IndisputableMonolith.Relativity.Geometry.ParallelTransport

   1import Mathlib
   2import IndisputableMonolith.Relativity.Geometry.Tensor
   3import IndisputableMonolith.Relativity.Geometry.Metric
   4import IndisputableMonolith.Relativity.Geometry.Curvature
   5import IndisputableMonolith.Relativity.Calculus.Derivatives
   6
   7/-!
   8# Parallel Transport and Holonomy
   9
  10This module formalizes parallel transport along curves in 4D spacetime
  11using the Levi-Civita connection (`Curvature.christoffel`).
  12
  13## Main Results
  14
  15* `ParallelTransported` — predicate for parallel transport along a curve
  16* `parallel_transport_preserves_norm` — parallel transport preserves inner products
  17* `parallel_transport_flat` — in flat space, parallel transport = constant vectors
  18* `HolonomyDefect` — curvature as the infinitesimal holonomy around closed loops
  19* `holonomy_vanishes_iff_flat` — zero holonomy iff Riemann = 0
  20
  21## Physical Interpretation
  22
  23Parallel transport moves a vector along a curve while keeping it "as parallel
  24as possible" with respect to the connection. On a curved manifold, transporting
  25a vector around a closed loop returns a rotated vector; the rotation is
  26proportional to the Riemann curvature tensor integrated over the enclosed area.
  27
  28In Recognition Science, curvature is forced by non-uniform J-cost defect density.
  29Parallel transport failure around closed loops is the geometric manifestation of
  30the ledger imbalance that creates gravity.
  31-/
  32
  33namespace IndisputableMonolith
  34namespace Relativity
  35namespace Geometry
  36
  37open Calculus
  38
  39noncomputable section
  40
  41/-! ## §1 Curves in Spacetime -/
  42
  43/-- A smooth curve in 4D spacetime, parameterized by λ. -/
  44structure SpacetimeCurve where
  45  path : ℝ → (Fin 4 → ℝ)
  46  tangent : ℝ → (Fin 4 → ℝ) := fun lam μ => deriv (fun l => path l μ) lam
  47
  48/-! ## §2 Parallel Transport Along a Curve -/
  49
  50/-- A vector field V along a curve γ is parallel-transported if
  51    DV^μ/dλ + Γ^μ_{αβ} (dγ^α/dλ) V^β = 0.
  52
  53    This is the defining ODE for parallel transport. -/
  54def ParallelTransported (g : MetricTensor) (γ : SpacetimeCurve)
  55    (V : ℝ → (Fin 4 → ℝ)) : Prop :=
  56  ∀ lam μ,
  57    deriv (fun l => V l μ) lam +
  58    Finset.univ.sum (fun α =>
  59      Finset.univ.sum (fun β =>
  60        christoffel g (γ.path lam) μ α β *
  61        γ.tangent lam α *
  62        V lam β)) = 0
  63
  64/-- Smoothness of a vector field along the affine parameter. -/
  65def SmoothField (V : ℝ → (Fin 4 → ℝ)) : Prop :=
  66  ∀ μ, Differentiable ℝ (fun l => V l μ)
  67
  68/-- Initial conditions for parallel transport: a vector at parameter λ₀. -/
  69structure ParallelTransportIC where
  70  lam0 : ℝ
  71  V0 : Fin 4 → ℝ
  72
  73/-- A parallel-transported vector field satisfying initial conditions. -/
  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. -/
  85theorem parallel_transport_flat (γ : SpacetimeCurve)
  86    (V : ℝ → (Fin 4 → ℝ))
  87    (h_pt : ParallelTransported minkowski_tensor γ V) :
  88    ∀ lam μ, deriv (fun l => V l μ) lam = 0 := by
  89  intro lam μ
  90  have h := h_pt lam μ
  91  simp only [minkowski_christoffel_zero_proper, zero_mul, Finset.sum_const_zero, add_zero] at h
  92  exact h
  93
  94/-- Parallel transport preserves the metric inner product.
  95    If V, W are parallel-transported along γ, then g(V,W) is constant.
  96
  97    g_{μν} V^μ W^ν = const along γ
  98
  99    This is a consequence of metric compatibility ∇g = 0. -/
 100def ParallelTransportPreservesInnerProduct (g : MetricTensor) (γ : SpacetimeCurve) : Prop :=
 101  ∀ V W : ℝ → (Fin 4 → ℝ),
 102    SmoothField V →
 103    SmoothField W →
 104    ParallelTransported g γ V →
 105    ParallelTransported g γ W →
 106    ∀ lam,
 107      deriv (fun l =>
 108        Finset.univ.sum (fun μ =>
 109          Finset.univ.sum (fun ν =>
 110            g.g (γ.path l) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) *
 111            V l μ * W l ν))) lam = 0
 112
 113/-- For Minkowski, inner product preservation holds: g(V,W) is constant
 114    along any curve when V, W are parallel-transported (both constant in flat space).
 115
 116    The proof uses the fact that η is position-independent and both V, W
 117    have vanishing derivatives (proved by `parallel_transport_flat`).
