New identities for linearized gravity on the Kerr spacetime

TL;DR

This paper derives new differential identities for linearized gravity on Kerr spacetime, revealing additional integrability conditions and gauge invariants, which enhance understanding of gravitational perturbations.

Contribution

It introduces a complex symmetric 2-tensor identity that extends classical Teukolsky-Starobinsky identities, providing new tools for analyzing linearized gravity on Petrov type D spacetimes.

Findings

New differential identity for linearized gravity on Kerr.

Additional integrability conditions beyond classical identities.

Construction of gauge invariants for linearized gravity.

Abstract

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which solves the linearized Einstein equations. The identity makes this manifest by relating $\mathcal{M}_{ab}$ to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of $\mathcal{M}_{ab}$ gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity.