Proposition: Geometry and local rods

The metric has signature $(-,+,+,+)$. Choose ${\partial }_T$ as future directed. A nonzero future-causal vector $a{\partial }_T+w$ satisfies

Static observers have proper time $\mathrm{d}\tau =\mathrm{d}T$, so the proper-time lapse is one.

On a sufficiently small lifted angular interval,

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

In the inverse-deck convention the peripheral affine gluing is

It has neither time nor axial translation.

Proof. The signature and causal inequality follow from the diagonal metric. Setting $w=0$ proves the proper-time statement. The local-isometry identity is

A positive angular lift around the apex advances through $2\pi -\mu$; the inverse deck map closes it by rotation through $\mu$. Its time and axial coordinates do not change.