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Spinless TD geodesic limit

Physical scope

The zero-spin invariant branch of the adopted neutral TD pole-dipole truncation on its local admissible region. It recovers a timelike geodesic with constant rest mass and makes no global-existence or empirical claim.

Example

Assume the neutral pole-dipole equations of Pole-dipole and closed multipole test-body models and the TD admissible region of TD spin and momentum norms: $p$ is future timelike, $K$ is invertible, $W = K^{- 1} p$ is future timelike, and the initial spin tensor satisfies $S^{a b} = 0$ . Then throughout the local solution the spin remains zero, $p$ is parallel transported, and $U = \frac{c}{μ} p, μ = \sqrt{- g (p, p)} = constant, D_{τ} U = 0.$ Thus the representative curve is a timelike geodesic with constant rest mass $M = μ / c$ in this restricted branch.

Facts & Assumptions

Given: The adopted neutral pole-dipole equations, the TD algebraic closure on its admissible region, and an initial zero spin tensor.

[A2]

The initial TD state has $S^{a b} = 0$ and lies in the open region where $K$ is invertible and $W = K^{- 1} p$ is future timelike.

[A1]

The neutral pole-dipole evolution is adopted as a finite-dimensional test-body truncation. Pole-dipole and closed multipole test-body models

[F1]

A solution satisfying TD throughout the local admissible domain has constant $μ$ and spin-tensor norm. If its initial spin tensor is zero, it remains zero and gives $p = (μ / c) U$ , $D_{τ} p = 0$ , and $D_{τ} U = 0$ . TD spin and momentum norms

[F2]

The variables $p, S$ are moments under the stated local body convention; this example adds the TD representative condition. Body moments and transport convention

[F3]

On the open region where $p$ and $W = K^{- 1} p$ are future timelike and $K$ is invertible, the normalized closure $U = c W / \sqrt{- g (W, W)}$ defines a smooth local ODE with the pole-dipole equations. A solution initialized with $S^{a b} p_{b} = 0$ preserves that constraint. Admissible TD pole-dipole local closure

Verification

technique · direct

1.1givenA1A2F1F3algebra

The zero initial tensor in [A2] gives $S^{a b} p_{b} = 0$ initially. The adopted equations [A1] and normalized TD closure are exactly the local ODE of [F3], so that supplier preserves the TD constraint throughout the local solution in the admissible region. The hypotheses of [F1] now hold: the zero initial tensor remains zero and $μ > 0$ is constant. The force in [A1] vanishes, hence $D_{τ} p = 0$ and $U = c p / μ$ . Covariant differentiation gives $D_{τ} U = (c / μ) D_{τ} p = 0$ .

2.1givenA2F1F2step 1.1algebra∎

The normalization $g (U, U) = - c^{2}$ and $p = (μ / c) U$ imply $g (p, p) = - μ^{2}$ and the positive rest-mass parameter is $M = μ / c$ . The conclusion holds only while the solution stays in the local TD admissible region [A2]; it makes no claim of global extension.

Remarks

The result is the zero-spin specialization of the adopted neutral TD model. It does not derive that model from the body-moment definition and does not assert completeness of the geodesic or the full spinning-body system.

Spinless TD geodesic limit

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources