Heat capacity of a self-gravitating spherical shell of radiations

We study the heat capacity of a static system of self-gravitating radiations analytically in the context of general relativity. To avoid the complexity due to a conical singularity at the center, we excise the central part and replace it with a regular spherically symmetric distribution of matters of which specifications we are not interested in. We assume that the mass inside the inner boundary and the locations of the inner and the outer boundaries are given. Then, we derive a formula relating the variations of physical parameters at the outer boundary with those at the inner boundary. Because there is only one free variation at the inner boundary, the variations at the outer boundary are related, which determines the heat capacity. To get an analytic form for the heat capacity, we use the thermodynamic identity delta S-rad = beta delta M-rad additionally, which is derived from the variational relation of the entropy formula with the restriction that the mass inside the inner boundary does not change. Even if the radius of the inner boundary of the shell goes to zero, in the presence of a central conical singularity, the heat capacity does not go to the form of the regular sphere. An interesting discovery is that another legitimate temperature can be defined at the inner boundary which is different from the asymptotic one beta(-1).