`Thermodynamics' of Minimal Surfaces and Entropic Origin of Gravity

Deformations of minimal surfaces lying in constant time slices in static space-times are studied. An exact and universal formula for a change of the area of a minimal surface under shifts of nearby point-like particles is found. It allows one to introduce a local temperature on the surface and represent variations of its area in a thermodynamical form by assuming that the entropy in the Planck units equals the quarter of the area. These results provide a strong support to a recent hypothesis that gravity has an entropic origin, the minimal surfaces being a sort of holographic screens. The gravitational entropy also acquires a definite physical meaning related to quantum entanglement of fundamental degrees of freedom across the screen.Comment: 12 pages, 1 figur

Statistical Mechanics on Axially-symmetric Space-times with the Killing Horizon and Entropy of Rotating Black Holes in Induced Gravity

We develop a method for computing the free-energy of a canonical ensemble of quantum fields near the horizon of a rotating black hole. We show that the density of energy levels of a quantum field on a stationary background can be related to the density of levels of the same field on a fiducial static space-time. The effect of the rotation appears in the additional interaction of the "static" field with a fiducial abelian gauge-potential. The fiducial static space-time and the gauge potential are universal, i.e., they are determined by the geometry of the given physical space-time and do not depend on the spin of the field. The reduction of the stationary axially symmetric problem to the static one leads to a considerable simplification in the study of statistical mechanics and we use it to draw a number of conclusions. First, we prove that divergences of the entropy of scalar and spinor fields at the horizon in the presence of rotation have the same form as in the static case and can be removed by renormalization of the bare black hole entropy. Second, we demonstrate that statistical-mechanical representation of the Bekenstein-Hawking entropy of a black hole in induced gravity is universal and does not depend on the rotation.Comment: 22 page

Thorny Spheres and Black Holes with Strings

We consider thorny spheres, that is 2-dimensional compact surfaces which are everywhere locally isometric to a round sphere $S^2$ except for a finite number of isolated points where they have conical singularities. We use thorny spheres to generate, from a spherically symmetric solution of the Einstein equations, new solutions which describe spacetimes pierced by an arbitrary number of infinitely thin cosmic strings radially directed. Each string produces an angle deficit proportional to its tension, while the metric outside the strings is a locally spherically symmetric solution. We prove that there can be arbitrary configurations of strings provided that the directions of the strings obey a certain equilibrium condition. In general this equilibrium condition can be written as a force-balance equation for string forces defined in a flat 3-space in which the thorny sphere is isometrically embedded, or as a constraint on the product of holonomies around strings in an alternative 3-space that is flat except for the strings. In the case of small string tensions, the constraint equation has the form of a linear relation between unit vectors directed along the string axes.Comment: 37 pages, 11 figure

Radial geodesics as a microscopic origin of black hole entropy. I: Confined under the Schwarzschild horizon

Causal radial geodesics with a positive interval in the Schwarzschild metric include a subset of trajectories completely confined under a horizon, which compose a thermal statistical ensemble with the Hawking-Gibbons temperature. The Bekenstein--Hawking entropy is given by an action at corresponding geodesics of particles with a summed mass equal to that of black hole in the limit of large mass.Comment: 16 pages, 12 eps-figures, iopart class, tought experiment (p.7) adde

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

Thermodynamics, Euclidean Gravity and Kaluza-Klein Reduction

The aim of this paper is to find out a correspondence between one-loop effective action $W_E$ defined by means of path integral in Euclidean gravity and the free energy $F$ obtained by summation over the modes. The analysis is given for quantum fields on stationary space-times of a general form. For such problems a convenient procedure of a "Wick rotation" from Euclidean to Lorentzian theory becomes quite non-trivial implying transition from one real section of a complexified space-time manifold to another. We formulate conditions under which $F$ and $W_E$ can be connected and establish an explicit relation of these functionals. Our results are based on the Kaluza-Klein method which enables one to reduce the problem on a stationary space-time to equivalent problem on a static space-time in the presence of a gauge connection. As a by-product, we discover relation between the asymptotic heat-kernel coefficients of elliptic operators on a $D$ dimensional stationary space-times and the heat-kernel coefficients of a $D-1$ dimensional elliptic operators with an Abelian gauge connection.Comment: latex file, 22 page