Lemma: Local rods, time orientation, and causal segments

There is a regular spatial neighborhood $U$ with a developing chart

such that $V$ is convex and

Choose ${\partial }_T$ future directed. Then:

1. a future-causal tangent $v=a{\partial }_T+w\ne 0$ satisfies

   $$a>0,|w{|}_h\le ca;$$

2. events $(T,x)$, $(T+\Delta T,x^{\prime })$ in this chart are joined by a future-causal straight segment whenever

   $$\Delta T>0,{\Vert x^{\prime }-x\Vert }_2\le c\Delta T;$$

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

Proof. On a sufficiently small lifted angular sector put

Its spatial metric is Euclidean. Restrict to the inverse image of a sufficiently small convex ball avoiding the apex and the sector boundary.

The causal inequality follows from

The straight segment has constant developed velocity $(x^{\prime }-x)/\Delta T$, proving the second assertion. For a static curve, $g=-c^2\mathrm{d}{T}^2$, proving the third.