Can One Understand Black Hole Entropy without Knowing Much about Quantum Gravity?

It is a common belief now that the explanation of the microscopic origin of the Bekenstein-Hawking entropy of black holes should be available in quantum gravity theory, whatever this theory will finally look like. Calculations of the entropy of certain black holes in string theory do support this point of view. In the last few years there also appeared a hope that an understanding of black hole entropy may be possible even without knowing the details of quantum gravity. The thermodynamics of black holes is a low energy phenomenon, so only a few general features of the fundamental theory may be really important. The aim of this review is to describe some of the proposals in this direction and the results obtained.Comment: 38 page

Spectral Geometry and One-loop Divergences on Manifolds with Conical Singularities

Geometrical form of the one-loop divergences induced by conical singularities of background manifolds is studied. To this aim the heat kernel asymptotic expansion on spaces having the structure $C_{\alpha}\times \Sigma$ near singular surface $\Sigma$ is analysed. Surface corrections to standard second and third heat coefficients are obtained explicitly in terms of angle $\alpha$ of a cone $C_{\alpha}$ and components of the Riemann tensor. These results are compared to ones to be already known for some particular cases. Physical aspects of the surface divergences are shortly discussed.Comment: preprint DSF-13/94, 13 pages, latex fil

Toroidal equilibria in spherical coordinates

The standard Grad-Shafranov equation for axisymmetric toroidal plasma equilibrium is customary expressed in cylindrical coordinates with toroidal contours, and through which benchmark equilibria are solved. An alternative approach to cast the Grad-Shafranov equation in spherical coordinates is presented. This equation, in spherical coordinates, is examined for toroidal solutions to describe low $\beta$ Solovev and high $\beta$ plasma equilibria in terms of elementary functions

Heat-kernel Coefficients and Spectra of the Vector Laplacians on Spherical Domains with Conical Singularities

The spherical domains $S^d_\beta$ with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with such a kind of singular points. In this paper the vector Laplacian on $S^d_\beta$ is considered and its spectrum is calculated exactly for any dimension $d$. This enables one to find the Schwinger-DeWitt coefficients of this operator by using the residues of the $\zeta$-function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on $S^d_\beta$ and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the one-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to the first order in the conical deficit angle.Comment: plain LaTeX, 23 pp., revised version, a misprint in expressions (1.8) and (4.38) of the second heat coefficient for the vector Laplacian is corrected. No other change

Finite Temperature Effective Potential for Gauge Models in de Sitter Space

The one-loop effective potential for gauge models in static de Sitter space at finite temperatures is computed by means of the $\zeta$--function method. We found a simple relation which links the effective potentials of gauge and scalar fields at all temperatures. In the de Sitter invariant and zero-temperature states the potential for the scalar electrodynamics is explicitly obtained, and its properties in these two vacua are compared. In this theory the two states are shown to behave similarly in the regimes of very large and very small radii a of the background space. For the gauge symmetry broken in the flat limit ($a \to \infty$) there is a critical value of a for which the symmetry is restored in both quantum states. Moreover, the phase transitions which occur at large or at small a are of the first or of the second order, respectively, regardless the vacuum considered. The analytical and numerical analysis of the critical parameters of the above theory is performed. We also established a class of models for which the kind of phase transition occurring depends on the choice of the vacuum.Comment: 23 pages, LaTeX, 5 figure.ep

Spectral Asymptotics of Eigen-value Problems with Non-linear Dependence on the Spectral Parameter

We study asymptotic distribution of eigen-values $\omega$ of a quadratic operator polynomial of the following form $(\omega^2-L(\omega))\phi_\omega=0$, where $L(\omega)$ is a second order differential positive elliptic operator with quadratic dependence on the spectral parameter $\omega$. We derive asymptotics of the spectral density in this problem and show how to compute coefficients of its asymptotic expansion from coefficients of the asymptotic expansion of the trace of the heat kernel of $L(\omega)$. The leading term in the spectral asymptotics is the same as for a Laplacian in a cavity. The results have a number of physical applications. We illustrate them by examples of field equations in external stationary gravitational and gauge backgrounds.Comment: latex, 20 page