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)

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Flat future mass shell has an explicit global orbit quotient

Physical scope

Explicit global reduction only for a free flat future massive or nonzero-null shell; no curved global quotient claim.

Statement

In affine Minkowski space with inertial coordinates $x^{0}, x^{i}$ measured in metres and metric $η = diag (- 1, 1, 1, 1)$ , identify each cotangent vector $p$ with its metric dual $p^{♯} \in V$ , the unique vector satisfying $η (p^{♯}, v) = p (v)$ for all $v \in V$ . In these coordinates $p^{♯} = (- p_{0}, p_{1}, p_{2}, p_{3})$ . Let $m \geq 0$ and let

$Σ_{m}^{+} = {(x, p) : - p_{0} = P^{0} = \sqrt{m^{2} c^{2} + ∣ p ∣^{2}}, p_{0} < 0},$

where $p \neq 0$ when $m = 0$ , and write $P^{i} = p_{i}$ for the spatial raised momentum components. The characteristic flow of the restricted canonical form is $(x, p) \mapsto (x + α P, p)$ , and its orbit space is globally diffeomorphic to $B_{m} = R_{a}^{3} \times R_{p}^{3}$ for $m > 0$ , or $R_{a}^{3} \times (R_{p}^{3} ∖ {0})$ for $m = 0$ , by

$a^{i} = x^{i} - \frac{P^{i}}{P^{0}} x^{0}, p_{i} = p_{i} .$

The reduced symplectic form is $d a^{i} \land d p_{i}$ . The quotient is Hausdorff and second countable and each full characteristic line meets each slice $x^{0} = c t$ exactly once. This is an explicit free flat quotient, not a conclusion about arbitrary first-class systems.

Facts & Assumptions

Given: The affine Minkowski model, its canonical cotangent form, and the specified future shell with $P^{0} > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Assume countable choice ACω, meaning every sequence of nonempty sets has a choice function; no full Axiom of Choice is assumed.

[F1]

Cotangent phase data, observables and action dimensions: $ω = \sum_{a} d x^{a} \land d p_{a}$ and $θ = p_{a} d x^{a}$ .

[F2]

Oriented time-oriented affine Minkowski space (the spacetime primitives): inertial affine coordinates have metric $η = diag (- 1, 1, 1, 1)$ , with fixed $c > 0$ .

[F3]

Canonical Poisson brackets and local Hamiltonian flow: a Hamiltonian depending only on momentum has characteristic field $(\partial H / \partial p_{a}) \partial_{x^{a}}$ .

[F4]

Regular first-class constraints admit local symplectic reduction: the characteristic kernel is the null direction of the restricted form; the present statement supplies explicit additional global coordinates.

[F5]

A regular level set is an embedded submanifold: a smooth real-valued constraint whose differential is nonzero at every point of its zero level has an embedded hypersurface as that level.

Proof

1.1givenA1F2F3F4F5algebra

Define $C_{m} (x, p) = \frac{1}{2} (- p_{0}^{2} + ∣ p ∣^{2} + m^{2} c^{2})$ . Solving $C_{m} = 0$ on the future branch gives $p_{0} = - P^{0}$ with $P^{0} = \sqrt{m^{2} c^{2} + ∣ p ∣^{2}} > 0$ . For $m > 0$ work on the full cotangent space; for $m = 0$ work on its open subset $U_{0} = {(x, p) : p \neq 0}$ . On the zero level in these domains, the vertical differential is $d_{p} C_{m} = p^{♯} \neq 0$ : for $m > 0$ the shell equation makes $p$ timelike, and for $m = 0$ nondegeneracy of $g^{- 1}$ and $p \neq 0$ give $p^{♯} \neq 0$ . Hence [F5] makes the zero level in the stated domain an embedded hypersurface, and $Σ_{m}^{+}$ is its open future branch. The single constraint is first class because ${C_{m}, C_{m}} = 0$ by skew-symmetry of the canonical Poisson bracket. Thus the regular embedded-level and first-class hypotheses of [F4] hold. Now $y = x^{0} / P^{0}$ and $a^{i} = x^{i} - (P^{i} / P^{0}) x^{0}$ are smooth shell coordinates, with inverse $x^{0} = y P^{0}, x^{i} = a^{i} + y P^{i}$ . These formulas prove a diffeomorphism $Σ_{m}^{+} ≅ R_{y} \times B_{m}$ .

2.1givenF1F3F4step 1.1algebra

Since the constraint depends only on $p$ , [F3] gives its characteristic field $X_{C_{m}} = P^{a} \partial_{x^{a}}$ ; [F4] applies by step 1.1 and identifies this direction with the characteristic leaves used for reduction. The flow sends $y \mapsto y + α$ while fixing $a, p$ . Hence two points are in the same full orbit exactly when their $a, p$ labels agree, and the quotient map is the product projection to $B_{m}$ .

2.2givenF1step 1.1algebra

In shell coordinates $x^{0} = y P^{0}, x^{i} = a^{i} + y P^{i}$ , so $d x^{0} = P^{0} d y + y d P^{0}$ and $d x^{i} = d a^{i} + P^{i} d y + y d P^{i}$ . Since $p_{0} = - P^{0}$ , expanding the canonical form gives $ω = d a^{i} \land d p_{i} + d y \land (P^{i} d p_{i} - P^{0} d P^{0}) + y d P^{i} \land d p_{i} .$ Differentiating $(P^{0})^{2} = m^{2} c^{2} + ∣ p ∣^{2}$ gives $P^{0} d P^{0} = P^{i} d p_{i}$ , so the (dy)-term vanishes. The matrix $\partial P^{i} / \partial p_{j} = δ^{i j}$ is symmetric, so $d P^{i} \land d p_{i} = 0$ . The restricted form is therefore the pullback of $d a^{i} \land d p_{i}$ , which is nondegenerate on $B_{m}$ .

3.1step 1.1step 2.1algebra

The spaces $R^{3}$ and $R^{3} ∖ {0}$ are Hausdorff second-countable manifolds; their product with $R_{a}^{3}$ has the same properties. Product projection is a smooth submersion with global section $y = 0$ , establishing the asserted global quotient directly.

4.1step 2.1step 1.1∎

Along every orbit $x^{0} (α) = x^{0} (0) + α P^{0}$ , with $P^{0} > 0$ , so each level $x^{0} = c t$ is attained at the unique parameter $α = (c t - x^{0} (0)) / P^{0}$ .

Flat future mass shell has an explicit global orbit quotient

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources