Lemma: Polyhedral graph criterion

Let $F_s$ be continuous functions on a Euclidean cone, piecewise affine in finitely many developed sectors. Suppose that every affine branch has spatial gradient of norm at most $\lambda <1$. Then the graph of $F_s$ is embedded and spacelike on each face, and

If, in addition,

for $s_2>s_1$ and $c>0$, then the graphs are strictly time ordered. If $F_s=H+s$ near the boundary of a compact spatial patch, they match the prescribed translating resting exterior there.

Proof. On a face with gradient $a$, the induced metric is

For every spatial tangent vector $v$,

Integrating the gradient bound along piecewise linear spatial paths and taking the infimum of path lengths gives the global Lipschitz bound. The projection of a graph to the cone is a homeomorphism, so the graph is embedded.

In the ultrastatic product, a causal curve between two points requires its time difference to be at least the spatial distance. The strict Lipschitz bound therefore also gives acausality between distinct graph points. At the apex, spacelikeness is understood in this uniformly Lipschitz, stratified sense.

The ordering and exterior statements follow directly from their respective hypotheses.