Restricted thermodynamic fluctuations and the Ruppeiner geometry of black holes

Thermodynamic fluctuation metrics in Ruppeiners formalism are worked out for Kerr-AdS black holes in the extended state space. The implications of constraints upon the state space geometry and their correspondence with thermodynamical ensembles are explicitly worked out in the most general setting. The state space scalar curvature for a given ensemble is found to be sensitive to the instabilities/phase transitions therein. In particular, it is found that the appropriate Ruppeiner scalar curvature does encode critical phenomena in the Kerr-AdS black holes. A detailed study is undertaken of the curvature contour of the state space of the 4d Kerr-AdS black hole and suitable inferences are drawn. In particular, thermodynamic geometry suggests an instability in the Schwarzschild-AdS limit for all the ensembles except the pressure ensemble which is equivalent to the unextended state space of the Kerr-AdS black holes. The extrinsic geometry of the ensemble hypersurfaces is introduced and its relevance to constrained thermodynamic fluctuations discussed. A new interpretation for the thermodynamic curvature of black hole systems is suggested.