Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Ambient spacetime

The marked cone and a restricted obstruction › Ambient spacetime · lines 189–256

VII-F@1#eq:J

Use Minkowski coordinates $z = (T, X, Y)$ and metric

η = diag (- 1, 1, 1) .

Fix

μ = \frac{π}{2}, β = 1 - \frac{μ}{2 π} = \frac{3}{4}, L = 2 π β = \frac{3 π}{2} .

The spatial cone $C_{μ}$ is obtained from

{(r, θ) : r \geq 0, 0 \leq θ \leq L}

by identifying $(r, 0)$ with $(r, L)$ and identifying all points with $r = 0$ . Its metric away from the apex is

d r^{2} + r^{2} d θ^{2} .

The fixed spacetime is

M = R_{T} \times C_{μ}, d s_{M}^{2} = - d T^{2} + d r^{2} + r^{2} d θ^{2} .

Its singular worldline is

W = {(T, 0) : T \in R} .

In the reference development, $W$ is the line ${(T, 0, 0)}$ .

For later use set

v_{i} = (cos (iπ /2) sin (iπ /2)), i = 0, 1, 2, 3.

The cone consists of three developed quadrants

S_{i} = {ξ v_{i} + η v_{i + 1} : ξ, η \geq 0}, i = 0, 1, 2.

The ray in direction $v_{3}$ is identified with the ray in direction $v_{0}$ . Indices on cone rays and cell labels are taken modulo three; the developed vectors $v_{0}, \dots, v_{3}$ are kept distinct.

The upper seam is mapped to the lower seam by

J = (R (μ), 0), R (μ) = 100001 0 - 1 0 .

Thus $J (T, 0, - r) = (T, r, 0)$ , and $J$ fixes $W$ pointwise.

Displayed formulas in this passage

unlabeled · line 192
$η = diag (- 1, 1, 1) .$
unlabeled · line 196
$μ = \frac{π}{2}, β = 1 - \frac{μ}{2 π} = \frac{3}{4}, L = 2 π β = \frac{3 π}{2} .$
unlabeled · line 204
${(r, θ) : r \geq 0, 0 \leq θ \leq L}$
unlabeled · line 209
$d r^{2} + r^{2} d θ^{2} .$
unlabeled · line 213
$M = R_{T} \times C_{μ}, d s_{M}^{2} = - d T^{2} + d r^{2} + r^{2} d θ^{2} .$
unlabeled · line 219
$W = {(T, 0) : T \in R} .$
unlabeled · line 225
$v_{i} = (cos (iπ /2) sin (iπ /2)), i = 0, 1, 2, 3.$
unlabeled · line 234
$S_{i} = {ξ v_{i} + η v_{i + 1} : ξ, η \geq 0}, i = 0, 1, 2.$
VII-F@1#eq:J · line 243
$J = (R (μ), 0), R (μ) = 100001 0 - 1 0 .$

Nearby in this section

remark: Remark: Wedge gluing is not a finite rotation quotient