BTZ Black Hole in Non-Commutative Spaces

In this letter we compute the corrections to the horizons, the horizon area and Hawking temperature of a BTZ black hole. These corrections stem from the space non-commutativity. We show that in non-commutative case, non-rotating BTZ black hole in contrast with commutative case has two horizons.     

变加速直线运动黑洞的温度和Dirac场的熵

采用Tortoise坐标变换,约化视界面附近Dirac场方程,得到Kinnersley黑洞的Hawking温度.用薄膜brick-wall模型,计算Kinnersley黑洞的熵,得到通过选择适当的截断因子和薄层,在视界面附近薄层上的熵就是黑洞的熵,结果表明黑洞熵与视界面积成正比.     

The real scalar field in extreme RNdS space

The real scalar field equation between the outer black hole horizon and the cosmological horizon is solved in the extreme Reissner-Nordström de Sitter (RNdS) space. We use an accurate approximation, the polynomial approximation, to approximate the tortoise coordinate x(r) in order to get the inverse function r = r(x) and then to solve the wave equation. The case where the two horizons are very close to each other is discussed in detail. We find that the wave function is harmonic only in the very small regions near the horizons, and the amplitude decreases remarkably near the potential peak because of the effect of the potential. Furthermore, it is found that the height of the potential increases as the cosmological constant Λ decreases, and the wave amplitude will decrease more remarkably with less Λ.     

Vaidya-Bonner-de Sitter黑洞对狄拉克粒子的热辐射

分析讨论了Vaidya-Bonner-de Sitter黑洞视界附近的狄拉克方程,准确地定出了Vaidya-Bonner-de Sitter黑洞的Hawking温度和辐射谱，同时计算出事件视界方程，所得结果与用零曲面方程得到的结果一致. 关键词： 黑洞 视界 狄拉克方程 热辐射     

Trapping Horizons in Sultana-Dyer Space-Time

The Sultana-Dyer space-time is suggested as a model describing a black hole embedded in an expanding universe. Recently, its global structure is analyzed and the trapping horizons are shown. In the paper, by directly calculating theexpansions of the radial null vector fields normal to the space-like two-spheres foliating the trapping horizons, we find that the trapping horizon outside the event horizon in the Sultana-Dyer space-time is a past trapping horizon. Further, we find that the past trapping horizon is an outer, instantaneously degenerate or inner trapping horizon accordingly when the radial coordinate isless than, equal to or greater than some value.     

Hawking effect and quantum nonthermal radiation of an arbitrarily accelerating charged black hole using a new tortoise coordinate transformation

Using a new tortoise coordinate transformation, this paper investigates the Hawking effect from an arbitrarily accelerating charged black hole by the improved Damour-Ruffini method. After the tortoise coordinate transformation, the Klein-Gordon equation can be written as the standard form at the event horizon. Then extending the outgoing wave from outside to inside of the horizon analytically, the surface gravity and Hawking temperature can be obtained automatically. It is found that the Hawking temperatures of different points on the surface are different. The quantum nonthermal radiation characteristics of a black hole near the event horizon is also discussed by studying the Hamilton-Jacobi equation in curved spacetime and the maximum overlap of the positive and negative energy levels near the event horizon is given. There is a dimensional problem in the standard tortoise coordinate and the present results may be more reasonable.     

String dynamics near a black hole

The dynamics of a string near a Kaluza-Klein black hole are studied. Solutions to the geodesic equations are obtained using the world sheet velocity of light as an expansion parameter. For a string falling into a magnetically charged black hole, it is shown that the compact dimension decreases with the world-sheet coordinate τ.     

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.     

On the Thermal Property of Arbitrarily Accelerating Charged Black Hole with a New Tortoise Coordinate Transformation

After a new tortoise coordinate transformation is adopted, the entropy and non-thermal radiation of an arbitrarily accelerating charged black hole are discussed as an example of non-stationary black holes. The same cut-off relation is chosen as static case, which is independent of space-time, and then the entropy of the non-stationary black hole is also proportional to the area of its event horizon. Meanwhile, the crossing of the particle energy levels near the event horizon is studied, the representative of the maximum value of the crossing energy levels is the same as the usual tortoise coordinate transformation.     

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.     

Area invariance of apparent horizons under arbitrary Lorentz boosts

It is a well known analytic result in general relativity that the 2-dimensional area of the apparent horizon of a black hole remains invariant regardless of the motion of the observer, and in fact is independent of the t = constant slice, which can be quite arbitrary in general relativity. Nonetheless the explicit computation of horizon area is often substantially more difficult in some frames (complicated by the coordinate form of the metric), than in other frames. Here we give an explicit demonstration for very restricted metric forms of (Schwarzschild and Kerr) vacuum black holes. In the Kerr–Schild coordinate expression for these spacetimes they have an explicit Lorentz-invariant form. We consider boosted versions with the black hole moving through the coordinate system. Since these are stationary black hole spacetimes, the apparent horizons are two dimensional cross sections of their event horizons, so we compute the areas of apparent horizons in the boosted space with (boosted) t = constant, and obtain the same result as in the unboosted case. Note that while the invariance of area is generic, we deal only with black holes in the Kerr–Schild form, and consider only one particularly simple change of slicing which amounts to a boost. Even with these restrictions we find that the results illuminate the physics of the horizon as a null surface and provide a useful pedagogical tool. As far as we can determine, this is the first explicit calculation of this type demonstrating the area invariance of horizons. Further, these calculations are directly relevant to transformations that arise in computational representation of moving black holes. We present an application of this result to initial data for boosted black holes.     

Spacetime Non-Commutativity Corrections to the Cardy-Verlinde Formula of Achúcarro-Ortiz Black Hole

In this letter we compute the corrections to the Cardy-Verlinde formula of Achúcarro-Ortiz black hole, which is the most general two-dimensional black hole derived from the three-dimensional rotating Banados-Teitelboim-Zanelli black hole. These corrections stem from the space non-commutativity. We show that in non-commutative case, non-rotating Achúcarro-Ortiz black hole in contrast with commutative case has two horizons.     

Rotating charged black holes in Einstein-Born-Infeld theories and their ADM mass

In this work, the solution of the Einstein equations for a slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties and horizons of this solution are analyzed. The conditions when the ADM mass (as in the nonlinear static cases) and the ADM angular momentum of the system have been modified by the non linear electromagnetic field of the black hole, are considered.     

On Thermodynamical Relation Between Rotating ChargedBTZ Black Holes and Effective String Theory

In this paper we study the first law of thermodynamics for the (2+1)-dimensional rotating charged BTZ black hole considering a pair of thermodynamical systems constructed with the two horizons of this solution. We show that these two systems are similar to the right and left movers of string theory and that the temperature associated with the black hole is the harmonic mean of the temperatures associated with these two systems.     

On the Quasinormal Modes for Gravitational Perturbations of the Bardeen Black Hole

We investigate gravitational perturbations on a regular black hole described by the Bardeen solution. Bardeen’s black hole is a solution of Einstein’s equations with no singularity at the origin of the radially symmetric system. Notwithstanding this regularity, the Bardeen solution still has event horizons dependent on its characteristic parameters. When a black hole is perturbed, it oscillates and gives rise to damped vibrational modes known as quasinormal modes. Here, we compute the quasinormal frequencies of a regular black hole to third order in the WKB approximation for gravitational perturbations.     

LETTER: Quantum Creation of BTZ Black Hole

The constrained instanton method is used to study quantum creation of a BTZ black hole. It is found that the relative creation probability is the exponential of the negative sum of the entropy associated with the outer and inner black hole horizons. The quantum creations of the 4- or higher dimensional versions of the BTZ black hole are also studied.     

Black Holes as Atoms

Stationary spacetimes containing a black hole have several properties akin to those of atoms. For instance, such spacetimes have only three classical degrees of freedom, or observables, which may be taken to be the mass, the angular momentum, and the electric charge of the hole. There are several arguments supporting a proposal originally made by Bekenstein that quantization of these classical degrees of freedom gives an equal spacing for the horizon area spectrum of black holes. We review some of these arguments and introduce a specific Hamiltonian quantum theory of black holes. Our Hamiltonian quantum theory gives, among other things, a discrete spectrum for the classical observables, and it produces an area spectrum which is closely related to Bekenstein's proposal. We also present a foamlike model of horizons of spacetime. In our model spacetime horizon consists of microscopic Schwarzschild black holes. Applying our Hamiltonian approach to this model we find that the entropy of any horizon is one quarter of its area.     

Entropy of a Black Hole with Distinct Surface Gravities

In gravitational thermodynamics, the entropy of a black hole with distinct surface gravities can be evaluated in a microcanonical ensemble. At the WKB level, the entropy becomes the negative of the Euclidean action of the constrained instanton, which is the seed for the black hole creation in the no-boundary universe. Using the Gauss-Bonnet theorem, we prove the quite universal formula in Euclidean quantum gravity that the entropy of a nonrotating black hole is one quarter the sum of the products of the Euler characteristics and the areas of the horizons. For Lovelock gravity, the entropy and quantum creation of a black hole are also studied.     

Quantum statistical entropy for Kerr－de Sitter black hole

Improving the membrane model by which the entropy of the black hole is studied, we study the entropy of the black hole in the non-thermal equilibrium state. To give the problem stated here widespread meaning, we discuss the (n 2)-dimensional de Sitter spacetime. Through discussion, we obtain that the black hole‘s entropy which contains two horizons (a black hole‘s horizon and a cosmological horizon) in the non-thermal equilibrium state comprises the entropy corresponding to the black hole‘s horizon and the entropy corresponding to the cosmological horizon. Furthermore, the entropy of the black hole is a natural property of the black hole. The entropy is irrelevant to the radiation field out of the horizon. This deepens the understanding of the relationship between black hole‘s entropy and horizon‘s area. A way to study the bosonic and fermionic entropy of the black hole in high non-thermal equilibrium spacetime is given.     

Expanding the Area of Gravitational Entropy

I describe how gravitational entropy is intimately connected with the concept of gravitational heat, expressed as the difference between the total and free energies of a given gravitational system. From this perspective one can compute these thermodyanmic quantities in settings that go considerably beyond Bekenstein's original insight that the area of a black hole event horizon can be identified with thermodynamic entropy. The settings include the outsides of cosmological horizons and spacetimes with NUT charge. However the interpretation of gravitational entropy in these broader contexts remains to be understood.