Theorem: Intrinsic finite propagation

Let $B=A_{F,\beta }+m^2$. For every compactly supported smooth spacetime section $f$,

The causal relation is the intrinsic relation (eq:causal), not an auxiliary completion relation.

Proof. We prove the domain-of-dependence statement for the homogeneous equation and then use Duhamel's formula.

Let $u(0)=u_0,\dot{u}(0)=u_1$, with smooth spectral data vanishing on $B_d(x_0,\varrho )$. For $t\ge 0$, choose a smooth nonincreasing ${\varphi }_{ϵ}$, equal to one on $(-\infty ,-ϵ]$ and zero on $[0,\infty )$, and put

The distance is Lipschitz and $|\nabla d|\le 1$ almost everywhere.

Multiplication by bounded Lipschitz functions with bounded gradient preserves the Friedrichs form domain $Q_F$. Indeed, approximate a form-domain section by compactly supported smooth sections, apply the Leibniz estimate, and smooth the multiplier on the compact regular supports. Thus all integrations by parts below are identities in the closed form, including when the support of ${\chi }_t$ approaches the apex.

Use the augmented local energy

The extra $|u{|}^2$ term also handles the massless case. The form identity for $B$ yields

Since ${\varphi }_{ϵ}^{\prime }\le 0$,

Also $2|u\dot{u}|\le |u{|}^2+|\dot{u}{|}^2\le e$. Hence

Gronwall's inequality gives $\mathcal{E}(t)=0$. Letting $ϵ\downarrow 0$ proves that $u(t)$ vanishes on $B_d(x_0,\varrho -t)$. Time reversal gives the negative-time statement.

It follows that $W_B(t)$ propagates spatial support by at most $|t|$. Duhamel's formula then gives (eq:finite-speed); closedness of ${J_M}^{\pm }(\mathrm{supp}f)$ was proved in Proposition prop:distance.