Proposition: Elementary geometric facts

The metric has signature $(-,+,+,+)$. Static observers have proper-time increment $\mathrm{d}\tau =\mathrm{d}T$. For a nonzero vector $v=a{\partial }_T+w$, future causality is equivalent to

On an angular interval with a fixed real lift of $\varphi$, the map

is a local isometry to $({\mathbb{R}}^{1,3},\eta )$, where $\eta =\mathrm{diag}(-1,1,1,1)$.

Proof. The metric matrix is diagonal with one negative and three positive entries. The causal inequality is $-c^2{a}^2+{\Vert w\Vert }_h^2\le 0$; a nonzero causal vector has $a\ne 0$. Along a static curve, $-g/c^2=\mathrm{d}{T}^2$. Finally,

which proves the isometry statement.