Quantization of Gravity in the Black Hole Background

TL;DR

This paper develops a covariant quantization method for perturbative gravity around a Schwarzschild black hole, addressing gauge fixing for different angular modes and analyzing ghost behavior, with implications for a ghost-free quantum gravity framework.

Contribution

It introduces a gauge fixing scheme for low multipole modes in black hole backgrounds and analyzes ghost properties, advancing covariant quantization of gravity in such spacetimes.

Findings

Ghosts are non-propagating for modes with l ≥ 2.

Low multipole ghosts have instantaneous propagators in Schwarzschild coordinates.

The approach suggests a possible ghost-free canonical quantization of gravity in black hole backgrounds.

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 $l \geq 2$, 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 non-propagating for the $l\geq2$ 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.