Excess inner-horizon entropy and the thermodynamics of Kerr formation

Abstract

\noindent We develop a charge-matched thermodynamic framework for the regular rotating black-hole family underlying an exact model of Kerr formation. Since the Kerr endpoint already possesses a Cauchy horizon, hidden entropy is identified with the excess inner-horizon area relative to Kerr at fixed mass $M$ and angular momentum $J=Ma$: $\Sexc=π[h_c^2-(r_-^{\Kerr})^2]$, where $h_c$ is the regular inner-horizon radius and $r_-^{\Kerr}=M-\sqrt{M^2-a^2}$. Under regularity and interior-profile assumptions, we prove $r_-^{\Kerr}<h_c<h$ and establish the sharp bound $0<\Sexc<4πM\sqrt{M^2-a^2}$. The Schwarzschild limit yields $\Sexc=πh_c^2$, while $\Sexc=0$ at the Kerr endpoint. The Kerr reference entropy satisfies $S_+^{\Kerr}+S_-^{\Kerr}=4πM^2$, so rotation redistributes its two-horizon contributions at fixed mass. For the time-dependent model, we derive exact evolution equations for the inner root and $\Sexc$, interpreted as root-bookkeeping identities unless the roots are established as dynamical horizons. In the minimal $N=1$ trajectory, the apparent pole at $n=2$ is removable in the mass profile, giving a logarithmic continuation of the root equation, although the stationary curvature is singular there. The stationary Ricci-scalar density is locally absolutely integrable for $-1<n\leq2$, with the Kerr endpoint $n=-1$ treated separately. Minkowski breaking occurs at $n=0$, independently of inner-horizon disappearance. Quasi-extremal regular configurations approach, but cannot attain, the entropy bound without losing a regular Killing horizon.