation of eigenstate thermalization. We apply this formalism to the AdS/CFT correspondence where the relative entropy measures the distinguishability between different black hole microstates. We find that black hole microstates are distinguishable even when the observer has arbitrarily small access to the quantum state, though the distinguishability is nonperturbatively small in Newton's constant. Finally, we interpret these results in the context of the subsystem Eigenstate Thermalization Hypothesis (sETH), concluding that holographic systems obey sETH up to subsystems half the size of the total system.
Using the exactness criteria of entropy from the first law of black hole thermodynamics, we study quantum corrections for axially symmetric black holes.
We study the corrections to the entropy of Einstein-Maxwell dilaton-axion black holes beyond semiclassical approximations. We consider the entropy of the black hole as a state variable and derive these corrections using the exactness criteria of the first law of thermodynamics. We note that from this general frame-work the entropy corrections for "simpler" black holes like Schwarzschild, Reissner-Nordström and anti-de Sitter-Schwarzschild black holes follow easily. This procedure gives us the modified area law as well.
It has been shown that the entropy function formalism is an efficient way to calculate the entropy of black holes in string theory. We check this formalism for the extremal charged dilaton black hole. We find the general four-derivative correction on the black hole entropy from the value of the entropy function at its extremum point.
We argue that the entanglement entropy offers us a useful coarse-grained entropy in time-dependent AdS/CFT. We show that the total von-Neumann entropy remains vanishing even when a black hole is created in a gravity dual, being consistent with the fact that its corresponding CFT is described by a time-dependent pure state. We analytically calculate the time evolution of entanglement entropy for a free Dirac fermion on a circle following a quantum quench. This is interpreted as a toy holographic dual of black hole creations and annihilations. It is manifestly free from the black hole informatio…
Based on a mathematical lemma related to the Vandermonde determinant and two theorems derived from the first law of black hole thermodynamics, we investigate the angular momentum independence of the entropy sum as well as the entropy product of general rotating Kaluza-Klein black holes in higher dimensions. We show that for both non-charged rotating Kaluza-Klein black holes and non-charged rotating Kaluza-Klein-AdS black holes, the angular momentum of the black holes will not be present in entropy sum relation in dimensions $d\geq4$, while the independence of angular momentum of the entropy pr…
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…
We have proposed the entropy formula of the black hole which is constructed by the intersecting D1-brane and D5-brane with no momentum, whose compactification radii are constrained by the surface gravities in ten-dimensions. We interpret the entropy of the black hole as the statistical entropy of the effective string living on the D5-brane. We further study the behavior of the absorption cross-section of the black hole using our entropy formula.
The study of black hole physics revealed a fundamental connection between thermodynamics, quantum mechanics, and gravity. Today, it is known that black holes are thermodynamical objects with well-defined temperature and entropy. Although black hole radiance gives us the mechanism from which we can associate a well-defined temperature to the black hole, the origin of its entropy remains a mystery. Here we investigate how the quantum fluctuations from the fields that render the black hole its temperature contribute to its entropy. By using the exact renormalization group equation for a self-inte…
Simple considerations about the fractal characteristic of the quantum-mechanical path give us the opportunity to derive the quantum black hole entropy in connection with the concept of fractal statistics. We show the geometrical origin of the numerical factor of four of the quantum black hole entropy expression and the statistics weight appears as a counting of the quanta of geometry.
We review some recently established connections between the mathematics of black hole entropy in string theory and that of multipartite entanglement in quantum information theory. In the case of N=2 black holes and the entanglement of three qubits, the quartic [SL(2)]^3 invariant, Cayley's hyperdeterminant, provides both the black hole entropy and the measure of tripartite entanglement. In the case of N=8 black holes and the entanglement of seven qubits, the quartic E_7 invariant of Cartan provides both the black hole entropy and the measure of a particular tripartite entanglement encoded in t…
We find the entropy of Kehagias-Sfetsos black hole in the deformed Hořava-Lifshitz gravity by using the first law of thermodynamics. When applying generalized uncertainty principle (GUP) to Schwarzschild black hole, the entropy $S=A/4+(π/ω)\ln(A/4)$ may be interpreted as the GUP-inspired black hole entropy. Hence, it implies that the duality in the entropy between the Kehagias-Sfetsos black hole and GUP-inspired Schwarzschild black hole is present.
