Black hole microstructures in the extremal limit

The microstructure of black holes is a mystery. There is yet no resolution of basic questions such as what the constituent particles are. We work here with black hole thermodynamics (BHT), and the metric geometry of thermodynamics, which connects to interparticle interactions via the invariant thermodynamic Ricci scalar curvature $R$. $R$ may be calculated with BHT. In ordinary thermodynamics (OT), $R$ is positive/negative for interparticle interactions repulsive/attractive. Its magnitude is the correlation length. The basic universality of thermodynamics leads us to expect similar relations for BHT. Our contribution here is motivated by a physical simplification that frequently occurs at low temperatures $T$ in OT: complicated interactions tend to freeze out, leaving only basic quantum statistical interactions, those of ideal Fermi or Bose gasses. Our hope is that similar simplification happens in black holes in the extremal limit, where the BHT temperature $T \to 0$. We evaluate the extremal $R$ for twelve BHT literature models, working with the independent variables mass, angular momentum, charge, and the cosmological constant, $\{M,J,Q,\Lambda\}$, respectively. We allowed only two of these variables to fluctuate at a time, with the other two fixed. $M$ always fluctuated, either $J$ or $Q$ fluctuated, and $\Lambda$ was always fixed. At constant average $M$, $R$ has limiting divergence $R=c\,T^{-1}$, with the nonsingular constant $c$ depending only on $M$ and the two fixed parameters. $c$ is positive for $11/12$ of the models we examined, and negative only for the tidal charged model. The positive sign for $R$ indicates a BHT microstructure composed of particles with repulsive (fermionic) interactions. The limiting BHT expression for $R$ resembles that for 2D and 3D ideal Fermi gasses at constant volume, which also have limiting divergence $R=c\,T^{-1}$, with positive $c$.