Thermodynamics of the 3-dimensional Einstein-Maxwell system

Abstract

Abstract Recently, I studied the thermodynamical properties of the Einstein-Maxwell system with a box boundary in 4-dimensions [1]. In this paper, I investigate those in 3-dimensions using the zero-loop saddle-point approximation and focusing only on a simple topology sector as usual. Similar to the 4-dimensional case, the system is thermodynamically well-behaved when Λ < 0 (due to the contribution of the “bag of gold” saddles). However, when Λ = 0, a crucial difference to the 4-dimensional case appears, i.e. the 3-dimensional system turns out to be thermodynamically unstable, while the 4-dimensional one is thermodynamically stable. This may offer two options for how we think about the thermodynamics of 3-dimensional gravity with Λ = 0. One is that the zero-loop approximation or restricting the simple topology sector is not sufficient for 3-dimensions with Λ = 0. The other is that 3-dimensional gravity is really thermodynamically unstable when Λ = 0.