Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume

Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran* Author to whom correspondence should be addressed; E-Mail: rbmann@sciborg.uwaterloo.ca;Tel.: +1-519-882-1211.Received: 9 January 2014; in revised form: 6 February 2014 / Accepted: 7 February 2014 /Published: 3 March 2014Abstract: In this review we summarize, expand, and set in context recent developmentson the thermodynamics of black holes in extended phase space, where the cosmologicalconstant is interpreted as thermodynamic pressure and treated as a thermodynamic variablein its own right. We specifically consider the thermodynamics of higher-dimensionalrotating asymptotically flat and AdS black holes and black rings in a canonical (fixedangular momentum) ensemble. We plot the associated thermodynamic potential—theGibbs free energy—and study its behavior to uncover possible thermodynamic phasetransitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdSblack holes exhibit a rich set of interesting thermodynamic phenomena analogous to the“every day thermodynamics” of simple substances, such as reentrant phase transitions ofmulticomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gasphase transitions of the van der Waals type. Furthermore, the reentrant phase transitions alsooccur for multiply-spinning asymptotically flat Myers–Perry black holes. These phenomenado not require a variable cosmological constant, though they are more naturally understoodin the context of the extended phase space. The thermodynamic volume, a quantity conjugateto the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy thereverse isoperimetric inequality; this provides a first example of calculation confirming the