The fate of Reissner--Nordstr\"om--de Sitter black holes: nonequilibrium discharge and evaporation

Abstract

We develop a semiclassical description of Reissner--Nordstr\"om--de Sitter (RN--dS) evaporation by combining a spherically reduced two-dimensional dilaton gravity model with Polyakov anomaly backreaction. The framework captures the causal and thermodynamic structure of the static patch and yields closed adiabatic evolution equations for the mass and charge. With an outward-oriented flux convention, the anomaly-induced Killing-energy flux is $\mathcal F=(N_{\rm eff}/48\pi)(\kappa_b^2-\kappa_c^2)$, while the full mass evolution is $\dot M=-\mathcal F+\Phi_b\dot Q$, with $\Phi_b=Q/r_b$. We prove analytically that along the entire sub-Nariai neutral Schwarzschild--de Sitter branch $\kappa_b>\kappa_c$, so neutral black holes lose mass monotonically. Schwinger pair production provides the discharge channel. In the rapid-discharge regime, controlled charged trajectories become effectively neutral on a timescale short compared with the anomaly-driven Hawking mass-loss time and then follow the neutral SdS channel toward empty de Sitter space. The classical lukewarm locus $T_b=T_c$ is only the nullcline of the anomaly-induced heat flux: the electromagnetic work term tilts the full semiclassical vector field away from this curve, so it is not an invariant trajectory. When sufficiently light charged species provide a rapid-discharge channel, the classical cold/extremal, charged Nariai, ultracold, and lukewarm loci are not semiclassical attractors for controlled nondegenerate trajectories. These results give an adiabatically backreacted derivation of the RN--dS evaporation endpoint in the regime controlled by anomaly-induced flux and rapid charge discharge, and provide the semiclassical background for generalized-second-law monotonicity and conservative quantum-extremal-surface/island estimates.