Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Moduli and (un)attractor black hole thermodynamics

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS ₂ × S ² and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen’s entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza–Klein black hole.     

广义测不准关系与三维BTZ黑洞熵

通过应用在量子引力中,由广义测不准关系得出的新的态密度方程,研究三维BTZ背景下黑洞的熵.当取广义测不准关系中引入的,具有Planck量级与空间维数有关的常数λ为特定值时,得到BTZ黑洞Bekenstein-Hawking 熵和修正项.由于利用新的态密度方程,在计算中不存在用brick-wall模型计算黑洞熵时出现的发散项和小质量近似.所得结论,从量子统计力学角度给出了黑洞Bekenstein-Hawking 熵的修正值,使人们对黑洞熵的修正值有更深入的认识. 关键词： 广义测不准关系 量子统计 BTZ黑洞熵     

Does black-hole entropy make sense?

Bekenstein and Hawking saved the second law of thermodynamics near a black hole by assigning to the hole an entropyS _h proportional to the area of its event horizon. It is tempting to assume thatS _h possesses all the features commonly associated with the physical entropy. Kundt has shown, however, thatS _h violates several reasonable physical expectations. We review his criticism, augmenting it as follows: (a)S _h is a badly behaved state function requiring knowledge of the hole's future history; and (b) close analogs of event horizons in other space-times do not possess an “entropy.” We also discuss these questions: (c) IsS _h suitable for all regions of a black-hole space-time? And (b) shouldS _h be attributed to the exterior of a white hole? One can retainS _h for the interior (respectively, exterior) of a black (respectively, white) hole, but we reject this as contrary to the information-theoretic derivation of horizon entropy given by Bekenstein. The total entropy defined by Kundt (all ordinary entropy on space-section cutting through the hole, no horizon term) and that of Bekenstein-Hawking (ordinary entropy outside horizon plus horizon term) appear to be complementary concepts with separate domains of validity. In the most natural choice, an observer inside a black hole will use Kundt's entropy, and one remaining outside that of Bekenstein-Hawking.     

Quantum geometry and entanglement entropy of a black hole

We have studied here black hole entropy in the framework of quantum geometry. It is pointed out that the black hole radiation consistent with Hawking spectrum can be realized as an effect of quantum geometry using a dynamical formalism for diffeomorphism invariance which envisages a discretized unit of time in the Planck scale. This formalism suggests that torsion acts within a quantized area unit (area bit) associated with a loop and this eventually forbids the Hamiltonian constraint to be satisfied for a finite loop size. We assign a spin with torsion in each area bit and entanglement entropy of a black hole is computed in terms of the entanglement entropy of this spin system. We have derived the Bekenstein-Hawking entropy along with a logarithmic correction term with a specific coefficient. Also we have shown that the Bekenstein-Hawking entropy can be formulated in terms of the Noether charge associated with a diffeomorphism invariant Lagrangian.     

Extremal black holes in the Hořava–Lifshitz gravity

We study the near-horizon geometry of extremal black holes in the z=3 Ho?ava–Lifshitz gravity with a flow parameter λ. For λ>1/2, near-horizon geometry of extremal black holes are AdS ₂×S ² with different radii, depending on the (modified) Ho?ava–Lifshitz gravity. For 1/3≤λ≤1/2, the radius v ₂ of S ² is negative, which means that the near-horizon geometry is ill-defined and the corresponding Bekenstein–Hawking entropy is zero. We show explicitly that the entropy function approach does not work for obtaining the Bekenstein–Hawking entropy of extremal black holes.     

