Entropy Of Scalar Field Research Articles

The entropy evolution of a noncommutative black hole under Hawking radiation

By calculating the entropy of a scalar field in the interior volume of noncommutative black holes and considering an infinitesimal process of Hawking radiation, a proportion function is constructed that reflects the evolution relation between the scalar field entropy and Bekenstein–Hawking entropy under Hawking radiation. Comparing with the case of Schwarzschild black holes, the new physics of this research can be expanded to the later stage of Hawking radiation. From the result, we find that the proportion function is still a constant in the earlier stage of Hawking radiation, which is identical to the case of Schwarzschild black holes. As Hawking radiation goes into the later stage, the behavior of the function will be dominated by the noncommutative effect. In this circumstance, the proportion function is no longer a constant and decreases with the evaporation process. When the noncommutative black hole evolves into its final state with Hawking radiation, the interior volume will converge to a certain value, which implies that the loss of information of the black hole during the evaporation process will finally be stored in the limited interior volume.

The entropy evolution of a Schwarzschild–(Anti) de Sitter black hole

Based on the definition of the interior volume of spherically symmetry black holes, the interior volume of Schwarzschild–(Anti) de Sitter black holes is calculated. It is shown that with the cosmological constant ([Formula: see text]) increasing, the changing behaviors of both the position of the largest hypersurface and the interior volume for the Schwarzschild–Anti de Sitter black hole are the same as the Schwarzschild–de Sitter black hole. Considering a scalar field in the interior volume and Hawking radiation with only energy, the evolution relation between the scalar field entropy and Bekenstein–Hawking entropy is constructed. The results show that the scalar field entropy is approximately proportional to Bekenstein–Hawking entropy during Hawking radiation. Meanwhile, the proportionality coefficient is also regarded as a constant approximately with the increasing [Formula: see text]. Furthermore, considering [Formula: see text] as a dynamical variable, the modified Stefan–Boltzmann law is proposed which can be used to describe the variation of both the mass and [Formula: see text] under Hawking radiation. Using this modified law, the evolution relation between the two types of entropy is also constructed. The results show that the coefficient for Schwarzschild–de Sitter black holes is closer to a constant than the one for Schwarzschild–Anti de Sitter black holes during the evaporation process. Moreover, we find that for Hawking radiation carrying only energy, the evolution relation is a special case compared with the situation that the mass and [Formula: see text] are both considered as dynamical variables.

Entropy Evolution in the Interior Volume of a Charged f(R) Black Hole**Supported by the National Natural Science Foundation of China under Grant No. 11235003

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.

Charged Hawking radiation and the entropy variation in a Reissner–Nordström black hole

Under charged Hawking radiation, we calculate the entropy of the scalar field in the interior volume of a Reissner–Nordström (RN) black hole and investigate the proportional relation between this entropy and the Bekenstein–Hawking entropy. Comparing with uncharged Hawking radiation, we find that whether the particles radiated from the black hole are charged cannot affect the proportional relation between the two types of entropy. Furthermore, the proportionality coefficient of the two types of entropy has been analyzed and discussed. The result shows that the variation of scalar field entropy inside the black hole is approximately proportional to the variation of Bekenstein–Hawking entropy in an infinitesimal evaporation process. Moreover, the proportional relation can degenerate to the Schwarzschild case when the charge of the black hole completely disappears. For a charged black hole, the extremal black hole will technically not prevent and stop the evaporation anymore under charged Hawking radiation.

Logarithmic corrections to the entropy of scalar field in BTZ black hole spacetime

The entanglement entropy correlates two quantum subsystems which are part of a larger system. A logarithmic divergence term present in the entanglement entropy is universal in nature and directly proportional to the conformal anomaly. We study this logarithmic divergence term of entropy for massive scalar field in [Formula: see text] dimensions by applying numerical techniques to entanglement entropy approach. This (2+1)-dimensional massive theory can be obtained from the (3+1)-dimensional massless scalar field via dimensional reduction. We also calculated mass corrections to entanglement entropy for scalar field. Finally, we observe that the area law contribution to the entanglement entropy is not affected by this mass term and the universal quantities depend upon the basic properties of the system.

Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Schwarzschild Black Hole

In this article, we will discuss a Lorentzian sector calculation of the entropy of a minimally coupled scalar field in the Schwarzschild black hole background using the brick wall model of 't Hooft. In the original article, the WKB approximation was used for the modes that are globally stationary. In a previous article, we found that the WKB quantization rule together with a proper counting of the states, leads to a new expression of the scalar field entropy which is not proportional to the area of the horizon. The expression of the entropy is logarithmically divergent in the brick wall cut-off parameter in contrast to an inverse power divergence obtained earlier. In this article, we will consider the entropy for a thin shell of matter field of a given thickness surrounding the black hole horizon. The thickness is chosen to be large compared with the Planck length and is of the order of the atomic scale. When expressed in terms of a covariant cut-off parameter, the entropy of a thin shell of matter field of a given thickness and surrounding the horizon in the Schwarzschild black hole background is given by an expression proportional to the area of the black hole horizon. This leading order divergent term in the cut-off parameter remains to be logarithmically divergent. The logarithmic divergence is expected from the nature of the solution in the near-horizon region. We will find that these discussions are significant in the context of the continuation to the Euclidean sector and the corresponding regularization schemes used to evaluate the thermodynamical properties of matter fields in curved spaces. These are related with the geometric aspects of curved spaces. The above discussions are also important in presence of cosmological event horizon.

Near horizon geometry, Brick wall model and the Entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds

In this article we will find the entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds using the brick wall model of t' Hooft. We will use the semi-classical WKB approximation. We will consider the modes which are globally stationary so that the WKB quantization rule used in the brick wall model remains valid. In the Schwarzschild black hole this consideration had led to a new expression of the entropy different from the conventional expression which is inversely divergent in the brick wall cut-off parameter and in terms of a proper distance cut-off parameter, is proportional to the area of the event horizon. The new expression of the scalar field entropy obtained in this article is logarithmically divergent in the brick wall cut-off parameter and is not proportional to the area of the black hole event horizon. For the extremal Reissner-Nordstrom black hole background the entropy of the scalar field is again divergent in the brick wall cut-off parameter and vanishes if the temperature of the Hawking radiation and the black hole is taken to be zero. We will next consider the entropy for a thin shell of matter field surrounding the black hole horizon. When expressed in terms of a covariant cut-off parameter, the entropy of a thin shell of matter field surrounding the horizon in the non-extreme Reissner-Nordstrom black hole background is given by an expression proportional to the area of the black hole horizon. We will briefly explain the significance of this result.

Statistical Entropy of Vaidya-de Sitter Black Hole to All Orders in Planck Length

Considering corrections to all orders in Planck length on the quantum state density from generalized uncertainty principle, we calculate the statistical entropy of scalar field near event horizon and cosmological horizon of Vaidya-de Sitter black hole without any artificial cutoff. It is shown that the entropy is linear sum of event horizon area and cosmological horizon area and there are similar proportional parameters related to changing rate of the horizon position. This is different from the static and stationary cases.

Statistical entropy of reissner-nordstrm-de sitter black hole by generalized uncertainty principle

Statistical entropy of scalar field outside Reissner-Nordstrm-de Sitter black hole is computed by the equation of state density corrected by the generalized uncertainty principle and by Wentzel-Kramers-Brillouin approximation method. The result shows that the entropy is proportional to the sum of the internal, the external and the cosmological horizon areas, which accords with these calculated by other methods. It shows internal relation between the entropy of black hole and horizon area. The entropy of black hole is the entropy of quantum state on horizon, which is a quantum effect.

Statistical Entropy of 5D Ricci-Flat Space-Time with Generalized Uncertainty Principle Revised

Considering corrections to all orders in the Planck length on the quantum state density from generalized uncertainty principle and using that quantum state density to all degrees of freedom including extra dimension, we calculate the statistical entropy of scalar field in the 5D Ricci-flat space-time without any artificial cutoff. Calculation shows that the entropy is the linear sum of the event horizon area and the cosmological horizon area.

