Revisiting the thermodynamics of the BTZ black hole with a variable gravitational constant* *Supported by the National Natural Science Foundation of China (12275216, 12105222, 12247103)

Abstract

The thermodynamics of BTZ black holes are revisited with a variable gravitational constant. A new pair of conjugated thermodynamic variables are introduced, including the central charge C and chemical potential μ. The first law of thermodynamics and the Euler relationship, instead of the Smarr relationship in the extended phase space formalism, are matched perfectly in the proposed formalism. Compatible with the standard extensive thermodynamics of an ordinary system, the black hole mass is verified to be a first order homogeneous function of the related extensive variables, and restores the role of internal energy. In addition, the heat capacity has also resulted in a first order homogeneous function using this formalism, and asymptotic behavior is demonstrated at the high temperature limit. The non-negativity of the heat capacity indicates that the rotating and charged BTZ black holes are thermodynamically stable.