Thermodynamic geometry of three-dimensional Einstein–Maxwell-dilaton black hole

The thermodynamics of a three-dimensional Einstein–Maxwell-dilaton black hole is investigated using the method of thermodynamic geometry. According to the definition of thermodynamic geometry and the first law of the black hole, two-dimensional Ruppeiner and Quevedo geometry are constructed respectively. Afterwards, both the scalar curvature and the extrinsic curvature of hypersurface at constant Q of the two-dimensional thermodynamic space are calculated. The results show that, the extrinsic curvature can play the role of heat capacity to locate the second-order critical point and determinate the stability of the black hole, which is much better than the scalar curvature. However, for values of the entropy below that for which the specific heat diverges, the curve of the extrinsic curvature has an extra divergent point. Considering the fluctuation of the AdS radius, we can modify the first law of thermodynamics and reconstruct the three-dimensional Quevedo geometry. In this geometry, the extrinsic curvature of the hypersurface at constant Q can replace the heat capacity to locate the second-order critical point and determinate the stability of the black hole near the critical point. In addition, the extra divergent point disappears. The results show that the AdS radius must be considered as a variable when the thermodynamics of an AdS black hole is investigated, so that the result can reflect the real physics