Manuscript XI-A-ST@1 · XI-A-ST.tex · assumption · c3f0929d

Assumption: Prescribed host and supported chart

The retained host and its three ledgers · lines 268–314

XI-A-ST@1#ass:host, XI-A-ST@1#eq:hostmetric, XI-A-ST@1#eq:signature

Retain the regular conical product

M^{\circ} = R_{T} \times C_{δ_{c}}^{\circ} \times R_{z}, α_{c} = 1 - \frac{δ _{c}}{2 π}, 0 < δ_{c} < 2 π,

with metric

G = - c_{h}^{2} d T^{2} + d r^{2} + α_{c}^{2} r^{2} d φ^{2} + d z^{2}, c_{h} > 0,

and future orientation given by increasing $T$ . Its completion has the worldsheet

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

The defect is a worldsheet, not a particle worldline.

Retain a regular developing chart

ι : V ⟶ L \subset M^{\circ}

from an open set $V$ in the declared completion

M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x} .

The signature and time-calibration ledger is

ι^{*} G = - η, t = c_{h} (T - T_{Σ}) .

Retain the observation region $O$ and the supported shadow $D_{*} ⋐ V$ declared for this laboratory instance.

Displayed formulas in this passage

XI-A-ST@1#eq:hostmetric · line 279
$G = - c_{h}^{2} d T^{2} + d r^{2} + α_{c}^{2} r^{2} d φ^{2} + d z^{2}, c_{h} > 0,$
XI-A-ST@1#eq:signature · line 305
$ι^{*} G = - η, t = c_{h} (T - T_{Σ}) .$
unlabeled · line 271
$M^{\circ} = R_{T} \times C_{δ_{c}}^{\circ} \times R_{z}, α_{c} = 1 - \frac{δ _{c}}{2 π}, 0 < δ_{c} < 2 π,$
unlabeled · line 287
$W = R_{T} \times {o} \times R_{z} .$
unlabeled · line 293
$ι : V ⟶ L \subset M^{\circ}$
unlabeled · line 297
$M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x} .$