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.
Using brick wall method the entropy of charged dilaton-axion black hole is determined for both asymptotically flat and non-flat cases. The entropy turns out to be proportional to the horizon area of the black hole confirming the Beckenstien, Hawking area-entropy formula for black holes. The leading order logarithmic corrections to the entropy are also derived for such black holes.
We show that in IIB string theory and for D1D5p black holes in ten dimensions the method of entropy function works. Despite the more complicated Wald formula for the entropy of D1D5p black holes in ten dimensions, their entropy is given by entropy function at its extremum point. We use this method for computing the entropy of the system both at the level of supergravity and for its higher order alpha'^3R^4 corrections.
The string-black hole correspondence is considered in the context of the correspondence principle proposed recently by Horowitz and Polchinski. We demonstrate that the entropy of string states and the entropy of a Schwarzschild black hole can be matched including the subleading terms which depend on mass logarithmically. We argue the necessity to include the string interaction (with coupling $g$) in the consideration and propose the $g^2$-dependent modification of the string entropy. The matching of it with the entropy of Schwarzschild black hole is analyzed.
We study the possibility that black hole entropy be identified as entropy of entanglement across the horizon of the vacuum of a quantum field in the presence of the black hole. We argue that a recent proposal for computing entanglement entropy using AdS/CFT holography implies that black hole entropy can be exactly equated with entanglement entropy. The implementation of entanglement entropy in this context solves all the problems (such as cutoff dependence and the species problem) typically associated with this identification.
We show that the entropy of asymptotically flat, nonextremal black holes can be computed at infinity. We provide a prescription for transforming these black holes to $AdS_2$ black holes with the same entropy by dimensional reduction to 2D and a Weyl transformation. We apply our prescription to Schwarzschild, 4D Reissner--Nordstrom and generic nonextremal black holes. In the transformed coordinates, the asymptotic regions contain a global $AdS_2$ whose entropy can be computed either as the holographic entanglement entropy or as the entanglement entropy of a pair of Rindler $AdS_2$ spaces in the…
We point out that the entropy of (near) extremal black holes can be interpreted as the entanglement entropy of dual conformal quantum mechanics via AdS2/CFT1. As an explicit example, we study near extremal BTZ black holes and derive this claim from AdS3/CFT2. We also analytically compute the entanglement entropy in the two dimensional CFT of a free Dirac fermion compactified on a circle at finite temperature. From this result, we clarify the relation between the thermal entropy and entanglement entropy, which is essential for the entanglement interpretation of black hole entropy.
We show that Schwarzschild black hole metrics are, asymptotically, Weyl equivalent to $AdS_2 \times S^{D-2}$ where the spherical factor is the horizon. The holographic entanglement entropy of $AdS_2$ exactly reproduces the Schwarzschild black hole entropy which implies that black hole degrees of freedom live at asymptotic infinity. In generalized theories of gravity, the same procedure reproduces Wald entropy.
We compute logarithmic corrections to the entropy of a magnetically charged extremal black hole in AdS4 x S7 using the quantum entropy function and discuss the possibility of matching against recently derived microscopic expressions.
The entropy of charged black hole is calculated by using the partition function evaluated at radial geodesics confined under horizons. We establish two quantum phase states inside the black hole and a transition between them.
In this work, we examine the entropy emission property of black holes. When the greybody factor is considered, it is found that Schwarzschild black hole is a one-dimensional entropy emitter, which is independent of the spacetime dimension and the spin of the emitted quanta. However, when generalized to other black holes with two or more parameters, the result shows that the one-dimensional entropy emission property will be violated. Thus our result implies that not all black holes behave as one-dimensional entropy emitters.
The entropy of a Bañados, Teitelboim, and Zanelli black hole in topologically massive gravity had been given with the form inconsistent with the Bekenstein-Hawking entropy. In the paper, we provide a consistent statistical interpretation for the entropy, and confirm it to be thermodynamic entropy by preserving the first and second laws. As a novel extension, the logarithmic correction to the black hole entropy is also discussed.
Geometric entropy and time-like entanglement entropy on a rotating BTZ black hole
The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；University of South China（南华大学）；Ningbo University（宁波大学）
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.
We explore the method of entanglement entropy applied to 2d black holes. We introduce a solvable model of a real scalar field with finite volume and lattice spacing in terms of $N$ coupled mechanical oscillators and compute its entanglement entropy in many cases. The large $N$ limit of this scheme, with finite lattice spacing, should reproduce the Bekenstein-Hawking formula for black hole entropy.
We consider the entropy of a black hole which has zero area horizon. The microstates appear as monopole solutions of the effective theory on the corresponding brane configurations.The resulting entropy formula coincides with the one expected from stringy calculation and agrees with U-duality invariance of entropy.
We make some observations regarding string/black hole correspondence with a view to understanding the nature of the quantum degrees of freedom of a black hole in string theory. In particular, we compare entropy change in analogous string and black hole processes in order to support the interpretation of the area law entropy as arising from stringy constituents.
We study entropy corrections due to thermal fluctuations for asymptotically AdS black holes in the grand canonical ensemble. To leading order, these can be expressed in terms of the black hole response coefficients via fluctuation moments. We also analyze entropy corrections due to mass and charge fluctuations of R-charged black holes, and our results indicate an universality in the logarithmic corrections to charged AdS black hole entropy in various dimensions.
Suggested correspondence between a black hole and a highly excited elementary string is explored. Black hole entropy is calculated by computing the density of states for an open excited string. We identify the square root of oscillator number of the excited string with Rindler energy of black hole to obtain an entropy formula which, not only agrees at the leading order with the Bekenstein-Hawking entropy, but also reproduces the logarithmic correction obtained for black hole entropy in the quantum geometry framework. This provides an additional supporting evidence for correspondence between bl…
The area formula for entropy is extended to the case of a dilatonic black hole. The entropy of a scalar field in the background of such a black hole is calculated semiclassically. The area and cutoff dependences are normal {\it except in the extremal case}, where the area is zero but the entropy nonzero.
I review a new (and still tentative) approach to black hole thermodynamics that seeks to explain black hole entropy in terms of microscopic quantum gravitational boundary states induced on the black hole horizon.
We generalize the entropy function formalism to five-dimensional and four-dimensional non-extremal black holes in string theory. In the near horizon limit, these black holes have BTZ metric as part of the spacetime geometry. It is shown that the entropy function formalism also works very well for these non-extremal black holes and it can reproduce the Bekenstein-Hawking entropy of these black holes in ten dimensions and lower dimensions.
Parikh-Wilczek tunnelling framework, which treats Hawking radiation as a tunnelling process, is investigated again. As the first order correction, the log-corrected entropy-area relation naturally emerges in the tunnelling picture if we consider the emission of a spherical shell. The second order correction of the emission rate for the Schwarzschild black hole is calculated too. In this level, the result is still in agreement with the unitary theory, however, the entropy of the black hole will contain three parts: the usual Bekenstein-Hawking entropy, the logarithmic term and the inverse area …
We study the quantum scalar fields in background of BTZ black hole spacetime. We calculate the entanglement entropy using the discretized model, which resembles a system of coupled harmonic oscillators. The leading term of the entropy formula is standard Bakenstein-Hawking entropy and sub-leading corresponds to quantum corrections to black hole entropy. We calculate the co-efficent of sub-leading logarithmic corrections numerically.
It is shown that the entropy of fourdimensional black holes in string theory compactified on weighted Calabi-Yau hypersurfaces shows scaling behavior in a certain limit. This leads to non-monotonic functions on the moduli space.
