Proposition: The broken ray and longitudinal matching

For $T=L_{\mathrm{ap}}>0$, the relation ${\mathcal{D}}_{\mathrm{ap}}^{+}$ is the endpoint relation of an incoming radial null segment and an outgoing radial null segment meeting the removed worldsheet at

The energy and longitudinal momentum agree across that meeting. The incoming and outgoing angular positions need not agree. Moreover,

consists exactly of these radial covector pairs at the angular transition $\delta \in {\mathcal{B}}_{\alpha }$.

Proof. Minimize the length of a two-segment spatial path through the worldsheet:

Equality of the two longitudinal slopes gives (eq:scatterpoint) and total length $L_{\mathrm{ap}}$. The corresponding null segments have common longitudinal velocity $Z/T$ and transverse speed $(r+r^{\prime })/T$. The covectors in (eq:Drelation) record the same energy and longitudinal momentum.

For a limiting geometric ray to have this length,

With $|{\theta }_k|\le \pi$, this means ${\theta }_k=\pi$ or $-\pi$, equivalently $\delta \in {\mathcal{B}}_{\alpha }$. At that point $\mathrm{d}{Q}_k=\mathrm{d}{F}_{\mathrm{ap}}$.