Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Geometry, local interpretation, and analytical continuation

Geometry, local interpretation, and analytical continuation · lines 210–227

V-B@1#eq:cone

Fix $c > 0$ , $0 < μ < 2 π$ , and $α = 1 - μ / (2 π)$ . Prescribe

M = R_{T} \times C_{μ}^{\circ} \times R_{z}, g ˉ = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2},

where $r > 0$ and $ϕ \in R / (2 π Z)$ . The missing apex gives the worldsheet

W = R_{T} \times {o} \times R_{z}

in the metric completion. It is not a worldline. Signature, lapse, future orientation, and deficit are inputs.

Displayed formulas in this passage

V-B@1#eq:cone · line 215
$M = R_{T} \times C_{μ}^{\circ} \times R_{z}, g ˉ = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2},$
unlabeled · line 222
$W = R_{T} \times {o} \times R_{z}$