Electrically charged spherical matter shells in higher dimensions: Entropy, thermodynamic stability, and the black hole limit

TL;DR

This paper investigates the thermodynamics and stability of electrically charged spherical shells in higher-dimensional spacetimes, deriving conditions under which these shells are stable and connecting their properties to black hole thermodynamics.

Contribution

It introduces a model for charged shells in higher dimensions with specific equations of state, linking shell thermodynamics to black hole limits and stability criteria.

Findings

Shell entropy proportional to gravitational area raised to power a

Stable configurations for 0<a≤(d-3)/(d-2), unstable otherwise

Black hole limit reproduces known d-dimensional Reissner-Nordström thermodynamics

Abstract

We study the thermodynamic properties of a static electrically charged spherical thin shell in $d$ dimensions by imposing the first law of thermodynamics on the shell. The shell is at radius $R$, inside it the spacetime is Minkowski, and outside it the spacetime is Reissner-Nordstr\"om. We obtain that the shell thermodynamics is fully described by giving two additional reduced equations of state, one for the temperature and another for the electrostatic potential. We choose the equation of state for the temperature as a power law in the gravitational radius $r_+$ with exponent $a$, such that the $a=1$ case gives the temperature of a shell with black hole thermodynamic properties, and for the electrostatic potential we choose an equation of state characteristic of a Reissner-Nordstr\"om black hole spacetime. The entropy of the shell is found to be proportional to $A_+^a$, where $A_+$ is the gravitational area corresponding to $r_+$, with $a>0$. We are then able to perform the black hole limit $R=r_+$, find the Smarr formula, and recover the thermodynamics of a $d$-dimensional Reissner-Nordstr\"om black hole. We study the intrinsic thermodynamic stability of the shell with the chosen equations of state. We obtain that for $0<a\leq \frac{d-3}{d-2}$ all the configurations of the shell are thermodynamically stable, for $\frac{d-3}{d-2}<a<1$ stability depends on the mass and electric charge, and for $a>1$ all the configurations are unstable, except for the shell at its own gravitational radius, which is marginally stable. We rewrite the stability conditions in terms of laboratory variables. We find that the sufficient condition for the stability of these shells is when the isothermal electric susceptibility $\chi_{p,T}$ is positive, marginal stability happening when this quantity is infinite, and instability arising for configurations with a negative electric susceptibility.