Corollary: Geometric and diffracted contributions

For the Friedrichs extension:

1. In a lift with $\vartheta =\psi$, the $k=0$ term is the direct flat term at $d(x,x^{\prime })=R_0$. Every additional geometric term arrives at $R_k\ge R_0$.

2. The diffracted contribution is supported in $t\ge {\rho }_0$. At a generic pair its massless front coefficient is

   $$\underset{t\downarrow {\rho }_0}{\lim }\sqrt{t^2-{{\rho }_0}^2}{G}_{+}^{\mathrm{dif}}(t;x,x^{\prime })=-\frac{{\Phi }_{\beta }(0,\vartheta )}{4{\pi }^2\alpha \sqrt{r{r}^{\prime }}}.$$

   This coefficient may vanish for a particular pair. A nonzero diffraction weight gives a late-time contribution, but not necessarily a nonzero value at every angle and every later time.

3. The diffraction weight vanishes identically precisely when

   $$\frac{1}{\alpha }=N\in \mathbb{N},N\beta \in \mathbb{Z}.$$

   In this case the kernel is the finite character-weighted $N$-image kernel of the corresponding cyclic quotient. For $\beta =0,\alpha =1/2$,

   $$G_{+}(t;x,x^{\prime })=F_m^{+}(t,R_0)+F_m^{+}(t,\overline{R}),$$

   where

   $${\overline{R}}^2=r^2+r^{\prime 2}+2r{r}^{\prime }\cos \psi +\Delta z^2.$$

4. Every term is supported in the intrinsic causal future. At an angular cut, two geometric fronts can lie on its boundary. Away from that cut, only the shortest geometric front does so.

Proof. The support assertions follow from (eq:ordered) and (eq:wave-master). Near $t={\rho }_0$,

which proves (eq:front).

If (eq:pure-images) holds, every $\sin (\pi {\nu }_n)$ vanishes. Conversely, if ${\Phi }_{\beta }$ vanishes identically, its absolutely convergent angular Fourier series at each $u>0$ has zero coefficients, so every ${\nu }_n$ is an integer. The successive positive orders differ by $1/\alpha$; therefore $1/\alpha$ is an integer, and $\beta /\alpha =N\beta$ is an integer. For a generic angle the geometric interval contains exactly $N$ indices. The remaining statements follow from the length comparison and Proposition prop:distance.