Manuscript VII-J@1 · VII-J.tex · corollary · d8c01b04

Corollary: A uniform conservative directional speed

The fixed cone and VII-F's slicing event › Physical ports, not abstract cell labels · lines 1491–1519 · proof included or linked

VII-J@1#cor:conespeed

Back to manuscript

A spatial route parametrized by $s$ , with metric speed at most

v_{*} = \frac{1}{1 + λ _{0}} = \frac{1}{1 + 2 2 /3},

lifts to a future causal curve $(F_{s} (x (s)), x (s))$ . In each regular developed chart, the Euclidean ball ${v : ∥ v ∥ \leq v_{*}}$ is an admissible conservative velocity set in the slicing parameter.

Proof. For a short increment $δ s > 0$ ,

F_{s + δ s} (x (s + δ s)) - F_{s} (x (s)) \geq δ s - λ_{0} d_{C} (x (s + δ s), x (s)) .

Taking the almost-everywhere metric derivative gives

\frac{d T}{d s} \geq 1 - λ_{0} ∣ x ˙ ∣.

The speed restriction makes this at least $∣ x ˙ ∣$ . Thus the lifted curve is causal.

Displayed formulas in this passage

unlabeled · line 1495
$v_{*} = \frac{1}{1 + λ _{0}} = \frac{1}{1 + 2 2 /3},$
unlabeled · line 1508
$F_{s + δ s} (x (s + δ s)) - F_{s} (x (s)) \geq δ s - λ_{0} d_{C} (x (s + δ s), x (s)) .$
unlabeled · line 1514
$\frac{d T}{d s} \geq 1 - λ_{0} ∣ x ˙ ∣.$

Corollary: A uniform conservative directional speed

Displayed formulas in this passage

Nearby in this section