$$|\mathbf S \times \mathbf v| \le |\mathbf S|\,v$$

Equation 41 · Evolutionary Physics

The Atlas That Refuses to Close

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

a shift transverse to both the spin and the relative velocity [ 13 ] . This is an exact identity of special relativity, not an approximation valid only for small v ; it holds all the way to v $\to$ c , at which point |$\Delta$$\mathbf x$| $\to$ |$\mathbf S$|/(Mc) = $\rho_M$ exactly, since |$\mathbf S$ $\times$ $\mathbf v$| $\le$ |$\mathbf S$|\,v saturates when the spin is perpendicular to the boost. No boosted observer, however extreme, can push a body’s own centroid further from its rest-frame value than $\rho_M$ . As u ranges over every inertial observer, $X^\mu(u,\Sigma)$ traces out a disk of exactly that radius, orthogonal to $\mathbf S$ — the world-tube first identified by Møller and reproduced in modern language…

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