Black holes and hot shells in the Euclidean path integral approach to quantum gravity

Abstract

We study a spherical black hole surrounded by a hot self-gravitating thin shell in the canonical ensemble, i.e. a black hole and a hot thin shell inside a heat reservoir acting as a boundary with its area and temperature fixed. To work out the quantum statistical mechanics partition function of this matter-black hole system, from which the thermodynamics of the system follows, we use the Euclidean path integral approach to quantum gravity that identifies the path integral of the gravitational system with the partition function itself. In a semiclassical evaluation of the path integral, one needs to compute the classical action of the system. From the action, one finds the result that the total entropy, given by the sum of black hole and matter entropies, is a function of the gravitational radius of the system alone. So, the black hole inside the shell has no direct influence on the total entropy. One also finds the free energy which is equal to the action times the temperature, the thermodynamic energy, and the temperature stratification along the system. Another important result is that the heat reservoir temperature is composed of a free function of the gravitational radius of the system, which acts as a reduced temperature equation of state, divided by the redshift function at the reservoir. Upon the specification of the reduced temperature, the solutions for the gravitational radii of the system compatible with the boundary data can be found. In addition, it is found that the black hole inside the shell has two possible horizon radii. The first law of thermodynamics is then identified, and it is shown that the first law is satisfied by the system as whole, it is realized by the matter in the hot shell, and it is also applicable to the black hole. The thermodynamic stability analysis is performed through the calculation of the system’s heat capacity. By specifying the available temperature free function as the Hawking temperature equation of state of the gravitational radius of the system, which itself is not a black hole, one finds a remarkable exact mechanical and thermodynamic solution. With the exact solution in hand one establishes that pure black hole spaces, hot shell with a black hole spaces, pure hot shell spaces, and hot flat spaces are phases that cohabit in the ensemble, with some of them acting as thermodynamic mimickers. This exact thermodynamic solution for a black hole with a self-gravitating hot shell is not only of interest in itself, but can also be seen as a model to situations involving black holes interacting with hot gravitons and other hot particles. The study of the high temperature limits for the system also reveals several important aspects.