General Tortoise Coordinate Transformation in a Dynamical Kerr-Newman Black Hole

Under the extended dynamical tortoise coordinate transformation, Damour-Ruffini method has been applied to calculate the charged particles’ Hawking radiation from the apparent horizon of a dynamical Kerr-Newman black hole. It is shown that Hawking radiation is still purely thermal black body spectrum. Moreover, the temperature of Hawking radiation is corresponding to the apparent horizon surface gravity and the first law of thermodynamics can also be constructed successfully on the apparent horizon in the dynamical Kerr-Newman black hole.     

Coalescing Black Hole Solution in the De-Sitter Universe

A new coalescing black hole solution of Einstein-Maxwell equation in general relativity is given. The new solution is also found to support the “Nernst Theorem” of thermodynamics in the case of black hole. Thus this solution poses to solve an outstanding problem of thermodynamics and black hole physics.     

Fluctuation around horizon on a Schwarzschild black hole using the null geodesic method

Using the null geodesic method, Hawking radiation from the horizon of a Schwarzschild black hole is calculated. The thermodynamics can be built successfully on the horizon where the apparent horizon and event horizon are coincident with each other. When a relativistic perturbation is given to the horizon, the first law of thermodynamics can also be constructed at a new supersurface near the horizon successfully. The expressions of the characteristic position and temperature are consistent with the previous result while the thermodynamics was built on the event horizon in a Vaidya black hole. Therefore, the thermodynamics of a dynamical black hole should be constructed on the apparent horizon exactly, and the event horizon thermodynamics is just one of the perturbations near the apparent horizon.     

Hawking Absorption and Planck Absolute Entropy of Kerr-Newman Black Hole

We find the existence of a quantum thermal effect, “Hawking absorption.” near the inner horizon of the Kerr–Newman black hole. Redefining the entropy, temperature, angular velocity, and electric potential of the black hole, we give a new formulation of the Bekenstein–Smarr formula. The redefined entropy vanishes for absolute zero temperature of the black hole and hence it is interpreted as the Planck absolute entropy of the KN black hole.     

Black Hole Entropy: Inside or Out?

A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy. Ted and Carlo support the idea that this entropy measures in some sense “the number of black hole microstates that can communicate with the outside world.” Don is critical of this approach, and discussion ensues, focusing on the question of whether the first law of black hole thermodynamics can be understood from a statistical mechanics point of view.     

The volume inside a black hole

The horizon (the surface) of a black hole is a null surface, defined by those hypothetical “outgoing” light rays that just hover under the influence of the strong gravity at the surface. Because the light rays are orthogonal to the spatial two-dimensional surface at one instant of time, the surface area of the black hole is the same for all observers (i.e. the same for all coordinate definitions of “instant of time”). This value is 4π(2Gm/c ²)² for nonspinning black holes, with G = Newton’s constant, c = speed of light, and m = mass of the black hole. The three-dimensional spatial volume inside a black hole, in contrast, depends explicitly on the definition of time, and can even be time dependent, or zero. We give examples of the volume found inside a standard, nonspinning spherical black hole, for several different standard time-coordinate definitions. Elucidating these results for the volume provides a new pedagogical resource of facts already known in principle to the relativity community, but rarely worked out.     

Hawking radiation from a Vaidya black hole by Hamilton-Jacobi method

Using the Hamilton-Jacobi method, Hawking radiation from the apparent horizon of a dynamical Vaidya black hole is calculated. The black hole thermodynamics can be built successfully on the apparent horizon. If a relativistic perturbation is given to the apparent horizon, a similar calculation can also lead to a purely thermal spectrum, which corresponds to a modified temperature from the former. The first law of thermodynamics can also be constructed successfully at a new supersurface which has a small deviation from the apparent horizon. When the event horizon is thought as such a deviation from the apparent horizon, the expressions of the characteristic position and temperature are consistent with the previous result that asserts that thermodynamics should be built on the event horizon. It is concluded that the thermodynamics should be constructed on the apparent horizon exactly while the event horizon thermodynamics is just one of the perturbations near the apparent horizon.     

