Critical Inter-Horizon Thermal Dynamics on the Lukewarm Reissner-Nordstr\"om-de Sitter Manifold

Abstract

We reinterpret the lukewarm sector of four-dimensional Reissner--Nordstr\"om--de Sitter black holes as the exact zero-dissipation thermal manifold of an effective two-horizon nonequilibrium system. In the fixed-charge sector, the inter-horizon thermal affinity controls the entropy production and vanishes precisely on the lukewarm branch. The corresponding linearized thermal mode is governed by an exact relaxation coefficient \(K_L(\rho)\), with \(\rho=r_+/r_c\), and changes stability at the critical ratio \[ \rho_*=\frac{1+\sqrt{3}-\sqrt{2}\,3^{1/4}}{2}\approx 0.4354, \] where the relaxation time diverges as \(\tau\sim |\rho-\rho_*|^{-1}\). We then encode this critical structure in a minimal Bragg--Williams functional and an Onsager--Machlup action for the effective trajectories of the thermal mode. In this way, the lukewarm branch is promoted from a geometric equal-temperature locus to a critical inter-horizon thermal manifold.