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

Proposition: Directional obstruction

Kinematics and causal cuts on a prescribed schedule › Directional propagation · lines 895–914 · proof included or linked

En03-A@1#prop:directional, En03-A@1#eq:velocity-diamond

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In a prescribed square-lattice model permitting one cardinal step of length $a$ , or waiting, per tick, the asymptotic velocity set is

V = {(v_{x}, v_{y}) : ∣ v_{x} ∣ + ∣ v_{y} ∣ \leq c}, c = a / τ .

An isotropic physical speed $c_{phys} = c$ is not dominated in every direction.

Proof. After $n$ ticks, $∣ Δ x ∣ + ∣ Δ y ∣ \leq na$ . Step proportions approximate every point of the displayed diamond. The velocity $(c / 2, c / 2)$ has Euclidean norm $c$ but $ℓ^{1}$ -norm $2 c$ , so is excluded.

Displayed formulas in this passage

En03-A@1#eq:velocity-diamond · line 899
$V = {(v_{x}, v_{y}) : ∣ v_{x} ∣ + ∣ v_{y} ∣ \leq c}, c = a / τ .$

Nearby in this section

remark: Remark: Conditional kinematics