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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.

Admissible TD pole-dipole local closure

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources