Rest magnetic spin rotation and independent centroid disk

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the canonical tangent-bundle and connection suppliers used by the BMT transport and charged-action equivalence. No full Axiom of Choice is assumed.

In a charged-body inertial frame of Inertial frames and inertial charts of the affine Minkowski model with $e_0=U\mathrm{/}c$, positively oriented spatial basis $(e_1,e_2,e_3)$, and constant coordinates for the stationary trajectory, let $U=(c,\mathbf{0})$, $\mathbf{E}=0$, and $\mathbf{B}=B{\mathbf{e}}_3$ with constant $B>0$ tesla. Let the adopted polarization law have $q$ in coulombs, $m>0$ kilograms, $\alpha =q\mathrm{/}m$, $G=g_s\mathrm{/}2$, and choose initial polarization $\mathbf{s}({\tau }_0)=s_0{\mathbf{e}}_1$ in kilogram metres squared per second. Independently, choose a separate momentum-rest inertial frame $(o^{\mathrm{\prime }};f_0,f_1,f_2,f_3)$ of Inertial frames and inertial charts of the affine Minkowski model with $f_0=p\mathrm{/}(Mc)$, spatial origin $o^{\mathrm{\prime }}$ at the stipulated body center, and positive spatial basis $(f_1,f_2,f_3)$. Let a flat material body have rest momentum components $p=(Mc,\mathbf{0})$ in this frame, $M>0$ kilograms, body angular momentum ${\mathbf{s}}_{\ast }=S_{\ast }{f}_3$ with $S_{\ast }>0$ in kilogram metres squared per second, and measurable densities $e:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ and $j:{\mathbb{R}}^3\to {\mathbb{R}}^3$ with each component of $j$ in $L^1({\mathbb{R}}^3)$, expressed in its spatial coordinates $\mathbf{x}$. Define ${\bar{B}}_{\mathrm{\ell }}(0):=\{\mathbf{x}:\mathrm{\mid }\mathbf{x}\mathrm{\mid }\le \mathrm{\ell }\}$ using this rest-space Euclidean norm. Assume $e$ and $j$ vanish almost everywhere outside this ball for $\mathrm{\ell }>0$ metres, $\mathrm{\mid }j\mathrm{\mid }$ is that norm and satisfies $\mathrm{\mid }j\mathrm{\mid }\le e\mathrm{/}c$ almost everywhere, $\int e{d}^{3}x=M{c}^2$, and the componentwise vector integral satisfies ${\mathbf{s}}_{\ast }=\int \mathbf{x}\times j{d}^{3}x$. Assume also that its spin tensor is the spatial dual of ${\mathbf{s}}_{\ast }$ and obeys the momentum SSC. Then the polarization and body-centroid results are

$\mathbf{s}(\tau )=s_0(\cos (\omega \mathrm{\Delta }\tau ){\mathbf{e}}_1-\sin (\omega \mathrm{\Delta }\tau ){\mathbf{e}}_2),\omega =\frac{qGB}{m},\mathrm{\Delta }\tau =\tau -{\tau }_0,$

$\mathrm{\ell }\ge \frac{S_{\ast }}{Mc},\mathit{\delta }=({\delta }_1,{\delta }_2,0)\text{ in the body-frame spatial components},{\delta }_1^2+{\delta }_2^2<{\left(\frac{S_{\ast }}{Mc}\right)}^2\mathrm{.}$

Every interior shift vector is realized by some future timelike observer. There is no identification between $s_0$ and $S_{\ast }$ in this example.

This is a model calculation. The field values are stipulated constants, not observations; the inequalities are conditional consequences for a separate body satisfying the stated material hypotheses.