Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

Spacetime conventions

Geometry, instruments, and admitted networks › Spacetime conventions · lines 178–213

En06-B@1#eq:causalcondition, En06-B@1#eq:Lipschitz

Fix a spatial dimension $D \geq 1$ and a constant $c > 0$ . On $R^{1 + D}$ , write points as $z = (t, x)$ and use

∥ (t, x) ∥_{c} = c^{2} t^{2} + ∥ x ∥_{2}^{2}, d_{c} (z, z^{'}) = ∥ z - z^{'} ∥_{c} .

All Hausdorff distances and neighborhoods in this paper use $d_{c}$ . For a set $K$ , let

K_{ρ} = {z : dist_{d_{c}} (z, K) \leq ρ} .

An edge from $z_{v} = (t_{v}, x_{v})$ to $z_{w} = (t_{w}, x_{w})$ is future causal when

t_{w} \geq t_{v}, ∥ x_{w} - x_{v} ∥_{2} \leq c (t_{w} - t_{v}) .

If its duration is zero, its endpoints coincide.

For $g \in P_{+}^{↑}$ , let $L_{g}$ be the operator norm of the linear part of $g$ with respect to $∥ \cdot ∥_{c}$ . Then

d_{c} (g z, g z^{'}) \leq L_{g} d_{c} (z, z^{'}), d_{H} (g K, g K^{'}) \leq L_{g} d_{H} (K, K^{'}) .

Translations do not enter $L_{g}$ .

Displayed formulas in this passage

unlabeled · line 182
$∥ (t, x) ∥_{c} = c^{2} t^{2} + ∥ x ∥_{2}^{2}, d_{c} (z, z^{'}) = ∥ z - z^{'} ∥_{c} .$
unlabeled · line 189
$K_{ρ} = {z : dist_{d_{c}} (z, K) \leq ρ} .$
En06-B@1#eq:causalcondition · line 195
$t_{w} \geq t_{v}, ∥ x_{w} - x_{v} ∥_{2} \leq c (t_{w} - t_{v}) .$
En06-B@1#eq:Lipschitz · line 206
$d_{c} (g z, g z^{'}) \leq L_{g} d_{c} (z, z^{'}), d_{H} (g K, g K^{'}) \leq L_{g} d_{H} (K, K^{'}) .$