Manuscript Oai282-A@1 · Oai282-A.tex · proposition · 5b6134ec

Proposition: Special conformal flows can leave the Minkowski chart

Stage 1: local currents, global actions, and bounded locality › Three statements with different mathematical types · lines 373–393 · proof included or linked

Oai282-A@1#prop:flow

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For

X_{b}^{μ} (x) = 2 (b \cdot x) x^{μ} - b^{μ} x^{2},

the corresponding flow need not be complete on $M$ .

Proof. Take $b = (b^{0}, 0, 0, 0)$ , $b^{0} > 0$ , and start on the time axis at $x^{0} (0) = t_{0} > 0$ . The flow remains on that axis and solves

\frac{d t}{d s} = b^{0} t^{2} .

Therefore

t (s) = \frac{t _{0}}{1 - s b ^{0} t _{0}},

which leaves the chart at $s = (b^{0} t_{0})^{- 1}$ .

Displayed formulas in this passage

unlabeled · line 376
$X_{b}^{μ} (x) = 2 (b \cdot x) x^{μ} - b^{μ} x^{2},$
unlabeled · line 385
$\frac{d t}{d s} = b^{0} t^{2} .$
unlabeled · line 389
$t (s) = \frac{t _{0}}{1 - s b ^{0} t _{0}},$

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section: Three statements with different mathematical types
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section: Three statements with different mathematical types