Manuscript VIII-ST@1 · VIII-ST.tex · section · fe979f3b

Prescribed conical products and supported presentations

Prescribed conical products and supported presentations · lines 246–292

VIII-ST@1#eq:interval, VIII-ST@1#eq:metric, VIII-ST@1#eq:worldsheet

Fix $c > 0$ , a length unit $ℓ > 0$ , and a time origin $T_{*}$ . Let

μ_{0} = \frac{π}{2}, I = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], α_{μ} = 1 - \frac{μ}{2 π} .

For each prescribed $μ \in I$ , define

M_{μ} = R_{T} \times C_{μ}^{\circ} \times R_{z}, g_{μ} = - c^{2} d T^{2} + d r^{2} + α_{μ}^{2} r^{2} d ϕ^{2} + d z^{2},

where $r > 0$ and $ϕ$ has period $2 π$ . The stipulated completion has singular locus

W_{μ} = R_{T} \times {o} \times R_{z} .

Thus the defect is a worldsheet. A fixed- $z$ section is a worldline. We do not assert a smooth Minkowski neighbourhood or a pointwise Einstein tensor on $W_{μ}$ .

The signature is $(-, +, +, +)$ . Static observers have proper time increment $d T$ ; the lapse in $T$ is one and the shift is zero. Use normalized coordinates

τ = \frac{c ( T - T _{*} )}{ℓ}, x = (X / ℓ, Y / ℓ), ζ = z / ℓ .

On a regular lifted angular interval,

(T, r, ϕ, z) ⟼ (c (T - T_{*}), r cos (α_{μ} ϕ), r sin (α_{μ} ϕ), z)

is a local Minkowski development. The geometry is an input, in the sense of I-B CheungIB.

Displayed formulas in this passage

VIII-ST@1#eq:interval · line 251
$μ_{0} = \frac{π}{2}, I = [\frac{π}{2} - \frac{π}{20}, \frac{π}{2} + \frac{π}{20}], α_{μ} = 1 - \frac{μ}{2 π} .$
VIII-ST@1#eq:metric · line 259
$M_{μ} = R_{T} \times C_{μ}^{\circ} \times R_{z}, g_{μ} = - c^{2} d T^{2} + d r^{2} + α_{μ}^{2} r^{2} d ϕ^{2} + d z^{2},$
VIII-ST@1#eq:worldsheet · line 267
$W_{μ} = R_{T} \times {o} \times R_{z} .$
unlabeled · line 279
$τ = \frac{c ( T - T _{*} )}{ℓ}, x = (X / ℓ, Y / ℓ), ζ = z / ℓ .$
unlabeled · line 284
$(T, r, ϕ, z) ⟼ (c (T - T_{*}), r cos (α_{μ} ϕ), r sin (α_{μ} ϕ), z)$

Prescribed conical products and supported presentations

Displayed formulas in this passage

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