Body moments and transport convention

Use the charged/spinning body-model conventions of the now-authored batch-24 carrier Charged and spinning test-body quantities: a smooth time-oriented Lorentzian four-manifold of signature $(-+++)$, coordinates $x^0=ct$ in metres, $c>0$, and a $C^2$ future timelike representative curve $\zeta :I\to M$ parametrized by proper time $\tau$ in seconds, with $U=d\zeta \mathrm{/}d\tau$ and $g(U,U)=-c^2$. The actual supplier defines these quantities and the SI electromagnetic-tensor convention used here. Assume Countable Choice for the GR smooth-structure, convex-normal and parallel-transport suppliers used below.

(a) Flat isolated-body moments. Choose an inertial frame $(o;e_0,e_1,e_2,e_3)$ and its chart $x:A\to {\mathbb{R}}^4$ as in Inertial frames and inertial charts of the affine Minkowski model, with the SI coordinate convention $x^0=ct$ and $x^i$ in metres of Charged and spinning test-body quantities. Thus $e_0$ is future unit timelike and $(e_1,e_2,e_3)$ is the positively oriented orthonormal spatial basis. Let $T$ be a $C^{\mathrm{\infty }}$ symmetric type-$(2,0)$ tensor field on the affine Minkowski spacetime $A$, whose chart components $T^{ab}$ are the model stress-energy densities with units $\mathrm{J}{\mathrm{m}}^{-3}$. The coordinate-time image $J:=\{x^0(\zeta (\tau ))\mathrm{/}c:\tau \in I\}$ is an open interval, and $\zeta (t)$ denotes the unique point of the curve with coordinate time $t$; this is well-defined because future timelike curves have strictly increasing inertial coordinate time. For a finite closed interval $[t_1,t_2]\subset \mathbb{R}$, define $K_{[t_1,t_2]}:=\stackrel{‾}{\{\mathbf{x}\in {\mathbb{R}}^3:\text{there are }t\in [t_1,t_2]\text{ and }a,b\in \{0,1,2,3\}\text{ with }{T}^{ab}(ct,\mathbf{x})\mathrm{\ne }0\}}\mathrm{.}$ Assume $K_{[t_1,t_2]}$ is compact for every such interval. For each $t\in \mathbb{R}$, define the spatial support by ${\mathrm{supp}}_{t}T:=\stackrel{‾}{\{\mathbf{x}\in {\mathbb{R}}^3:\text{there are }a,b\in \{0,1,2,3\}\text{ with }{T}^{ab}(ct,\mathbf{x})\mathrm{\ne }0\}}\mathrm{.}$ At a fixed time $t$, the spatial slice has coordinates $\mathbf{x}=(x^1,x^2,x^3)\in {\mathbb{R}}^3$ and three-dimensional Lebesgue measure $d^{3}x$; define $p^a(t):=\frac{1}{c}{\int }_{{\mathbb{R}}^3}{T}^{0a}(ct,\mathbf{x})d^{3}x,$ and, for the representative event $\zeta (t)$ on that time slice, define $S_{\zeta (t)}^{ab}(t):=\frac{1}{c}{\int }_{{\mathbb{R}}^3}((x-\zeta (t){)}^a{T}^{0b}(ct,\mathbf{x})-(x-\zeta (t){)}^b{T}^{0a}(ct,\mathbf{x}))d^{3}x\mathrm{.}$ Here $(x-\zeta (t){)}^0=0$ on the chosen slice, $p$ has units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and $S_{\zeta (t)}$ has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. These are definitions of moments; no conservation is asserted by clause (a).

(b) Local curved-body moments. Let $z\in M$ and let $U_0$ be a convex normal neighborhood containing $z$. Let $\mathrm{\Sigma }=i(S)\subset U_0$ be a $C^1$ embedded spacelike hypersurface through $z$, where $S$ is a three-dimensional $C^1$ manifold without boundary and $i:S\to U_0$ is an injective $C^1$ immersion that is a homeomorphism onto its image. Thus each tangent space $T_x\mathrm{\Sigma }$ is three-dimensional; require the restriction of $g_x$ to it to be positive definite. Let $n_x$ be its continuous future unit normal, meaning $g_x(n_x,n_x)=-1$ and $g_x(n_x,w)=0$ for every $w\in T_x\mathrm{\Sigma }$. For a local parametrization $X:V\subseteq {\mathbb{R}}^3\to \mathrm{\Sigma }$, put $g_{ij}^{\mathrm{\Sigma }}(y):=g_{X(y)}({\mathrm{\partial }}_{i}X,{\mathrm{\partial }}_{j}X)$ and define $d\mathrm{\Sigma }:=\sqrt{\det (g_{ij}^{\mathrm{\Sigma }}(y))}d{y}^{1}d{y}^{2}d{y}^3\mathrm{.}$ The positive-definite condition makes this density positive, and the coordinate-change rule makes the resulting measure independent of the parametrization; $d{y}^{1}d{y}^{2}d{y}^3$ is Lebesgue measure on $V$. Let $T$ be a smooth symmetric type-$(2,0)$ stress-energy tensor field on $U_0$, with components in an orthonormal tetrad in joules per cubic metre; only its restriction $T{\mathrm{\mid }}_{\mathrm{\Sigma }}$ is used. With $(n_x{)}_b:=g_{bc}(x)n_x^c$, define ${\mathrm{supp}}_{\mathrm{\Sigma }}(T{\mathrm{\mid }}_{\mathrm{\Sigma }}):={\overline{\{x\in \mathrm{\Sigma }:T_x\mathrm{\ne }0\}}}^{\mathrm{\Sigma }}\mathrm{.}$ Assume this restricted support is a compact subset of $U_0$ and that all displayed integrands are absolutely integrable with respect to $d\mathrm{\Sigma }$. For every $x\in \mathrm{\Sigma }$, let $v(x):={\exp }_z^{-1}(x)\in T_{z}M$ be the unique joining position vector supplied by Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points and The geodesic initial-value problem and the exponential map of the metric-selected connection. Let $H_{x\to z}:T_{x}M\to T_{z}M$ be parallel transport along that unique geodesic, with existence, uniqueness and metric preservation as in Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed. Define $T^{\mathrm{♯}}(n_x{)}^a:=T^{ab}(x)(n_x{)}_b$ and $p_z:=\frac{1}{c}{\int }_{\mathrm{\Sigma }}{H}_{x\to z}\text{\hspace{0.17em}⁣}(-T^{\mathrm{♯}}(n_x))d\mathrm{\Sigma },S_z:=\frac{1}{c}{\int }_{\mathrm{\Sigma }}v(x)\wedge H_{x\to z}\text{\hspace{0.17em}⁣}(-T^{\mathrm{♯}}(n_x))d\mathrm{\Sigma },$ where $(v\wedge w{)}^{ab}:=v^a{w}^b-v^b{w}^a$ and the finite-dimensional vector and tensor integrals are taken componentwise in any basis of $T_{z}M$. The resulting $p_z$ and $S_z$ are a vector and an antisymmetric tensor at $z$; their units are $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$ and $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. “Support on $\mathrm{\Sigma }$” here means only compact support of the restriction $T{\mathrm{\mid }}_{\mathrm{\Sigma }}$; it does not assert that the ambient smooth field $T$ has spacetime support contained in $\mathrm{\Sigma }$.

• The curved definition is local: all support and transport paths remain in the stated convex normal neighborhood. A different hypersurface or transport convention can define different moments; no slice-independence theorem is asserted here.
• In the flat branch, if additionally ${\mathrm{\partial }}_a{T}^{ab}=0$ on a finite slab and the tensor has zero side flux through a closed axis-parallel box containing its spatial support, Finite-box stress balance and the flux form of conservation supplies the corresponding finite-box conservation of $p$. That theorem makes no noncompact-slice or arbitrary-hypersurface claim.
• The equations of motion in Pole-dipole and closed multipole test-body models are separately adopted; they are not consequences of this definition. The AC${}_{\omega }$ dependency is inherited through the cited GR smooth structure and parallel-transport interfaces.