Proposition: Geometry, rods, and time orientation

The metric (eq:cone) has signature $(-,+,+,+)$. Choose the orientation for which ${\partial }_T$ is future directed. A nonzero future-causal vector $a{\partial }_T+w$ satisfies

Static observers have proper-time increment $\mathrm{d}\tau =\mathrm{d}T$.

On a sufficiently small lifted angular interval,

is a local isometry to Minkowski space. Its three spatial coordinates are local rods.

Proof. The diagonal metric matrix has one negative and three positive entries. Its causal inequality is $-c^2{a}^2+|w{|}_{\mathrm{spatial}}^2\le 0$. A nonzero causal vector cannot have $a=0$, and the chosen orientation selects $a>0$. The static proper-time formula follows by setting $w=0$. Finally,