Restricted thermodynamic fluctuations and the Ruppeiner geometry of black holes

Understand

Thermodynamic fluctuation metrics in Ruppeiner's 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.