Extremal black hole entropy from conformal string sigma model

We present a string-theory derivation of the semiclassical entropy of extremal dyonic black holes in the approach based on the conformal sigma model (NS-NS embedding of the classical solution). We demonstrate (resolving some puzzles existing in previous related discussions) that the degeneracy responsible for the entropy is due to string oscillations in four transverse dimensions ‘intrinsic’ to the black hole: four non-compact directions of the D = 5 black hole case and three non-compact and one compact (responsible for embedding of magnetic charges) dimensions in the D = 4 black hole case. Oscillations in other compact internal dimensions give subleading contributions to statistical entropy in the limit when all charges are large. The dominant term in the statistical entropy is thus universal (i.e. is the same in type II and heterotic string theory) and agrees with the Bekenstein-Hawking expression.     

Holographic Entanglement Entropy of the BTZ Black Hole

We investigate quantum entanglement of gravitational configurations in 3D AdS gravity using the AdS/CFT correspondence. We derive explicit formulas for the holographic entanglement entropy (EE) of the BTZ black hole, conical singularities and regularized AdS₃. The leading term in the large temperature expansion of the holographic EE of the BTZ black hole reproduces exactly its Bekenstein-Hawking entropy S _BH , whereas the subleading term behaves as ln S _BH . We also show that the leading term of the holographic EE for the BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus. This result indicates that black hole EE is not a fundamental feature of the underlying theory of quantum gravity but emerges when the semiclassical notion of spacetime geometry is used to describe the black hole.     

Entropy of an extremal regular black hole

We introduce a magnetically charged extremal regular black hole in the coupled system of Einstein gravity and nonlinear electrodynamics. Its near horizon geometry is given by AdS₂×S²${\mathit{AdS}}_2\times S^2$. It turns out that the entropy function approach does not automatically lead to a correct entropy of the Bekenstein–Hawking entropy. This contrasts to the case of the extremal Reissner–Norström black hole in the Einstein–Maxwell theory. We conclude that the entropy function approach does not work for a magnetically charged extremal regular black hole without singularity, because of the nonlinearity of the entropy function.     

Rediscussion of charged dilaton-axion black hole entropy

We rediscuss the entropy of a charged dilaton-axion black hole for both the asymptotically flat and non-flat cases by using the thin film brick-wall model. This improved method avoids some drawbacks in the original brick-wall method such as the small mass approximation, neglecting the logarithm term, and taking the term L ³ as the contribution of the vacuum surrounding the black hole. The entropy we obtain turns out to be proportional to the horizon area of the black hole, conforming to the Bekenstein-Hawking area-entropy formula for black holes.      

Black Hole Entropy with Modified Dispersion Relations

There is much interest in resolving the quantum corrections to Bekenstein-Hawking entropy with a large length scale limit. The leading correction term is given by the logarithm of black hole area with a model-dependent coefficient. Recently the research for quantum gravity implies the emergence of a modification of theenergy-momentum dispersion relation (MDR), which plays an importantrole in the modified black hole thermodynamics. In this paper, we investigate the quantum corrections to Bekenstein-Hawking entropy in four-dimensional Schwarzschild black hole and Reissner-Nordström black hole respectively based on MDR.     

Statistical entropies of extremal Kaluza-Klein AdS black holes in arbitrary dimensions

We have investigated the microscopic interpretations of the entropies for the four-dimensional extremal Kaluza-Klein AdS black hole and its higher-dimensional generalizations by using the Kerr/CFT correspondence. These newly-found Kaluza-Klein AdS black holes are charged rotating asymptotically AdS black hole solutions of gauged supergravity in four and higher dimensions. With suitable boundary conditions on the perturbations of the near-horizon geometry, it is shown that the asymptotic symmetry generators form a two-dimensional Virasoro algebra with a central term. By utilizing the central charge and the temperature of the dual conformal field theory, Cardy formula reproduces the expected Bekenstein-Hawking entropy. Directly working on the ordinary metrics of the extremal Kaluza-klein AdS black holes without taking the near-horizon limit, we also re-derive their microscopic entropies.     

