Manuscript En03-A@1 · En03-A.tex · proposition · 5398614e

Proposition: Spacelike graph and time order

External conical geometry and certified local moves › Four affine faces · lines 1521–1540 · proof included or linked

En03-A@1#prop:R1, En03-A@1#eq:time-ordering

The graph of $F_{s}$ is continuous, piecewise affine, and $p$ -Lipschitz with $p < 1$ . Every face is spacelike. For $s_{2} > s_{1}$ ,

F_{s_{2}} (x, y) - F_{s_{1}} (x, y) \geq a (s_{2} - s_{1}) > 0.

Proof. The spatial gradients have norm $p$ or $q$ ; their maximum retains the common Lipschitz bound. A face tangent satisfies

η ((v_{i} \cdot u, u), (v_{i} \cdot u, u)) = ∣ u ∣^{2} - (v_{i} \cdot u)^{2} > 0

for $u \neq = 0$ . Every face function increases by at least $a (s_{2} - s_{1})$ , so their maximum does too.

Displayed formulas in this passage

En03-A@1#eq:time-ordering · line 1525
$F_{s_{2}} (x, y) - F_{s_{1}} (x, y) \geq a (s_{2} - s_{1}) > 0.$
unlabeled · line 1534
$η ((v_{i} \cdot u, u), (v_{i} \cdot u, u)) = ∣ u ∣^{2} - (v_{i} \cdot u)^{2} > 0$