Dynamical Black Hole Thermodynamics in Modified Gravity

Abstract

We study the dynamical and thermodynamic evolution of a Schwarzschild black hole in Modified Gravity (MOG) under a scalar gravitational wave breathing mode. The time-dependent apparent horizon reveals that both the scalar strain velocity and the repulsive vector charge modulate the effective surface gravity and the instantaneous dynamical temperature in a quasi-adiabatic way. As a result, this regime breaks the semiclassical adiabatic approximation and triggers explicit non-thermal particle creation. We resolve a thermodynamic paradox by decoupling first-order reversible kinematic-horizon fluctuations from second-order irreversible entropy growth, using the Raychaudhuri equation. Consequently, the Generalized Second Law remains preserved. We apply these results to address the black hole information paradox across two timescales. Short-term non-thermal emission opens a dynamical channel for the escape of correlated geometric information. On long timescales, the massive vector field halts evaporation as mass approaches the extremal bound, $M_G \to Q_G$. This yields a stable, zero-temperature remnant. These signals provide a framework for probing scalar-tensor-vector modifications to general relativity with next-generation gravitational-wave observatories

Figures (2)

• Figure 1: Entropy production rates for a dynamically perturbed Schwarzschild MOG black hole. The dashed curve tracks the reversible first order geometric rate $\mathcal{O}(\dot{h_b})$, while the dotted curve traces the positive second order radiation flux $\mathcal{O}(\dot{h_b^2})$. The instantaneous apparent total (Eq. \ref{['eq:gsl_total']}, solid red line) exhibits transient entropy deficits (shaded regions) during horizon contraction. Resolving this paradox, the Raychaudhuri equation isolates the true secular evolution (solid black line). Driven strictly by the $\mathcal{O}(\dot{h_b^2})$ shear squared and energy flux, the physical thermodynamic rate remains unconditionally positive, robustly preserving the Generalized Second Law.
• Figure 2: The MOG resolution to the black hole information paradox. Panel (a) illustrates the dynamic regime governed by Eq. (\ref{['eq:temperature_dynamic']}), where the effective Hawking temperature is quasi-adiabatically modulated by the transient scalar breathing mode, opening a non-thermal channel for geometric information to leak. Panel (b) models the secular evaporation regime. While GR predicts a divergent temperature singularity as shown in the dotted gray curve, MOG in the solid indigo curve predicts a smooth transition to $T_0 = 0$ as dictated by Eq. (\ref{['eq:extremal_limit']}), leaving a cold, stable remnant that permanently houses the initial quantum state.