Two-horizon sector thermodynamics and NUT bi-hair in Kerr-Newman-NUT spacetimes

Abstract

First law consistency alone does not determine the black hole state space of NUT-charged spacetimes, because several inequivalent macroscopic formulations can yield consistent thermodynamics. We develop a two-horizon sector diagnostic for the electrically charged Kerr-Newman-NUT family by organizing the inner- and outer-horizon entropies and inverse temperatures into sum ($Σ$) and difference ($Δ$) sectors. The charged rotating geometry gives a more stringent test of the uncharged Taub-NUT sector result, since the mass $M$, ordinary electric charge $Q$, ordinary angular momentum $J$, NUT charge $N$, and the NUT secondary hair $J_N=MN$ all enter the same two-horizon thermodynamics. For the physically anchored minimal homogeneous state spaces defined and tested here, we find a clear sector separation: the $Σ$ sector closes with the charge-like variables $Q$ and $N$, whereas the $Δ$ sector additionally requires the rotation-like variables $J$ and $J_N$. Natural reductions that eliminate either $J_N$ or $N$ fail to reproduce the first law coefficient fixed by the sector temperature, and their associated Bekenstein-Smarr relations do not close with the assigned scaling weights. Within this class, electric charge and ordinary rotation therefore do not absorb the NUT secondary response, and the charged rotating family supports the NUT bi-hair organization: $N$ has a charge-like sector role, while $J_N=MN$ is a rotation-like thermodynamic secondary hair. The latter is defined off shell in the homogeneous equation of state, not as a new metric parameter or asymptotic conserved charge.