The cone, its regular part, and its completion

Fix

Let $C_{\mu }$ be the completion of

with spatial line element

Write $o$ for its apex. The circumference of the circle $r=R$ is $(2\pi -\mu )R$, so the opening angle is $2\pi -\mu$.

Set

For any smooth positive lapse $N$ on $M$, prescribe

The main model uses

In geometrized units $c=1$, this is the metric in the proposed conical-product branch. Equivalently, with $\theta =\alpha \varphi$, the angular coordinate has period $2\pi -\mu$ and the angular term is $r^2\mathrm{d}{\theta }^2$.

The singular completion locus is

It is two-dimensional. For fixed $z_0$, the subset ${\mathcal{W}}_{z_0}={\mathbb{R}}_T\times \{o\}\times \{z_0\}$ is a worldline.

The smooth Lorentzian manifold in our theorem is $M$, not $\overline{M}$ equipped with a purported smooth metric at its apex. The completion records the conical defect. No smooth chart or ordinary pointwise Einstein tensor is asserted on $\mathcal{W}$.