Corollary: A nonzero delayed singularity on a static restriction

Suppose

Then, near $u-u^{\prime }=\pm 2r_0$,

The restricted covariance therefore has a genuine delayed singularity at each of those nonzero proper-time separations.

For example, $\alpha =2/3,\beta =0$ gives ${\mathcal{A}}_{\alpha ,\beta }(0)=1$, and the hypotheses hold.

Proof. The pullback exists by Proposition prop:pullbacks. On this worldline, $\delta =Z=0$ and $F_{\mathrm{ap}}=4{r_0}^2-(u-u^{\prime }{)}^2$. The first condition in (eq:echoconditions) excludes an angular transition at $\delta =0$. Every geometric travel time at this fixed spatial point is strictly below $2r_0$ unless it is a transition, so no other front meets a sufficiently small neighbourhood of either displayed separation. Lemma lem:diffnormal and (eq:diffcoefficient) now give (eq:echo). Nonvanishing of the coefficient and Lemma lem:sqrtWF prove the singularity.

For the example, $c=3\pi /2$, so