The finite-time choice

This paper makes one choice throughout: harmonic coordinates are defined by a finite-time Cauchy problem, not by a Feynman inverse. The interaction switch is compactly supported, while the harmonic-coordinate equation is unswitched. Compatibility is obtained by restricting the admitted gauge transformations to a declared interaction plateau.

The effective-field-theory motivation is distinct from a claim of ultraviolet completion. Perturbative, locally covariant treatments of gravity provide an appropriate broader setting BFR. The present paper proves only a classical compatibility statement needed before attempting quantum products.

The principal result is the following, stated informally here and precisely in Theorem thm:second:

For the switched scalar action specified below, the scalar solution with fixed free Cauchy data and the unswitched harmonic-coordinate map transform by the same pullback under plateau-supported formal gauge transformations. Their relational composition is therefore invariant through second order.

The proof is off shell in the metric coefficients. Consequently it remains valid on any invariant restriction of those coefficients, including the Einstein–scalar jet defined in Section sec:einstein.