Thermodynamic curvature and phase transitions in Kerr-Newman black holes

Singularities in the thermodynamics of Kerr-Newman black holes are commonly associated with phase transitions. However, such interpretations are complicated by a lack of stability and, more significantly, by a lack of conclusive insight from microscopic models. Here, I focus on the later problem. I use the thermodynamic Riemannian curvature scalar $R$ as a try to get microscopic information from the known thermodynamics. The hope is that this could facilitate matching black hole thermodynamics to known models of statistical mechanics. For the Kerr-Newman black hole, the sign of $R$ is mostly positive, in contrast to that for ordinary thermodynamic models, where $R$ is mostly negative. Cases with negative $R$ include most of the simple critical point models. An exception is the Fermi gas, which has positive $R$. I demonstrate several exact correspondences between the two-dimensional Fermi gas and the extremal Kerr-Newman black hole. Away from the extremal case, $R$ diverges to $+\infty$ along curves of diverging heat capacities $C_{J,\Phi }$ and $C_{\Omega ,Q}$, but not along the Davies curve of diverging $C_{J,Q}$. Finding statistical mechanical models with like behavior might yield additional insight into the microscopic properties of black holes. I also discuss a possible physical interpretation of $|R|$.

Figure 1

Some characteristic curves for the Kerr-Newman black hole; see the Appendix. The curve along which $C_{J,Q}$ diverges is the Davies curve. $R$ diverges at the extremal limit and along curves corresponding to a change of stability, which have diverging $C_{J,\Phi }$ and $C_{\Omega ,Q}$.

Figure 2

The event horizon broken up into Planck area pixels. The dark pixels are portrayed as somehow correlated. I propose that $|R|$ measures the average number of correlated pixels.

Figure 3

$R(M/M_p{)}^2$ as a function of $J/M^2$ and $Q/M$ for $(M,J,Q)$ fluctuations. $R$ is real, positive, and regular in the physical regime, and diverges as $T^{-1}$ at the extremal limit.

Figure 4

$R(M/M_p{)}^2$ as a function of $J/M^2$ and $Q/M$ for $(J,Q)$ fluctuations. $R$ is real, positive, and regular in the physical regime, and diverges as $T^{-1}$ at the extremal limit.

Figure 5

Stable fluctuation regime for $(M,Q)$ fluctuating at fixed $J$ is indicated by $+$ signs. The case with $J=0$ corresponds to the Reissner-Nordström black hole, which lies entirely out of the stable regime.

Figure 6

$R(M/M_p{)}^2$ as a function of $J/M^2$ and $Q/M$ for $(M,Q)$ fluctuations. $R$ is real everywhere in the physical regime. It is mostly positive, but there are two regimes of negative values near $Q/M=\pm 1$. $R$ diverges at both limits of stability.

Figure 7

Stable fluctuation regime for $(M,J)$ fluctuating at fixed $Q$ is indicated by $+$ signs. The case with $Q=0$ corresponds to the Kerr black hole, which lies entirely out of the stable regime.

Figure 8

$R(M/M_p{)}^2$ as a function of $J/M^2$ and $Q/M$ for $(M,J)$ fluctuations. $R$ is real and positive everywhere in the physical regime. $R$ diverges at both limits of stability.