Frozen gauge group

Admit spacetime diffeomorphisms with compact support in $\mathcal{M}$, preserving orientation and the perturbative timelike branch. They are the identity in the spatial asymptotic region and near the time ends of their support. They therefore preserve the fixed asymptotic rod labels.

The group acts by pullback on $g,\varphi$. The background is held as the reference for the perturbative expansion, not treated as a newly dynamical quantum variable.