Ruppeiner thermodynamic geometry for the Schwarzschild-AdS black hole

Due to the nonindependence of entropy and thermodynamic volume for spherically symmetric black holes in the anti--de Sitter (AdS) spacetime, when applying the Ruppeiner thermodynamic geometry theory to these black holes, we often encounter an unavoidable problem of the singularity about the line element of thermodynamic geometry. In this paper, we propose a basic and natural scheme for dealing with the thermodynamic geometry of spherically symmetric AdS black holes. We point out that enthalpy, not internal energy, is the fundamental thermodynamic characteristic function for the Ruppeiner thermodynamic geometry. Based on this fact, we give the specific forms of the line element of thermodynamic geometry for the Schwarzschild AdS (SAdS) black hole in different phase spaces, and the results show that the thermodynamic curvatures obtained in different phase spaces are equivalent. It is shown that the thermodynamic curvature is negative which may be related to the information of attractive interaction between black hole molecules for the SAdS black hole. Meanwhile we also give an approximate expression of the thermodynamic curvature of the Schwarzschild black hole which shows that the black hole may be dominated by repulsion on the low temperature region and by attraction on the high temperature region phenomenologically or qualitatively.