Prescribed spacetime, support, and the incidence input

Fix a length unit $\ell >0$, a speed $c>0$, and a time origin $T_{\ast }$. Write

The regular background is

where $r>0$, $\varphi$ has period $2\pi$, and the stipulated completion has singular locus

The defect is a worldsheet. Only its fixed-$z$ sections are worldlines. No smooth chart, pointwise Einstein tensor, or field boundary condition is asserted on $\mathcal{W}$.

On a lifted regular angular interval,

is a local Minkowski development. Static observers have $\mathrm{d}{\tau }_{\mathrm{proper}}=\mathrm{d}T$. The metric lapse in $T$ is one. The spinless seam gluing fixes $T,z$ and rotates the transverse developed plane by $\pi /2$ in the chosen gluing convention. These are the prescribed I-B data CheungIB.