Proposition: Two useful pullbacks

The equal-time restriction of $W_{\beta }$ exists and equals the spatial covariance $1/(2H_{\beta })$. For any smooth timelike curve $\gamma$ in $M$, the pullback

also exists as a bidistribution on its parameter domain.

Proof. A covector normal to an equal-time embedding has only temporal components at its endpoints. A nonzero null covector cannot be purely temporal, so the normal set is disjoint from $\mathrm{WF}(W_{\beta })$. The spectral restriction is therefore legitimate and gives $1/(2H_{\beta })$.

A nonzero null covector cannot annihilate a timelike tangent. Hence neither endpoint of a wavefront pair can lie in the annihilator required to obstruct the timelike pullback. The wavefront pullback theorem applies HormanderI.