Outer Event Horizon Research Articles

Topological black holes in Lovelock-Born-Infeld gravity

In this paper, we present topological black holes of third order Lovelock gravity in the presence of cosmological constant and nonlinear electromagnetic Born-Infeld field. Depending on the metric parameters, these solutions may be interpreted as black hole solutions with inner and outer event horizons, an extreme black hole or naked singularity. We investigate the thermodynamics of asymptotically flat solutions and show that the thermodynamic and conserved quantities of these black holes satisfy the first law of thermodynamic. We also endow the Ricci flat solutions with a global rotation and calculate the finite action and conserved quantities of these class of solutions by using the counterterm method. We compute the entropy through the use of the Gibbs-Duhem relation and find that the entropy obeys the area law. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta, and the charge, and compute temperature, angular velocities, and electric potential and show that these thermodynamic quantities coincide with their values which are computed through the use of geometry. Finally, we perform a stability analysis for this class of solutions in both the canonical and the grand-canonical ensemble and show that the presence of a nonlinear electromagnetic field and higher curvature terms has no effect on the stability of the black branes, and they are stable in the whole phase space.

THERMODYNAMICS OF ROTATING BLACK BRANES IN GAUSS–BONNET–BORN–INFELD GRAVITY

Considering both the Gauss–Bonnet and the Born–Infeld terms, which are on a similar footing with regard to the string corrections on the gravity and electrodynamics sides respectively, we present a new class of rotating solutions in Gauss–Bonnet–Born–Infeld gravity with k rotation parameters. Although these solutions are not real in the whole space–time, they can be made real by a suitable transformation. Depending on the metric parameters, these real solutions may be interpreted as black brane solutions with inner and outer event horizons, an extreme black brane or naked singularity. We calculate the finite action and conserved quantities of the solutions by using the counterterm method, and find that these quantities do not depend on the Gauss–Bonnet parameter. We also compute the temperature, the angular velocities, the electric charge and the electric potential. Then, we calculate the entropy of the black brane through the use of the Gibbs–Duhem relation and show that it obeys the area law of entropy. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta and the charge, and show that the conserved and thermodynamic quantities satisfy the first law of thermodynamics. Finally, we perform a stability analysis in both the canonical and the grand-canonical ensemble and show that the presence of a nonlinear electromagnetic field has no effect on the stability of the black branes, which are stable in the whole phase space.

Thermodynamics of black holes in (n+1)-dimensional Einstein-Born-Infeld-dilaton gravity

We construct a new class of ($n+1$)-dimensional ($n\ensuremath{\ge}3$) black hole solutions in Einstein-Born-Infeld-dilaton gravity with Liouville-type potential for the dilaton field and investigate their properties. These solutions are neither asymptotically flat nor (anti)-de Sitter. We find that these solutions can represent black holes, with inner and outer event horizons, an extreme black hole, or a naked singularity provided the parameters of the solutions are chosen suitably. We compute the thermodynamic quantities of the black hole solutions and find that these quantities satisfy the first law of thermodynamics. We also perform a stability analysis and investigate the effect of dilaton on the stability of the solutions.

Thermodynamics of rotating black branes in (n+1)-dimensional Einstein-Born-Infeld gravity

We construct a new class of charged rotating solutions of (n+1)-dimensional Einstein-Born-Infeld gravity with cylindrical or toroidal horizons in the presence of cosmological constant and investigate their properties. These solutions are asymptotically (anti)-de Sitter and reduce to the solutions of Einstein-Maxwell gravity as the Born-Infeld parameters goes to infinity. We find that these solutions can represent black branes, with inner and outer event horizons, an extreme black brane or a naked singularity provided the parameters of the solutions are chosen suitably. We compute temperature, mass, angular momentum, entropy, charge and electric potential of the black brane solutions. We obtain a Smarr-type formula and show that these quantities satisfy the first law of thermodynamics. We also perform a stability analysis by computing the heat capacity and the determinant of Hessian matrix of mass of the system with infinite boundary with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles, and show that the system is thermally stable in the whole phase space. Also, we find that there exists an unstable phase when the finite size effect is taken into account.

