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

Lemma: Axis and trace information

External conical geometry and certified local moves › Deficit and particle–slice data · lines 1314–1332 · proof included or linked

En03-A@1#lem:axis, En03-A@1#eq:particle-axis

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For $L = C R (μ) C^{- 1}$ , $C \in SO^{+} (2, 1)$ , and $0 < μ < 2 π$ ,

Fix (L) = R C e_{0}, p_{L} = C e_{0}, e_{0} = (1, 0, 0) .

The future unit fixed vector is unique, and

Tr L = 1 + 2 cos μ .

The trace alone does not distinguish $μ$ from $2 π - μ$ .

Proof. The spatial rotation has no nonzero fixed vector in its spatial plane. Conjugation preserves trace and future orientation and carries the fixed line to $R C e_{0}$ .

Displayed formulas in this passage

En03-A@1#eq:particle-axis · line 1317
$Fix (L) = R C e_{0}, p_{L} = C e_{0}, e_{0} = (1, 0, 0) .$
unlabeled · line 1322
$Tr L = 1 + 2 cos μ .$

Nearby in this section

section: Deficit and particle–slice data
proposition: Proposition: Gauge invariance of marked-slice data