Numerator: mathbf S × mathbf v

The passage around this formula

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…

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