Proposition: Full coefficient action

The action (eq:gauge) is

It preserves ${\mathcal{J}}_{R,b}$ and the coefficient initial data.

Proof. Expand the exponential and collect powers of $\kappa$.

The Levi–Civita connection of a pullback metric is the pulled-back connection: it is torsion free and compatible with that metric, which characterizes it uniquely. Its curvature and Einstein tensor therefore pull back. The gradient-divergence definition of the scalar wave operator gives, by the chain rule,

The stress tensor has the same covariance by its displayed tensor formula. Taking coefficients proves preservation of (eq:jet1)–(eq:jet5). All changes vanish on a neighborhood of the initial surface, including their initial time derivatives.