Weinhold geometry and Ruppeiner geometry of black holes with conformal anomaly

Abstract

The thermodynamic geometry of black holes with conformal anomaly has been investigated in this paper. We study two classical kinds of thermodynamic geometry. Namely, Weinhold geometry and Ruppeiner geometry. It is shown that the condition when Weinhold scalar curvature diverges is the same as the phase transition condition characterized by the divergence of specific heat. It is also shown that Ruppeiner scalar curvature not only reveals the phase structure but also contains the information of Hawking temperature. In a word, both Weinhold metric and Ruppeiner metric can correctly reproduce the phase structure of black holes even when conformal anomaly is taken into consideration.