Quantum aspects of black hole entropy

This survey intends to cover recent approaches to black hole entropy which attempt to go beyond the standard semiclassical perspective. Quantum corrections to the semiclassical Bekenstein-Hawking area law for black hole entropy, obtained within the quantum geometry framework, are treated in some detail. Their ramification for the holographic entropy bound for bounded stationary spacetimes is discussed. Four dimensional supersymmetric extremal black holes in string-based N=2 supergravity are also discussed, albeit more briefly.     

Hamiltonian evolution and quantization for extremal black holes

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes.     

砖墙模型不能给出黑洞熵

砖墙模型被广泛用于静态或稳态黑洞熵的计算,但为了避免发散,砖墙模型需要引入一紫外截断因子. 截断因子的引入至今没有给以合理的解释. 有工作表明,用砖墙模型或薄膜模型计算黑洞熵时,若采用广义不确定关系则可以去掉截断因子. 证明了将广义不确定关系用于砖墙模型计算Schwarzschild黑洞熵时,由于砖墙模型给出熵的第一项既是Bekenstein-Hawking项又含有截断因子,因此在去掉截断因子的同时也丢掉了Bekenstein-Hawking项,将得不到黑洞熵. 关键词： 黑洞熵 砖墙模型 截断因子 广义不确定关系     

Corrected Entropy of BTZ Black Holes

In this paper, corrected entropy of a class of BTZ black holes, include charge and rotation, studied. We obtain corrected Bekenstein-Hawking entropy and find that effect of charge viewed at order A ^?2, and effect of rotation viewed at order A ^?6, therefore Q and J don’t have contribution in corrected entropy lower than the second order. We also write the first law of black hole thermodynamics for the case of charged rotating BTZ black hole.     

Generalized Uncertainty Principle and Correction Value to the Kerr Black Hole Entropy

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coefficient of the logarithmic term of the black hole entropy correction term. In this paper, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty principle and obtain the correction term caused by the generalized uncertainty principle. Because in our calculation we think that the Bekenstein-Hawking area theorem is still valid after considering the generalized uncertainty principle, we derive that the coefficient of the logarithmic term of the black hole entropy correction term is positive. This result is different from the known result at present. Our method is valid not only for single horizon spacetime but also for spin axial symmetric spacetimes with double horizons. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the entropy correction of the complicated spacetime.     

Generalized Uncertainty Principle and Thermodynamic Quantities of SAdS5 Black Hole

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking black hole entropy. The different correction leading terms are obtained by the different methods. In this paper, we calculate the correction to SAdS₅ black hole thermodynamic quantity due to the generalized uncertainty principle. Furthermore we derive that the black hole entropy obeys Bekenstein-Hawking area theorem. The entropy has infinite correction terms. And every term is finite and calculable. The corrected Cardy-Verlinde formula is derived. In our calculation, Bekenstein-Hawking area theorem still holds after considering the generalized uncertainty principle. We have not introduced any hypothesis. The calculation is simple. Physics meaning is clear. We note that our results are quite general. It is not only valid for four-dimensional spacetime but also for higher-dimensional SAdS spacetime.     

Black hole entropy function,attractors and precision counting of microstates

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.     

Curvature-corrected dilatonic black holes and black hole—string transition

We investigate extremal charged black hole solutions in the four-dimensional string frame Gauss-Bonnet gravity with the Maxwell field and the dilaton. Without curvature corrections, the extremal electrically charged dilatonic black holes have singular horizon and zero Bekenstein entropy. When the Gauss-Bonnet term is switched on, the horizon radius expands to a finite value provided curvature corrections are strong enough. Below a certain threshold value of the Gauss-Bonnet coupling the extremal black hole solutions cease to exist. Since decreasing Gauss-Bonnet coupling corresponds to decreasing string coupling g _s , the situation can tentatively be interpreted as classical indication on the black hole—string transition. Previously the extremal dilaton black holes were studied in the Einstein-frame version of the Gauss-Bonnet gravity. Here we work in the string frame version of the theory with the S-duality symmetric dilaton function as required by the heterotic string theory. The article is published in the original.