Using brick wall method and thin film brick wall method to calculate the statistical-mechanical entropy of scalar field in Gibbons-Maeda spacetime

摘要 利用砖墙方法和薄层方法计算了Gibbons-Maeda黑洞背景时空中标量场的统计力学熵.用砖墙方法求得的统计力学熵有两项:其中一项与Gibbons-Maeda黑洞视界面积成正比，并且当截断因子满足一定的关系时，熵为其视界面积的四分之一;另一项是对数发散项.利用薄层方法所求得的熵只有与Gibbons-Maeda黑洞视界面积成正比的项，对数发散项被自然消去. 关键词: 砖墙方法 / 薄层方法 / Gibbons-Maeda黑洞 / 统计力学熵 Abstract The statistical-mechanical entropy of scalar field is calculated by using brick wall method and thin film brick-wall method in Gibbons-Maeda black hole spacetime. The entropy obtained from brick-wall method has two terms. One term is proportional to the event horizon area, and the proportional coefficient is 1/4 when the cutoff factor satisfies a suitable condition. The other term is logarithmic-divergent. The entropy obtained from thin film brick-wall method has only one term which is proportional to the event horizon area, and the logarithmic divergence vanishes. Keywords: brick-wall method / thin film brick-wall method / Gibbons-Maeda black hole / statistical-mechanical entropy 作者及机构信息 贺锋, 赵凡, 颜千 1. 湖南科技大学物理学院，湘潭 411201 基金项目: 湖南科技大学研究生创新基金（批准号: S080111）资助的课题. Authors and contacts He Feng, Zhao Fan, Yan Qian 1. 湖南科技大学物理学院，湘潭 411201 参考文献 [1] ［1］Hawking S W 1975 Commun. Math. Phys 43 199 [2] ［2］Bekenstein J D 1973 Phys. Rev. D 7 2333 [3] ［3］G’t Hooft 1985 Nucl. Phys. B 256 727 [4] ［4］Luo Z J, Zhu J Y 1999 Acta Phys. Sin. 48 395(in Chinese)［罗智坚、朱建阳 1999物理学报 48 395］ [5] ［5］Liu W B, Zhao Z 2000 Journal of Beijing Normal University(Natural Science) 36 626 (in Chinese) ［刘文彪、赵峥 2000北京师范大学学报36 626］ [6] ［6］Zhao R, Zhang L C 2002 Acta Phys. Sin. 51 1167(in Chinese) ［赵仁、张丽春 2002 物理学报 51 1167］ [7] ［7］Liu C Z, Li X, Zhao Z 2004 General Ralativity and Gravitation 36 1135 [8] ［8］Ding C K, Jing J L 2007 Chin. Phys. 16 3610 [9] ［9］Yang B 2008 Acta Phys. Sin. 57 2614(in Chinese)［杨波 2008物理学报57 2614］ [10] ］Wang B B 2008 Chin. Phys. B 17 467［11］Liu C Z 2005 Acta Phys. Sin. 54 1977(in Chinese)［刘成周 2005物理学报54 1977］ [11] ］Jing J L, Yan M L 2001 Phys. Rev. D 64 064015 [12] ］Jing J L, Yan M L 2001 Phys. Rev. D 63 084028 [13] ］Li X 2001 Phys. Rev. D 65 084005 [14] ］Wei Y H, Wang C H, Zhao Z 2002 Phys. Rev. D 65 124023 [15] ］Ghosh T, SenGupta S 2008 Phys. Rev. D 78 024045 [16] ］Li X, Zhao Z 2000 Phys. Rev. D 62 104001 [17] ］He F, Zhao Z, Kim S W 2001 Phys. Rev. D 64 044025 [18] ］Gao C J, Shen Y G 2002 Phys. Rev. D 65 084043 [19] ］G W Gibbons, K Maeda 1988 Nucl. Phys. B 298 741 施引文献

Scalar field entanglement entropy of a Schwarzschild black hole from the Schmidt decomposition viewpoint

Scalar field entropy generated with the black hole horizon is considered in the light of the Schmidt decomposition to internal and external states of the field. Within the frame of this approach, the unitary character of the black hole evolution appears to be natural. We estimated the ratio of radiation field entropy to the Bekenstein–Hawking entropy, which has appeared to be small; in the zeroth order the ratio is constant. The obtained corrections to the massive radiation component are found to have both negative and positive values, hence witnessing the possibility of the information outflow from the black holes.

Statistical Entropy of Reissner-Nordstrom Black Hole Computed by Generalized Uncertainty Principle

The equation of state density is corrected by the generalized uncertainty principle. Statistical entropy of scalar field outside Reissner-Nordstrom black hole is computed by WKB approximation method. The result shows that this black hole entropy is proportional to its horizon area, which is the same as that given by brick-wall method. The difference from the brick-wall method is that the present result is convergent without any cutoff.

Statistical entropy of Gibbons-Maeda black hole computed by generalized uncertainty principle

Statistical entropy of scalar field outside Gibbons-Maeda black hole is computed by the equation of state density corrected by the generalized uncertainty principle and by WKB approximation method. The result shows that the black hole entropy is proportional to its horizon area, which is the same as that given by Brick-Wall method. The difference from the Brick-Wall method is that the present result is convergent without any cutoff.

Generalized area law under multiparameter rotating black hole spacetime

We study the statistical mechanics of quantum scalar fields under multiparameter rotating black hole spacetime in arbitrary D dimensions. The method of analysis is general because the metric does not depend on explicit black hole solutions. A generalized Stefan–Boltzmann law for scalar fields is derived by properly considering the allowed energy region. Then a generalized area law for scalar field entropy is derived by introducing an invariant regularization parameter in Rindler spacetime. The derived area law is applied to Kerr–AdS black holes in four and five dimensions. The thermodynamic implications are also discussed.

Quantum entropy of the non-static black hole in（2+1）dimensions

In the tortoise coordinates，the quantum entropy of the non-static black hole in (2+1)dimensions due to scalar fields has been investigated by using the brick-wall model.It is shown that the entropy of scalar fields within the thin region near the horizon has the same structure as that of the black ho1e.In particular,the quantum entropy satisfies the perimeter law of the entropy for the static case.

Entropy of Horowitz–Strominger black holes due to arbitrary spin fields

Using the membrane model which is based on brick-wall model, we calculated the free energy and entropy of Horowitz–Strominger black holes due to arbitrary spin fields. The result shows that the entropy of scalar field and fermions field has similar formulas. There is only a coefficient difference between them.

The entropy of Garfinkle-Horne dilaton black hole due to arbitrary spin fields

Using the membrane model which is based on brick wall model, we calculated the free energy and entropy of Garfinkle-Horne dilatonic black hole due to arbitrary spin fields.The result shows that the entropy of scalar field and the entropy of Fermionic field have similar formulas.There is only a coefficient between them

THE ENTROPY CALCULATED VIA BRICK-WALL METHOD COMES FROM A THIN FILM NEAR THE EVENT HORIZON

The brick-wall method put forward by 't Hooft has contributed a great deal to the understanding and calculating of the entropy of a black hole. However, there are some drawbacks in it such as little mass approximation, neglecting logarithm terms, and taking the term including L3as a contribution of the vacuum surrounding the black hole. Moreover, the fundamental problem is why the entropy of scalar field or Dirac field surrounding a black hole is the entropy of the black hole itself. It is well known that the event horizon is the characteristic of a black hole. The entropy calculation of a black hole should be only related to its horizon. Due to this analysis, we improve the brick-wall model by taking that the entropy of a black hole is only contributed by a thin film near the event horizon. This improvement not only gives us a satisfied result, but also avoids the drawbacks in the original brick-wall method. It is found that there is an intrinsic relation between the event horizon and the entropy. We also calculate the entropy of Schwarzschild–de Sitter space–time via the improved method, which can hardly be resolved via the original model.

Entropy of Dilatonic Black Hole due to Arbitrary Spin Fields

Using the membrane model based on the brick-wall model, we calculate the free energy and entropy of dilatonic black hole due to arbitrary spin fields. The result shows that the entropy of scalar field and the entropy of Fermionic field have similar formulas. There is only a numerical coefficient between them.