Phase-Transition Theory of Kerr Black Holes in Electromagnetic Field

For a Kerr black hole (KBH) with spin $J$ and mass $M$ in a steady electromagnetic field, a special Wald vacuum solution (WVS) has been found in the case of no-source uniform field. For WVS, the Meissner effect (ME) occurs only in the the extreme KBH where $M^2/J=1$, in this case, the magnetic field is totally excluded from the event horizon (EH) of KBH. However, WVS does not consider the Hawking radiation (HR) but treats KBH as an absolutely black body. If HR is added , researchers believe that the condition is not so restricted and it is possible for ME to occur in less extreme case. How less is the "less extreme case"? This paper tries to answer this question. Since the Hawking temperature $T_H$ of KBH defined by HR is proportional to the surface gravity $\kappa$ at the EH, this question is actually about the so-called existence/non-existence of ME (ME/NME) or superconducting phase transition. In this paper, we study the connection between the superconductivity of KBH-EH and the existence of Weyl Fermion (WF). Using thermodynamic formulas and the KBH state equation, we prove that the inherent-parameter condition for ME to occur is $M^2/J\leq \epsilon_c=1.5$ in force-free fields whether it be in the simple axisymmetric vacuum zero source case or in the non-zero source case which can be described by the nonlinear Grad-Shafranov (G-S) equation. We suggest that this is a second-order phase transition and we calculate the critical exponents $\delta=1$ and $\eta=1/2$ for the specific heat diverging at constant $J$, and the critical point $(M_c, \Omega_c)$, which equals $(1.22\sqrt{ J}, 0.16/\sqrt{ J})$ where $\Omega$ is the angular velocity of KBH. Furthermore we draw the phase diagrams in both $(M, J)$ and $(M,\Omega)$ coordinates.