Proposition: Local geometry

The signature is $(-,+,+,+)$, with future orientation ${\partial }_T$. Static observers have $\mathrm{d}\tau =\mathrm{d}T$. On a lifted regular angular interval,

is a local Minkowski isometry. A future-causal curve satisfies

The map $J$ fixes $\mathcal{W}$ pointwise and has neither time nor axial translation.

Proof. The first two assertions follow directly from (eq:metric). The identity

proves local isometry. The causal inequality follows by integrating the pointwise causal-speed bound. Equation (eq:J) proves the last assertion.