The Positive Mass Theorem, Stability, and Phase Transitions

Abstract

This chapter explains the notions of mass in general relativity. The positive mass theorem, when it exists, guarantees stability. We also introduce the idea of Wick rotation to imaginary time, and explain how to use this technique to study possible phase transitions between various gravitational configurations. This includes, specifically, the Hawking-Page phase transition between a Schwarzschild-AdS black hole and a “thermal AdS” spacetime; and a similar phase transition discovered by Horowitz and Myers, between toral AdS black holes and a type of soliton; which configurations are preferred is dictated by their free energy.