The Phase Transition of Reissner-Nordstr\"om Black Holes

Under the framework of thermodynamics, the phase transition of the black hole is a general issue in general relativity. In this work, the phase transition of charged black holes is discussed carefully. The metric tensor of thermodynamics is redefined in the charged black hole, based on the Ruppeiner geometry. With the well-defined metric tensor of thermodynamics, the scalar curvature of the charged black hole is obtained. It is indicated that the scalar curvature is diverged and infinite when the mass M or charge Q are set by some values, and it is shown that the charged black hole suffers from a phase transition. At the same time, there is a phase transition from small mass to large mass or from small to high charged state. It is shown that the phase transition of a charged black hole is a common and general process and this work is meaningful for the construction of microscopic states of black holes.