Killing compatibility complex on Kerr spacetime

The Killing operator $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$ on a Lorentzian spacetime $(M,g)$ plays an important role in General Relativity (GR): it generates infinitesimal gauge symmetries of the theory. Gauge symmetry invariants play the role of physical observables. In PDE language, this translates to the following: the components of a compatibility operator for $K_{ab}$ generate all local observables for linearized GR on the background $(M,g)$. In arXiv:1910.08756 we have explicitly constructed such a compatibility operator (indeed, a full compatibility complex) on the astrophysically interesting Kerr spacetime of a rotating black hole. I will motivate and explain our approach and describe the complexity of the construction.