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

Proposition: Exact reachable set and earliest arrival

A spatially varying causal benchmark · lines 996–1024 · proof included or linked

VII-J@1#prop:varying, VII-J@1#eq:varyingreach

Communication from $(s, x)$ to $(t, y)$ , $t \geq s$ , is possible exactly when

∣ y - x ∣ \leq c_{0} lo g \frac{1 + t}{1 + s} .

Its earliest arrival is

T_{arr} (x, y; s) = (1 + s) exp (∣ y - x ∣ / c_{0}) - 1,

provided that value lies in the declared time window.

Proof. Integrating the velocity bound proves necessity. For sufficiency, put

x (r) = x + (y - x) \frac{\int _{s} ^{r} c _{0} ( 1 + u ) ^{- 1} d u}{\int _{s} ^{t} c _{0} ( 1 + u ) ^{- 1} d u} .

Its speed is bounded by $c_{0} / (1 + r)$ , and it reaches $y$ . Solving the equality in (eq:varyingreach) gives the arrival formula.

Displayed formulas in this passage

VII-J@1#eq:varyingreach · line 1000
$∣ y - x ∣ \leq c_{0} lo g \frac{1 + t}{1 + s} .$
unlabeled · line 1006
$T_{arr} (x, y; s) = (1 + s) exp (∣ y - x ∣ / c_{0}) - 1,$
unlabeled · line 1016
$x (r) = x + (y - x) \frac{\int _{s} ^{r} c _{0} ( 1 + u ) ^{- 1} d u}{\int _{s} ^{t} c _{0} ( 1 + u ) ^{- 1} d u} .$