Quantization of gravity in the black hole background

Abstract

We perform a covariant (Lagrangian) quantization of perturbative gravity in the background of a Schwarzschild black hole. The key tool is a decomposition of the field into spherical harmonics. We fix Regge-Wheeler gauge for modes with angular momentum quantum number l≥2, while for low-multipole modes with l=0 or 1-for which Regge-Wheeler gauge is inapplicable-we propose a set of gauge-fixing conditions which are 2D background covariant and perturbatively well defined. We find that the corresponding Faddeev-Popov ghosts are nonpropagating for the l≥2 modes, but are in general nontrivial for the low-multipole modes with l=0, 1. However, in Schwarzschild coordinates, all time derivatives acting on the ghosts drop from the action and the low-multipole ghosts have instantaneous propagators. Up to possible subtleties related to quantizing gravity in a space with a horizon, Faddeev's theorem suggests the possibility of an underlying canonical (Hamiltonian) quantization with a manifestly ghost-free Hilbert space.