Dynamical Selection of Horizon-BMS Goldstone Modes in Evaporating Black Holes

Abstract

We investigate the dynamics of horizon soft degrees of freedom associated with near-horizon BMS supertranslations in an evaporating Vaidya-Schwarzschild black hole spacetime. Considering the dynamic nature of the supertranslation parameter, we derive its effective action directly from the Einstein-Hilbert action. The resulting Goldstone sector is intrinsically coupled to the evolving black hole dynamics, with the mass function entering the evolution of the Goldstone modes while the Goldstone configuration contributes to the dynamical evolution of the black hole mass. Azimuthal periodicity and regularity at the poles select the physically admissible angular sector and yields a mass-dependent $selection\ rule$ on the azimuthal modes. Consequently, the spectrum of horizon-supported Goldstone modes evolves with the black hole mass; as the horizon shrinks during evaporation, the horizon is left with a progressively reduced set of lower-order modes. The evaporation thus induces a dynamical filtering of the horizon soft sector, arising intrinsically from the black hole mass dynamics. These results provide an effective gravitational framework linking near-horizon BMS symmetry, dynamical Goldstone modes, and black hole evaporation, suggesting a direct connection between macroscopic horizon evolution with the microscopic organization of the gravitational soft degrees of freedom residing on the horizon.