Bundles, pairing, and operators

Let

All tensor operators below have constant coefficients. Indices are raised and lowered with $\eta$.

Let $H={\mathrm{Sym}}^2{T}^{\ast }M$. A symmetric contravariant tensor distribution $C$ is identified with its lowered tensor $c$. Its pairing with a smooth covariant tensor $h$ is

In the smooth case this means

Compact support of $c$ makes this pairing meaningful for arbitrary smooth $h$, without a decay assumption on $h$.

Define trace reversal, the gauge operator, and the de Donder operator by

In four dimensions, $T^2=1$. Trace reversal is self-adjoint for the stated pairing.

The linearized Einstein operator used in this paper is

This equation fixes the normalization and sign convention. The formal gauge adjoint is

We call $c$ conserved when $K^{\ast }c=0$.

Define three spaces of smooth tensors:

They are not identified with any manuscript's named solution space without an additional, explicit domain identification.