Thermodynamics of charged rotating black branes in Brans-Dicke theory with quadratic scalar field potential

We construct a class of charged rotating solutions in (n+1)-dimensional Maxwell-Brans-Dicke theory with flat horizon in the presence of a quadratic potential and investigate their properties. These solutions are neither asymptotically flat nor (anti)-de Sitter. We find that these solutions can present black brane, with inner and outer event horizons, an extreme black brane or a naked singularity provided the parameters of the solutions are chosen suitably. We compute the finite Euclidean action through the use of counterterm method, and obtain the conserved and thermodynamic quantities by using the relation between the action and free energy in grand-canonical ensemble. We find that these quantities satisfy the first law of thermodynamics, and the entropy does not follow the area law.

Thermodynamics of rotating solutions in(n+1)-dimensional Einstein-Maxwell-dilaton gravity

We construct a class of charged, rotating solutions of $(n+1)$-dimensional Einstein-Maxwell-dilaton gravity with cylindrical or toroidal horizons in the presence of Liouville-type potentials and investigate their properties. These solutions are neither asymptotically flat nor (anti)-de Sitter. We find that these solutions can represent black brane, with inner and outer event horizons, an extreme black brane or a naked singularity provided the parameters of the solutions are chosen suitably. We also compute temperature, entropy, charge, electric potential, mass and angular momentum of the black brane solutions, and find that these quantities satisfy the first law of thermodynamics. We find a Smarr-type formula and perform a stability analysis by computing the heat capacity in the canonical ensemble. We find that the system is thermally stable when the coupling constant between the dilaton and matter field $\ensuremath{\alpha}\ensuremath{\le}1$, while for $\ensuremath{\alpha}>1$ the system has an unstable phase. This shows that the dilaton field makes the solution unstable, while it is stable even in Lovelock gravity.

Thermodynamics of rotating solutions in Gauss-Bonnet-Maxwell gravity and the counterterm method

By a suitable transformation, we present the $(n+1)$-dimensional charged rotating solutions of Gauss-Bonnet gravity with a complete set of allowed rotation parameters which are real in the whole spacetime. We show that these charged rotating solutions present black hole solutions with two inner and outer event horizons, extreme black holes, or naked singularities provided the parameters of the solutions are chosen suitable. Using the surface terms that make the action well defined for Gauss-Bonnet gravity and the counterterm method for eliminating the divergences in action, we compute finite action of the solutions. We compute the conserved and thermodynamical quantities through the use of free energy and the counterterm method, and find that the two methods give the same results. We also find that these quantities satisfy the first law of thermodynamics. Finally, we perform a stability analysis by computing the heat capacity and the determinant of Hessian matrix of mass with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles, and show that the system is thermally stable. This is commensurate with the fact that there is no Hawking-Page phase transition for black objects with zero curvature horizon.

Thermodynamics of rotating charged black branes in third order lovelock gravity and the counterterm method

We generalize the quasilocal definition of the stress-energy tensor of Einstein gravity to the case of third order Lovelock gravity, by introducing the surface terms that make the action well-defined. We also introduce the boundary counterterm that removes the divergences of the action and the conserved quantities of the solutions of third order Lovelock gravity with zero curvature boundary at constant $t$ and $r$. Then, we compute the charged rotating solutions of this theory in $n+1$ dimensions with a complete set of allowed rotation parameters. These charged rotating solutions present black hole solutions with two inner and outer event horizons, extreme black holes or naked singularities provided the parameters of the solutions are suitably chosen. We compute temperature, entropy, charge, electric potential, mass and angular momenta of the black hole solutions, and find that these quantities satisfy the first law of thermodynamics. We find a Smarr-type formula and perform a stability analysis by computing the heat capacity and the determinant of Hessian matrix of mass with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles, and show that the system is thermally stable. This is commensurate with the fact that there is no Hawking-Page phase transition for black objects with zero curvature horizon.

Thermodynamics of asymptotically flat charged black holes in third order Lovelock gravity

We present a new class of asymptotically flat charge static solutions in third order Lovelock gravity. These solutions present black hole solutions with two inner and outer event horizons, extreme black holes, or naked singularities provided the parameters of the solutions are chosen suitable. We find that the uncharged asymptotically flat solutions can present black holes with two inner and outer horizons. This kind of solution does not exist in Einstein or Gauss-Bonnet gravity, and it is a special effect in third order Lovelock gravity. We compute temperature, entropy, charge, electric potential, and mass of the black hole solutions, and find that these quantities satisfy the first law of thermodynamics. We also perform a stability analysis by computing the determinant of the Hessian matrix of the mass with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles, and show that there exists only an intermediate stable phase.

Total Quantum Statistical Entropy of Reissner–Nordstrom Black Holes:in Dirac Field Case

The total quantum statistical entropy of Reissner–Nordstromblack holes in Dirac field case is evaluated in this article.The space-time of the black holes is divided into three regions:region 1 (r>ro), region 2 (ro>r>ri), andregion 3 (ri>r>0), where ro is theradius of the outer event horizon, and ri is the radius of theinner event horizon. The total quantum statistical entropy ofReissner–Nordstrom black holes is S = S1+S2+S3,where Si (i = 1,2,3) is the entropy, contributed by regions1,2,3. The detailed calculation shows that S2 is neglectfullysmall. S1 = wt(π2/45)kb(Ao/ϵ2β3), S3 = −wt(π2/45)kb(Ai/ϵ2β3),where Ao and Ai are, respectively, the areas of theouter and inner event horizons, wt = 2s[1−2−(s+1)],s = d/2, d is the space-time dimension, here d = 4, s = 2. Asri approaches ro in the extreme case the totalquantum statistical entropy of Reissner–Nordstrom black holes approaches zero.

Gravitational energy of Kerr and Kerr anti-de Sitter space-times in the teleparallel geometry

In the context of the Hamiltonian formulation of the teleparallel equivalent of general relativity we compute the gravitational energy of Kerr and Kerr Anti-de Sitter (Kerr-AdS) space-times. The present calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. In each case, the energy is exactly computed for finite and arbitrary spacelike two-spheres, without any restriction on the metric parameters. In particular, we evaluate the energy at the outer event horizon of the black holes.

Charged rotating black branes in anti–de Sitter Einstein-Gauss-Bonnet gravity

We present a new class of charged rotating solutions in the Einstein-Gauss-Bonnet gravity with a negative cosmological constant. These solutions may be interpreted as black brane solutions with two inner and outer event horizons or an extreme black brane depending on the value of the mass parameter m. We also find that the Killing vectors are the null generators of the event horizon. The physical properties of the brane such as the temperature, the angular velocity, the entropy, the electric charge and potential are computed. We also compute the action and the Gibbs potential as a function of temperature and angular velocity for the uncharged solutions, and compute the angular momentum and the mass of the black brane through the use of the Gibbs potential. We show that these thermodynamic quantities satisfy the first law of thermodynamics. We also perform a local stability analysis of the asymptotically anti--de Sitter (AdS) uncharged rotating black brane in various dimensions and show that they are locally stable for the whole phase space both in the canonical and grand-canonical ensemble. We find that the thermodynamic properties of Gauss-Bonnet rotating black branes are completely the same as those without the Gauss-Bonnet term, although the two solutions are quite different.

Quantum Statistical Entropy of Non-extreme and Nearly ExtremeBlack Holes in Higher-Dimensional Space-Time

The free energy and entropy of Reissner-Nordstrom black holes inhigher-dimensional space-time are calculated by the quantum statistic methodwith a brick wall model. The space-time of the black holes is dividedinto three regions: region 1, (r > ro); region 2, (ro > r > ri);and region 3, (ri > r > 0), where ro is the radius of the outerevent horizon, and ri is the radius of the inner event horizon.Detailed calculation shows that the entropy contributed by region 2is zero, the entropy contributed by region 1 is positive andproportional to the outer event horizon area, the entropy contributed byregion 3 is negative and proportional to the inner event horizon area.The total entropy contributed by all the three regions is positive andproportional to the area difference between the outer and innerevent horizons. As ri approaches ro in the nearly extremecase, the total quantum statistical entropy approaches zero.

Total Quantum Statistical Entropy of Reissner-Nordstrom BlackHoles: Scalar Field Case

The total quantum statistical entropy of Reissner-Nordstrom (RN) blackholes is evaluated. The spacetime of the black holes is divided into threeregions - region 1, (r > ro); region 2, (ro > r > ri); and region 3, (ri > r > 0) - where ro is theradius of the outer event horizon, and ri is the radius of theinner event horizon. The total quantum statistical entropy of RN blackholes is S = S1 + S2 + S3, where Si (i = 1,2,3) is theentropy, contributed by region Si (i = 1,2,3). The detailedcalculation shows that S2≈0. S1 = (π2/45)[kbAo/ϵ2β3], S3 = -(π2/45)[kbAi/ϵ2β3], where Ao andAi are, respectively, the area of the outer and inner eventhorizons. Thus, as ri approaches ro, in the extreme casethe total quantum statistical entropy of RN black holes approaches zero.

ON THE QUANTIZATION OF CHARGED BLACK HOLES

The Wheeler–DeWitt equation for the wave function Ψ of the Schwarzschild black hole has been derived by Tomimatsu in the form of a Schrödinger equation, valid on the apparent horizon, using the two-dimensional Hamiltonian formalism of Hajicek and the radiating Vaidya metric. Here, the analysis is generalized to the Reissner–Nordström black hole. At constant charge Q, the evaporation rate is calculated from the solution for Ψ to be [Formula: see text], where k is a constant and [Formula: see text] are the radii of the outer event horizon and inner Cauchy horizon. In the extremal limit M → Q, however, the Hawking temperature [Formula: see text] tends to zero, suggesting, when the back reaction is taken into account, that the evaporation cannot occur this way and in agreement with the known discharging process of the hole via the Schwinger electron–positron pair-production mechanism. The more general charged dilaton black holes obtained from the theory L4 = [R4 - 2 (∇ Φ)2 - e-2aΦF2 ]/16π are also discussed, and it is explained why this quantization procedure cannot be applied when a is non-zero.

General rotating black holes in string theory: Greybody factors and event horizons

We derive the wave equation for a minimally coupled scalar field in the background of a general rotating five-dimensional black hole. It is written in a form that involves two types of thermodynamic variables, defined at the inner and outer event horizon, respectively. We model the microscopic structure as an effective string theory, with the thermodynamic properties of the left and right moving excitations related to those of the horizons. Previously known solutions to the wave equation are generalized to the rotating case, and their regime of validity is sharpened. We calculate the greybody factors and interpret the resulting Hawking emission spectrum microscopically in several limits. We find a U-duality invariant expression for the effective string length that does not assume a hierarchy between the charges. It accounts for the universal low-energy absorption cross-section in the general non-extremal case.

The classical stability of charged (1 + 1)-dimensional black holes

We discuss the classical stability of charged, two-dimensional, black holes. Here is the dilaton - Maxwell coupling parameter. We introduce a tachyon field, which arises from string theories, as a test field. One finds barrier - well type potentials surrounding the extremal black holes, which induce the classical instability. For the non-extremal black holes, one finds the barrier - well type potentials between the inner (Cauchy) horizon and outer (event) horizon, while potential barriers appear outside the event horizon. This implies that the inner horizons are unstable and the outer horizons are stable. However, this means that the non-extremal black holes themselves are stable because the stability of a black hole depends only on the outer horizon. Finally we mention that the instability of extremal black holes may originate from the unstable inner horizon of non-extremal black holes.

LIOUVILLE AND TODA SOLITONS IN M-THEORY

We study the general form of the equations for isotropic single-scalar, multi-scalar and dyonic p-branes in superstring theory and M-theory, and show that they can be cast into the form of Liouville, Toda (or Toda-like) equations. The general solutions describe non-extremal isotropic p-branes, reducing to the previously-known extremal solutions in limiting cases. In the non-extremal case, the dilatonic scalar fields are finite at the outer event horizon.

Behaviour of scalar perturbations of a Reissner-Nordström black hole inside the event horizon

This paper considers general scalar perturbations of a Reissner-Nordstrdöm black hole and examines the qualitative behaviour of these perturbations in the region between and on the inner and outer horizons ( r - ≼ r ≼ r + ). Initial data are specified in terms of the ingoing radiation crossing the outer (event) horizon. The only essential restriction on these data is that the radiation should not die away too slowly on this horizon. The resultant perturbations are shown to be bounded and continuous. It is also shown that if ũ is any retarded null coordinate such that ũ = 0 on the event horizon, then the perturbations tend to zero along lines of constant radius as ũ ↓ 0. In particular, all these properties hold for pertur­bations on the inner horizon. For certain types of scalar field (including the zero rest mass scalar field) perturbations vanish at the crossover point on the inner horizon.