Proposition: Causal plateau condition

Assume $P_m\varphi =0$ and that $g=1$ on a neighborhood of

Then

If also $A\gamma =0$ on a neighborhood of $\mathrm{supp}\xi$, then

Proof. The source $D{s}_{\xi }$ is supported above $\epsilon$, so ${\rho }_{\xi }$ and its derivatives vanish before that time. Consequently the coordinate ramp is one wherever $K{\rho }_{\xi }$ can contribute. The coordinate equation gives

with zero past data.

For any smooth vector $v$, direct differentiation of (eq:completion-legs) with $g=1$ gives

On the free scalar shell this is $-P_m(v^{\mu }{\partial }_{\mu }\varphi )$. The support condition (eq:causal-plateau-condition) makes the switched and unswitched scalar sources agree throughout the causal past relevant to $K$. Finite propagation and zero past data therefore give

on $K$. The two legs in (eq:Aone) cancel.

If $A\gamma$ vanishes near $\mathrm{supp}\xi$, the Einstein-shell source in (eq:plateau-identity) vanishes. The final assertion follows.