What does this equation mean?

Fix the slicing convention that anchors the centroid to the body’s own rest frame — the choice, standard in the spinning-body literature, in which the spin tensor about the centroid satisfies $S^{\alpha\beta}$$P_\beta$ = 0 — and ask how $X^\mu(u,\Sigma)$ moves as u is varied away from the body’s own rest-frame observer to some other inertial observer boosted at velocity $\mathbf v$ relative to it. Costa and Natário’s synthesis of the classical spinning-body literature gives the answer in closed form: the centroid measured by the boosted observer is displaced from the rest-frame centroid by $\Delta \mathbf x = \frac{\mathbf S \times \mathbf v}{Mc^2}$. a shift transverse to both the spin and the relative velocity [ 13 ] . This is an exact identity of special relativity, not an approximation valid only for small v ; it holds all the way to v $\to$ c , at which point |$\Delta$$\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 by Costa and Natário’s review of spin supplementary conditions [ 13 ] .