Thermodynamics and entropy of self-gravitating matter shells and black holes in $d$ dimensions

Abstract

The thermodynamic properties of self-gravitating spherical thin matter shells an black holes in $d>4$ dimensions are studied, extending previous analysis for $d=4$. The shell joins a Minkowski interior to a Tangherlini exterior, i.e., a Schwarzschild exterior in $d$ dimensions, with $d\geqslant4$, The junction conditions alone together with the first law of thermodynamics enable one to establish that the entropy of the thin shell depends only on its own gravitational radius. Endowing the shell with a power-law temperature equation of state allows to establish a precise form for the entropy and to perform a thermodynamic stability analysis for the shell. An interesting case is when the shell's temperature has the Hawking form, i.e., it is inversely proportional to the shell's gravitational radius. It is shown in this case that the shell's heat capacity is positive, and thus there is stability, for shells with radii in-between their own gravitational radius and the photonic radius, i.e., the radius of circular photon orbits, reproducing unexpectedly York's thermodynamic stability criterion for a $d=4$ black hole in the canonical ensemble. Additionally, the Euler equation for the matter shell is derived, the Bekenstein and holographic entropy bounds are studied, and the large $d$ limit is analyzed. Within this formalism the thermodynamic properties of black holes can be studied too. Putting the shell at its own gravitational radius, i.e., in the black hole situation, obliges one to choose precisely the Hawking temperature for the shell which in turn yields the Bekenstein-Hawking entropy. The stability analysis implies that the black hole is thermodynamically stable substantiating that in this configuration our system and York's canonical ensemble black hole are indeed the same system. Also relevant is the derivation in a surprising way of the Smarr formula for black holes in $d$ dimensions.