Compatibility complex for black hole spacetimes

TL;DR

This paper establishes a complete set of local gauge invariant quantities for linearized gravity on Kerr spacetime, providing a foundational tool for analyzing gravitational perturbations in black hole physics.

Contribution

It constructs a complete compatibility complex for the Killing operator, proving the completeness of the gauge invariants previously introduced for Kerr spacetime.

Findings

Gauge invariants can be expressed in terms of derivatives of a complete set.

The compatibility complex links gauge invariants with the Killing operator.

The results facilitate analysis of linearized gravity on Kerr spacetime.

Abstract

The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (S.A, T.B.) in (arXiv:1803.05341) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from (arXiv:1803.05341) with the first compatibility operator from that complex.