Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Developing coordinates and support margins

Prescribed geometry and the local analytical domain › Developing coordinates and support margins · lines 376–410

III-B@1#eq:flat-chart, III-B@1#eq:margin

Choose a regular spatial neighborhood $U$ with a developing chart

χ : U ⟶ B_{R} (0) \subset R^{3}

whose image is a Euclidean ball. Locally it is obtained from

(r, ϕ, z) ⟼ (r cos (α ϕ), r sin (α ϕ), z),

followed by a rigid Euclidean coordinate change and restriction.

Fix a reference time $T_{0}$ and put

s = c (T - T_{0}), x = χ (r, ϕ, z) .

Then

g ˉ = - d s^{2} + i = 1 \sum^{3} (d x^{i})^{2} = η_{μν} d x^{μ} d x^{ν}, x^{0} = s, η = diag (- 1, 1, 1, 1) .

Choose $a, r_{*} > 0$ with

r_{*} + a < R, I = (- a, a) .

Displayed formulas in this passage

III-B@1#eq:flat-chart · line 396
$g ˉ = - d s^{2} + i = 1 \sum^{3} (d x^{i})^{2} = η_{μν} d x^{μ} d x^{ν}, x^{0} = s, η = diag (- 1, 1, 1, 1) .$
III-B@1#eq:margin · line 405
$r_{*} + a < R, I = (- a, a) .$
unlabeled · line 380
$χ : U ⟶ B_{R} (0) \subset R^{3}$
unlabeled · line 385
$(r, ϕ, z) ⟼ (r cos (α ϕ), r sin (α ϕ), z),$
unlabeled · line 392
$s = c (T - T_{0}), x = χ (r, ϕ, z) .$