Lemma: Local causal certificates

In the developing chart:

1. a future-causal displacement satisfies

   $$\Delta T>0,|\Delta x|\le c\Delta T;$$

2. a straight segment with this inequality is future causal;

3. a static observer has $\mathrm{d}\tau =\mathrm{d}T$.

Proof. The tangent inequality is $-c^2{\dot{T}}^2+|\dot{x}{|}^2\le 0$, with the future orientation $\dot{T}>0$. Apply it to a constant-velocity segment. For a static curve, $\bar{g}=-c^2\mathrm{d}{T}^2$.