Lemma: A local causal certificate

Let $x,y\in U$ and $T_1\ge T_0$. If

then $(T_1,y)\in J_g^{+}((T_0,x))$. If equality holds and $x\ne y$, the straight developed segment, traversed at constant speed, is null.

Proof. Convexity keeps the straight spatial segment in $V$. Its lift has constant coordinate velocity of norm at most $c$, so (eq:tangent-cone) applies with $N=1$. Equality makes its tangent null.