Consistent Gauge Conditions for Dust-Shell Dynamics in Effective Quantum Gravity

Abstract

Previous analyses of shocks generated by shell-crossing singularities are affected by inappropriate gauge choices, and no systematic method is available for selecting a consistent gauge. To address this issue, we focus on the shock dynamics, which can be effectively described by a thin dust shell interacting with the surrounding dust. As a first step toward the full shell-crossing problem, we neglect this interaction and study an isolated thin shell, for which we develop a systematic method for constructing consistent gauges in generally covariant effective gravity. We apply this method to a generally covariant effective Hamiltonian model of quantum gravity characterized by a quantum parameter $\zeta$, with classical GR recovered in the limit $\zeta\to0$. In this classical limit, the resulting shell dynamics reproduces the Israel junction conditions, providing a nontrivial validation of our method, whereas for $\zeta\neq0$, it exhibits genuine quantum-gravity corrections. We also show that gauges such as the Painlev\'e-Gullstrand and Schwarzschild ones are incompatible with the presence of a dust shell when imposed on the whole spatial slice. This explains the difficulties in previous treatments. The framework developed here provides a basis for studying shell-crossing singularities and shock dynamics in generally covariant effective black-hole models.