Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

In this overview of selected aspects of the black hole attractor mechanism, after introducing the necessary foundations, we examine the relationship between two ways to describe the attractor phenomenon in four‐dimensional N = 2 supergravity: the entropy function and the black hole potential. We also exemplify their practical application to finding solutions to the attractor equations for a conifold prepotential. Next we describe an extension of the original definition of the entropy function to a class of rotating black holes in five‐dimensional N = 2 supergravity based on cubic polynomials, exploiting a connection between four‐ and five‐dimensional black holes. This link allows further the derivation of five‐dimensional first‐order differential flow equations governing the profile of the fields from infinity to the event horizon and construction of non‐supersymmetric interpolating solutions in four dimensions by dimensional reduction. Finally, since four‐dimensional extremal black holes in N = 2 supergravity can be viewed as certain two‐dimensional string compactifications with fluxes, we discuss implications of the conifold example in the context of the entropic principle, which postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes.     

On the Møller Energy Associated with Black Holes

In this paper, we consider both Einstein's theory of general relativity and the teleparallel gravity (the tetrad theory of gravitation) analogs of the energy-momentum definition of Møller in order to explicitly evaluate the energy distribution (due to matter and fields including gravity) associated with a general black hole model which includes several well-known black holes. To calculate the special cases of energy distribution, here we consider eight different types of black hole models such as anti-de Sitter Cmetric with spherical topology, charged regular black hole, conformal scalar dyon black hole, dyadosphere of a charged black hole, regular black hole, charged topological black hole, charged massless black hole with a scalar field, and the Schwarzschild-de Sitter space-time. Our teleparallel gravitational result is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in teleparallel equivalent of general relativity but also in any teleparallel model. This paper also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the Møller energy-momentum complex is the powerful concept to calculate energy distribution in a given space-time.     

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.     

Notes on Black Hole Phase Transitions

In these notes we present a summary of existing ideas about phase transitions of black hole spacetimes in semiclassical gravity and offer some thoughts on three possible scenarios or mechanisms by which these transitions could take place. We begin with a review of the thermodynamics of a black hole system and emphasize that the phase transition is driven by the large entropy of the black hole horizon. Our first theme is illustrated by a quantum atomic black hole system, generalizing to finite-temperature a model originally offered by Bekenstein. In this equilibrium atomic model, the black hole phase transition is realized as the abrupt excitation of a high energy state, suggesting analogies with the study of two-level atoms. Our second theme argues that the black hole system shares similarities with the defect-mediated Kosterlitz–Thouless transition in condensed matter. These similarities suggest that the black hole phase transition may be more fully understood by focusing upon the dynamics of black holes and white holes, the spacetime analogy of vortex and antivortex topological defects. Finally, we compare the black hole phase transition to another transition driven by an (exponentially) increasing density of states, the Hagedorn transition first found in hadron physics in the context of dual models or the old string theory. In modern string theory the Hagedorn transition is linked by the Maldacena conjecture to the Hawking–Page black hole phase transition in Anti-de Sitter (AdS) space, as observed by Witten. Thus, the dynamics of the Hagedorn transition may yield insight into the dynamics of the black hole phase transition. We argue that characteristics of the Hagedorn transition are already contained within the dynamics of classical string systems. Our third theme points to carrying out a full nonperturbative and nonequilibrium analysis of the large N behavior of classical SU(N) gauge theories to understand its Hagadorn transition. By invoking the Maldacena conjecture we can then gain valuable insight into black hole phase transitions in AdS space.     

ROTATING RINDLER SPACE TIME WITH CONSTANT ANGULAR VELOCITY

A new space time metric is derived from Kerr metric if its mass and location approach to infinite in an appropriate way. The new space-time is an infinitesimal neighborhood nearby one of the two horizon poles of an infinite Kerr black hole. In other words, it is the second order infinitesimal neighborhood nearby one of the two horizon poles of a Kerr black hole. It is flat and has event horizon and infinite red shift surface. We prove that it is a rotating Rindler space time with constant angular velocity.     

Fine micro-thermal structures for Reissner-Nordstr?m black hole

We solve the condundrum on whether the molecules of the Reissner-Nordstr?m black hole interact through the Ruppeiner thermodynamic geometry, basing our study on the concept of the black hole molecule proposed in [Phys. Rev. Lett. 115 (2015) 111302] and choosing the appropriate extensive variables. Our results show that the Reissner-Nordstr?m black hole is indeed an interaction system that may be dominated by repulsive interaction. More importantly, with the help of a novel quantity, namely the thermal-charge density, we describe the fine micro-thermal structures of the Reissner-Nordstr?m black hole in detail. Three different phases are presented, namely the free, interactive, and balanced phases. The thermal-charge density plays a role similar to the order parameter, and the back hole undergoes a new phase transition between the free phase and interactive phase. The competition between the free phase and interactive phase exists, which leads to extreme behavior of the temperature of the Reissner-Nordstr?m black hole. For the extreme Reissner-Nordstr?m black hole, the entire system is completely in the interactive phase. More importantly, we provide the thermodynamic micro-mechanism for the formation of the naked singularity of the Reissner-Nordstr?m black hole.     

