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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.

SSC choices and hidden momentum

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources