The independent causal geometry

Let $C_{\pi /2}$ be the spatial Euclidean cone of total opening $3\pi /2$, defined by gluing the boundary rays of that wedge, not by quotienting the whole plane by a finite rotation group. Let $d_C$ be its intrinsic length distance. Define

Away from the axis the metric is

At the axis we use its metric-length completion.

A future causal curve is a locally absolutely continuous curve whose spatial metric speed is at most its nonnegative time derivative.