Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holes

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem,it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime.By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

Topology in Entropy of Schwarzschild Black Hole

In the light of -mapping method and the relationship between the entropy and the Euler characteristic, the inner topological structure of the entropy of Schwarzschild black hole is studied. By introducing an entropy density, it is shown that the entropy of Schwarzschild black hole is determined by the singularities of the timelike Killing vector field of spacetime and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally. Taking account of the statistical meaning of entropy in physics, the entropy of Schwarzschild black hole is merely the sum of the Hopf indices, which will give the increasing law of entropy of black holes.     

Comments on the Statistical Origin of Black Hole Entropy

This paper shows that the black hole entropy can be interpreted as emerging as a result of missing information about the exact state of the matter from which the black hole was formed.     

Black Hole Radiation and Volume Statistical Entropy

The simplest possible equation for Hawking radiation and other black hole radiated power is derived in terms of black hole density, ρ . Black hole density also leads to the simplest possible model of a gas of elementary constituents confined inside a gravitational bottle of Schwarzchild radius at tremendous pressure, which yields identically the same functional dependence as the traditional black hole entropy S _bh∝ (kAc ³)/ℏ G. Variations of S _bh can be obtained which depend on the occupancy of phase space cells. A relation is derived between the constituent momenta and the black hole radius R _H, p = which is similar tothe Compton wavelength relation.     

Topological Structure of Entropy of 4-Dimensional Axisymmetric Black Holes

Using the relationship between the entropy and the Euler characteristic, an entropy density is introduced to describe the inner topological structure of the entropy of 4-dimensional axisymmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of spacetime, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, which are topological invariants. At last, Kerr–Newman black hole as an example of axisymmetric black holes is given. What’s more, the entropy and the latent heat of the topological phase transition of the black hole mentioned above are calculated and the latent heat just lies in the range of the energy of gamma ray bursts. This work is supported in part by the NSFs of China under Grant No. 10575068 and of Shanghai Municipal Committee of Science and Technology under Grant No. 04ZR14059 and Shanghai Leading Academic Discipline Project under Project Number: T0104.     

Entropy of Higher-Dimensional Charged de Sitter Black Holes and Phase Transition

From a new perspective, we discuss the thermodynamic entropy of (n+2)-dimensional Reissner-Nordströmde Sitter (RNdS) black hole and analyze the phase transition of the effective thermodynamic system. Considering the correlations between the black hole event horizon and the cosmological horizon, we conjecture that the total entropy of the RNdS black hole should contain an extra term besides the sum of the entropies of the two horizons. In the lukewarm case, the effective temperature of the RNdS black hole is the same as that of the black hole horizon and the cosmological horizon. Under this condition, we obtain the extra contribution to the total entropy. With the corrected entropy, we derive other effective thermodynamic quantities and analyze the phase transition of the RNdS black hole in analogy to the usual thermodynamic system.     

Entropy Correction for Kerr Black Hole

In this paper, we discuss leading-order corrections to the entropy of Kerr black hole due to thermal fluctuations in the finite cavity. Then temperature is constant, the solution of the black hole is obtained within a cavity, that is, the solution of the spacetime after considering the radiation of the black hole. Therefore, we derive that the location of the black hole horizon and specific heat are the functions of temperature and the radius of the cavity.Corrections to entropy also are related to the radius of the cavity. Through calculation, we obtain conditions of taking the value of the cavity‘s radius. We provide a new way for studying the corrections of complicated spacetimes.     

Entropy Correction for Kerr Black Hole

In this paper, we discuss leading-order corrections to the entropy of Kerr black hole due to thermal fluctuations in the finite cavity. Then temperature is constant, the solution of the black hole is obtained within a cavity, that is, the solution of the spacetime after considering the radiation of the black hole. Therefore, we derive that the location of the black hole horizon and specific heat are the functions of temperature and the radius of the cavity. Corrections to entropy also are related to the radius of the cavity. Through calculation, we obtain conditions of taking the value of the cavity＇s radius. We provide a new way for studying the corrections of complicated spacetimes.     

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 ＆ 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 the energy-momentum dispersion relation （MDR）, which plays an important role in the modified black hole thermodynamics. In this paper, we investigate the quantum corrections to Bekenstein-Hawking entropy in four-dimensional Sehwarzschild black hole and Reissner-Nordstrom black hole respectively based on MDR.     

Entropy Quantization of Schwarzschild Black Hole

The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.     

Canonical Entropy of Reissner-Nordstrom Black Hole

Recently, Hawking radiation of the black hole has been studied using the tunnel effect method. It is found the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How the departure from pure thermal spectrum affects the entropy? This is a very interesting problem. In this paper, we calculate the partition function by energy spectrum obtained by tunnel effect. Using the relation between the partition function and entropy, we derive the expression of entropy the general charged black hole. In our calculation, we not only consider the correction to the black hole entropy due to fluctuation of energy but also consider the effect of the change of the black hole charges on entropy. We discuss Reissner-Nordstrom black hole and obtain that Reissner-Nordstrom black hole cannot approach the extreme black hole by changing its charges.     

Entropy of a Uniformly Accelerating Black Hole

Kinnersley has discussed the space–time of an arbitrarily accelerating point mass. We select a simple case in which the black hole is uniformly accelerated and the mass does not vary with time. We adopt thin film brick-wall model to calculate the entropy of black hole. We find that both the temperature and the entropy density of black hole can be calculated at every point on the horizon. This result indicates that the conclusion that black hole entropy is proportional to its area can be applied to horizon not only globally, but also locally.     

The Quantum Entropy in Horowitz-Strominger Black Hole Background

Using 't Hooft's brick wall model and Newman-Penrose's spinor analysis, the expression of the quantum entropy is derived in the Horowitz-Strominger black hole background. The calculations show us that the Fermionic entropy is 7/2 times the Bosonic entropy.     

Entropy of Dilatonic Black Hole

By using the method of quantum statistics, we directly derive the partition function of bosonic and fermionic field in dilatonic black hole and obtain the integral expression of the black hole's entropy, which avoids the difficulty in solving the wave equationof various particles. Then via the improved brick-wall method, membrane model, we obtain that we can choose proper parameter in order to let the thickness of film tend to zero and have it approach the surface of its horizon. Consequently the entropy of the black hole is proportional to the area of its horizon. In our result, the stripped term and the divergent logarithmic term in the original brick-wall method no longer exist. In the whole process, physics idea is clear; calculation is simple. We offer a new simple and direct way of calculating the entropy of different complicated 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 Entropy of Horowitz—Strominger Black Hole

The partition functions of bosonic and fermionic fields in Horowitz-Strominger black hole are derived directly by quantum statistical method.Then via the improved brick-wall method (membrane model),the statistical entropy of black hole is obtained.If a proper parameter is chosen in our result,it is found out that the entropy is proportional to the area of horizon.The stripped term and the divergent logarithmic term in the original brick-wall method no longer exist.The difficulty in solving the wave equations of scalar and Dirac fields is avoided.A new neat way of calculating the entropy of various complicated black holes is offered.     

Entropy of N-Dimensional Spherically Symmetric Charged Black Hole

By using the method of quantum statistics, we derive directly the partition functions of bosonic andfermionic fields in the N-dimensional spherically symmetric charged black hole space-time. The statistical entropy ofblack hole is obtained by an improved brick-wall method. When we choose proper parameters in our results, we canobtain that the entropy of black hole is proportional to the area of horizon. In our result, there do not exist neglectedterm and divergent logarithmic term given in the original brick-wall method. We avoid the difficulty in solving the waveequation of scalar and Dirac fields. We offer a simple and direct way of studying entropy of the higher-dimensional black hole.     

Thermodynamics and Phase Transition of Topological Dilatonic Lifshitz-Like Black Holes

It is known that scalar-tensor gravity models can be studied in Einstein and Jordan frames. In this paper, a model of scalar-tensor gravity in Einstein's frame is considered to calculate the Lifshitz-like black hole solutions with different horizon topologies. Thermodynamic properties and first order van der Waals-like phase transition are studied, and it is found that the Lifshitz parameter affects the phase structure. In addition, thermal stability is investigated by using the behavior of heat capacity and various methods of geometrical thermodynamics.     

LETTER: Entropy Using Path Integrals for Quantum Black Hole Models

Canonical quantum gravity has been used in the search for eigenvalue equations that could describe black holes. In this paper we choose one of the simplest of these quantum equations to show how the usual Feynman's path integral approach can be applied to get the corresponding statistical properties. We get a logarithmic correction to the Bekenstein–Hawking entropy as already obtained by other authors by other means.