Manuscript En06-A@1 · En06-A.tex · section · e04c9892

Euclidean cone convergence and microscopic compatibility

Euclidean cone convergence and microscopic compatibility · lines 1163–1185

Use the spacetime comparison metric

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

This is a topology and error metric, not a Lorentz-invariant metric.

Events embed as $(s Δ t_{n}, x)$ . Define the full discrete metric past of an apex $a_{n} = (T_{n} Δ t_{n}, A_{n})$ by

J_{n}^{-} (a_{n}) = {(s Δ t_{n}, x) : s \leq T_{n}, d_{n} (x, A_{n}) \leq T_{n} - s} .

Finite experiments use its truncation below their starting tick.

Local convergence means that, on every compact comparison set, points of either set have distance tending to zero from the other full set. It does not require approximating points to remain inside the same compact set.

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unlabeled · line 1167
$d_{c} ((t, x), (t^{'}, x^{'})) = max (c ∣ t - t^{'} ∣, ∥ x - x^{'} ∥_{2}) .$
unlabeled · line 1175
$J_{n}^{-} (a_{n}) = {(s Δ t_{n}, x) : s \leq T_{n}, d_{n} (x, A_{n}) \leq T_{n} - s} .$

Euclidean cone convergence and microscopic compatibility

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