Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Killing particle charge and force balance

Physical scope

Conditional scalar balance for a test particle and a specified Killing field; energy interpretation only for a future unit time-translation field, with no global symmetry assumption.

Statement

Let $Ω \subseteq M$ be an open region of the prescribed background and let $K$ be a smooth Killing field on $Ω$ . For a massive force-postulate trajectory $x : I \to M$ , parameterized by proper time on a nonempty open interval $I \subseteq R$ with $x (I) \subseteq Ω$ , constant $m > 0$ , $P = m U$ , and $D_{τ} P = f$ , the scalar $g (K, P)$ obeys

$\frac{d}{d τ} g (K, P) = g (K, f) .$

Thus it is constant when the body is free ( $f = 0$ ). For a free null geodesic $y : J \to M$ on a nonempty open interval $J \subseteq R$ , with $y (J) \subseteq Ω$ , affine parameter $λ$ , and tangent momentum $P = d y / d λ$ , $g (K, P)$ is constant along $λ$ as well. If $K$ is a future unit timelike observer field along the applicable curve, then $- c g (K, P)$ has energy units and is the energy defined by that observer; for a general Killing field it is only a conserved scalar, not automatically an energy.

Facts & Assumptions

Given: The smooth metric-compatible torsion-free connection, Killing field, force law, and null branch in the Statement.

[F1]

A Killing field satisfies $L_{K} g = 0$ (Killing fields of a Lorentzian metric).

[F2]

Along a curve, $\frac{d}{d τ} g (V, W) = g (D_{τ} V, W) + g (V, D_{τ} W)$ (Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed).

[F3]

On the massive branch $P = m U$ and $D_{τ} P = f$ with $g (U, f) = 0$ are the adopted constant-mass force assumptions (Mechanical momentum and observer energy, Prescribed constant-mass force law).

[F4]

The null sector adopts independent einbein variation (Massless test-particle action and momentum scale); its nonzero future-null branch satisfies $P = T / e$ and $D_{λ} P = 0$ (Positive-einbein variation and massive/null sector equations).

[F5]

The Levi-Civita connection is metric-compatible and torsion-free; in a chart these properties give $\partial_{c} g_{a b} = Γ^{d}_{c a} g_{d b} + Γ^{d}_{c b} g_{a d}$ and $Γ^{d}_{a b} = Γ^{d}_{b a}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Torsion freedom and metric compatibility in chart components).

[F6]

A differentiable scalar function with zero derivative on an interval is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

1.1givenF1F2F3F5algebra

In any chart, differentiating the flow pullback that defines $L_{K} g$ gives $(L_{K} g)_{a b} = K^{c} \partial_{c} g_{a b} + g_{c b} \partial_{a} K^{c} + g_{a c} \partial_{b} K^{c}$ : in the coordinate expression $(Φ_{t}^{*} g)_{a b} = g_{c d} (Φ_{t} (x)) \partial_{a} Φ_{t}^{c} \partial_{b} Φ_{t}^{d}$ , use $Φ_{t}^{c} (x) = x^{c} + t K^{c} (x) + o (t)$ and $\partial_{a} Φ_{t}^{c} = δ_{a}^{c} + t \partial_{a} K^{c} + o (t)$ . On the other hand, $K_{b} = g_{b c} K^{c}$ gives $\nabla_{a} K_{b} + \nabla_{b} K_{a} = \partial_{a} (g_{b c} K^{c}) + \partial_{b} (g_{a c} K^{c}) - 2 Γ^{d}_{a b} g_{d c} K^{c}$ . Expanding and substituting [F5], the terms with $Γ^{d}_{a b}$ cancel by torsion-freeness, and the remaining metric-derivative terms become $K^{c} \partial_{c} g_{a b}$ by metric compatibility. Thus $\nabla_{a} K_{b} + \nabla_{b} K_{a} = (L_{K} g)_{a b}$ on this region, without an orientability hypothesis. Since $K$ is Killing by [F1], this symmetrized covariant derivative vanishes. Now metric compatibility [F2] gives $\frac{d}{d τ} g (K, P) = g (D_{τ} K, P) + g (K, D_{τ} P)$ . With $P = m U$ , the first term is $m g (D_{U} K, U) = \frac{m}{2} (\nabla_{a} K_{b} + \nabla_{b} K_{a}) U^{a} U^{b} = 0$ . Substituting [F3] proves the forced balance $d g (K, P) / d τ = g (K, f)$ .

2.1step 1.1F2F4F6algebra

If $f = 0$ , step 1.1 gives zero derivative, so [F6] makes the scalar constant on the connected parameter interval. On a free null curve, [F2] and [F4] give $d g (K, P) / d λ = g (D_{P} K, P)$ ; the same locally derived identity from step 1.1 with $X = Y = P$ makes this zero, hence the null scalar is constant.

3.1F3given∎

If $K$ is a future unit observer, [F3] gives $E_{K} (P) = - c g (P, K) = - c g (K, P)$ by symmetry of $g$ . With $P$ in kg m s $^{- 1}$ and $c$ in m s $^{- 1}$ , this has units joules. For other Killing fields that observer definition does not apply, so the conserved quantity remains only a scalar.

Killing particle charge and force balance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources