Two-dimensional quantum-corrected black hole in a finite size cavity

We consider the gravitation-dilaton theory (not necessarily exactly solvable), whose potentials represent a generic linear combination of an exponential and linear functions of the dilaton. A black hole, arising in such theories, is supposed to be enclosed in a cavity, where it attains thermal equilibrium, whereas outside the cavity the field is in the Boulware state. We calculate quantum corrections to the Hawking temperature $T_{H}$, with the contribution from the boundary taken into account. Vacuum polarization outside the shell tend to cool the system. We find that, for the shell to be in the thermal equilibrium, it cannot be placed too close to the horizon. The quantum corrections to the mass due to vacuum polarization vanish in spite of non-zero quantum stresses. We discuss also the canonical boundary conditions and show that accounting for the finiteness of the system plays a crucial role in some theories (e.g., CGHS), where it enables to define the stable canonical ensemble, whereas consideration in an infinite space would predict instability.Comment: 21 pages. In v.2 misprints corrected. To appear in Phys. Rev.

General approach to potentials with two known levels

We present the general form of potentials with two given energy levels $E_{1}$, $E_{2}$ and find corresponding wave functions. These entities are expressed in terms of one function $\xi (x)$ and one parameter $\Delta E=E_{2}$-$E_{1}$. We show how the quantum numbers of both levels depend on properties of the function $\xi (x)$. Our approach does not need resorting to the technique of supersymmetric (SUSY) quantum mechanics but automatically generates both the potential and superpotential.Comment: 14 pages, REVTeX 3.0. In v.2 misprints and inaccuracies in presentation corrected, discussion of 3-dim. case added. In v.3 misprint in eq. 41, several typos and inaccuracies in English corrected. To be published in J. of Phys. A: Math. Ge

Quasi-exactly solvable quartic Bose Hamiltonians

We consider Hamiltonians, which are even polynomials of the forth order with the respect to Bose operators. We find subspaces, preserved by the action of Hamiltonian These subspaces, being finite-dimensional, include, nonetheless, states with an \QTR{it}{infinite} number of quasi-particles, corresponding to the original Bose operators. The basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labeling the representation. In some particular degenerate cases they turn (up to the power factor) into the trigonometric or hyperbolic functions, Bessel functions or combinations of the exponent and Hermit polynomials. We find explicitly the relationship between coefficients at different powers of Bose operators that ensure quasi-exact solvability of Hamiltonian.Comment: 21 pages, REVTeX 3.0, no figures. In v.2 couple of misprints in English corrected. To be published in J. Phys. A: Math. Ge

Acceleration of particles by rotating black holes: near-horizon geometry and kinematics

Nowadays, the effect of infinite energy in the centre of mass frame due to near-horizon collisions attracts much attention.We show generality of the effect combining two seemingly completely different approaches based on properties of a particle with respect to its local light cone and calculating its velocity in the locally nonrotaing frame directly. In doing so, we do not assume that particles move along geodesics. Usually, a particle reaches a horizon having the velocity equals that of light. However, there is also case of "critical" particles for which this is not so. It is just the pair of usual and critical particles that leads to the effect under discussion. The similar analysis is carried out for massless particles. Then, critical particles are distinguishable due to the finiteness of local frequency. Thus, both approach based on geometrical and kinematic properties of particles moving near the horizon, reveal the universal character of the effect.Comment: 8 page

Unified approach to the entropy of an extremal rotating BTZ black hole: Thin shells and horizon limits

Using a thin shell, the first law of thermodynamics, and a unified approach, we study the thermodymanics and find the entropy of a (2+1)-dimensional extremal rotating Ba\~{n}ados-Teitelbom-Zanelli (BTZ) black hole. The shell in (2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and rotating, with the inner region being a ground state of the anti-de Sitter (AdS) spacetime and the outer region being the rotating BTZ spacetime. The extremal BTZ rotating black hole can be obtained in three different ways depending on the way the shell approaches its own gravitational or horizon radius. These ways are explicitly worked out. The resulting three cases give that the BTZ black hole entropy is either the Bekenstein-Hawking entropy, $S=\frac{A_+}{4G}$, or it is an arbitrary function of $A_+$, $S=S(A_+)$, where $A_+=2\pi r_+$ is the area, i.e., the perimeter, of the event horizon in (2+1) dimensions. We speculate that the entropy of an extremal black hole should obey $0\leq S(A_+)\leq\frac{A_+}{4G}$. We also show that the contributions from the various thermodynamic quantities, namely, the mass, the circular velocity, and the temperature, for the entropy in all three cases are distinct. This study complements the previous studies in thin shell thermodynamics and entropy for BTZ black holes. It also corroborates the results found for a (3+1)-dimensional extremal electrically charged Reissner-Nordstr\"om black hole.Comment: 8 pages, 1 table, no figur

Quasiblack holes with pressure: General exact results

A quasiblack hole is an object in which its boundary is situated at a surface called the quasihorizon, defined by its own gravitational radius. We elucidate under which conditions a quasiblack hole can form under the presence of matter with nonzero pressure. It is supposed that in the outer region an extremal quasihorizon forms, whereas inside, the quasihorizon can be either nonextremal or extremal. It is shown that in both cases, nonextremal or extremal inside, a well-defined quasiblack hole always admits a continuous pressure at its own quasihorizon. Both the nonextremal and extremal cases inside can be divided into two situations, one in which there is no electromagnetic field, and the other in which there is an electromagnetic field. The situation with no electromagnetic field requires a negative matter pressure (tension) on the boundary. On the other hand, the situation with an electromagnetic field demands zero matter pressure on the boundary. So in this situation an electrified quasiblack hole can be obtained by the gradual compactification of a relativistic star with the usual zero pressure boundary condition. For the nonextremal case inside the density necessarily acquires a jump on the boundary, a fact with no harmful consequences whatsoever, whereas for the extremal case the density is continuous at the boundary. For the extremal case inside we also state and prove the proposition that such a quasiblack hole cannot be made from phantom matter at the quasihorizon. The regularity condition for the extremal case, but not for the nonextremal one, can be obtained from the known regularity condition for usual black holes.Comment: 18 pages, no figures; improved introduction, added references, calculations better explaine

Entropy of an extremal electrically charged thin shell and the extremal black hole

There is a debate as to what is the value of the the entropy $S$ of extremal black holes. There are approaches that yield zero entropy $S=0$, while there are others that yield the Bekenstein-Hawking entropy $S=A_+/4$, in Planck units. There are still other approaches that give that $S$ is proportional to $r_+$ or even that $S$ is a generic well-behaved function of $r_+$. Here $r_+$ is the black hole horizon radius and $A_+=4\pi r_+^2$ is its horizon area. Using a spherically symmetric thin matter shell with extremal electric charge, we find the entropy expression for the extremal thin shell spacetime. When the shell's radius approaches its own gravitational radius, and thus turns into an extremal black hole, we encounter that the entropy is $S=S(r_+)$, i.e., the entropy of an extremal black hole is a function of $r_+$ alone. We speculate that the range of values for an extremal black hole is $0\leq S(r_+) \leq A_+/4$.Comment: 11 pages, minor changes, added references, matches the published versio