Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

TL;DR

This paper constructs a universal compatibility complex for the Killing operator in overdetermined PDEs, with explicit examples in cosmological and black hole spacetimes, advancing gauge-invariant analysis in linearized gravity.

Contribution

It provides a systematic method to explicitly construct the full compatibility complex for the Killing operator on arbitrary backgrounds, including new examples in cosmological and black hole geometries.

Findings

Constructed the Killing compatibility operator on arbitrary backgrounds.

Explicitly applied the method to FLRW spacetimes in any dimension.

Extended the approach to generalized Schwarzschild-Tangherlini black holes.

Abstract

In linearized gravity, two linearized metrics are considered gauge-equivalent, $h_{ab} \sim h_{ab} + K_{ab}[v]$, when they differ by the image of the Killing operator, $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$. A universal (or complete) compatibility operator for $K$ is a differential operator $K_1$ such that $K_1 \circ K = 0$ and any other operator annihilating $K$ must factor through $K_1$. The components of $K_1$ can be interpreted as a complete (or generating) set of local gauge-invariant observables in linearized gravity. By appealing to known results in the formal theory of overdetermined PDEs and basic notions from homological algebra, we solve the problem of constructing the Killing compatibility operator $K_1$ on an arbitrary background geometry, as well as of extending it to a full compatibility complex $K_i$ ($i\ge 1$), meaning that for each $K_i$ the operator $K_{i+1}$ is its universal compatibility operator. Our solution is practical enough that we apply it explicitly in two examples, giving the first construction of full compatibility complexes for the Killing operator on these geometries. The first example consists of the cosmological FLRW spacetimes, in any dimension. The second consists of a generalization of the Schwarzschild-Tangherlini black hole spacetimes, also in any dimension. The generalization allows an arbitrary cosmological constant and the replacement of spherical symmetry by planar or pseudo-spherical symmetry.