Entropy of thin shells in a ($2+1$)-dimensional asymptotically AdS spacetime and the BTZ black hole limit

The thermodynamic equilibrium states of a static thin ring shell in a ($2+1$)-dimensional spacetime with a negative cosmological constant are analyzed. Inside the ring, the spacetime is pure anti–de Sitter, whereas outside it is a Bañados-Teitelbom-Zanelli spacetime and thus asymptotically anti–de Sitter. The first law of thermodynamics applied to the thin shell, plus one equation of state for the shell’s pressure and another for its temperature, leads to a shell’s entropy, which is a function of its gravitational radius alone. A simple example for this gravitational entropy, namely, a power law in the gravitational radius, is given. The equations of thermodynamic stability are analyzed, resulting in certain allowed regions for the parameters entering the problem. When the Hawking temperature is set on the shell and the shell is pushed up to its own gravitational radius, there is a finite quantum backreaction that does not destroy the shell. One then finds that the entropy of the shell at the shell’s gravitational radius is given by the Bekenstein-Hawking entropy.