We introduce a new microscopic quantity $\varepsilon $ that describes the contribution of a single black hole molecule to black hole entropy and give a key relation that this microscopic quantity is proportional to the macroscopic quantity -- entropy surface density $σ_{S}$, thus connecting a microscopic quantity to a macroscopic quantity of black holes. Such a connection provides a new probe for understanding the black hole microstructure from the macroscopic perspective. We also classify black holes in terms of entropy surface density. When its entropy surface density is larger (smaller) tha…
According to the widely accepted statistical interpretation of black hole entropy the mean separation between energy levels of black hole should be exponentially small. But this sharply disagrees with the value obtained from the quantization of black hole area. It is shown that the new statistical interpretation of black hole entropy proposed in my paper arXiv:0911.5635 gives the correct value.
The classical Bekenstein entropy of a black hole is argued to arise from configurations of strings with ends which are frozen on the horizon. Quantum corrections to this entropy are probably finite unlike the case in quantum field theory. Finally it is speculated that all black holes are single string states. The level density of strings is of the right order of magnitude to reproduce the Bekenstein entropy.
We show that the exact entropy and the temperature (including coeffecient) of non-extremal black p-brane are calculated by maximizing the entropy under several assumptions. We argue the relation of those assumptions and certain Dp-D(p-2) system in compactified space.
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…
Heating processes inside large black holes can produce tremendous amounts of entropy. Locality requires that this entropy adds on space-like surfaces, but the resulting entropy (10^10 times the Bekenstein-Hawking entropy in an example presented in the companion paper) exceeds the maximum entropy that can be accommodated by the black hole's degrees of freedom. Observer complementarity, which proposes a proliferation of non-local identifications inside the black hole, allows the entropy to be accommodated as long as individual observers inside the black hole see less than the Bekenstein-Hawking …
We clarify the relationship between black hole entropy and the number of degrees of freedom in the dual QFT with a cut-off. We show that simple gravity arguments predict the correct cut-off procedure.
We study higher derivative corrections to the statistical entropy function and the statistical entropy for five dimensional BMPV black holes by doing the asymptotic expansion of the partition function. This enables us to evaluate entropy for a large range of charges, even out of Cardy (Farey tail) limit.
String propagation on a cone with deficit angle $2π(1-{1\over N})$ is considered for the purpose of computing the entropy of a large mass black hole. The entropy computed using the recent results on condensation of twisted-sector tachyons in this theory is found to be in precise agreement with the Bekenstein-Hawking entropy.
The spectrum of the area operator for a Schwarzschild black hole in loop quantum gravity is fixed by the demand that the entropy of a black hole is maximum. This paper has been withdrawn by the author, due a crucial error in the derivation.
We examine in details the entropy function formalism for non-extremal D3, M2, and M5-branes that their throat approximation is given by Schwarzschild black hole in AdS_{p+2} \times S^{D-(p+2)}. We show that even though there is no attractor mechanism in the non-extremal black holes/branes, the entropy function formalism does work and the entropy is given by the entropy function at its saddle point.
For extremal black holes, the thermodynamic entropy is not proportional to the area. The general form allowed by thermodynamics is worked out for three classes of extremal black hole solutions of string theory and shown to be consistent with the entropy calculated from the density of elementary string states. On the other hand, the entanglement entropy does not in general agree with these results.
The island rule for the entanglement entropy is applied to an eternal Reissner-Nordström black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The ent…
The quantum corrections to the entropy of charged black holes are calculated. The Reissner-Nordstrem and dilaton black holes are considered. The appearance of logarithmically divergent terms not proportional to the horizon area is demonstrated. It is shown that the complete entropy which is sum of classical Bekenstein-Hawking entropy and the quantum correction is proportional to the area of quantum-corrected horizon.
In a recent letter, Cadoni and Mignemi proposed a formulation for the statistical computation of the 2D black holes entropy. We present a criticism about their formulation.
