Quantum electron levels in the field of a charged black hole

Stationary solutions of the Dirac equation in the metric of the charged Reissner–Nordstrom black hole are found. In the case of an extremal black hole, the normalization integral of the wave functions is finite, and the regular stationary solution is physically self-consistent. The presence of quantum electron levels under the Cauchy horizon can have an impact on the final stage of the Hawking evaporation of the black hole, as well as on the particle scattering in the field of the black hole.     

直线加速运动动态黑洞的熵

选取超前爱丁顿坐标,采用薄膜brickwall模型,计算Kinnersley度规表述的直线加速运动动态黑洞的熵.通过此方法,可以给出视界面上每一点的温度和熵密度.这一结果表明,熵与视界面积成正比的结论,不仅适用于整个视界,也适用于视界面上的局部;不仅适用于稳态黑洞,也适用于动态黑洞.在薄膜趋于视界面时,其厚度也趋于零,薄膜本身成为视界面,黑洞熵就是视界面上量子态的熵 关键词： 熵 加速黑洞 薄膜brickwall模型     

The Inner Structure of Black Holes

We study the gravitational collapse of a self-gravitating charged scalar-field. Starting with a regular spacetime, we follow the evolution through the formation of an apparent horizon, a Cauchy horizon and a final central singularity. We find a null, weak, mass-inflation singularity along the Cauchy horizon, which is a precursor of a strong, spacelike singularity along the r = 0 hypersurface. The inner black hole region is bounded (in the future) by singularities. This resembles the classical inner structure of a Schwarzschild black hole and it is remarkably different from the inner structure of a charged static Reissner-Nordström or a stationary rotating Kerr black holes.     

Gibbons-Maeda dilaton黑洞的全息熵

依据全息原理,通过计算Gibbons-Maeda dilaton黑洞事件视界上量子场的统计熵,得到了该黑洞的全息熵和Bekenstein-Hawking熵.计算中利用非对易量子场论,克服了普通量子场论中态密度在视界上的发散困难,避免了黑洞熵热气体方法中紫外截断的引入.用留数定理克服了计算中的积分困难,所得的结果定量成立.研究表明,黑洞熵可以视为其视界上量子场的熵;通过计算视界上量子态的统计熵可以得到黑洞熵,计算中可以且应该避免视界外量子态的影响. 关键词： 黑洞熵 全息原理 事件视界 非对易量子场论     

Black holes in general relativity

It is assumed that the singularities which occur in gravitational collapse are not visible from outside but are hidden behind an event horizon. This means that one can still predict the future outside the event horizon. A black hole on a spacelike surface is defined to be a connected component of the region of the surface bounded by the event horizon. As time increase, black holes may merge together but can never bifurcate. A black hole would be expected to settle down to a stationary state. It is shown that a stationary black hole must have topologically spherical boundary and must be axisymmetric if it is rotating. These results together with those of Israel and Carter go most of the way towards establishing the conjecture that any stationary black hole is a Kerr solution. Using this conjecture and the result that the surface area of black holes can never decrease, one can place certain limits on the amount of energy that can be extracted from black holes.     

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.     

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.     

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.     

On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case. Received: 4 November 1998 / Accepted: 13 February 1999     

Entropy of Non-stationary and Slowly Changing Reissner-Nordström Black Hole

Simplifying Dirac equation near the horizon, Hawking temperature is obtained by applying a new tortoise coordinate transformation. Using the improved thin film brick-wall model and WKB approximation, the entropy of Dirac field in the non-stationary and slowly changing Reissner-Nordström black hole is calculated. The result shows that the entropy of the black hole is still proportional to the horizon area, and black hole entropy is just identical to the entropy of the quantum state at the horizon. In addition, the new tortoise coordinate transformation can make the cut-off parameter introduced in solving the entropy of non-stationary black hole simplified to the same as that in the static and stationary cases.     

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.     

广义不确定关系与整体单极黑洞Dirac场的熵

将广义不确定关系引入新的态密度方程，采用WKB近似方法，对含整体单极黑洞Dirac场的熵进行了直接计算，所得黑洞熵与它的视界面积成正比，以此揭示了黑洞熵是其视界面处量子态的熵.与brick-wall模型方法不同，该结果不需要取任何截断.同时表明，用此方法不仅可以计算黑洞标量场的熵，而且可以计算Dirac场的熵.     

黑洞的统计熵

运用量子统计的方法,直接求解Schwarzschild时空背景下玻色场和费米场的配分函数,得到熵的积分表达式.按照最近的研究结果,认为黑洞的Hawking辐射过程是隧道效应过程,在考虑黑洞隧道效应产生过程中黑洞能量发生变化的基础上,给出积分的下限为黑洞的视界位置.由此得到黑洞熵的主要项为视界面积的1/4.不存在使人疑惑的紫外截断因子,并且由此可得黑洞辐射粒子的能量与辐射温度成正比的结论. 关键词： 黑洞熵 量子统计 隧道效应 反作用     

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.     

Quantum Entropy of Black Hole with Internal Global Monopole

Using the generalized uncertainty relation, the new equation of state density is obtained, and then the entropy of black hole with an internal global monopole is discussed. The divergence that appears in black hole entropy calculation through original brick-wall model is overcome. The result of the direct proportion between black hole entropy and its event horizon area is drawn and given. The result shows that the black hole entropy must be the entropy of quantum state near the event horizon.     

A rotating black ring solution in five dimensions

The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1xS2. It describes a rotating "black ring." This is the first example of a stationary asymptotically flat vacuum solution with an event horizon of nonspherical topology. The existence of this solution implies that the uniqueness theorems valid in four dimensions do not have simple five-dimensional generalizations. It is suggested that increasing the spin of a spherical black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.     

Entropy bound of horizons for accelerating,rotating and charged Plebanski–Demianski black hole

We first review the accelerating, rotating and charged Plebanski–Demianski (PD) black hole, which includes the Kerr–Newman rotating black hole and the Taub-NUT spacetime. The main feature of this black hole is that it has 4 horizons like event horizon, Cauchy horizon and two accelerating horizons. In the non-extremal case, the surface area, entropy, surface gravity, temperature, angular velocity, Komar energy and irreducible mass on the event horizon and Cauchy horizon are presented for PD black hole. The entropy product, temperature product, Komar energy product and irreducible mass product have been found for event horizon and Cauchy horizon. Also their sums are found for both horizons. All these relations are dependent on the mass of the PD black hole and other parameters. So all the products are not universal for PD black hole. The entropy and area bounds for two horizons have been investigated. Also we found the Christodoulou–Ruffini mass for extremal PD black hole. Finally, using first law of thermodynamics, we also found the Smarr relation for PD black hole.     

Correction to Hawking Pure Thermal Spectrum of Stationary Kaluza-Klein Black Hole

Applying Parikh's quantum tunneling method, the tunneling characteristics of stationary Kaluza-Klein black hole is researched. The result shows that the tunneling rate across the event horizon of the black hole is relevant to the change of Bekenstein-Hawking entropy and the derived radiation spectrum deviates from pure thermal when the self-gravitation, energy conservation and angular momentum conservation are taken into consideration. Finally, we use the obtained results to reduce to stationary Kerr black hole and static Swarzschild black hole, and find that only ignoring the spectrum at higher energies the tunneling radiation spectrum is consistent with Hawking pure thermal one. PACS:97.60.Lf,04.70._s