“Mass inflation” with lightlike branes

We discuss properties of a new class of p-brane models, describing intrinsically lightlike branes for any world-volume dimension, in various gravitational backgrounds of interest in the context of black hole physics. One of the characteristic features of these lightlike p-branes is that the brane tension appears as an additional nontrivial dynamical world-volume degree of freedom. Codimension one lightlike brane dynamics requires that bulk space with a bulk metric of spherically symmetric type must possess an event horizon which is automatically occupied by the lightlike brane while its tension evolves exponentially with time. The latter phenomenon is an analog of the well known “mass inflation” effect in black holes.      

Effective spacetime and Hawking radiation from a moving domain wall in a thin film of 3He-A

An event horizon for “relativistic” fermionic quasiparticles can be constructed in a thin film of superfluid ³He-A. The quasiparticles see an effective “gravitational” field which is induced by a topological soliton of the order parameter. Within the soliton the “speed of light” crosses zero and changes sign. When the soliton moves, two planar event horizons (black hole and white hole) appear, with a curvature singularity between them. Aside from the singularity, the effective spacetime is incomplete at future and past boundaries, but the quasiparticles cannot escape there because the nonrelativistic corrections become important as the blueshift grows, yielding “superluminal” trajectories. The question of Hawking radiation from the moving soliton is discussed but not resolved. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 11, 833–838 (10 December 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Horizons,Constraints, and Black Hole Entropy

Black hole entropy appears to be “universal”—many independent calculations, involving models with very different microscopic degrees of freedom, all yield the same density of states. I discuss the proposal that this universality comes from the behavior of the underlying symmetries of the classical theory. To impose the condition that a black hole be present, we must partially break the classical symmetries of general relativity, and the resulting Goldstone-boson-like degrees of freedom may account for the Bekenstein–Hawking entropy. In particular, I demonstrate that the imposition of a “stretched horizon” constraint modifies the algebra of symmetries at the horizon, allowing the use of standard conformal field theory techniques to determine the asymptotic density of states. The results reproduce the Bekenstein–Hawking entropy without any need for detailed assumptions about the microscopic theory.     

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.     

Horizon Entropy

Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here causal horizons. The root of this generalization is the local notion of horizon entropy density.     

On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions

All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [24] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein’s equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain “diophantine condition,” which holds except for a set of measure zero.     

Tunneling Effect of Two Horizons from a Reissner-Nordstrom Black Hole

The understanding of possible role played by the inner horizon of black holes in black hole thermodynamics is still somewhat incomplete. By adopting Damour-Ruffini method and the thin film model which is developed on the base of brick wall model suggested by ’t Hooft, we calculate the temperature and the entropy of the inner horizon of a R-N black hole. We conclude that the temperature of inner horizon is positive and the entropy of the inner horizon is proportional to the area of the inner horizon. In addition, the cut-off factor is 90β, which is same in calculation of the entropy of the outer horizon. So, we prove the existence of thermal characters of the inner horizon. Then, we discuss that if the contribution of the inner horizon is taken into account to the total entropy of the black hole, the Nernst theorem can be satisfied. At last, we study the tunneling effect including the inner horizon of the Reissner-Nordstrom black hole. We calculate the tunneling rate of the outer horizon Γ₊ and the inner horizon Γ₋. The total tunneling rate Γ should be the product of the rates of the outer and inner horizon, Γ=Γ₊⋅Γ₋. We find that the total tunneling rate is in agreement with the Parikh’s standard result, Γ→exp (ΔS _BH ), and there is no information loss.     

How many black holes fit on the head of a pin?

The Bekenstein–Hawking entropy of certain black holes can be computed microscopically in string theory by mapping the elusive problem of counting microstates of a strongly gravitating black hole to the tractable problem of counting microstates of a weakly coupled D-brane system, which has no event horizon, and indeed comfortably fits on the head of a pin. We show here that, contrary to widely held beliefs, the entropy of spherically symmetric black holes can easily be dwarfed by that of stationary multi-black-hole “molecules” of the same total charge and energy. Thus, the corresponding pin-sized D-brane systems do not even approximately count the microstates of a single black hole, but rather those of a zoo of entropically dominant multicentered configurations. Fourth Award in the 2007 Essay Competition of the Gravity Research Foundation.     

Non-Existence of Black Hole Solutions¶for a Spherically Symmetric, Static Einstein–Dirac–Maxwell System

We consider for j=?, … a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein–Dirac–Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner–Nordstr?m solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a “cloud” of spin-?-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon. Received: 2 November 1998 / Accepted: 23 February 1999     

Black Hole Thermodynamics of Hořava-Lifshitz and IR Modified Ho řava-Lifshitz Gravity Theory

Recently, a renormalizable model of gravity has been proposed by Hoř ava, which might be an ultraviolet completion of general relativity and it reduces to Einstein gravity with a non-vanishing cosmological constant in infrared approximation. Kehagias and Sfetsos have added a relevant operator proportional to the 3D geometry Ricci scalar to the original Hoř ava-Lifshitz theory action, which “softly” deviated from detailed-balance. This does not modify the ultraviolet properties of the theory. However, it modifies the infrared approximation and the Minkowski vacuum can be allowed in the infrared Hořava-Lifshitz gravity theory. The static spherical symmetric black hole solutions have been obtained in the Hořava-Lifshitz and infrared Hořava-Lifshitz gravity theory. Based on the metric of the black holes, Hawking radiation of massless scalar particles is investigated using Damour-Ruffini method. Then the black hole thermodynamics property will also be discussed.     

Conformal scalar propagation inside the Schwarzschild black hole

The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle–Hawking vacuum state for small values of the Schwarzschild radial coordinate above r = 2M is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below r = 2M and is an analytic expression which manifestly features its dependence on the background space–time geometry. This feature as well as the absence of any assumptions and prerequisites in the derivation render this Hartle–Hawking scalar propagator in the interior of the Schwarzschild black-hole geometry distinct from previous results. The two propagators obtained in the interior and in the exterior region of the Schwarzschild black hole are matched across the event horizon. The result of that match is a massless conformal scalar propagator in the Hartle–Hawking vacuum state which is shown to describe particle production by the Schwarzschild black hole.

“The future is not what it used to be!” From Alan Parker’s film “Angel Heart”     

Power-Law entropy corrected holographic dark energy model

Among various scenarios to explain the acceleration of the universe expansion, the holographic dark energy (HDE) model has got a lot of enthusiasm recently. In the derivation of holographic energy density, the area relation of the black hole entropy plays a crucial role. Indeed, the power-law corrections to entropy appear in dealing with the entanglement of quantum fields in and out the horizon. Inspired by the power-law corrected entropy, we propose the so-called “power-law entropy-corrected holographic dark energy” (PLECHDE) in this Letter. We investigate the cosmological implications of this model and calculate some relevant cosmological parameters and their evolution. We also briefly study the so-called “power-law entropy-corrected agegraphic dark energy” (PLECADE).     

Forks in the road, on the way to quantum gravity

In seeking to arrive at a theory of “quantum gravity,” one faces several choices among alternative approaches. I list some of these “forks in the road” and offer reasons for taking one alternative over the other. In particular, I advocate the following: the sum-over-histories framework for quantum dynamics over the “observable and state-vector” framework; relative probabilities over absolute ones; spacetime over space as the gravitational “substance” (4 over 3+1); a Lorentzian metric over a Riemannian (“Euclidean”) one; a dynamical topology over an absolute one; degenerate metrics over closed timelike curves to mediate topology change; “unimodular gravity” over the unrestricted functional integral; and taking a discrete underlying structure (the causal set) rather than the differentiable manifold as the basis of the theory. In connection with these choices, I also mention some results from unimodular quantum cosmology, sketch an account of the origin of black hole entropy, summarize an argument that the quantum mechanical measurement scheme breaks down for quantum field theory, and offer a reason why the cosmological constant of the present epoch might have a magnitude of around 10⁻¹²⁰ in natural units. This paper is the text of a talk given at the symposium on Directions in General Relativity held at the University of Maryland, College Park, Maryland, in May 1993 in honor of Dieter Brill and Charles Minser.