Theorem: Shell descent forces triviality

Suppose a rule in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ defines (eq:delta) and preserves ${\mathcal{N}}_{\mathrm{sol}}$. Then, for every $u\in {\mathcal{C}}_{\mathrm{sol}}$,

Consequently its induced action is

Proof. Apply ideal preservation to all compactly smeared linear equation functionals. The metric and scalar tests give

The auxiliary relation gives

These identities follow because the compact tests separate smooth functions and ${\delta }_{\xi }E$ vanishes on the shell whenever $E\in {\mathcal{N}}_{\mathrm{sol}}$.

By Lemma lem:causal, the metric and scalar variations vanish, with their first time derivatives, on a Cauchy surface preceding $S_{\xi }$. Smooth homogeneous wave uniqueness therefore makes them zero. The auxiliary identity then gives ${Y_{\xi }}^b[u]=0$. For Grassmann-valued fields the same argument applies to every coefficient BGP (Chapter 3).

Finally, evaluation of (eq:delta) on $u$ pairs each derivative with a zero variation. Thus ${\delta }_{\xi }G$ vanishes on ${\mathcal{C}}_{\mathrm{sol}}$ for every admitted $G$, proving (eq:trivial-rep).