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

Support and collar

An explicit seam-compatible slicing family › Support and collar · lines 528–561

VII-F@1#eq:particle-time

Let

K = {x : ρ (x) \leq 4}, K_{s} = {(F_{s} (x), x) : x \in K} .

The moving core is ${ρ < 3}$ . Choose the collar

C_{s} = {(2 + s, x) : 7/2 \leq ρ (x) \leq 4} .

It is an annulus, disjoint from $W$ , and agrees with the fixed resting exterior translated in time by $s$ .

Here and below, “fixed exterior” means fixed spatial cellulation, fixed transition data, and the prescribed equation $T = 2 + s$ . It cannot mean an identical embedded subset of spacetime for every $s$ , since the slices are strictly ordered.

The particle intersection is

P_{s} = Σ_{s} \cap W = {(s, 0), (2 s, 0), s \leq 0, s \geq 0.

Thus $P_{s} \in K_{s}$ for the entire window, including $s = 0$ . The worldline itself has not changed.

Displayed formulas in this passage

VII-F@1#eq:particle-time · line 551
$P_{s} = Σ_{s} \cap W = {(s, 0), (2 s, 0), s \leq 0, s \geq 0.$
unlabeled · line 531
$K = {x : ρ (x) \leq 4}, K_{s} = {(F_{s} (x), x) : x \in K} .$
unlabeled · line 538
$C_{s} = {(2 + s, x) : 7/2 \leq ρ (x) \leq 4} .$