Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Regular part and completion

The prescribed conical geometry › Regular part and completion · lines 339–375

II-B@1#eq:metric

Fix

0 < μ < 2 π, α = 1 - \frac{μ}{2 π}, c > 0.

Let

C_{μ}^{\circ} = (0, \infty) \times (R /2 π Z)

with spatial metric

h_{C} = d r^{2} + α^{2} r^{2} d ϕ^{2} .

Its metric completion has an apex $o$ . Set

M = R_{T} \times C_{μ}^{\circ} \times R_{z}, M = R_{T} \times C_{μ} \times R_{z},

and prescribe

g = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2} .

The singular completion locus is

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

It is a worldsheet. Fixing $z$ gives one worldline.

We work on the smooth regular part $M$ . We do not assign an ordinary pointwise curvature tensor at the apex. This is the prescribed-background convention of I-B CheungIB.

Displayed formulas in this passage

II-B@1#eq:metric · line 361
$g = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2} .$
unlabeled · line 342
$0 < μ < 2 π, α = 1 - \frac{μ}{2 π}, c > 0.$
unlabeled · line 347
$C_{μ}^{\circ} = (0, \infty) \times (R /2 π Z)$
unlabeled · line 351
$h_{C} = d r^{2} + α^{2} r^{2} d ϕ^{2} .$
unlabeled · line 355
$M = R_{T} \times C_{μ}^{\circ} \times R_{z}, M = R_{T} \times C_{μ} \times R_{z},$
unlabeled · line 366
$W = R_{T} \times {o} \times R_{z} .$

Regular part and completion

Displayed formulas in this passage

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