Manuscript Oai282-B@1 · Oai282-B.tex · proposition · f2956de5

Proposition: A special-conformal flow leaves the chart

What the local result does not integrate · lines 1801–1827 · proof included or linked

Oai282-B@1#prop:flow

The local flow of

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

g_{s} (x)^{μ} = \frac{x ^{μ} - s b ^{μ} x ^{2}}{1 - 2 s b \cdot x + s ^{2} b ^{2} x ^{2}}

where the denominator is nonzero. It need not define a flow $R \times M \to M$ on the entire Minkowski chart.

Proof. Differentiating the rational formula verifies the flow equation and the initial condition. For $b = (1, 0, 0, 0)$ and $x = (t, 0, 0, 0)$ , $t \neq = 0$ , it reduces to

g_{s} (x)^{0} = \frac{t}{1 - s t} .

It has a pole at $s = 1/ t$ . A compactification or another chart is required to discuss continuation through that value.

Displayed formulas in this passage

unlabeled · line 1804
$X_{b}^{μ} (x) = 2 (b \cdot x) x^{μ} - b^{μ} x^{2}$
unlabeled · line 1808
$g_{s} (x)^{μ} = \frac{x ^{μ} - s b ^{μ} x ^{2}}{1 - 2 s b \cdot x + s ^{2} b ^{2} x ^{2}}$
unlabeled · line 1822
$g_{s} (x)^{0} = \frac{t}{1 - s t} .$

Proposition: A special-conformal flow leaves the chart

Displayed formulas in this passage

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