Phase transition of modified thermodynamics of the Kerr-AdS black hole

Abstract: We investigate the critical phenomena of Kerr-AdS black holes in the modified first law of thermodynamics. The modified black hole thermodynamics exhibits the van der Waals-like phase structure. All the critical exponents are calculated, and the swallowtail diagram of free energy is plotted. Comparing with existing results, the main difference is the correspondence between thermal quantities of the Kerr-AdS black holes and the van der Waals system. In previous work[16], the correspondence was $(\Omega_HJ)\to (V, P)$. In our work, the correspondence is $(J,{\hat\Omega}_H)\to (V, P)$. This difference results from rotating effect. The modified black hole thermodynamics is associated with rotating observers. The free energy in such reference contains an extra rotating energy. This extra part induces a Legendre transformation in $({\hat\Omega}_H, J)$ cross-section, which yields the different correspondence.