Manuscript IX-A-ST@1 · IX-A-ST.tex · section · 6e99d1b2

Prescribed cone and a causally saturated developing laboratory

Prescribed cone and a causally saturated developing laboratory · lines 254–281

IX-A-ST@1#eq:cone

Fix $0 < μ < 2 π$ , $c > 0$ , and

α = 1 - \frac{μ}{2 π} .

The regular background and its metric are

M_{μ} = R_{T} \times C_{μ}^{\circ} \times R_{z}, g_{μ} = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, r > 0,

with $φ$ of period $2 π$ . The stipulated completion has singular locus

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

It is a worldsheet. Only its fixed- $z$ sections are worldlines. No smooth metric chart or pointwise Einstein tensor is asserted on $W$ .

The signature is $(-, +, +, +)$ . The lapse in $T$ is one, and static observers have proper-time increment $d T$ . These are prescribed I-B data CheungIB.

Displayed formulas in this passage

IX-A-ST@1#eq:cone · line 262
$M_{μ} = R_{T} \times C_{μ}^{\circ} \times R_{z}, g_{μ} = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, r > 0,$
unlabeled · line 258
$α = 1 - \frac{μ}{2 π} .$
unlabeled · line 270
$W = R_{T} \times {o} \times R_{z} .$

Nearby in this section

proposition: Proposition: Non-global hyperbolicity