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

Proposition: The family is embedded, spacelike, and ordered

An explicit seam-compatible slicing family › Spacelikeness, ordering, and embedding · lines 484–526 · proof included or linked

VII-F@1#prop:spacelike

The functions (eq:family) descend through the seam and define embedded spacelike slices. Uniform constants are

λ = \frac{2 2}{3} < 1, 1 - λ^{2} = \frac{1}{9}, c = 1.

More precisely, for $s_{2} > s_{1}$ ,

s_{2} - s_{1} \leq F_{s_{2}} (x) - F_{s_{1}} (x) \leq 2 (s_{2} - s_{1})

at every spatial point.

Proof. Single-valuedness was established before (eq:family). The nonconstant gradients are

(\frac{2}{3}, \frac{2}{3}), (\frac{3}{5}, 0), (0, \frac{3}{5}),

up to sector rotations. Their norms are bounded by $22 /3$ .

For fixed $x$ , the function

s ⟼ min {2, max {h (x), s + k (x)}}

is nondecreasing and $1$ -Lipschitz. Adding $s$ gives the displayed two-sided ordering bound. Lemma lem:graph applies.

Displayed formulas in this passage

unlabeled · line 488
$λ = \frac{2 2}{3} < 1, 1 - λ^{2} = \frac{1}{9}, c = 1.$
unlabeled · line 496
$s_{2} - s_{1} \leq F_{s_{2}} (x) - F_{s_{1}} (x) \leq 2 (s_{2} - s_{1})$
unlabeled · line 509
$(\frac{2}{3}, \frac{2}{3}), (\frac{3}{5}, 0), (0, \frac{3}{5}),$
unlabeled · line 520
$s ⟼ min {2, max {h (x), s + k (x)}}$