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Killing scalar conservation in cotangent shell form

Statement

Assume ${A C}_{ω}$ for the cotangent supplier. Let $(M, g)$ be a smooth Lorentzian manifold with canonical cotangent coordinates $(x^{a}, p_{a})$ and $C_{m} (x, p) = \frac{1}{2} (g^{a b} p_{a} p_{b} + m^{2} c^{2})$ for fixed $m \geq 0$ . For any smooth vector field $ξ$ , set $F_{ξ} = p_{a} ξ^{a} (x)$ . With canonical bracket ${F, G} = F_{x} G_{p} - F_{p} G_{x}$ ,

${F_{ξ}, C_{m}} = - \frac{1}{2} (L_{ξ} g^{- 1})^{a b} p_{a} p_{b} .$

If $ξ$ is Killing, this bracket vanishes for all cotangent covectors, and $F_{ξ}$ is constant along every constrained free-shell trajectory generated by $e C_{m}$ with any smooth multiplier $e$ on its actual interval. No global existence or empirical conservation is asserted.

Facts & Assumptions

Given: A smooth nondegenerate Lorentzian metric, cotangent coordinates, constants $m \geq 0$ , $c > 0$ , and a smooth vector field $ξ : M \to T M$ with $ξ (x) \in T_{x} M$ . A trajectory generated by $e C_{m}$ means a smooth curve $z : I \to T^{*} M$ on an interval $I$ , with $C_{m} (z) = 0$ and $z^{'} (λ) = e (λ) X_{C_{m}} (z (λ))$ for a smooth parameter-only multiplier $e : I \to R$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only for the cotangent supplier; no full axiom of choice is used.

[L2]

Canonical Poisson brackets and local Hamiltonian flow: the current H0 draft fixes the canonical bracket $F_{x} G_{p} - F_{p} G_{x}$ and $d F / d t = F_{t} + {F, H}$ on actual solution intervals, with no completeness claim. Its H25 Step 1 record is ready/hash-current, but the carrier remains draft/unbound with ai-altered proof provenance and no independent proof approval.

[F1]

Here the Lie derivative of the metric is $L_{ξ} g = {\frac{d}{d s} ∣}_{s = 0} Φ_{s}^{*} g$ , where $Φ_{s}$ is the local flow of $ξ$ . In coordinates its components are $K_{a b} : = (L_{ξ} g)_{a b} = ξ^{c} \partial_{c} g_{a b} + g_{c b} \partial_{a} ξ^{c} + g_{a c} \partial_{b} ξ^{c}$ . A Killing field means $K_{a b} = 0$ in every chart. This definition requires neither orientation nor time orientation nor dimension four. The inverse metric has Lie derivative $(L_{ξ} g^{- 1})^{a b} = ξ^{c} \partial_{c} g^{a b} - g^{c b} \partial_{c} ξ^{a} - g^{a c} \partial_{c} ξ^{b}$ .

Proof

technique · direct tensor differentiation

Given: The smooth metric, canonical bracket [L2], shell function and generated trajectories specified above.

1.1givenL2F1algebra

In coordinates, $\partial_{x^{c}} F_{ξ} = p_{a} \partial_{c} ξ^{a}$ , $\partial_{p_{c}} F_{ξ} = ξ^{c}$ , $\partial_{p_{c}} C_{m} = g^{c b} p_{b}$ , and $\partial_{x^{c}} C_{m} = \frac{1}{2} \partial_{c} g^{a b} p_{a} p_{b}$ . Therefore [L2] gives ${F_{ξ}, C_{m}} = p_{a} (\partial_{c} ξ^{a}) g^{c b} p_{b} - \frac{1}{2} ξ^{c} (\partial_{c} g^{a b}) p_{a} p_{b} = - \frac{1}{2} (L_{ξ} g^{- 1})^{a b} p_{a} p_{b}$ , by [F1] and symmetry of $p_{a} p_{b}$ .

2.1F1step 1.1algebra

Differentiating $g^{a c} g_{c b} = δ_{b}^{a}$ gives $\partial_{e} g^{a b} = - g^{a c} g^{b d} \partial_{e} g_{c d}$ . Substitution in the coordinate formula of [F1] gives $(L_{ξ} g^{- 1})^{a b} = - g^{a c} g^{b d} K_{c d}$ . If $ξ$ is Killing, $K_{c d} = 0$ by definition, so $L_{ξ} g^{- 1} = 0$ . Step 1.1 then gives ${F_{ξ}, C_{m}} = 0$ at every cotangent covector.

3.1givenL2A1step 2.1∎

Along any generated trajectory in the stated sense, the chain rule and [L2] give $\frac{d}{d λ} F_{ξ} = d F_{ξ} (e X_{C_{m}}) = e {F_{ξ}, C_{m}}$ because $F_{ξ}$ has no explicit parameter dependence. This vanishes by step 2.1 throughout the solution's actual interval, and hence $F_{ξ}$ is constant there. The calculation is local in charts and thus applies along the whole interval, with no completeness assumption.

Killing scalar conservation in cotangent shell form

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources