Pole-dipole and closed multipole test-body models

Assume Countable Choice as inherited from Body moments and transport convention. For a neutral test body, let $I\subseteq \mathbb{R}$ be an open interval of proper-time values and let $z:I\to M$ be a $C^2$ future timelike curve, with $U=dz\mathrm{/}d\tau$. Let $p^a$ and $S^{ab}=-S^{ba}$ be the momentum and spin moments of Body moments and transport convention. For a $C^1$ vector or rank-two tensor field $V$ along $z$, define $D_{\tau }V:=(z^{\ast }\mathrm{\nabla }{)}_{\mathrm{\partial }\mathrm{/}\mathrm{\partial }\tau }V$ using the pullback of the metric-selected connection. In a chart, with $\mathrm{\Gamma }$ the coefficients of $\mathrm{\nabla }$, this is

Adopt the pole-dipole evolution equations $D_{\tau }{p}^a=-\frac{1}{2}{R}^a{}_{bcd}{U}^b{S}^{cd},D_{\tau }{S}^{ab}=p^a{U}^b-p^b{U}^a\mathrm{.}$ A spin-supplementary condition (SSC) is an additional equation $S^{ab}{n}_b=0$ for a specified future unit timelike field $n$ along $z$; this postulate does not select one SSC, and $p$ is an independently evolved moment rather than a primitive assumption $p=mU$.

For a chosen finite multipole model, let $Q_1,\dots ,Q_r\to M$ be the declared tensor bundles of its additional moments. Its state bundle is the fiber product over $M$ of $TM$ for $U$, $TM$ for $p$, ${\mathrm{\Lambda }}^{2}TM$ for $S$, and the bundles $Q_1,\dots ,Q_r$, restricted to states with $U$ future timelike and $g(U,U)=-c^2$ and to the stated algebraic constraints on the moments. Fix an open admissible state domain $\mathcal{D}$ in this bundle. The extra force and torque are specified smooth bundle maps over the identity, $f_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}:\mathcal{D}\to TM$ and $N_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}:\mathcal{D}\to {\mathrm{\Lambda }}^{2}TM$, so at a state based at $x\in M$ their values lie in $T_{x}M$ and ${\mathrm{\Lambda }}^2{T}_{x}M$, respectively. Their units are $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}$ and $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-2}$. For a retained quadrupole or higher-multipole model, adopt $D_{\tau }{p}^a=-\frac{1}{2}{R}^a{}_{bcd}{U}^b{S}^{cd}+f_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}^a,D_{\tau }{S}^{ab}=p^a{U}^b-p^b{U}^a+N_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}^{ab},$ where $f_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}$ is a specified smooth force map along the state and $N_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}^{ab}=-N_{\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{a}}^{ba}$ is a specified smooth torque map. A quadrupole convention that may be adopted is $f_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}^a=-\frac{1}{6}{J}^{bcde}{\mathrm{\nabla }}^a{R}_{bcde},N_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}^{ab}=\frac{4}{3}{R}^{[a}{}_{cde}{J}^{b]cde},$ with $J^{bcde}$ satisfying the algebraic symmetries of the curvature tensor, carrying units $\mathrm{k}\mathrm{g}{\mathrm{m}}^4{\mathrm{s}}^{-2}$, and brackets including the factor $1\mathrm{/}2$. Any evolved $J$ or higher moment needs its own transport or constitutive law; without it the multipole equations are not a closed initial-value model.

The pole-dipole and multipole equations are adopted finite-dimensional truncations, not derivations from the moment integrals above. Their use as approximations for a material body is restricted to regimes in which its size is small compared with the relevant curvature and background-variation lengths and omitted moment effects are negligible on the stated finite interval. These conditions are qualitative applicability assumptions here: this postulate supplies no uniform error bound, no convergence of truncations, and no global existence claim. No experimental result or measurement is asserted.

The pole-dipole clause is neutral and test-body only. The optional multipole clause applies only when every extra force, torque, and evolved moment has explicit smooth constitutive data. No electric or magnetic dipole force is included in the BMT translation model, and no radiation reaction, self-force, singular-field evaluation, or quantum-spin claim is included.