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Conditional particle trajectories from Hamilton-Jacobi characteristics

Physical scope

Adopted free test-particle/massless model on the actual time-pulled-back chart domain and invertible characteristic projection; no global graph continuation is asserted.

Statement

Assume ${A C}_{ω}$ for the cotangent/H7/H9 supplier chain. Adopt the massive free action or separate massless shell action in Free massive test-particle action and Separate massless shell action. Let $I_{0} \subseteq R$ be the open first-coordinate interval, measured in metres, of a temporal product chart, and let $Ω \subseteq R^{3}$ be its open spatial coordinate range. Let $(N, β, h)$ be the smooth temporal-chart data on $I_{0} \times Ω$ of Temporal chart with lapse, shift and positive slice metric, with $N > 0$ and $h_{i j}$ positive definite. Take $m \geq 0$ in kg and $c > 0$ in m/s, and define the physical-time domain $I_{t} : = {t \in R : c t \in I_{0}}$ and pulled-back data $\hat{N} (t, q) : = N (c t, q)$ , ${\hat{β}}^{i} (t, q) : = β^{i} (c t, q)$ , ${\hat{h}}_{i j} (t, q) : = h_{i j} (c t, q)$ . Write ${\hat{h}}^{i j}$ for the inverse matrix. On the future shell with $x^{0} = c t$ , define $κ (t, q, p) = \sqrt{m^{2} c^{2} + {\hat{h}}^{i j} (t, q) p_{i} p_{j}}$ , $p_{0} = {\hat{β}}^{i} p_{i} - \hat{N} κ$ , and the positive-root reduced Hamiltonian $H (t, q, p) = - c p_{0} = c (\hat{N} κ - {\hat{β}}^{i} p_{i})$ from A curved temporal chart gives a local reduced Hamiltonian. Its open domain is $U = I_{t} \times T^{*} Ω$ for $m > 0$ , and $U = I_{t} \times (T^{*} Ω ∖ {p = 0})$ for $m = 0$ . Here $t$ is measured in seconds, $q$ in metres, $p$ in kg m/s, $H$ in joules, and $S$ in joule-seconds. Let $D \subseteq I_{t} \times Ω$ be open and let $S \in C^{2} (D, R)$ (all partial derivatives through order two are continuous) satisfy $S_{t} + H (t, q, S_{q}) = 0$ wherever $(t, q, S_{q}) \in U$ . If $S$ is the smooth local solution supplied by Local Hamilton-Jacobi solution by invertible characteristics from an initial exact-graph point in $U$ , and $J = [t_{1}, t_{2}] \subset I_{t}$ with $t_{1} < t_{2}$ is an interval on which its characteristic projection remains invertible, let $q : J \to Ω$ be a $C^{1}$ characteristic with $\dot{q} = H_{p} (t, q, S_{q})$ . Then $p (t) : = S_{q} (t, q (t))$ satisfies $\dot{p} = - H_{q}$ , so $(q, p)$ obey Hamilton's equations. The smooth ODE makes this characteristic smooth; by A curved temporal chart gives a local reduced Hamiltonian these reduced equations are exactly the equations of the future-shell first-order system of First-order shell action yields massive and null geodesics. Under the matching adopted model postulate, this curve is a trajectory of the massive or separate massless model. The conclusion holds only on the chart and the actual invertible projection interval; no global Hamilton--Jacobi graph continuation is asserted. It is a conditional mathematical prediction, not a reported observation.

The characteristic graph is restricted to points with $(t, q, S_{q} (t, q)) \in U$ . The action $S$ is measured in joule-seconds.

Facts & Assumptions

Given: One of the stated free-particle model postulates, the temporal chart and future-shell domain, a smooth HJ solution, and a characteristic curve on a compact interval of its invertible projection.

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only for the cotangent/H7 supplier chain; no full axiom of choice is used.

[A1]

Free massive test-particle action: the massive action is adopted only for future timelike curves and fixed $m > 0$ with the stated regularity and endpoint conditions. This H5 carrier remains a draft model postulate with no pipeline_run binding; its current Step 1 record is ready/hash-current, which does not make it an empirical report or independent proof approval.

[A2]

Separate massless shell action: the massless model separately adopts nonzero future null momentum and a positive multiplier; it has no massive proper-time parameterization. This H5 carrier remains a draft model postulate with no pipeline_run binding; its current Step 1 record is ready/hash-current and is not an empirical report or independent proof approval.

[L1]

First-order shell action yields massive and null geodesics: the current H7 draft assumes a prescribed smooth time-oriented Lorentzian four-manifold, a compact chart interval, smooth $x, p, e$ , $e > 0$ , $m \geq 0$ , and the future massive or separate nonzero future-null branch. It supplies $C_{m} = 0$ , $x'^{a} = e g^{a b} p_{b}$ , $p'_{a} = - \frac{e}{2} \partial_{a} g^{b c} p_{b} p_{c}$ , and the unparameterized geodesic interpretation. This is a mathematical variational supplier for explicit functionals and selected future branches; adoption of particle dynamics is a separate physical premise.

[L2]

Local Hamilton-Jacobi solution by invertible characteristics: when the initial exact-graph point lies in its open Hamiltonian domain, it supplies a smooth local HJ solution while the characteristic projection remains invertible. Its proof uses the current draft canonical-flow supplier's stated local Hamiltonian equations; that supplier's Step 1 readiness is not independent proof approval.

[L3]

Complete integrals and covariant versus time Hamilton-Jacobi: for a nonsingular mixed Hessian, the HJ parameters and their conjugates give local canonical integration coordinates; the item states the exact chart and projection-invertibility limitation.

[L4]

A curved temporal chart gives a local reduced Hamiltonian: at the point $x^{0} = c t$ in the chart, under $N > 0$ , positive-definite $h$ , and nonzero spatial momentum on the massless branch, the positive future-shell root gives $p_{0} = β^{i} (c t, q) p_{i} - N (c t, q) κ$ , $H = - c p_{0}$ , and $C_{p_{0}} = κ / N > 0$ for $C = \frac{1}{2} (g^{a b} p_{a} p_{b} + m^{2} c^{2})$ . Its gauge-reduced action has Hamilton equations on the chart interval. This current H9 supplier is a draft; the use here is conditional on that exact statement.

Proof

technique · differentiate the HJ equation along characteristics

Given: The model premises [A1] or [A2], an HJ solution on its invertible domain, and the exact conditional H7 interface [L1].

1.1givenA3algebra

Put $p (t) = S_{q} (t, q (t))$ . From $S_{t} + H (t, q, S_{q}) = 0$ , differentiation with respect to $q$ gives $S_{q t} + H_{q} + H_{p} S_{q q} = 0$ . Since $\dot{q} = H_{p}$ , the chain rule yields $\dot{p} = S_{q t} + S_{q q} \dot{q} = - H_{q}$ . Thus $(q, p)$ satisfies both Hamilton equations.

2.1A3L1L4step 1.1algebra

Write $C_{m} (x^{0}, q, p_{0}, p_{i}) = \frac{1}{2} (g^{a b} p_{a} p_{b} + m^{2} c^{2})$ and lift the characteristic by $x^{0} = c t$ , $x^{i} = q^{i}$ , with $p_{0} (t, q, p) = {\hat{β}}^{i} (t, q) p_{i} - \hat{N} (t, q) κ (t, q, p)$ from [L4]. All derivatives of $C_{m}$ below are evaluated at this lifted point. Set $e : = c / C_{p_{0}} = c \hat{N} / κ > 0$ . Implicit differentiation of $C_{m} (c t, q, p_{0} (t, q, p), p) = 0$ gives $H_{p_{i}} = c C_{p_{i}} / C_{p_{0}} = e C_{p_{i}}$ and $- H_{q^{i}} = - c C_{q^{i}} / C_{p_{0}} = - e C_{q^{i}}$ . Thus ${\dot{x}}^{0} = c = e C_{p_{0}}$ , ${\dot{x}}^{i} = e C_{p_{i}}$ , and ${\dot{p}}_{i} = - e C_{q^{i}}$ . Differentiating $C_{m} = 0$ along the lifted curve makes the $q^{i}, p_{i}$ terms cancel and gives $C_{p_{0}} {\dot{p}}_{0} = - c C_{x^{0}}$ , hence ${\dot{p}}_{0} = - e C_{x^{0}}$ . The lift therefore satisfies the complete H7 first-order shell system. The smooth HJ characteristic and $e$ meet H7's regularity conditions on the compact chart interval.

3.1A1A2A3L1L4step 2.1

By [A1] or [A2], the massive or separate massless action is adopted on exactly its stated future branch; H7's variational equivalence then identifies the shell trajectory of step 2.1 with a trajectory of that adopted model. This conclusion remains conditional on those physical postulates.

4.1L2L3A3step 1.1∎

The inverse projection and complete-integral coordinates from [L2]–[L3] apply only where their Jacobians remain nonsingular. If that projection loses invertibility, the single-valued graph $p = S_{q}$ used in step 1.1 is no longer supplied; therefore no continuation beyond the actual invertible projection interval or chart domain follows.

Conditional particle trajectories from Hamilton-Jacobi characteristics

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources