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BMT constraint and restricted-ansatz algebra

1 · Prerequisites

2 · Summary

This page develops a conditional flat and local-curved framework for classical polarization transport, finite-size charge estimates, observer-dependent centroids, and neutral pole-dipole dynamics. The mathematical carriers prove the restricted BMT transport algebra, positive-energy spin–size inequality, componentwise charge-multipole remainder, flat centroid disk, differentiated SSC identity, and local TD closure only on its explicitly admissible region. The physical items separately define body moments and adopt MPD and BMT models before deriving their conditional consequences.

The charged-spin definition and BMT transport law are current batch-24 suppliers on rpm-external-charge; their inspected interface fixes the $(-, +, +, +)$ signature, $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , the magnetic-moment parameter, and the curved BMT law as an adopted model extension. These assumptions do not prove quantum-spin ontology, empirical validity, or a curved Maxwell theory. The MPD equations are finite-dimensional neutral test-body truncations, not arbitrary stress-energy moment theorems. The TD closure is local and can fail when its matrix becomes singular or its algebraically determined tangent ceases to be timelike.

The body angular-momentum vector $s_{*}$ and the BMT polarization $s$ remain distinct unless an additional identification is adopted. The positive-energy bound requires $e \geq 0$ and $∣ j ∣ \leq e / c$ in the body rest frame. The Coulomb-exterior support-radius bound additionally assumes an exact Coulomb exterior and nonnegative remaining energy; it does not infer the field profile from spherical symmetry alone. Multipole estimates are componentwise on a fixed smooth trivialization and do not establish a global MPD approximation error.

Prerequisite frameworks: cm-math-dynamics, em-electrostatics-in-vacuum, gr-gr13-convex-normal-neighborhoods, rpm-external-charge, and sr-accelerated-observers.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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BMT transport constraint and restricted-ansatz algebra

Statement

Assume Countable Choice for the canonical tangent-bundle smooth structure supplied by the GR items below. Let $(M, g, N)$ be a smooth time-oriented Lorentzian four-manifold with its metric-compatible connection $\nabla$ , let $c > 0$ , and let $I \subseteq R$ be a nonempty open interval of proper-time values. Let $z : I \to M$ be a $C^{2}$ future timelike curve and set $U : = d z / d τ$ , with $g (U, U) = - c^{2}$ . For a $C^{1}$ vector field $V$ along $z$ , define its covariant derivative by the pullback connection $D_{τ} V : = (z^{*} \nabla)_{\partial / \partial τ} V$ ; in a chart this means $(D_{τ} V)^{a} = d V^{a} / d τ + Γ^{a}_{b c} (z (τ)) U^{b} V^{c}$ . Let $F_{τ}$ be a continuous alternating two-form along $z$ . Define $F_{τ}^{♯}$ by $g (F_{τ}^{♯} (v), w) = F_{τ} (w, v) .$ Then $F_{τ}^{♯}$ is skew-adjoint with respect to $g$ .

For constants $α, a, b \in R$ , consider a $C^{1}$ vector field $s$ along $z$ satisfying the restricted linear transport equations $D_{τ} U = α F^{♯} (U), D_{τ} s = α (a F^{♯} (s) + \frac{b}{c^{2}} U κ), κ : = g (s, F^{♯} (U)) .$ For every $τ_{0} \in I$ and initial value $s (τ_{0})$ there is a unique solution on each compact subinterval of $I$ containing $τ_{0}$ . Along any such solution, $\frac{d}{d τ} g (U, s) = α (1 - a - b) κ .$ Consequently, the orthogonality constraint $g (U, s) = 0$ is preserved when $a + b = 1$ , and on that constraint $g (s, s)$ is constant. For the coefficient comparison, fix a parameter value $τ$ and put $n : = U (τ) / c$ . Then $n$ is an observer at $z (τ)$ and its rest space is the three-dimensional positive-definite subspace $E_{τ} : = n^{⊥} = U (τ)^{⊥}$ (the observer and projection conventions are those of Observers, rest spaces and the spatial projection at an event and The observer rest-space decomposition and the spatial projection). Choose an orientation of $E_{τ}$ and a positively oriented orthonormal basis $(e_{1}, e_{2}, e_{3})$ ; set $e_{0} : = n$ . For $v \in E_{τ}$ , write $v = (g (v, e_{1}), g (v, e_{2}), g (v, e_{3}))$ , use $(u \times v)_{i} = ϵ_{i j k} u_{j} v_{k}$ , and define $B$ by $F_{i j} : = F (e_{i}, e_{j}) = ϵ_{i j k} B_{k}$ . Let $P_{U} (v) : = v + g (U, v) U / c^{2}$ , the projection onto $E_{τ}$ , and fix a target coefficient $G \in R$ . The rest-space matching condition means the pointwise equality $(P_{U} (D_{τ} s))^{i} = α G (s \times B)_{i} (i = 1, 2, 3)$ for every $s, B \in R^{3}$ , when $s (τ) \in E_{τ}$ . Thus it compares spatial components of the covariant derivative at the event, not the derivative of components in an unspecified moving tetrad. For $α \neq 0$ , if the coefficients both preserve $g (U, s) = 0$ for every initially orthogonal datum and every allowed alternating field and satisfy this matching condition, then $a = G$ and $b = 1 - G$ . Conversely, these coefficients satisfy both requirements. This uniqueness is only within the displayed two-term ansatz; no uniqueness among all covariant spin-transport laws is asserted. If $α = 0$ , the rest-space matching condition does not identify $a$ or $b$ .

Facts & Assumptions

Given: The stated Lorentzian manifold, its metric-compatible connection, $c > 0$ , a $C^{2}$ future timelike proper-time curve, a continuous alternating two-form along that curve, real coefficients $α, a, b$ , and an initial vector $s (τ_{0})$ .

[A1]

Countable Choice, inherited from the GR smooth-structure supplier The Axiom of Countable Choice ( ${A C}_{ω}$ ), is the only choice assumption used by the GR connection interface.

[A2]

The restricted transport equations, real coefficients, continuous alternating two-form, proper-time normalization, and initial spin vector are exactly the data in the Statement.

[F1]

The metric-selected connection is metric-compatible: along a curve, the covariant derivative differentiates the metric pairing by the Leibniz rule and the metric derivative vanishes. Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

[F2]

A continuous locally Lipschitz vector field on an open Euclidean domain has a unique local solution through each initial point, and solutions with the same initial data agree on common intervals. Picard-Lindelöf local existence and uniqueness for first-order systems

[F3]

At a point of the curve, $U / c$ is an observer; its orthogonal complement is a three-dimensional positive-definite rest space with projection $h_{U / c}$ , as specified in Observers, rest spaces and the spatial projection at an event and proved in The observer rest-space decomposition and the spatial projection. The coordinate cross product in a chosen positively oriented orthonormal basis is $(u \times v)_{i} = ϵ_{i j k} u_{j} v_{k}$ . The cross product in $R^{3}$

[F4]

The coefficient comparison uses the pointwise projected rest-space matching condition in the Statement for every $s, B$ ; it is a matching condition, not a claim about measured magnetic moments.

Proof

technique · direct

1.1givenA1F1algebra

Alternation gives $g (F^{♯} v, w) = F (w, v) = - F (v, w) = - g (v, F^{♯} w)$ , so $F^{♯}$ is $g$ -skew. By [F1] and the product rule, $\frac{d}{d τ} g (U, s) = g (D_{τ} U, s) + g (U, D_{τ} s) = α κ + α (a g (U, F^{♯} s) + b \frac{g (U, U)}{c^{2}} κ) .$ Skew-adjointness gives $g (U, F^{♯} s) = - κ$ , and $g (U, U) = - c^{2}$ ; therefore the derivative equals $α (1 - a - b) κ$ .

1.2givenF2algebra

In any chart along a compact subinterval, the spin equation is a linear system $\dot{s} = A (τ) s$ with continuous coefficients: the connection coefficients, $U$ , $F$ , and $g$ are continuous on the compact curve image. Its right-hand side is locally Lipschitz in $s$ , so [F2] gives local existence and uniqueness. The coefficient norm is bounded on each compact chart subinterval, say by $C$ ; iterating the integral equation gives $∥ s (τ) ∥ \leq ∥ s (τ_{0}) ∥ \sum_{n \geq 0} (C ∣ τ - τ_{0} ∣)^{n} / n! = ∥ s (τ_{0}) ∥ e^{C ∣ τ - τ_{0} ∣}$ . Hence no finite endpoint blow-up occurs there, and finitely many chart subintervals cover any compact part of $I$ ; uniqueness patches the solutions across chart overlaps.

2.1givenF1step 1.1algebra

If $a + b = 1$ , step 1.1 makes $g (U, s)$ constant, so its zero initial value remains zero. On this constraint, $\frac{1}{2} \frac{d}{d τ} g (s, s) = g (s, D_{τ} s) = α (a g (s, F^{♯} s) + \frac{b}{c^{2}} g (s, U) κ) = 0,$ since $F^{♯}$ is skew and $g (s, U) = 0$ .

3.1givenA2F3F4step 1.1algebra∎

At the fixed event, $e_{0} = U / c$ and $g (U, s) = 0$ give $s = \sum_{j} s^{j} e_{j}$ . The projection $P_{U}$ kills the term proportional to $U$ , while for every spatial index $i$ , $g (F^{♯} (s), e_{i}) = F (e_{i}, s) = F_{i j} s^{j} = ϵ_{i j k} B_{k} s^{j} = (s \times B)_{i}$ by [F3]. Thus the projected equation is $(P_{U} (D_{τ} s))^{i} = α a (s \times B)_{i}$ . For $α \neq 0$ , matching [F4] for all $s, B$ forces $a = G$ (choose a nonzero cross product). Independently, requiring preservation for every initially orthogonal datum and every alternating $F$ forces $a + b = 1$ by step 1.1: nonzero $κ$ is attainable, for example by taking $s = e_{1}$ and an alternating $F$ with $F (e_{1}, e_{0}) \neq 0$ . Hence $b = 1 - G$ . The equations do not identify $a, b$ when $α = 0$ .

Remarks

The proof is a conditional calculation for an already specified smooth connection and transport equation. The Countable Choice requirement is inherited through the supplied GR tangent-bundle and connection construction; no choice is used in the coefficient algebra.

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Positive-energy spin–size inequality

Statement

Fix $c > 0$ and a future unit observer $n$ in an affine Minkowski spacetime as in Oriented time-oriented affine Minkowski space (the spacetime primitives) and Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. Identify the positive-definite rest space $n^{⊥}$ isometrically with $R^{3}$ , with its Euclidean norm $∣ \cdot ∣$ and Lebesgue measure $d^{3} x$ . Define the closed Euclidean ball by ${\overline{B}}_{ℓ} (0) : = {x \in R^{3} : ∣ x ∣ \leq ℓ} .$ Let $e : R^{3} \to [0, \infty)$ be measurable with $e \in L^{1} (R^{3})$ , and let $j : R^{3} \to R^{3}$ be measurable with each component $j_{i} \in L^{1} (R^{3})$ for $i = 1, 2, 3$ . Thus $j$ is vector-valued integrable and every vector integral below is the componentwise Lebesgue integral. Assume $e$ and $j$ are both zero almost everywhere outside ${\overline{B}}_{ℓ} (0)$ for some $ℓ > 0$ , and that $∣ j (x) ∣ \leq e (x) / c$ almost everywhere. Define $E_{0} : = \int_{R^{3}} e (x) d^{3} x > 0, M : = E_{0} / c^{2}, s_{*} : = \int_{R^{3}} x \times j (x) d^{3} x .$ Then $s_{*}$ is absolutely integrable and $∣ s_{*} ∣ \leq ℓ M c, equivalently ℓ \geq \frac{∣ s_{*} ∣}{M c} .$ The hypotheses are the full mathematical content of this estimate; it is not a bound for a quantum point particle or for a source whose energy density or momentum density fails them.

Facts & Assumptions

Given: The affine Minkowski spacetime, a future unit observer and its Euclidean rest space, $c > 0$ , and integrable $e, j$ satisfying the stated positivity, support and pointwise bound.

[A1]

The rest densities are Lebesgue-integrable, vanish almost everywhere outside ${\overline{B}}_{ℓ} (0)$ , satisfy $e \geq 0$ and $∣ j ∣ \leq e / c$ almost everywhere, and have $E_{0} = \int e > 0$ .

[F1]

The affine Minkowski model has signature $(- + + +)$ ; the orthogonal complement of a future unit timelike vector is a positive-definite Euclidean rest space, with the Euclidean inner product and norm of $R^{3}$ . Oriented time-oriented affine Minkowski space (the spacetime primitives); Norms of four-velocity and four-acceleration, and the observer rest-space decomposition; The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

[F2]

The cross product has components $(x \times y)_{1} = x_{2} y_{3} - x_{3} y_{2}$ , $(x \times y)_{2} = x_{3} y_{1} - x_{1} y_{3}$ , $(x \times y)_{3} = x_{1} y_{2} - x_{2} y_{1}$ . The cross product in $R^{3}$

[F3]

$L^{1}$ means measurable with finite integral of the modulus; real function integrals are defined by positive and negative parts. A vector-valued map into $R^{3}$ is integrable when each coordinate function is in $L^{1}$ , and its integral is the componentwise integral. The class $L^{1} (μ)$ of integrable functions; Integrable real and complex functions, and their integrals

[F4]

The nonnegative integral is monotone and positively homogeneous, the Lebesgue integral is linear on $L^{1}$ , and $∣ \int f ∣ \leq \int ∣ f ∣$ for $f \in L^{1}$ . Monotonicity and nonnegative homogeneity of the nonnegative integral; The Lebesgue integral is linear on $L^{1} (μ)$ ; The modulus of an integral is bounded by the integral of the modulus

[F5]

The Euclidean inner product satisfies $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

Proof

technique · direct

1.1givenA1F1F2F3F4algebra

By the Statement, $e$ and each component of $j$ belong to $L^{1}$ ; the norm and cross-product conventions are those of [F1] and [F2]. The component formula of [F2] expands to $∣ x \times y ∣^{2} = ∣ x ∣^{2} ∣ y ∣^{2} - (x \cdot y)^{2} \leq ∣ x ∣^{2} ∣ y ∣^{2} .$ For almost every $x$ in the support ball, $∣ x ∣ \leq ℓ$ and $∣ j (x) ∣ \leq e (x) / c$ , so $∣ x \times j (x) ∣ \leq ℓ e (x) / c$ . From $e, j \in L^{1}$ in [F3], monotonicity and homogeneity in [F4] give $\int ∣ x \times j ∣ d^{3} x \leq ℓ E_{0} / c < \infty$ , establishing absolute integrability of $s_{*}$ .

2.1givenA1F3F4F5step 1.1algebra

If $s_{*} = 0$ the bound is immediate. Otherwise set $u = s_{*} / ∣ s_{*} ∣$ , so $∣ u ∣ = 1$ . Linearity, the integral triangle inequality [F4] and Cauchy–Schwarz [F5] give $∣ s_{*} ∣ = u \cdot s_{*} = \int u \cdot (x \times j (x)) d^{3} x \leq \int ∣ u \cdot (x \times j (x)) ∣ d^{3} x \leq \int ∣ x \times j (x) ∣ d^{3} x \leq \frac{ℓ E_{0}}{c} = ℓ M c .$ This proves the first inequality.

3.1givenstep 2.1algebra∎

Since $M = E_{0} / c^{2} > 0$ and $c > 0$ , divide the inequality of step 2.1 by $M c$ to obtain $ℓ \geq ∣ s_{*} ∣ / (M c)$ .

Remarks

This is a conditional material inequality. Positivity of $e$ and the pointwise energy-current bound $∣ j ∣ \leq e / c$ are load-bearing hypotheses, not consequences of the definition of a quantum or point particle.

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Finite-size charge multipole remainder

Statement

Throughout this item, use one-based coordinate labels: for a vector $x \in R^{n}$ (a function on ${0, \dots, n - 1}$ in The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), write $x_{i} : = x (i - 1)$ for $1 \leq i \leq n$ . For a vector-valued map $f$ , write $f_{i} (y) : = f (y) (i - 1)$ with the same convention.

Let $ℓ > 0$ , $z \in R^{3}$ , write $d^{3} y$ for three-dimensional Lebesgue measure on $R^{3}$ , and interpret $L^{1} (R^{3})$ relative to this measure. Define the closed Euclidean ball ${\overline{B}}_{ℓ} (z) : = {y \in R^{3} : ∣ y - z ∣ \leq ℓ},$ using the norm of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . Let $ρ \in L^{1} (R^{3})$ be real and zero almost everywhere outside ${\overline{B}}_{ℓ} (z)$ . Write $∥ ρ ∥_{1} : = \int_{R^{3}} ∣ ρ (y) ∣ d^{3} y$ . Let $U \subseteq R^{3}$ be open with ${\overline{B}}_{ℓ} (z) \subset U$ , and let $b : U \to R^{m}$ be $C^{2}$ for a finite integer $m \geq 1$ . For each component $b_{k}$ , let $M_{k} : = \sup_{x \in {\overline{B}}_{ℓ} (z)} ∥ D^{2} b_{k} (x) ∥_{o p} < \infty,$ where, for a real bilinear map $B : R^{3} \times R^{3} \to R$ , define explicitly $∥ B ∥_{o p} : = \sup {∣ B [u, v] ∣ : ∣ u ∣ \leq 1, ∣ v ∣ \leq 1}$ using the Euclidean vector norm. This norm is finite, and the continuous Hessian coefficients on the compact ball make each $M_{k}$ finite. Bilinearity gives $∣ B [u, v] ∣ \leq ∥ B ∥_{o p} ∣ u ∣ ∣ v ∣$ for all $u, v$ . Define $Q : = \int ρ d^{3} y$ , $d : = \int (y - z) ρ (y) d^{3} y$ , and $B : = \int ρ (y) b (y) d^{3} y$ , with all three integrals over ${\overline{B}}_{ℓ} (z)$ and vector integrals taken componentwise. Then $B_{k} = Q b_{k} (z) + D b_{k} (z) d + R_{k}, ∣ R_{k} ∣ \leq \frac{1}{2} M_{k} ℓ^{2} ∥ ρ ∥_{1} (k = 1, \dots, m) .$ Thus every fixed component in a smooth finite-component trivialization has the stated charge multipole remainder. A tensor-valued component may be treated only after a fixed smooth trivialization over the ball; derivatives of that trivialization enter $M_{k}$ , and no covariant tensor remainder without this data is asserted.

Facts & Assumptions

Given: A compactly supported real $ρ \in L^{1} (R^{3})$ , $ℓ > 0$ , $z \in R^{3}$ , and a $C^{2}$ finite-component map $b$ on a neighborhood of the closed ball, with the component Hessian bounds in the Statement.

[A1]

The charge density is real and integrable with support in the stated ball; the map $b$ is $C^{2}$ on its neighborhood and each component Hessian has the stated finite bound.

[F1]

$L^{1}$ is the class of functions whose absolute value has finite Lebesgue integral, and the Lebesgue integral is linear on $L^{1}$ . The class $L^{1} (μ)$ of integrable functions; Integrable real and complex functions, and their integrals; The Lebesgue integral is linear on $L^{1} (μ)$

[F2]

The nonnegative integral is monotone and positively homogeneous; for an integrable real function, $∣ \int f ∣ \leq \int ∣ f ∣$ . Monotonicity and nonnegative homogeneity of the nonnegative integral; The modulus of an integral is bounded by the integral of the modulus

[F3]

A closed bounded Euclidean ball is compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). A continuous function on a nonempty compact metric space is bounded, and a continuous function on a compact interval is Riemann integrable. A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value; A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion

[F4]

The chain rule along $t \mapsto z + t (y - z)$ and the one-dimensional fundamental theorem of calculus hold for the stated $C^{2}$ maps. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ; The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$

[F5]

The Euclidean inner product and norm are the usual positive-definite ones. The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

[F6]

For a bounded real function $f$ on a compact interval and a partition $P$ , the lower and upper Darboux sums use the subinterval infima and suprema weighted by the positive subinterval lengths. For an integrable function, $\int f = \sup_{P} L (f, P) = \inf_{P} U (f, P)$ . For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$ ; The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$

Proof

technique · direct

1.1givenA1F1F2F3F5

For each component $k$ , write $D^{2} b_{k} (x) [u, v] = \sum_{i, j = 1}^{3} a_{i j} (x) u_{i} v_{j}$ . The coefficients $a_{i j}$ are continuous because $b$ is $C^{2}$ . The ball is compact by [F3], so the finitely many coefficients have a common bound $C_{k} < \infty$ there. Since $∣ u_{i} ∣ \leq ∣ u ∣$ and $∣ v_{j} ∣ \leq ∣ v ∣$ , the sum has magnitude at most $9 C_{k} ∣ u ∣ ∣ v ∣$ , hence the displayed bilinear supremum and $M_{k}$ are finite. For nonzero $u, v$ , apply that supremum to $u / ∣ u ∣, v / ∣ v ∣$ and multiply by $∣ u ∣ ∣ v ∣$ ; when either vector is zero the bilinear expression is zero. This proves the homogeneous bound used below. Since $∣ y - z ∣ \leq ℓ$ on the support, monotonicity and homogeneity in [F2] show that each moment is absolutely integrable: $∣ Q ∣ \leq ∥ ρ ∥_{1}$ and $∣ d_{i} ∣ \leq ℓ ∥ ρ ∥_{1}$ . By compactness of the ball and continuity of each $b_{k}$ , [F3] bounds $b_{k}$ there, so $ρ b_{k}$ is integrable as well.

2.1givenA1F3F4F5F6step 1.1algebra

Fix $y \in {\overline{B}}_{ℓ} (z)$ and component $k$ , and set $ϕ (t) = b_{k} (z + t (y - z))$ on $[0, 1]$ . The segment stays in the closed ball, and the chain rule [F4] gives $ϕ^{'} (0) = D b_{k} (z) (y - z)$ and $ϕ^{''} (t) = D^{2} b_{k} (z + t (y - z)) [y - z, y - z]$ , hence $∣ ϕ^{''} (t) ∣ \leq C : = M_{k} ∣ y - z ∣^{2}$ . Set $H (t) = (1 - t) ϕ^{'} (t) + ϕ (t)$ and $f (t) : = H^{'} (t) = (1 - t) ϕ^{''} (t)$ . This derivative is continuous and Riemann integrable by [F3], so [F4] gives $r_{k} (y) : = b_{k} (y) - b_{k} (z) - D b_{k} (z) (y - z) = \int_{0}^{1} f (t) d t$ . To bound this Riemann integral, let $h_{\pm} (t) : = \pm C (1 - t)$ . The pointwise inequalities $h_{-} \leq f \leq h_{+}$ imply $L (h_{-}, P) \leq L (f, P)$ and $U (f, P) \leq U (h_{+}, P)$ for every partition, by the infimum, supremum and positive weights in [F6]. Taking the corresponding supremum and infimum gives $\int h_{-} \leq \int f \leq \int h_{+}$ . The continuous functions $h_{\pm}$ have primitives $\pm C (t - t^{2} / 2)$ , so [F4] evaluates their integrals as $\pm C / 2$ . Consequently $∣ r_{k} (y) ∣ \leq \frac{1}{2} M_{k} ∣ y - z ∣^{2}$ .

3.1givenA1F1F2step 2.1algebra∎

The function $r_{k}$ defined in step 2.1 is continuous on the ball. Its bound there makes $ρ r_{k}$ absolutely Lebesgue integrable by [F2]. Multiply that component expansion by $ρ (y)$ and integrate over ${\overline{B}}_{ℓ} (z)$ . Linearity [F1] gives $B_{k} = Q b_{k} (z) + D b_{k} (z) d + R_{k}$ , with $R_{k} = \int_{{\overline{B}}_{ℓ} (z)} ρ (y) r_{k} (y) d^{3} y$ . By the Lebesgue integral triangle inequality and domination [F2], $∣ R_{k} ∣ \leq \frac{1}{2} M_{k} ℓ^{2} \int_{{\overline{B}}_{ℓ} (z)} ∣ ρ ∣ d^{3} y \leq \frac{1}{2} M_{k} ℓ^{2} ∥ ρ ∥_{1}$ .

Remarks

This is a local component estimate. It does not differentiate the remainder or identify it with a force; a tensor estimate requires the stated fixed trivialization and its derivatives.

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Exact flat centroid translation and disk

Statement

Let $(A, V, η, C^{+})$ be an oriented, time-oriented affine Minkowski space, with signature $(- + + +)$ , and fix $c > 0$ . Let $p$ be any future timelike vector and define $M : = \sqrt{- η (p, p)} / c > 0$ . Choose an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ of Inertial frames and inertial charts of the affine Minkowski model with $e_{0} = p / (M c)$ , so its ordered orthonormal basis is future and positively oriented and $p$ has components $(M c, 0)$ in this frame. Its positive-definite rest space is $e_{0}^{⊥} = span (e_{1}, e_{2}, e_{3})$ ; use $(e_{1}, e_{2}, e_{3})$ for spatial components and the induced right-handed symbol $ϵ_{i j k}$ . Let $S^{a b} = - S^{b a}$ satisfy the momentum supplementary condition $S^{a b} p_{b} = 0$ , and set $s_{*}^{k} : = \frac{1}{2} ϵ_{k i j} S^{i j}$ in this oriented orthonormal rest-space basis. Every future unit observer has the components $n = γ (e_{0} + \sum_{i = 1}^{3} \frac{v^{i}}{c} e_{i}), γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}, ∣ v ∣ < c,$ by Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. For every such observer, define $δ^{a} : = \frac{S^{a b} n_{b}}{p \cdot n}, S^{' a b} : = S^{a b} - (δ^{a} p^{b} - δ^{b} p^{a}) .$ Then $p \cdot n \neq 0$ , $δ \cdot p = δ \cdot n = 0$ , and $S^{' a b} n_{b} = 0$ . Let $z \in A$ be the representative event. For $u, v \in V$ , define $(u \land v)^{a b} : = u^{a} v^{b} - u^{b} v^{a}$ . If the representative point changes by $z \mapsto z + δ$ , the total angular momentum about the frame origin $o$ , $J_{o} : = (z - o) \land p + S,$ is unchanged by this stipulated spin-shift rule. In the rest space, $δ = \frac{s_{*} \times v}{M c^{2}},$ If $s_{*} \neq 0$ , the shifts of all future timelike observers fill the open disk in $s_{*}^{⊥}$ of radius $∣ s_{*} ∣ / (M c)$ ; the boundary is a supremum and is not attained. If $s_{*} = 0$ , only the zero shift occurs. This is an exact affine-flat algebraic result.

Facts & Assumptions

Given: The oriented, time-oriented affine Minkowski model, a fixed $c > 0$ , a future timelike momentum $p = (M c, 0)$ with $M > 0$ , an antisymmetric $S$ , the stated supplementary condition, and a future unit observer with $∣ v ∣ < c$ .

[A1]

The momentum supplementary condition $S^{a b} p_{b} = 0$ and the displayed spin-shift rule for the representative point are algebraic hypotheses of this flat model.

[F1]

Minkowski spacetime is an oriented time-oriented affine space with Lorentz form of signature $(- + + +)$ ; an inertial frame is a future, positively oriented orthonormal frame with its affine chart. Oriented time-oriented affine Minkowski space (the spacetime primitives); Inertial frames and inertial charts of the affine Minkowski model

[F2]

Relative to the inertial observer $e_{0}$ , a future unit observer has spatial velocity $v \in e_{0}^{⊥}$ , $∣ v ∣ < c$ , and components $γ (1, v / c)$ with $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ ; $e_{0}^{⊥}$ is positive definite. Norms of four-velocity and four-acceleration, and the observer rest-space decomposition

[F3]

The spatial cross product is $(u \times v)_{i} = ϵ_{i j k} u_{j} v_{k}$ . The cross product in $R^{3}$

Proof

technique · direct

1.1givenA1F1F2algebra

Since $S^{a b} p_{b} = 0$ and $p = (M c, 0)$ , antisymmetry gives $S^{0 i} = 0$ . The observer decomposition [F2] gives $p \cdot n = - M c γ \neq 0$ , while $n_{a} n_{b} S^{a b} = 0$ and $p_{a} S^{a b} = 0$ give $δ \cdot n = δ \cdot p = 0$ . The definition $s_{*}^{k} = \frac{1}{2} ϵ_{k i j} S^{i j}$ inverts to $S^{i j} = ϵ_{i j k} s_{*}^{k}$ : contracting the right-hand side with $\frac{1}{2} ϵ_{ℓ i j}$ gives $s_{*}^{ℓ}$ by $\frac{1}{2} ϵ_{ℓ i j} ϵ_{i j k} = δ_{ℓ k}$ .

2.1givenA1step 1.1algebra

Contracting the spin-shift rule with $n_{b}$ yields $S^{' a b} n_{b} = S^{a b} n_{b} - δ^{a} (p \cdot n) + p^{a} (δ \cdot n) = 0,$ because $δ^{a} (p \cdot n) = S^{a b} n_{b}$ by definition and step 1.1 gives $δ \cdot n = 0$ . Also $(z + δ - o) \land p + S^{'} = (z - o) \land p + S$ , which proves the asserted invariance of $J_{o}$ .

2.2givenA1step 1.1F3algebra

In rest coordinates, $n_{j} = γ v_{j} / c$ , $p \cdot n = - M c γ$ , and step 1.1 gives $S^{i j} = ϵ_{i j k} s_{*}^{k}$ . Therefore $δ^{i} = \frac{ϵ_{i j k} s_{*}^{k} γ v_{j} / c}{- M c γ} = \frac{ϵ_{i j k} s_{*}^{j} v_{k}}{M c^{2}} = \frac{(s_{*} \times v)_{i}}{M c^{2}}, δ^{0} = 0.$ The second equality uses antisymmetry of $ϵ_{i j k}$ when interchanging $j, k$ .

3.1givenA1step 2.2F3algebra∎

Expanding the coordinate definition [F3] gives $∣ u \times v ∣^{2} = \sum_{i < j} (u_{i} v_{j} - u_{j} v_{i})^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - (u \cdot v)^{2}$ and $u \times (v \times w) = v (u \cdot w) - w (u \cdot v)$ . Hence $∣ δ ∣ \leq ∣ s_{*} ∣ ∣ v ∣ / (M c^{2}) < ∣ s_{*} ∣ / (M c)$ for $s_{*} \neq 0$ . Conversely, for any $δ ⊥ s_{*}$ with $∣ δ ∣ < ∣ s_{*} ∣ / (M c)$ , choose $v = (M c^{2} / ∣ s_{*} ∣^{2}) (δ \times s_{*})$ . The expanded triple-product identity gives $s_{*} \times v = M c^{2} δ$ , and the norm identity gives $∣ v ∣ = M c^{2} ∣ δ ∣ / ∣ s_{*} ∣ < c$ . Thus every interior point occurs; equality in the radius would require $∣ v ∣ = c$ , excluded for a timelike observer. When $s_{*} = 0$ , the formula of step 2.2 gives only $δ = 0$ .

Remarks

The spin-shift rule is part of the algebraic hypotheses. This item does not assert a curved-spacetime translation law, a stress-integral theorem, or a material-body model.

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Differentiated SSC momentum–velocity identity

Statement

Assume Countable Choice as inherited by the supplied Lorentzian smooth-structure and connection results. Let $I \subseteq R$ be a nonempty open interval and let $z : I \to M$ be a $C^{2}$ future timelike worldline in a time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , parametrized by proper time in the convention of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. Set $U : = d z / d τ$ , so $g (U, U) = - c^{2}$ . For finite regularity, a $C^{1}$ tensor field along $z$ means a tensor in the fibre over $z (τ)$ whose components in every smooth chart are $C^{1}$ functions of $τ$ . Define its derivative in that chart by differentiating each component and adding $+ Γ^{a}_{b c} (z) U^{b}$ on each upper index and $- Γ^{c}_{b a} (z) U^{b}$ on each lower index. In particular, $(D_{τ} V)^{a} = {\dot{V}}^{a} + Γ^{a}_{b c} (z) U^{b} V^{c}, (D_{τ} n)_{a} = {\dot{n}}_{a} - Γ^{c}_{b a} (z) U^{b} n_{c},$ $(D_{τ} S)^{a b} = {\dot{S}}^{a b} + Γ^{a}_{c d} (z) U^{c} S^{d b} + Γ^{b}_{c d} (z) U^{c} S^{a d} .$ These formulas extend the smooth-curve derivative of Covariant derivative along a curve to the stated $C^{2} / C^{1}$ data. Let $p$ be a $C^{1}$ vector field along $z$ , let $S^{a b} = - S^{b a}$ and $T^{a b} = - T^{a b}$ be $C^{1}$ bivector fields along $z$ , and let $n$ be a $C^{1}$ future unit timelike field along $z$ . Suppose $S^{a b} n_{b} = 0, D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} + T^{a b} .$ Define $γ : = - g (n, U) / c$ and $E_{n} : = - c g (p, n)$ . Then $γ > 0$ , and $c γ p^{a} = \frac{E_{n}}{c} U^{a} + S^{a b} D_{τ} n_{b} + T^{a b} n_{b} .$ Here $T$ denotes an arbitrary antisymmetric torque; setting it to zero gives the torque-free case.

Facts & Assumptions

Given: The Lorentzian manifold, its metric-compatible connection, the stated future timelike proper-time worldline, vector and bivector fields along it, a future unit field $n$ , the SSC and the spin evolution equation.

[A1]

Countable Choice is inherited through the supplied GR smooth-structure and connection items. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A2]

Along the same proper-time curve, the observer satisfies $S^{a b} n_{b} = 0$ and the antisymmetric fields obey $D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} + T^{a b}$ , with the stated regularity.

[F1]

Proper-time parametrization of a future timelike Lorentzian worldline gives $g (U, U) = - c^{2}$ . Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold

[F2]

The covariant derivative along a curve satisfies the product rule on a pairing, and metric compatibility gives $D_{τ} g = 0$ . Covariant derivative along a curve; Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

[F3]

For a future unit timelike observer $n$ and future timelike $U / c$ , $- g (n, U / c) > 0$ . Pairing of an observer with future causal vectors

Proof

technique · direct

1.1givenA1A2F2algebra

The chain rule and the connection coefficient transformation law cancel derivatives of the chart transition matrices in the displayed formulas, so $D_{τ}$ is independent of the chart. The ordinary component product rule gives the tensor product and contraction rules; the coordinate identity $\partial_{c} g_{a b} = Γ^{d}_{c a} g_{d b} + Γ^{d}_{c b} g_{a d}$ from metric compatibility gives $D_{τ} g = 0$ . These arguments require only $C^{1}$ tensor components along a $C^{2}$ curve. Differentiate the SSC in [A2] along $z$ using the inherited smooth-structure assumption and covariant product rule, then substitute its spin evolution equation: $0 = D_{τ} (S^{a b} n_{b}) = p^{a} g (U, n) - U^{a} g (p, n) + T^{a b} n_{b} + S^{a b} D_{τ} n_{b} .$ Every term is defined at the same point of $z$ , so the covariant product rule applies without comparing different tangent fibres.

2.1givenA2step 1.1F1F3algebra∎

The definitions give $g (U, n) = - c γ$ and $g (p, n) = - E_{n} / c$ . Substitution into step 1.1 and rearrangement yield $c γ p^{a} = \frac{E_{n}}{c} U^{a} + T^{a b} n_{b} + S^{a b} D_{τ} n_{b} .$ Positivity of $γ$ follows from [F3].

Remarks

This is tensor algebra along one specified curve. It does not select an SSC, prove closure of the MPD equations, or identify $p$ with $m U$ .

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Admissible TD pole-dipole local closure

Statement

Assume Countable Choice and use a smooth time-oriented Lorentzian four-manifold $(M, g, N)$ of signature $(-, +, +, +)$ with its metric-compatible Levi–Civita connection. Use SI units: coordinates are metres, proper time is seconds, $c > 0$ is in metres per second, $p$ is in kilogram metres per second, $S$ is in kilogram metres squared per second, and $R$ has inverse-square-metre components. Let $z : I \to M$ be a $C^{2}$ future timelike worldline parametrized by proper time, with $U = d z / d τ$ and $g (U, U) = - c^{2}$ . For $C^{1}$ tensor fields along this $C^{2}$ curve, use the chart-independent derivative defined in Differentiated SSC momentum–velocity identity: differentiate the components, add $+ Γ^{a}_{b c} (z) U^{b}$ on each upper index, and add $- Γ^{c}_{b a} (z) U^{b}$ on each lower index. It obeys the product and contraction rules and $D_{τ} g = 0$ , as proved there. Let $p$ be a $C^{1}$ future timelike vector field along $z$ , let $S^{a b} = - S^{b a}$ be a $C^{1}$ bivector field, and suppose the displayed evolution equations hold:

$\begin{matrix} D_{τ} p^{a} = f^{a} : = - \frac{1}{2} R^{a}_{b c d} U^{b} S^{c d}, D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} . & (1) \end{matrix}$

Put $μ : = \sqrt{- g (p, p)} > 0$ , $m : = - g (p, U) / c^{2} > 0$ , $C^{a} : = S^{a b} p_{b}$ , and define the linear map

$K^{a}_{b} : = μ^{2} δ_{b}^{a} - \frac{1}{2} S^{a c} R_{c b d e} S^{d e} .$

At every state where $K$ is invertible and $W : = K^{- 1} p$ is future timelike, the TD condition $C = 0$ and (1) imply

$\begin{matrix} K U = m c^{2} p, U = \frac{c W}{\sqrt{- g (W, W)}} . & (2) \end{matrix}$

Conversely, on the open state region where $p$ and $W$ are future timelike and $K$ is invertible, the normalized formula for $U$ in (2), together with (1), defines a smooth local ODE. A solution initialized with $C = 0$ preserves $C = 0$ . While it remains in that admissible region, $μ$ and $S_{a b} S^{a b} / 2$ are constant. If $S = 0$ , then $p = (μ / c) U$ , $D_{τ} p = 0$ , and $D_{τ} U = 0$ . These are local statements; no continuation past loss of invertibility of $K$ or timelikeness of $W$ is asserted. Equations (1) are hypotheses of this mathematical lemma, not a physical derivation of a stress-energy moment model.

Facts & Assumptions

Given: The stated Lorentzian background, proper-time curve, future timelike $p$ , antisymmetric $S$ , constant $c > 0$ , curvature convention, and equations (1). For the converse, the initial state satisfies $C = 0$ , $K$ is invertible and $W = K^{- 1} p$ is future timelike.

[A1]

Countable Choice ${A C}_{ω}$ is inherited by the smooth-structure and connection constructions used by the GR suppliers. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A2]

The proper-time curve and antisymmetric fields satisfy the neutral pole-dipole equations (1), and the TD constraint is $C^{a} = S^{a b} p_{b} = 0$ at the initial state.

[F1]

Proper-time parametrization gives $g (U, U) = - c^{2}$ ; the Levi–Civita connection is metric-compatible and its derivative along a curve obeys the covariant product rule. Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold; Covariant derivative along a curve; Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

[F2]

The curvature convention is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ , with $R_{a b c d} = g_{a e} R^{e}_{b c d}$ ; in particular $R_{a b c d} = - R_{b a c d}$ . The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors; The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors

[F3]

For any future unit timelike $n$ and future unit timelike $U / c$ , $- g (n, U / c) > 0$ . Pairing of an observer with future causal vectors

[F4]

If $S^{a b} n_{b} = 0$ and $D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a}$ , then with $γ = - g (n, U) / c$ and $E_{n} = - c g (p, n)$ , $c γ p^{a} = \frac{E_{n}}{c} U^{a} + S^{a b} D_{τ} n_{b} .$ Differentiated SSC momentum–velocity identity

[F5]

A locally Lipschitz vector field on an open Euclidean domain has a unique local solution through each initial state, with uniqueness on common intervals. Picard-Lindelöf local existence and uniqueness for first-order systems

[F6]

The background is a smooth Lorentzian manifold and its metric-selected connection is the smooth Levi–Civita connection. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; Metric compatibility, torsion freedom and the metric-selected connection class

Proof

technique · direct

1.1givenA1A2F3F4algebra

Under [A2]'s TD condition $C = 0$ , take $n = p / μ$ . Then $n$ is a future unit observer, $S^{a b} n_{b} = 0$ , $γ = - g (p, U) / (μ c) = m c / μ$ , and $E_{n} = - c g (p, p / μ) = c μ$ . Also $D_{τ} n = D_{τ} p / μ - p D_{τ} μ / μ^{2}$ ; contracting with $S$ removes the second term because $S^{a b} p_{b} = 0$ . Substitution in [F4] and multiplication by $μ$ give $m c^{2} p = μ^{2} U + S f$ . The definition of $f$ in (1) now gives exactly $K U = m c^{2} p$ . Positivity of $m$ follows from [F3].

2.1givenA2F1step 1.1algebra

Since $K$ is invertible, step 1.1 gives $U = m c^{2} W$ . The coefficient is positive, so $W$ is future timelike. Its normalization is $- c^{2} = g (U, U) = m^{2} c^{4} g (W, W)$ , hence $m = 1 / (c \sqrt{- g (W, W)})$ and the second formula in (2) follows. Conversely, on $C = 0$ , contraction of $K^{a}_{b}$ with $p_{a}$ gives $p_{a} K^{a}_{b} = μ^{2} p_{b}$ because $p_{a} S^{a c} = - C^{c} = 0$ . Multiplying by $W^{b}$ yields $g (p, W) = - 1$ , so the normalized formula has $m = - g (p, U) / c^{2} = 1 / (c \sqrt{- g (W, W)})$ and satisfies the first formula in (2).

3.1givenA2F1F2F5F6step 2.1algebra

On the open region stated in the theorem, the inverse $K^{- 1}$ and the positive square root $\sqrt{- g (W, W)}$ depend smoothly on the state coordinates. The smooth Lorentzian background, curvature and metric-selected connection [F2, F6] make the coordinate form of $z^{'} = U$ , (1) a smooth, hence locally Lipschitz, vector field. [F5] gives unique local solutions. For constraint preservation, set $λ : = c / \sqrt{- g (W, W)}$ , so $K U = λ p$ throughout this open region. The product rule [F1] gives $D_{τ} C^{a} = p^{a} g (p, U) + μ^{2} U^{a} + S^{a b} f_{b} = p^{a} (g (p, U) + λ) .$ Moreover, $p_{a} K^{a}_{b} U^{b} = μ^{2} g (p, U) - C^{b} f_{b} = λ g (p, p) = - λ μ^{2}$ , so $μ^{2} (g (p, U) + λ) = C^{b} f_{b}$ . Therefore $D_{τ} C^{a} = \frac{p^{a} f_{b}}{μ^{2}} C^{b} .$ This is a homogeneous linear ODE for $C$ along the solution. Its unique solution with $C (τ_{0}) = 0$ is $C \equiv 0$ , proving preservation.

4.1givenA2F1F2step 1.1step 3.1algebra

On the preserved constraint, step 1.1 gives $μ^{2} U = m c^{2} p - S f$ . The curvature antisymmetry [F2] yields $g (U, f) = - \frac{1}{2} R_{a b c d} U^{a} U^{b} S^{c d} = 0$ . Contract the displayed relation with $f_{a}$ : the left side is zero and $f_{a} S^{a b} f_{b} = 0$ by antisymmetry of $S$ , so $m c^{2} g (p, f) = 0$ . Since step 1.1 gives $m > 0$ and $c > 0$ , $g (p, f) = 0$ , hence $D_{τ} g (p, p) = 2 g (p, f) = 0$ and $μ$ is constant. Also $\frac{1}{2} D_{τ} (S_{a b} S^{a b}) = S_{a b} (p^{a} U^{b} - p^{b} U^{a}) = 2 S_{a b} p^{a} U^{b} = 0$ by $C = 0$ from step 3.1.

5.1givenA2F1step 2.1step 4.1algebra∎

If $S = 0$ , then $f = 0$ , $K = μ^{2} Id$ , and $W = p / μ^{2}$ with $g (W, W) = - 1 / μ^{2}$ . The normalized formula (2) gives $U = c p / μ$ , equivalently $p = (μ / c) U$ . Equation (1) gives $D_{τ} p = 0$ ; since $μ$ is constant by step 4.1, $D_{τ} U = 0$ .

Remarks

The matrix $K$ need not be invertible on every TD state, and $K^{- 1} p$ need not remain timelike. The theorem supplies a local closure only on the explicitly stated admissible region; it proves neither global existence nor physical adequacy of a multipole truncation.

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Body moments and transport convention

Physical scope

Definition of flat and local curved stress-energy moments for a smooth compact test body. It supplies moment and transport conventions only; it does not derive MPD equations, a multipole truncation, a matter constitutive law, self-gravity, or an empirical body model.

Definition

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 $x^{0} = c t$ in metres, $c > 0$ , and a $C^{2}$ future timelike representative curve $ζ : I \to M$ parametrized by proper time $τ$ in seconds, with $U = d ζ / d τ$ 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 $x : A \to R^{4}$ as in Inertial frames and inertial charts of the affine Minkowski model, with the SI coordinate convention $x^{0} = c t$ 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 $C^{\infty}$ symmetric type- $(2, 0)$ tensor field on the affine Minkowski spacetime $A$ , whose chart components $T^{a b}$ are the model stress-energy densities with units $J m^{- 3}$ . The coordinate-time image $J : = {x^{0} (ζ (τ)) / c : τ \in I}$ is an open interval, and $ζ (t)$ 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 $[t_{1}, t_{2}] \subset R$ , define $K_{[t_{1}, t_{2}]} : = \overline{{x \in R^{3} : there are t \in [t_{1}, t_{2}] and a, b \in {0, 1, 2, 3} with T^{a b} (c t, x) \neq 0}} .$ Assume $K_{[t_{1}, t_{2}]}$ is compact for every such interval. For each $t \in R$ , define the spatial support by ${supp}_{t} T : = \overline{{x \in R^{3} : there are a, b \in {0, 1, 2, 3} with T^{a b} (c t, x) \neq 0}} .$ At a fixed time $t$ , the spatial slice has coordinates $x = (x^{1}, x^{2}, x^{3}) \in R^{3}$ and three-dimensional Lebesgue measure $d^{3} x$ ; define $p^{a} (t) : = \frac{1}{c} \int_{R^{3}} T^{0 a} (c t, x) d^{3} x,$ and, for the representative event $ζ (t)$ on that time slice, define $S_{ζ (t)}^{a b} (t) : = \frac{1}{c} \int_{R^{3}} ((x - ζ (t))^{a} T^{0 b} (c t, x) - (x - ζ (t))^{b} T^{0 a} (c t, x)) d^{3} x .$ Here $(x - ζ (t))^{0} = 0$ on the chosen slice, $p$ has units $k g m s^{- 1}$ , and $S_{ζ (t)}$ has units $k g m^{2} s^{- 1}$ . 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 $Σ = i (S) \subset U_{0}$ 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 $T_{x} Σ$ 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 $w \in T_{x} Σ$ . For a local parametrization $X : V \subseteq R^{3} \to Σ$ , put $g_{i j}^{Σ} (y) : = g_{X (y)} (\partial_{i} X, \partial_{j} X)$ and define $d Σ : = \sqrt{\det (g_{i j}^{Σ} (y))} d y^{1} d y^{2} d y^{3} .$ 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 $T ∣_{Σ}$ is used. With $(n_{x})_{b} : = g_{b c} (x) n_{x}^{c}$ , define ${supp}_{Σ} (T ∣_{Σ}) : = {\overline{{x \in Σ : T_{x} \neq 0}}}^{Σ} .$ Assume this restricted support is a compact subset of $U_{0}$ and that all displayed integrands are absolutely integrable with respect to $d Σ$ . For every $x \in Σ$ , 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 $T^{♯} (n_{x})^{a} : = T^{a b} (x) (n_{x})_{b}$ and $p_{z} : = \frac{1}{c} \int_{Σ} H_{x \to z} ⁣ (- T^{♯} (n_{x})) d Σ, S_{z} : = \frac{1}{c} \int_{Σ} v (x) \land H_{x \to z} ⁣ (- T^{♯} (n_{x})) d Σ,$ where $(v \land w)^{a b} : = v^{a} w^{b} - v^{b} w^{a}$ 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 $k g m s^{- 1}$ and $k g m^{2} s^{- 1}$ . “Support on $Σ$ ” here means only compact support of the restriction $T ∣_{Σ}$ ; it does not assert that the ambient smooth field $T$ has spacetime support contained in $Σ$ .

Remarks

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 $\partial_{a} T^{a b} = 0$ 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 $_{ω}$ dependency is inherited through the cited GR smooth structure and parallel-transport interfaces.

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Pole-dipole and closed multipole test-body models

Physical scope

Adopted neutral pole-dipole and explicitly closed multipole test-body truncations on a prescribed smooth Lorentzian background. They exclude self-gravity, self-force, arbitrary stress-integral derivation, and any unqualified error or convergence theorem.

Postulate

Assume Countable Choice as inherited from Body moments and transport convention. For a neutral test body, let $I \subseteq R$ be an open interval of proper-time values and let $z : I \to M$ be a $C^{2}$ future timelike curve, with $U = d z / d τ$ . Let $p^{a}$ and $S^{a b} = - S^{b a}$ 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 $D_{τ} V : = (z^{*} \nabla)_{\partial / \partial τ} V$ using the pullback of the metric-selected connection. In a chart, with $Γ$ the coefficients of $\nabla$ , this is

(D_{τ} p)^{a} = \frac{d p^{a}}{d τ} + Γ^{a}_{b c} U^{b} p^{c}, (D_{τ} S)^{a b} = \frac{d S^{a b}}{d τ} + Γ^{a}_{c d} U^{c} S^{d b} + Γ^{b}_{c d} U^{c} S^{a d} .

Adopt the pole-dipole evolution equations $D_{τ} p^{a} = - \frac{1}{2} R^{a}_{b c d} U^{b} S^{c d}, D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} .$ A spin-supplementary condition (SSC) is an additional equation $S^{a b} n_{b} = 0$ 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 $p = m U$ .

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 $T M$ for $U$ , $T M$ for $p$ , $Λ^{2} T M$ 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 $D$ in this bundle. The extra force and torque are specified smooth bundle maps over the identity, $f_{e x t r a} : D \to T M$ and $N_{e x t r a} : D \to Λ^{2} T M$ , so at a state based at $x \in M$ their values lie in $T_{x} M$ and $Λ^{2} T_{x} M$ , respectively. Their units are $k g m s^{- 2}$ and $k g m^{2} s^{- 2}$ . For a retained quadrupole or higher-multipole model, adopt $D_{τ} p^{a} = - \frac{1}{2} R^{a}_{b c d} U^{b} S^{c d} + f_{e x t r a}^{a}, D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} + N_{e x t r a}^{a b},$ where $f_{e x t r a}$ is a specified smooth force map along the state and $N_{e x t r a}^{a b} = - N_{e x t r a}^{b a}$ is a specified smooth torque map. A quadrupole convention that may be adopted is $f_{q u a d}^{a} = - \frac{1}{6} J^{b c d e} \nabla^{a} R_{b c d e}, N_{q u a d}^{a b} = \frac{4}{3} R^{[a}_{c d e} J^{b] c d e},$ with $J^{b c d e}$ satisfying the algebraic symmetries of the curvature tensor, carrying units $k g m^{4} s^{- 2}$ , and brackets including the factor $1 / 2$ . 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.

Physical scope

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.

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BMT orthogonality and conserved spin length

Physical scope

Conditional constraint preservation and restricted-ansatz coefficient matching for the adopted classical BMT polarization model on a prescribed external field. No measured value of the gyromagnetic ratio, quantum-spin derivation, or empirical validation is asserted.

Statement

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

Adopt the charged-body quantities and SI tensor convention of Charged and spinning test-body quantities and The electromagnetic two-form, its observer decomposition and the four-current, the polarization transport postulate Classical BMT polarization model, and its translational equation obtained from Stationary charged action and Lorentz force. On a local potential domain, assume the finite curve segments are stationary for the adopted action. Let $z$ be a $C^{2}$ future timelike proper-time curve, $U = d z / d τ$ , $g (U, U) = - c^{2}$ , with prescribed continuous alternating field $F$ along it. Let $m > 0$ , $q \in R$ , $g_{s} \in R$ , and an initial $C^{1}$ spin vector satisfy $g (s, U) = 0$ . Put $α = q / m$ , $G = g_{s} / 2$ , and $κ = g (s, F^{♯} (U))$ . The adopted equations and action result give

$D_{τ} U = α F^{♯} (U), D_{τ} s = α [G F^{♯} (s) + (1 - G) \frac{U}{c^{2}} κ] .$

Then $g (s, U) = 0$ and $g (s, s)$ are constant along every solution. In the restricted ansatz $D_{τ} s = α [a F^{♯} (s) + b U κ / c^{2}]$ , require preservation of orthogonality for every initially orthogonal datum and every allowed alternating field. For $α \neq 0$ this gives $a + b = 1$ . If, in addition, the pointwise projected rest-space matching condition of BMT transport constraint and restricted-ansatz algebra holds with target coefficient $G$ , that lemma gives $a = G$ , and hence $b = 1 - G$ . This coefficient conclusion is only within that ansatz and the stated joint conditions. If $α = 0$ , the rest-space matching condition does not identify $a, b$ .

Facts & Assumptions

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure and inherited BMT/action-equivalence supplier hypotheses.

Given: The adopted classical BMT model, $c > 0$ , constant $m > 0, q, g_{s}$ , a future timelike proper-time curve, a prescribed alternating field, and an initially orthogonal spin vector.

[A1]

The adopted BMT transport and Lorentz-force translation equations, with $G = g_{s} / 2$ and $κ = g (s, F^{♯} (U))$ . Classical BMT polarization model

[A2]

The trajectory is stationary under the adopted action on a local potential domain, and the initial spin satisfies $g (U, s) = 0$ .

[F1]

The SI field tensor has $F_{0 i} = - E_{i} / c$ and $F_{i j} = ϵ_{i j k} B_{k}$ in the named inertial convention; the charged-body definition uses the same convention. The electromagnetic two-form, its observer decomposition and the four-current; Charged and spinning test-body quantities

[F2]

For the restricted two-term transport ansatz, metric compatibility and skew-adjointness give $D_{τ} g (U, s) = α (1 - a - b) κ$ ; if $a + b = 1$ , orthogonality and spin norm are preserved, and the pointwise projected rest-space matching condition defined in the supplier determines $a = G$ when $α \neq 0$ . BMT transport constraint and restricted-ansatz algebra

[F3]

Under [A3], stationarity of the adopted local charged-particle action implies $m D_{τ} U = q F^{♯} (U)$ on the stated finite fixed-endpoint potential-domain model. Stationary charged action and Lorentz force

Proof

technique · direct

1.1givenA3A1A2F2F3algebra

In the lemma [F2], set $a = G$ and $b = 1 - G$ . The postulate [A1] has exactly this displayed transport equation because $- (G - 1) = 1 - G$ , and [F3] gives the matching translation equation $D_{τ} U = α F^{♯} (U)$ . Thus $a + b = 1$ in the lemma; its derivative identity gives $D_{τ} g (U, s) = 0$ . The initial value is zero by [A2], hence orthogonality is preserved.

2.1givenA3A2F2step 1.1algebra

With $g (U, s) = 0$ from step 1.1 and [A2], the same lemma gives $\frac{1}{2} D_{τ} g (s, s) = g (s, D_{τ} s) = 0$ : the $F^{♯}$ term vanishes by skew-adjointness, and the term proportional to $U$ vanishes by orthogonality. Thus $g (s, s)$ is constant.

3.1givenA3F1F2step 1.1algebra∎

In the general ansatz, [F2] shows that for $α \neq 0$ , preservation for every initially orthogonal datum and every allowed field requires $a + b = 1$ ; an orthogonal rest spin $s = e_{1}$ with an alternating field satisfying $F (e_{1}, e_{0}) \neq 0$ gives an allowed instance with $κ \neq 0$ . The pointwise spatial projection in [F2] gives $a = G$ under the stipulated rest-space matching condition for every $s, B$ ; substituting into $a + b = 1$ gives $b = 1 - G$ . If $α = 0$ , both transport equations vanish and impose no coefficient condition. This establishes uniqueness only under the joint conditions and within the stated ansatz.

Remarks

This is a conditional deduction inside the adopted classical transport model. The original BMT treatment is flat and homogeneous-field; it does not prove the separately adopted curved or varying-field law, and this theorem makes no quantum or empirical claim.

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Coulomb-exterior support-radius energy bound

Physical scope

A conditional static charge-size inequality for a body with the stated rest-frame support radius, an exactly Coulomb vacuum exterior, and total rest energy equal to that exterior field energy plus nonnegative remaining energy. It is not a universal radius theorem for arbitrary charge distributions, and it makes no mass inference for an ideal point charge.

Statement

Let an extended static charged body have a nonempty compact rest-frame stress-energy support ${supp}_{0} T$ in the flat-body moment convention of Body moments and transport convention, total charge $q \in R$ coulombs, and finite total rest energy $M c^{2}$ with $M > 0$ kilograms. Use a momentum-rest inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ of Inertial frames and inertial charts of the affine Minkowski model, with $p = M c e_{0}$ and $o$ the stipulated body center. Its rest spatial chart has coordinates $x = (x^{1}, x^{2}, x^{3})$ ; define $r : = ∣ x ∣ : = \sqrt{(x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}}$ , $\hat{r} : = x / r$ for $r > 0$ , and ${\overline{B}}_{ℓ} (0) : = {x : r \leq ℓ}$ . Define the body support radius $ℓ : = \max_{x \in {supp}_{0} T} ∣ x ∣ > 0$ , in metres. Thus the material support lies in ${\overline{B}}_{ℓ} (0)$ ; $ℓ$ is its exact radius about $o$ , and the exterior region below is defined from this support, not inferred from spherical symmetry. Use the SI constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, including $c$ in metres per second and $ε_{0} > 0$ . On the open exterior spatial region $Ω_{ℓ} : = {x \in R^{3} : r > ℓ}$ , the static field is assumed to obey $E (x) = \frac{q}{4 π ε_{0} r^{2}} \hat{r}, B = 0.$ Let $u (x)$ be the time-independent vacuum energy density of Vacuum electromagnetic energy density and energy flux (the Poynting vector) on this exterior region and define $E_{e x t} : = \int_{Ω_{ℓ}} u (x) d^{3} x$ . Assume the total rest energy is the sum of this exterior field energy and a remaining material-plus-interior contribution $E_{r e m} \geq 0$ . Then $M c^{2} \geq \frac{q^{2}}{8 π ε_{0} ℓ}, ℓ \geq \frac{q^{2}}{8 π ε_{0} M c^{2}} .$ The exact Coulomb exterior and nonnegative energy accounting are hypotheses. The theorem does not infer an exterior field from spherical symmetry alone. No identification of field energy with an ideal point particle's rest mass is made.

Facts & Assumptions

Given: The static rest-frame body, radius $ℓ > 0$ , total charge $q$ , total rest mass $M > 0$ , exact exterior field stated above, and $M c^{2} = E_{e x t} + E_{r e m}$ with $E_{r e m} \geq 0$ .

[A1]

The rest-energy accounting is adopted for this item: $M c^{2} = E_{e x t} + E_{r e m}$ with the remaining material-plus-interior energy $E_{r e m} \geq 0$ .

[A2]

The exterior electric field is the Coulomb field of charge $q$ for every $r > ℓ$ , with no exterior magnetic field; this is an adopted physical condition of the present theorem, not a consequence of spherical symmetry alone. The localized Newtonian potential and the ideal point-charge Coulomb field

[F1]

In vacuum, the electromagnetic energy density is $u = ε_{0} ∣ E ∣^{2} / 2 + ∣ B ∣^{2} / (2 μ_{0})$ , with the positive SI constant $ε_{0}$ . Vacuum electromagnetic energy density and energy flux (the Poynting vector); Framework-local SI units, the vacuum constants and the exact/measured status of their values

[F2]

For the point-charge Coulomb field centered at spatial coordinate $x_{0}$ , the energy in the finite shell $δ \leq ∣ x - x_{0} ∣ \leq R$ is $q^{2} (1 / δ - 1 / R) / (8 π ε_{0})$ . Here $x_{0} = 0$ , the spatial coordinate of the inertial-frame origin event $o$ . Divergence of the point-charge field energy and the limits of mass inference

[F3]

If nonnegative measurable functions $f_{n}$ increase pointwise to $f$ , then $\int f_{n}$ increases to $\int f$ (Monotone convergence for the integral).

Proof

technique · direct

1.1givenA2F1F2F3algebra

From [A2] and [F1], the exterior energy density is $u (x) = ε_{0} ∣ E (x) ∣^{2} / 2 \geq 0$ on $Ω_{ℓ}$ . For each positive integer $n$ , set $δ_{n} : = ℓ + ℓ / n$ , $R_{n} : = ℓ (n + 2)$ , and $A_{n} : = {x : δ_{n} \leq r \leq R_{n}}$ . These measurable shells increase to $Ω_{ℓ}$ , and their nonnegative energy integrands increase pointwise to $u 1_{Ω_{ℓ}}$ . By [F3], $E_{e x t} = \lim_{n \to \infty} \int_{A_{n}} u d^{3} x$ . Applying [F2] with $δ = δ_{n}$ and $R = R_{n}$ gives $\int_{A_{n}} u d^{3} x = q^{2} (1 / δ_{n} - 1 / R_{n}) / (8 π ε_{0})$ , whose limit is $q^{2} / (8 π ε_{0} ℓ)$ , including $q = 0$ . This applies only under the stated exact exterior-field and energy-accounting hypotheses, not to every spherical profile.

2.1givenA1step 1.1algebra∎

By [A1], $M c^{2} = E_{e x t} + E_{r e m} \geq E_{e x t}$ . Substitute step 1.1 to get $M c^{2} \geq q^{2} / (8 π ε_{0} ℓ)$ . Since $M, c, ℓ, ε_{0}$ are positive, multiplication by $ℓ$ and division by $M c^{2}$ yield $ℓ \geq q^{2} / (8 π ε_{0} M c^{2})$ .

Remarks

The energy is a field contribution on the exterior region only. Nonnegative interior/material energy and the specified total-energy accounting are load-bearing assumptions. The point-charge divergence result Divergence of the point-charge field energy and the limits of mass inference explicitly does not identify this field energy with particle rest mass.

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SSC choices and hidden momentum

Physical scope

Conditional momentum–velocity identities for the adopted neutral pole-dipole test-body equations under Pirani, parallel-observer (OKS), and Tulczyjew–Dixon supplementary conditions. The identity distinguishes the momentum, curve tangent, and observer; no universal identification of momentum with mass times velocity is asserted.

Statement

Assume the neutral pole-dipole equations adopted in Pole-dipole and closed multipole test-body models for the local moments of Body moments and transport convention: along a $C^{2}$ future timelike proper-time curve $z$ , with $U = d z / d τ$ , $p$ future timelike and $S^{a b} = - S^{b a}$ , $D_{τ} p^{a} = - \frac{1}{2} R^{a}_{b c d} U^{b} S^{c d}, D_{τ} S^{a b} = p^{a} U^{b} - p^{b} U^{a} .$ For any $C^{1}$ future unit observer field $n$ along the curve satisfying the SSC $S^{a b} n_{b} = 0$ , put $γ = - g (n, U) / c > 0$ and $E_{n} = - c g (p, n)$ . Then $c γ p^{a} = \frac{E_{n}}{c} U^{a} + S^{a b} D_{τ} n_{b} .$ In the Pirani branch $n = U / c$ , define $m = - g (p, U) / c^{2} > 0$ and $a = D_{τ} U$ ; then $p^{a} = m U^{a} + \frac{1}{c^{2}} S^{a b} a_{b} .$ In the parallel-observer (OKS) branch $D_{τ} n = 0$ , $p$ is a positive scalar multiple of $U$ . In the TD branch $n = p / μ$ , $μ = \sqrt{- g (p, p)}$ , define $m : = - g (p, U) / c^{2} > 0$ as in the Pirani branch. The derivative term generally remains and the identity gives $m c^{2} p = μ^{2} U + S D_{τ} p$ ; it does not by itself imply $p ∥ U$ .

Facts & Assumptions

Given: The adopted neutral MPD pole-dipole equations, a proper-time timelike worldline, future timelike $p$ , antisymmetric spin $S$ , and one of the stated future unit observer SSC choices.

[A2]

The selected future unit observer field satisfies $S^{a b} n_{b} = 0$ ; the Pirani, parallel-observer (OKS), and TD branches use the choices in the Statement.

[A1]

The neutral pole-dipole moment equations are adopted on the stated finite-dimensional test-body model; no arbitrary-body stress-integral derivation is assumed. Pole-dipole and closed multipole test-body models

[F1]

The moments $p, S$ and their units are those defined by the local body-moment convention. Body moments and transport convention

[F2]

Differentiating an observer SSC under the torque-free spin equation gives $c γ p^{a} = (E_{n} / c) U^{a} + S^{a b} D_{τ} n_{b}$ , where $γ = - g (n, U) / c$ and $E_{n} = - c g (p, n)$ . Differentiated SSC momentum–velocity identity

[F3]

For any two future timelike vectors, their Lorentzian pairing is strictly negative in the $(-, +, +, +)$ convention. Pairing of an observer with future causal vectors

Proof

technique · direct

1.1givenA1A2F2

Under [A1] and [A2], the spin equation has the zero-torque form required in [F2]. Applying that lemma to the chosen observer $n$ yields the displayed general momentum–velocity identity.

2.1givenA2F1F3step 1.1algebra

For the Pirani choice in [A2], $n = U / c$ ; proper-time normalization gives $γ = 1$ , while $E_{n} = - g (p, U) = m c^{2}$ and $D_{τ} n = a / c$ . Positivity of $m$ follows from [F3]. Substituting these values in step 1.1 gives $c p = m c U + S a / c$ ; division by $c > 0$ yields $p = m U + S a / c^{2}$ . The second term is the hidden-momentum contribution relative to the Pirani centroid.

2.2givenA2F2F3step 1.1algebra

For the parallel-observer choice in [A2], $D_{τ} n = 0$ , so step 1.1 reduces to $c γ p = (E_{n} / c) U$ and hence $p = E_{n} U / (c^{2} γ)$ . Both $E_{n}$ and $γ$ are positive by [F3] because $p, n$ and $U / c$ are future timelike, so the proportionality factor is positive.

3.1givenA1A2F2F3step 1.1algebra∎

In the TD case in [A2], use $m = - g (p, U) / c^{2} > 0$ , whose positivity follows from [F3]. Then $n = p / μ$ is future unit, $E_{n} = c μ$ , and $γ = - g (p, U) / (μ c) = m c / μ$ . Differentiation gives $D_{τ} n = (D_{τ} p) / μ - p (D_{τ} μ) / μ^{2}$ ; the last term vanishes after contraction with $S$ by $S^{a b} p_{b} = 0$ . Substitution in step 1.1 and multiplication by $μ$ give $m c^{2} p = μ^{2} U + S D_{τ} p$ . This leaves a curvature-dependent derivative term, so proportionality does not follow from this identity alone.

Remarks

The SSC selects a representative curve and the observer used to define the centroid. The momentum $p$ , tangent $U$ , and observer $n$ remain distinct model quantities; only the parallel-observer branch above forces their displayed proportionality.

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Rest-frame spin transport and finite-body size bounds

Physical scope

Conditional consequences of the adopted classical BMT polarization law in free and uniform-rest-magnetic-field branches, together with separate flat-body centroid and positive-energy size bounds. The BMT polarization vector and body angular-momentum vector are distinct unless an additional identification is explicitly adopted.

Statement

Adopt the charged-body quantities of Charged and spinning test-body quantities, the SI field convention of The electromagnetic two-form, its observer decomposition and the four-current, and the classical BMT law Classical BMT polarization model. For a trajectory stationary under the local action postulate, the translational force equation is supplied by Stationary charged action and Lorentz force. In flat Minkowski spacetime let $m > 0$ , $q, g_{s} \in R$ , $α = q / m$ , $G = g_{s} / 2$ , and let $U, s$ satisfy $D_{τ} U = α F^{♯} (U), D_{τ} s = α [G F^{♯} (s) - (G - 1) \frac{U}{c^{2}} g (s, F^{♯} (U))], g (U, U) = - c^{2}, g (U, s) = 0.$ The tensor equation is covariant under proper orthochronous Lorentz transformations of the complete field and worldline data. If $F = 0$ , then $D_{τ} U = 0$ and $D_{τ} s = 0$ . For any acceleration $a = D_{τ} U$ within the Lorentz-force model, the $G = 0$ branch is Fermi–Walker transport on the orthogonal spin subspace: $D_{τ} s = \frac{1}{c^{2}} (U g (a, s) - a g (U, s)) = \frac{U}{c^{2}} g (a, s) .$ For the uniform-field branch, choose an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ of Inertial frames and inertial charts of the affine Minkowski model with $e_{0} = U / c$ for the stationary trajectory and with the positively oriented orthonormal spatial basis $(e_{1}, e_{2}, e_{3})$ . Let $E = 0$ and let the constant spatial field be $B \neq 0$ ; its norm and cross product are those of the frame's positive-definite spatial structure. Then $U = (c, 0)$ is stationary, $\hat{B} : = B / ∣ B ∣$ , and for $s_{0} : = s (τ_{0})$ define $s_{∥, 0} : = (s_{0} \cdot \hat{B}) \hat{B}$ and $s_{⊥, 0} : = s_{0} - s_{∥, 0}$ . The $G$ branch has the exact solution $s (τ) = s_{∥, 0} + \cos (ω Δ τ) s_{⊥, 0} - \sin (ω Δ τ) \hat{B} \times s_{⊥, 0}, ω = \frac{q G}{m} ∣ B ∣, Δ τ = τ - τ_{0} .$ The $F = 0$ , $G = 0$ , and uniform-field claims are deductions within the adopted model, not experimental statements. For the independent finite-body branch, use the moment and unit convention of Body moments and transport convention; the density-to-spin identity below is stipulated data.

Separately, let a flat body have rest momentum $p$ , and define $M : = \sqrt{- g (p, p)} / c > 0$ . Choose its momentum-rest inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ of Inertial frames and inertial charts of the affine Minkowski model with $e_{0} = p / (M c)$ and spatial origin at the stipulated body center. Define ${\overline{B}}_{ℓ} (0) : = {x \in R^{3} : ∣ x ∣ \leq ℓ}$ using the Euclidean norm of its positive-definite spatial rest space. In this frame $p$ has components $(M c, 0)$ ; let $S^{a b} = - S^{b a}$ be its angular-momentum tensor, with $S^{0 i} = 0$ (hence $S^{a b} p_{b} = 0$ ), and let $e : R^{3} \to [0, \infty)$ and $j : R^{3} \to R^{3}$ be measurable rest densities, with each component of $j$ in $L^{1} (R^{3})$ ; their units are joules per cubic metre and kilogram per square metre per second, respectively. Assume $e$ and $j$ vanish almost everywhere outside ${\overline{B}}_{ℓ} (0)$ , $∣ j ∣$ is the Euclidean norm and satisfies $∣ j ∣ \leq e / c$ almost everywhere, $\int e d^{3} x = M c^{2}$ , and the stipulated componentwise spin-density identity $s_{*} : = \int_{R^{3}} x \times j (x) d^{3} x = \frac{1}{2} ϵ_{k i j} S^{i j} e_{k},$ where the vector integral is componentwise in the positively oriented spatial basis of this inertial frame. Then $ℓ \geq \frac{∣ s_{*} ∣}{M c} .$ If $s_{*} \neq 0$ , every observer-induced centroid shift $δ$ of the flat SSC translation law satisfies $∣ δ ∣ < ∣ s_{*} ∣ / (M c)$ , every point of the open disk in $s_{*}^{⊥}$ with that radius is attained, and its boundary is a supremum not attained by a timelike observer. If $s_{*} = 0$ , the only shift is $δ = 0$ . Under proper orthochronous Lorentz transformations of the complete body data and its momentum-rest hypersurface, the scalars $M$ , $ℓ$ , $∣ s_{*} ∣$ , and $∣ s_{*} ∣ / (M c)$ are unchanged; $s_{*}$ itself transforms as a spatial vector, so its components depend on the oriented rest-space basis. No equality between $s$ and $s_{*}$ is assumed, so the size inequalities do not bound the BMT polarization unless such an identification is separately adopted.

Facts & Assumptions

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure and inherited BMT/action-equivalence supplier hypotheses.

Given: The adopted classical BMT and Lorentz-force equations, the flat Lorentz metric and SI field convention, an initially orthogonal polarization, and, independently, flat-body rest data with the stated support, positivity and energy-current bound.

[A2]

The polarization begins orthogonal to $U$ ; the uniform-field clause uses $E = 0$ and constant $B$ in the inertial rest chart; independently, the body data satisfy the stated rest-frame angular-momentum, support, positive-energy and energy-current hypotheses.

[A1]

The charged-body definition treats $m, q, g_{s}$ as model parameters, $s$ as a polarization vector orthogonal to $U$ , and $F$ as a prescribed field; the BMT transport and translation laws are separately adopted. Charged and spinning test-body quantities; Classical BMT polarization model

[F1]

In the stated SI convention, $F_{0 i} = - E_{i} / c$ and $F_{i j} = ϵ_{i j k} B_{k}$ ; $F^{♯}$ is defined by $g (F^{♯} (v), w) = F (w, v)$ . The electromagnetic two-form, its observer decomposition and the four-current; Charged and spinning test-body quantities

[F2]

The restricted transport algebra preserves $g (U, s) = 0$ and $g (s, s)$ when the coefficients are $G$ and $1 - G$ , gives the pointwise rest-space projected operator $α G (s \times B)$ in its stated basis convention, and supplies uniqueness for the continuous-coefficient linear spin transport on compact intervals. BMT transport constraint and restricted-ansatz algebra

[F3]

Proper orthochronous Lorentz maps preserve the Lorentz form and future cone. Oriented time-oriented affine Minkowski space (the spacetime primitives); The proper orthochronous Lorentz group: boosts, rotations, unique factorization and invariants

[F4]

Under the explicit rest-frame hypotheses $e \geq 0$ , $∣ j ∣ \leq e / c$ , support radius $ℓ$ , and $M c^{2} = \int e > 0$ , the body spin satisfies $∣ s_{*} ∣ \leq ℓ M c$ . Positive-energy spin–size inequality

[F5]

For an antisymmetric flat spin tensor with $S^{a b} p_{b} = 0$ and $p = (M c, 0)$ , observer-induced shifts fill the open disk in $s_{*}^{⊥}$ of radius $∣ s_{*} ∣ / (M c)$ ; the boundary is not attained. Exact flat centroid translation and disk

[F6]

Under [A3], stationarity of the adopted local charged-particle action yields $D_{τ} U = (q / m) F^{♯} (U)$ . Stationary charged action and Lorentz force

Proof

technique · direct

1.1givenA3A1A2F1F6algebra

If $F = 0$ , [F6] and the BMT postulate [A1] make both right-hand sides zero, so $D_{τ} U = 0$ and $D_{τ} s = 0$ . For $G = 0$ and general field, the spin equation reduces to $D_{τ} s = α U g (s, F^{♯} (U)) / c^{2}$ . Since [F6] gives $a = α F^{♯} (U)$ , this equals $U g (a, s) / c^{2}$ . The full Fermi–Walker expression in the Statement has the additional term $- a g (U, s) / c^{2}$ , which is zero by [A2]; thus the stated branch is exactly Fermi–Walker transport.

1.2givenA3A2F1F3algebra

Let $Λ$ be a proper orthochronous Lorentz map and transform $U, s, F$ as vector and two-form data. Metric preservation [F3] and the defining identity of $F^{♯}$ imply $F^{'}^{♯} (Λ v) = Λ F^{♯} (v)$ : pair both sides with an arbitrary $Λ w$ and use $F^{'} (Λ w, Λ v) = F (w, v)$ . The scalar $g (s, F^{♯} (U))$ and constraint $g (U, s) = 0$ are preserved. Therefore applying $Λ$ to each BMT and translation equation gives the same equation for the transformed data.

1.3givenA3A1A2F1F2F6algebra

In the uniform rest magnetic field in [A2], [F6] gives $F^{♯} (U) = 0$ because $E = 0$ , so $U = (c, 0)$ remains at rest and $κ = g (s, F^{♯} (U)) = 0$ . The cross-product convention [F1,F2] gives $d s / d τ = ω s \times \hat{B}$ . With $θ : = ω Δ τ$ , direct differentiation of the displayed formula gives $\frac{d s}{d τ} = - ω \sin (θ) s_{⊥, 0} - ω \cos (θ) \hat{B} \times s_{⊥, 0} = ω s (τ) \times \hat{B},$ and at $τ = τ_{0}$ the formula equals $s_{0}$ . Uniqueness of the continuous-coefficient linear spin transport on compact intervals, supplied by [F2], makes this the exact solution. If $q G = 0$ the formula reduces to constant spin.

2.1givenA3A2F3F4F5algebra∎

By the stipulated rest-density identity in the Statement and the hypotheses [A2], the data meet [F4], so $∣ s_{*} ∣ \leq ℓ M c$ ; divide by $M c > 0$ for the first inequality. The hypotheses of [F5] are exactly the rest momentum and spin condition in [A2], so every timelike observer shift lies inside the disk of radius $∣ s_{*} ∣ / (M c)$ and the boundary is not attained. In the case $s_{*} = 0$ , [F5] gives only zero shift and [F4] gives the trivial lower bound $ℓ \geq 0$ . In the momentum rest frame, $S^{0 i} = 0$ and $S^{i j} = ϵ_{i j k} s_{*}^{k}$ , so direct contraction gives $S_{a b} S^{a b} / 2 = ∣ s_{*} ∣^{2}$ ; together with $M = \sqrt{- g (p, p)} / c$ , this shows the radius is a Lorentz scalar. A proper orthochronous Lorentz map sends the momentum-rest hyperplane isometrically to the transformed momentum-rest hyperplane, so the support radius $ℓ$ is unchanged for the transformed complete rest-frame data. The BMT vector $s$ has not entered either estimate; only a separately adopted identification $s = s_{*}$ would allow its magnitude to be substituted.

Remarks

The BMT polarization vector belongs to the charged test-body transport model. The body spin $s_{*}$ is the dual of an angular-momentum moment in a momentum-rest frame. Their units agree, but their physical identification is additional input. The Lorentz-covariance claim applies to the full tensor model, while the magnetic sine–cosine expression and size assumptions are stated in their specified rest frames.

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TD spin and momentum norms

Physical scope

Conditional local consequences of the adopted neutral TD pole-dipole truncation: constant momentum norm and spin-tensor norm on the admissible domain, including the zero-spin geodesic branch. No continuation beyond loss of invertibility or timelikeness, global existence, or empirical adequacy is asserted.

Statement

Adopt the neutral pole-dipole equations and local body moments of Pole-dipole and closed multipole test-body models and Body moments and transport convention. Suppose a solution satisfies the TD condition $S^{a b} p_{b} = 0$ , with future timelike $p$ , and remains in the local admissible region of Admissible TD pole-dipole local closure where $K$ is invertible and $W = K^{- 1} p$ is future timelike. Then, on its local interval of existence, $μ : = \sqrt{- g (p, p)} and \frac{1}{2} S_{a b} S^{a b}$ are constant. If the initial spin tensor is zero, it remains zero and the solution obeys $p = \frac{μ}{c} U, D_{τ} p = 0, D_{τ} U = 0,$ so the representative curve is a timelike geodesic with constant rest mass $M = μ / c$ . The claims stop if the local TD admissible region ends; no global extension follows.

Facts & Assumptions

Given: The adopted neutral pole-dipole system, a future timelike momentum satisfying the TD supplementary condition, and a solution within the open region where the algebraically determined tangent is future timelike.

[A2]

The TD condition holds initially, $K$ remains invertible, $W = K^{- 1} p$ remains future timelike, and the solution stays in the local admissible region stated in the Statement; for the spinless clause, $S (τ_{0}) = 0$ .

[A1]

The neutral test-body pole-dipole equations are adopted as a finite-dimensional truncation, not derived from arbitrary stress-energy moments. Pole-dipole and closed multipole test-body models

[F1]

On the TD admissible region, the local closure lemma preserves the TD constraint, momentum norm and spin-tensor norm; at zero spin it gives $p = (μ / c) U$ , $D_{τ} p = 0$ , and $D_{τ} U = 0$ . Admissible TD pole-dipole local closure

[F2]

The variables $p, S$ are the local body moments defined by the stated transport convention; the theorem adds the TD condition. Body moments and transport convention

[F3]

For a future unit observer $n$ , the rest space $n^{⊥}$ is three-dimensional and positive definite, with spatial projection $h_{n}$ . Observers, rest spaces and the spatial projection at an event; The observer rest-space decomposition and the spatial projection

Proof

technique · direct

1.1givenA1A2F1F2

The equations in [A1] act on the moment variables defined in [F2] and are exactly the pole-dipole system assumed in [F1]. On the region [A2], [F1] gives preservation of $S^{a b} p_{b} = 0$ and constancy of $μ$ and $S_{a b} S^{a b} / 2$ on the local solution. This is a conditional deduction within the adopted truncation.

2.1givenA2F1F2F3step 1.1algebra∎

Suppose also $S (τ_{0}) = 0$ . Step 1.1 preserves $C^{a} = S^{a b} p_{b} = 0$ and the scalar $\frac{1}{2} S_{a b} S^{a b}$ , while the admissible region keeps $p$ future timelike. At each $τ$ , set $μ = \sqrt{- g (p, p)} > 0$ , $e_{0} = p / μ$ , and $E : = e_{0}^{⊥}$ ; [F3] says $E$ is a three-dimensional positive-definite inner-product space. Take any basis $v_{1}, v_{2}, v_{3}$ of $E$ and form $e_{1} : = \frac{v_{1}}{\sqrt{g (v_{1}, v_{1})}}, w_{2} : = v_{2} - g (v_{2}, e_{1}) e_{1}, e_{2} : = \frac{w_{2}}{\sqrt{g (w_{2}, w_{2})}}, w_{3} : = v_{3} - g (v_{3}, e_{1}) e_{1} - g (v_{3}, e_{2}) e_{2}, e_{3} : = \frac{w_{3}}{\sqrt{g (w_{3}, w_{3})}} .$ The denominators are positive: positive definiteness gives $g (v_{1}, v_{1}) > 0$ , and linear independence of the $v_{i}$ makes $w_{2}, w_{3}$ nonzero. These formulas give an orthonormal basis of $E$ ; with $e_{0}$ they give an orthonormal basis of $T_{z (τ)} M$ . The TD constraint says $S^{a b} p_{b} = 0$ ; since $p_{b} = (- μ, 0, 0, 0)$ in this basis, it gives $S^{a 0} = 0$ , and antisymmetry also gives $S^{0 a} = 0$ . Therefore $\frac{1}{2} S_{a b} S^{a b} = \frac{1}{2} \sum_{i, j = 1}^{3} (S^{i j})^{2} \geq 0,$ with equality if and only if every component $S^{i j}$ vanishes, that is, $S = 0$ . Its conserved value is initially zero, so $S$ remains zero throughout the local solution. The zero-spin branch in [F1] now yields $p = (μ / c) U$ and $D_{τ} p = 0$ . Since [F1] also gives constant $μ > 0$ , covariant differentiation of $U = (c / μ) p$ gives $D_{τ} U = 0$ . Therefore $z$ is a timelike geodesic in proper-time parametrization and $M = μ / c$ is constant. No continuation outside the admissible region is supplied.

Remarks

The conserved quantity is the tensor norm $S_{a b} S^{a b} / 2$ on the local TD branch. For nonzero spin, conservation of that scalar alone does not assert that the tensor components are constant. The zero-spin conclusion uses the separate zero-spin solution branch, not an inference from a vanishing norm.

5 · Examples, experiments and counterexamples

None yet.

BMT constraint and restricted-ansatz algebra

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

BMT transport constraint and restricted-ansatz algebra

Statement

Facts & Assumptions

Proof

Remarks

Positive-energy spin–size inequality

Statement

Facts & Assumptions

Proof

Remarks

Finite-size charge multipole remainder

Statement

Facts & Assumptions

Proof

Remarks

Exact flat centroid translation and disk

Statement

Facts & Assumptions

Proof

Remarks

Differentiated SSC momentum–velocity identity

Statement

Facts & Assumptions

Proof

Remarks

Admissible TD pole-dipole local closure

Statement

Facts & Assumptions

Proof

Remarks

Body moments and transport convention

Definition

Remarks

Pole-dipole and closed multipole test-body models

Postulate

Physical scope

BMT orthogonality and conserved spin length

Statement

Facts & Assumptions

Proof

Remarks

Coulomb-exterior support-radius energy bound

Statement

Facts & Assumptions

Proof

Remarks

SSC choices and hidden momentum

Statement

Facts & Assumptions

Proof

Remarks

Rest-frame spin transport and finite-body size bounds

Statement

Facts & Assumptions

Proof

Remarks

TD spin and momentum norms

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources