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

Understand

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.