Proposition: Local geometry and proper time

The metric (eq:cone) has signature $(-,+,+,+)$. Choose the future orientation defined by ${\partial }_T$. For a nonzero future-causal tangent $a{\partial }_T+w$,

At a static port, $\mathrm{d}\tau =\mathrm{d}T$.

On a sufficiently small lifted angular interval,

is a local isometry to Minkowski space. The global conical gluing remains nontrivial.

Proof. The metric matrix has one negative and three positive diagonal entries. The causal inequality is $-c^2{a}^2+|w{|}_{\mathrm{spatial}}^2\le 0$, and a nonzero causal tangent cannot have $a=0$. For $w=0$, the proper-time formula follows directly.

Differentiation of the developing map gives

A deck shift $\varphi \mapsto \varphi +2\pi$ changes the developed angle by $2\pi -\mu$. In the inverse-deck convention this is rotation by $\mu$. Thus local flatness does not eliminate the conical identification.