Compatibility Complex for Black Hole Spacetimes

The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (Aksteiner and Backdahl in Phys Rev Lett 121:051104, 2018) 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 Aksteiner and Backdahl (Phys Rev Lett 121:051104, 2018) with the first compatibility operator from that complex.