A divergent volume for black holes calls for no ‘firewall’

The presumption that Hawking radiations are thermally distributed can be considered to result from their entanglement with the internal degrees of freedom for a black hole. This leads to the‘firewall' paradox if unitary evolution continues into Page's time when a black hole evaporates away half of its initial entropy. However, if the interior of a black hole houses sufficient degrees of freedom to maintain entanglement with the outside at all times, unitarity can be preserved during the complete radiation process and no firewall will be required. This paper proposes a scenario that rescinds the firewall by introducing the concept of volume for a black hole. Based on the operational definition by Christodoulou and Rovelli(Christodoulou and Rovelli 2015 Phys. Rev. D 91 064046), we show that the volume and its associated entropy for a collapsed black hole diverges if the final evaporation stage is treated using noncommutative space. This implicates the interior of a black hole possesses adequate space to store information for a black hole of any mass, like the inside of a ‘magician's bag’.     

Thermodynamics of Gauss–Bonnet black holes revisited

We investigate the Gauss–Bonnet black hole in five dimensional anti-de Sitter spacetimes (GBAdS). We analyze all thermodynamic quantities of the GBAdS, which is characterized by the Gauss–Bonnet coupling c and mass M, comparing with those of the Born–Infeld-AdS (BIAdS), Reissner–Norstr?m-AdS black holes (RNAdS), Schwarzschild-AdS (SAdS), and BTZ black holes. For c<0 we cannot obtain the black hole with positively definite thermodynamic quantities of mass, temperature, and entropy, because the entropy does not satisfy the area law. On the other hand, for c>0, we find the BIAdS-like black hole, showing that the coupling c plays the role of a pseudo-charge. Importantly, we could not obtain the SAdS in the limit of c→0, which means that the GBAdS is basically different from the SAdS. In addition, we clarify the connections between thermodynamic and dynamical stability. Finally, we also conjecture that if a black hole is big and thus globally stable, its quasi-normal modes may take on analytic expressions.     

R-Charged Black Hole and Holographic Superfluid

In this paper we consider an interesting model of superfluid and use AdS/CFT correspondence to extract sound modes. We assume that dual picture of superfluid is the R-charged black hole with two equal charges. By using hydrodynamic variables of such a black hole we obtain first, second and fourth sound modes as a function of black hole charge.     

Thermodynamics and phase transition of the Reissner–Nordstr?m black hole surrounded by quintessence

We investigate thermodynamics and phase transition of the Reissner–Nordstr?m black hole surrounded by quintessence. Using thermodynamical laws of black holes, we derive the expressions of some thermodynamics quantities for the Reissner–Nordstr?m black hole surrounded by quintessence. The variations of the temperature and heat capacity with the entropy were plotted for different values of the state parameter related to the quintessence, ω _q , and the normalization constant related to the density of quintessence c. We show that when varying the entropy of the black hole a phase transition is observed in the black hole. Moreover, when increasing the density of quintessence, the transition point is shifted to lower entropy and the temperature of the black hole decreases.     

Reversible Carnot cycle outside a black hole

A Carnot cycle outside a Schwarzschild black hole is investigated in detail. We propose a reversible Carnot cycle with a black hole being the cold reservoir. In our model, a Carnot engine operates between a hot reservoir with temperature T₁ and a black hole with Hawking temperature T_H. By naturally extending the ordinary Carnot cycle to the black hole system, we show that the thermal efficiency for a reversible process can reach the maximal efficiency 1-T_H/T₁. Consequently, black holes can be used to determine the thermodynamic temperature by means of the Carnot cycle. The role of the atmosphere around the black hole is discussed. We show that the thermal atmosphere provides a necessary mechanism to make the process reversible.     

Letter: Quantum Statistical Entropy of d-Dimensional Horowitz–Strominger Black Hole

By using the method of quantum statistics, we derive directly the partition functions of bosonic and fermionic field in the d-dimensional Horowitz-Strominger black hole. The statistical entropy of black hole is obtained by an improved brick—wall method. When we choose proper parameter in our results, we can obtain that the entropy of the black hole is proportional to the area of the horizon. In our result, there don't exist the left out term and divergent logarithmic term given in the original brick—wall method. We avoid the difficulty in solving the wave equation of scalar and Dirac field. And we offer a simple and direct way of studying entropy of the higher-dimensional complicated black hole.     

A black hole spectral signature

An accreting black hole is, by definition, characterized by the drain. Namely, the matter falls into a black hole much the same way as water disappears down a drain — matter goes in and nothing comes out. As this can only happen in a black hole, it provides a way to see “a black hole”, an unique observational signature. The accretion proceeds almost in a free-fall manner close to the black hole horizon, where the strong gravitational field dominates the pressure forces. In this paper we present analytical calculations and Monte-Carlo simulations of the specific features of X-ray spectra formed as a result of upscattering of the soft (disk) photons in the converging inflow (CI) into the black hole. The full relativistic treatment has been implemented to reproduce these spectra. We show that spectra in the soft state of black hole systems (BHS) can be described as the sum of a thermal (disk) component and the convolution of some fraction of this component with the CI upscattering spread (Greens) function. The latter boosted photon component is seen as an extended power-law at energies much higher than the characteristic energy of the soft photons. We demonstrate the stability of the power spectral index over a wide range of the plasma temperature 0 – 10 keV and mass accretion rates (higher than 2 in Eddington units). We also demonstrate that the sharp high energy cutoff occurs at energies of 200–400 keV which are related to the average energy of electrons m_ec² impinging upon the event horizon. The spectrum is practically identical to the standard thermal Comptonization spectrum when the CI plasma temperature is getting of order of 50 keV (the typical ones for the hard state of BHS). In this case one can see the effect of the bulk motion only at high energies where there is an excess in the CI spectrum with respect to the pure thermal one. Furthermore we demonstrate that the change of spectral shapes from the soft X-ray state to the hard X-ray state is clearly to be related with the temperature of the bulk flow. In other words the effect of the bulk Comptonization compared to the thermal one is getting stronger when the plasma temperature drops below 10 keV. We clearly demonstrate that these spectra emerging from the converging inflow are a inevitable stamp of the BHS where the strong gravitational field dominates the pressure forces.     

Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Gravitational radiation from an extreme Kerr black hole

We analytically investigate gravitational radiation induced by a test particle falling into an extreme Kerr black hole. Assuming the radiation is dominated by the infinite sequence of quasi-normal modes which has the limiting frequencym/(2M), wherem is an azimuthal eigenvalue andM is the mass of the black hole. we find the radiated energy diverges logarithmically in time. Then we evaluate the back reaction to the black hole by appealing to the energy and angular momentum conservation laws. We find the radiation has a tendency to increase the ratio of the angular momentum to mass of the black hole, which is completely different from the non-extreme case, while the contribution of the test particle is to decrease it.     

Availability of the thermodynamics and weak cosmic censorship conjecture for a charged AdS black hole in the large dimension limit

The thermodynamics and the weak cosmic censorship conjecture (WCCC) in a high dimensional RN ? AdS_d+?1 black hole with energy-momentum relation are investigated by absorbing a charged particle in the phase space. In the RN ? AdS_d+?1 space-time, the cosmological constant Λ is treated as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume. We use the energy-momentum relation of the absorbed particle to discuss the thermodynamics of the RN ? AdS_d+?1 black hole and to prove the WCCC in the phase space. Based on this assumption, we find that the first law and the second law of thermodynamics are satisfied in normal phase space. On the other hand, in the extend phase space, the first law is satisfied and the second law is violated. Then we study the WCCC in the phase space, we find that the WCCC is satisfied for an extreme black and a near-extreme black hole in the normal phase space. In the extend phase space, the WCCC is satisfied for an extreme black hole and unidentified for a near-extreme black hole.

     

Black hole physics in globally hyperbolic space-times

The usual definition of a black hole is modified to make it applicable in a globally hyperbolic space-time. It is shown that in a closed globally hyperbolic universe the surface area of a black hole must eventually decrease. The implications of this breakdown of the black hole area theorem are discussed in the context of thermodynamics and cosmology. A modified definition of surface gravity is also given for non-stationary universes. The limitations of these concepts are illustrated by the explicit example of the Kerr-Vaidya metric.     

A new class of exact hairy black hole solutions

We present a new class of black hole solutions with a minimally coupled scalar field in the presence of a negative cosmological constant. We consider an one-parameter family of self-interaction potentials parametrized by a dimensionless parameter g. When g = 0, we recover the conformally invariant solution of the Martinez–Troncoso–Zanelli (MTZ) black hole. A non-vanishing g signals the departure from conformal invariance. Thermodynamically, there is a critical temperature at vanishing black hole mass, where a higher-order phase transition occurs, as in the case of the MTZ black hole. Additionally, we obtain a branch of hairy solutions which undergo a first-order phase transition at a second critical temperature which depends on g and it is higher than the MTZ critical temperature. As g → 0, this second critical temperature diverges.     

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.     

W-hairs of the black holes in three-dimensional spacetime

In a previous publication, we claimed that a black hole can be considered as a topological insulator. A direct consequence of this claim is that their symmetries should be related. In this paper, we give a representation of the near-horizon symmetry algebra of the BTZ black hole using the W1+∞ symmetry algebra of the topological insulator in three-dimensional spacetime. Based on the W1+∞ algebra, we count the number of the microstates of the BTZ black holes and obtain the Bekenstein-Hawking entropy.