ParallelTransportSolution
ParallelTransportSolution packages a vector field V along a spacetime curve that obeys the parallel transport ODE and matches a prescribed initial vector at parameter lambda0. Researchers constructing holonomy defects or curvature correspondences in Recognition Science cite it as the basic solution object. The declaration is a plain structure definition with three fields and no proof obligations.
claimLet $g$ be a metric tensor on 4D spacetime and let $γ$ be a smooth curve parameterized by $λ$. A parallel transport solution with initial condition $ic = (λ_0, V_0)$ is a map $V: ℝ → ℝ^4$ such that the covariant derivative of $V$ along $γ$ vanishes (i.e., $dV^μ/dλ + Γ^μ_{αβ} (dγ^α/dλ) V^β = 0$ for each component $μ$) and $V(λ_0) = V_0$.
background
The module formalizes parallel transport along curves in 4D spacetime using the Levi-Civita connection. A SpacetimeCurve consists of a smooth path together with its tangent vector. ParallelTransportIC records the starting parameter value and the initial vector. ParallelTransported is the predicate that a vector field $V$ along the curve satisfies the first-order ODE obtained from the Christoffel symbols of $g$ vanishing the covariant derivative.
proof idea
This is a structure definition that directly assembles the vector field, the ParallelTransported predicate, and the initial-condition equality. No lemmas are invoked; the declaration encodes the mathematical content of a solution to the transport equation.
why it matters in Recognition Science
It supplies the basic data type for the downstream HolonomyDefect, which quantifies the vector mismatch after transport around a closed loop and thereby realizes the geometric meaning of the Riemann tensor. Within Recognition Science the construction connects non-uniform J-cost defect density to observable gravitational effects through the failure of parallel transport on curved manifolds.
scope and limits
- Does not establish existence of solutions for arbitrary metrics or curves.
- Does not address uniqueness of the transported vector field.
- Does not compute explicit holonomy or curvature integrals.
- Does not incorporate higher-order curvature effects beyond the first-order transport ODE.
formal statement (Lean)
74structure ParallelTransportSolution (g : MetricTensor) (γ : SpacetimeCurve)
75 (ic : ParallelTransportIC) where
76 V : ℝ → (Fin 4 → ℝ)
77 is_parallel : ParallelTransported g γ V
78 initial_condition : V ic.lam0 = ic.V0
79
80/-! ## §3 Properties of Parallel Transport -/
81
82/-- In flat Minkowski spacetime, parallel transport is trivial:
83 the Christoffel symbols vanish, so DV/dλ = dV/dλ = 0,
84 meaning V is constant along any curve. -/