Entropies of a Toroidal Black Hole Due to Scalar and Dirac Fields

Using the thin film brick-wall model, the entropies of a toroidal black hole due to scalar and Dirac fields are investigated. The entropy due to the scalar field is one fourth of the horizon area, and that due to the Dirac field is seven eighth of the area. These results are similar to that in black holes with horizon topology S ². The cutoff in toroidal black hole is chosen as the same as one in black holes with horizon topology S ², which seems to mean that the thin film brick-wall model is universal.     

Reissner-Nordstrom几何中标量场的统计熵与能斯特定理

从Reissner-Nordstrom时空背景下的Klein-Gordon方程出发,利用改进的brick-wall方法膜模型,计算黑洞背景下标量场的自由能和熵.得到标量场的熵是由两部分组成的,根据熵是广延量的性质,得到黑洞熵是由两个子热力学系统贡献的.在此基础上给出了新的Bekenstein-Smarr公式.结果表明,用两个子热力学系统表达的熵,当黑洞的辐射温度趋于绝对零度时,黑洞的熵也趋于零,它满足能斯特定理,可视为黑洞的普朗克绝对熵. 关键词： brick-wall方法 膜模型 黑洞熵 能斯特定理     

Entropy Evolution in the Interior Volume of a Charged $f(R)$ Black Hole *

Based on the 4-dimensional black hole solution of $f(R)$ theory coupled to a nonlinear Maxwell field, we calculate the interior volume of a charged $f(R)$ black hole using the method proposed by Christodoulou and Rovelli. Considering massless scalar field in the interior volume and Hawking radiation carrying only energy, we calculate the entropy of the scalar field inside a charged $f(R)$ black hole and investigate the evolution of the entropy under Hawking radiation. In the meantime, the evolution of the Bekenstein-Hawking entropy under Hawking radiation has also been calculated. Based on these results, the proportional relation is obtained between the evolution of the scalar field entropy and the evolution of Bekenstein-Hawking entropy under Hawking radiation. According to the result, we investigate and discuss how the modified coefficient $b$ in $f(R)$ gravity theory affects the evolution relation between the two types of entropy. It is shown that the radiation rate for Hawking radiation of a charged $f(R)$ black hole can increase with the modified coefficient $b$.     

动态广义球对称含荷黑洞的量子熵

计算了广义球对称含荷黑洞视界上标量场的量子态数和自由能，得到了黑洞熵与视界面积成 正比的结论，表明黑洞熵就是其视界上的量子态的熵．考虑广义不确定原理对黑洞熵的影响 ，采用二维膜模型，克服了brick-wall模型中的发散困难，计算中无须任何截断，且brick- wall模型中的小质量近似也可以避免．对视界外二维膜上的量子场的熵做了级数展开讨论， 得到了一些值得探讨的结论． 关键词： 广义不确定原理 黑洞熵 视界 截断     

Vacuum polarization near the horizon of a black hole in the presence of boundaries

The renormalized expectation value in the Hartle-Hawking vacuum of the square of the field operator of a massless scalar field is discussed near the horizon of a Schwarzschild black hole inside a spherical cavity. For small radii of the cavity the expectation value becomes negative.     

环面黑洞背景下量子场的熵

利用薄层（改进的brick-wall模型），通过分别求解标量场方程和Dirac场方程，计算了环面黑洞事件视界附近的标量场和Dirac场的量子统计熵.按薄层模型的观点，在视界面附近薄层上的量子场的熵就是黑洞的熵.结果表明，黑洞熵正比于事件视界的面积，遵循Beken-stein-Hawking面积熵公式. 关键词： 熵 环面黑洞 薄层模型 量子场     

Black Hole Entropies of the Thin Film Model and the Membrane Model Without Cutoffs

Taking into account the effect of the generalized uncertainty principle on the generalized black hole entropy and tacking the thin film brick-wall model, we calculate the entropy of the quantum scalar field in generalized static black hole. The Bekenstein–Hawking entropies of all well-known static black holes are obtained. The entropy of 2-D membrane just at the event horizon of static black hole is also calculated, and the result of the black hole entropy proportional to the event horizon area can be obtained more easily and generally. This discussion shows that black hole entropy is just identified with the entropy of the quantum field on the event horizon. The difference from the original brick-wall model is that the present result is convergent without any cutoff and the little mass approximation is removed. With residue theorem, the integral difficulty in the calculation of black hole entropy is overcome.     

Black Hole and Cosmic Entropy for Schwarzschild–de Sitter Space–Time

We calculate the free energy and the entropy of a scalar field in terms of the brick-wall method in the background of the Schwarzschild–de Sitter space–time. We obtain the entropy of a black hole and the cosmic entropy at nonasymptotic flat space. When the cut-off satisfies the proper condition, the entropy of a black hole is proportional to the area of a black hole horizon, and the cosmic entropy is proportional to the cosmic horizon area.     

高自旋场对静态球对称黑洞熵的贡献

利用改进后的brick wall模型，研究具有高自旋的引力场对静态球对称黑洞熵的贡献.结果表明：在静态球对称黑洞中，自旋为2的引力场的量子熵仍与视界面积成正比.当选择与标量场相同的截断因子时，其量子熵为标量场的两倍，为Dirac场的4/7. 关键词： 黑洞熵 brick wall模型 自旋场 Teukolsky型主方程     

Nernst Theorem and Statistical Entropy of 5-Dimensional Rotating Black Hole

In this paper, by using quantum statistical method, we obtain the partition function of Bose field and Fermi field on the background of the 5-dimensional rotating black hole. Then via the improved brick-wall method and membrane model, we calculate the entropy of Bose field and Fermi field of the black hole. And it is obtained that the entropy of the black hole is not only related to the area of the outer horizon but also is the function of inner horizon‘s area. In our results, there are not the left out term and the divergent logarithmic term in the original brick-wall method.The doubt that why the entropy of the scalar or Dirac field outside the event horizon is the entropy of the black hole in the original brick-wall method does not exist. The influence of spinning degeneracy of particles on entropy of the black hole is also given. It is shown that the entropy determined by the areas of the inner and outer horizons will approach zero,when the radiation temperature of the black hole approaches absolute zero. It satisfies Nernst theorem. The entropy can be taken as the Planck absolute entropy. We provide a way to study higher dimensional black hole.     

Nernst Theorem and Statistical Entropy of 5-Dimensional Rotating Black Hole

In this paper, by using quantum statistical method, we obtain the partition function of Bose field and Fermi field on the background of the 5-dimensional rotating black hole. Then via the improved brick-wall method and membrane model, we calculate the entropy of Bose field and Fermi field of the black hole. And it is obtained that the entropy of the black hole is not only related to the area of the outer horizon but also is the function of inner horizon‘s area. In our results, there are not the left out term and the divergent logarithmic term in the original brick-wall method.The doubt that why the entropy of the scalar or Dirac field outside the event horizon is the entropy of the black hole in the original brick-wall method does not exist. The influence of spinning degeneracy of particles on entropy of the black hole is also given. It is shown that the entropy determined by the areas of the inner and outer horizons will approach zero,when the radiation temperature of the black hole approaches absolute zero. It satisfies Nernst theorem. The entropy can be taken as the Planck absolute entropy. We provide a way to study higher dimensional black hole.     

Quantum entropy of Dirac field in toroidal black hole

The quantum entropy of Dirac field in toroidal black hole is considered and the effects of the spins of the Dirac particles on the entropy are investigated using brick-wall model. It is shown that the quantum entropy has both linearly and logarithmically divergent terms and it has the different structure from that of the black hole. It is also shown that the contribution of the subleading logarithmic term which relates to the spins of the Dirac particles are always positive.     

LETTER: The Nernst Theorem and the Entropy of the Reissner–Nordström Black Hole

We calculate the free energy and the entropy of a scalar field in terms of the brick-wall method on the background of the Reissner–Nordström black hole. We obtain the entropy of a scalar field is not only related to the location of an outer horizon but also is the function of the location of an inner horizon. In the approximation, the entropy is only proportional to the area of an outer horizon. The entropy expressed by location parameter of outer and inner horizon approaches zero, when the radiation temperature of a black hole approaches absolute zero. It satisfies the Nernst theorem.     

Sen黑洞熵与能斯特定理

避开求解黑洞背景下波动方程的困难,应用量子统计方法,直接求解轴对称Sen黑洞背景下Bose场和Fermi场的配分函数.然后利用改进的 brick-wall 方法-膜模型,计算黑洞背景下Bose场和Fermi场的熵.得到黑洞熵不但与黑洞的外视界面积有关，而且也是内视界面积的函数.在所得结论中不存在对数发散项与舍去项,也不存在黑洞视界外标量场或Dirac场为什么是黑洞熵疑难,并且给出粒子的自旋简并度对黑洞熵的影响. 当黑洞的辐射温度趋于绝对零度时,由黑洞内外视界面积决定的黑洞熵也趋于零,它满足能斯特定理,可视 关键词： 膜模型 黑洞熵 能斯特定理     

利用广义不确定关系计算 Reissner-Nordström-de Sitter黑洞的统计力学熵

利用广义不确定关系修正的态密度方程并采用Wentzel-Kramers-Brillouin (WKB) 近似方法, 计算了Reissner-Nordström-de Sitter (RNdS) 黑洞时空中标量场的统计力学熵. 结果表明, 由这种方法得到的黑洞熵与它的内、外视界面积和宇宙视界面积之和成正比, 这与采用其他方法所得的结果一致, 从而揭示了黑洞熵与视界面积之间的内在联系, 也进一步表明了黑洞熵是视界面上量子态的熵, 是一种量子效应.     

Distribution of entropy of Bardeen regular black hole with corrected state density

We consider corrections to all orders in the Planck length on the quantum state density, and calculate the statistical entropy of the scalar field on the background of the Bardeen regular black hole numerically. We obtain the distribution of entropy which is inside the horizon of black hole and the contribution of the vicinity of horizon takes a great part of the whole entropy.     

Holographic Entropy Bound of a Nonstationary Black Hole

In accordance with the holographic principle, by counting the states of the scalar field just at the event horizon of the Vaidya-Bonner black hole, the holographic entropy bound of the black hole is calculated and the Bekenstein- Hawking formula is obtained, With the generalized uncertainty principle, the divergence of state density at event horizon in the ordinary quantum field theory is removed, With the residue theorem, the integral trouble in the calculation is overcome. The present result is quantitatively tenable and the holographic principle is realized by applying the quantum field theory to the black hole entropy problem. Compared with some previous works, it is suggested that the quantum states contributing to black hole entropy should be restricted on the event horizon.     

The Quantum Corrections to the Entropy of Stationary Axisymmetric Einstein-Maxwell-Dilaton-Axion Black Holes

By using 't Hooft's brick wall model, thecorrections for a massless quantum scalar field to theblack hole entropy are studied in a stationaryaxisymmetric Einstein-Maxwell-dilaton-axion black holespace-time. The free energy and entropy for this case arecalculated; in Hartle-Hawking states, the derivedquantum entropy is composed of the part that has ageometric feature and the part that is logarithmically divergent, and it turns out that thelogarithmic part is related to the characteristicquantities of a black hole.     

Bosonic and fermionic entropy of black holes with different temperatures on horizon surface

By using the method of quantum statistics, we derive directly the partition functions of bosonic and fermionic field in the black hole space－time with different temperatures on horizon surface. The statistical entropy of the black hole is obtained by an improved brick-wall method. When we choose a proper parameter in our results, we can obtain that the entropy of the black hole is proportional to the area of horizon. In our result, there do not exist any neglected term or divergent logarithmic term as given in the original brick-wall method. We have avoided the difficulty in solving the wave equation of the scalar and Dirac field. A simple and direct way of studying entropy of the black hole is given.     

利用广义不确定关系计算Reissner-Nordstrm-de Sitter黑洞的统计力学熵

利用广义不确定关系修正的态密度方程并采用Wentzel-Kramers-Brillouin(WKB)近似方法,计算了Reissner-Nordstrm-de Sitter(RNdS)黑洞时空中标量场的统计力学熵.结果表明,由这种方法得到的黑洞熵与它的内、外视界面积和宇宙视界面积之和成正比,这与采用其他方法所得的结果一致,从而揭示了黑洞熵与视界面积之间的内在联系,也进一步表明了黑洞熵是视界面上量子态的熵,是一种量子效应.