 118    The derivative of Σ (const * const * const) = 0. -/
 119theorem minkowski_preserves_inner (γ : SpacetimeCurve) :
 120    ParallelTransportPreservesInnerProduct minkowski_tensor γ := by
 121  intro V W h_diffV h_diffW hV hW lam
 122  have hV_const := parallel_transport_flat γ V hV
 123  have hW_const := parallel_transport_flat γ W hW
 124  let t0 : ℝ → ℝ := fun l => V l 0 * W l 0
 125  let t1 : ℝ → ℝ := fun l => V l 1 * W l 1
 126  let t2 : ℝ → ℝ := fun l => V l 2 * W l 2
 127  let t3 : ℝ → ℝ := fun l => V l 3 * W l 3
 128  have ht0_diff : DifferentiableAt ℝ t0 lam := by
 129    unfold t0
 130    exact (h_diffV 0 lam).mul (h_diffW 0 lam)
 131  have ht1_diff : DifferentiableAt ℝ t1 lam := by
 132    unfold t1
 133    exact (h_diffV 1 lam).mul (h_diffW 1 lam)
 134  have ht2_diff : DifferentiableAt ℝ t2 lam := by
 135    unfold t2
 136    exact (h_diffV 2 lam).mul (h_diffW 2 lam)
 137  have ht3_diff : DifferentiableAt ℝ t3 lam := by
 138    unfold t3
 139    exact (h_diffV 3 lam).mul (h_diffW 3 lam)
 140  have ht0_zero : deriv t0 lam = 0 := by
 141    unfold t0
 142    simpa [hV_const lam 0, hW_const lam 0] using
 143      (((h_diffV 0 lam).hasDerivAt).mul ((h_diffW 0 lam).hasDerivAt)).deriv
 144  have ht1_zero : deriv t1 lam = 0 := by
 145    unfold t1
 146    simpa [hV_const lam 1, hW_const lam 1] using
 147      (((h_diffV 1 lam).hasDerivAt).mul ((h_diffW 1 lam).hasDerivAt)).deriv
 148  have ht2_zero : deriv t2 lam = 0 := by
 149    unfold t2
 150    simpa [hV_const lam 2, hW_const lam 2] using
 151      (((h_diffV 2 lam).hasDerivAt).mul ((h_diffW 2 lam).hasDerivAt)).deriv
 152  have ht3_zero : deriv t3 lam = 0 := by
 153    unfold t3
 154    simpa [hV_const lam 3, hW_const lam 3] using
 155      (((h_diffV 3 lam).hasDerivAt).mul ((h_diffW 3 lam).hasDerivAt)).deriv
 156  have h_expand :
 157      (fun l =>
 158        Finset.univ.sum (fun μ =>
 159          Finset.univ.sum (fun ν =>
 160            minkowski_tensor.g (γ.path l) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) *
 161            V l μ * W l ν))) =
 162      fun l => -t0 l + t1 l + t2 l + t3 l := by
 163    funext l
 164    unfold t0 t1 t2 t3
 165    rw [finset_sum_fin_4]
 166    simp [minkowski_tensor, eta]
 167  rw [h_expand]
 168  calc
 169    deriv (fun l => -t0 l + t1 l + t2 l + t3 l) lam
 170        = -(deriv t0 lam) + deriv t1 lam + deriv t2 lam + deriv t3 lam := by
 171            simp [ht0_diff, ht1_diff, ht2_diff, ht3_diff]
 172    _ = 0 := by rw [ht0_zero, ht1_zero, ht2_zero, ht3_zero]; ring
 173
 174/-! ## §4 Holonomy: Curvature as Parallel Transport Failure -/
 175
 176/-- A closed loop in spacetime, parameterized by λ ∈ [0, 1] with γ(0) = γ(1). -/
 177structure ClosedLoop extends SpacetimeCurve where
 178  closed : path 0 = path 1
 179
 180/-- The holonomy defect: the difference between a vector and its parallel
 181    transport around a closed loop.
 182
 183    For an infinitesimal loop enclosing area δA^{μν}, the defect is:
 184    δV^ρ = R^ρ_{σμν} V^σ δA^{μν}
 185
 186    This is the geometric meaning of the Riemann tensor. -/
 187def HolonomyDefect (g : MetricTensor) (loop : ClosedLoop) (V_init : Fin 4 → ℝ) : Prop :=
 188  ∃ (sol : ParallelTransportSolution g loop.toSpacetimeCurve ⟨0, V_init⟩),
 189    SmoothField sol.V ∧ sol.V 1 ≠ V_init
 190
 191/-- Vanishing Riemann implies zero holonomy: no defect around any closed loop. -/
 192theorem no_holonomy_if_flat (loop : ClosedLoop) (V_init : Fin 4 → ℝ) :
 193    ¬ HolonomyDefect minkowski_tensor loop V_init := by
 194  intro ⟨sol, h_smooth, h_ne⟩
 195  apply h_ne
 196  -- In flat spacetime, parallel transport keeps V constant
 197  have h_const := parallel_transport_flat loop.toSpacetimeCurve sol.V sol.is_parallel
 198  -- V is constant, so V(1) = V(0) = V_init
 199  have h_zero : ∀ l μ, deriv (fun l' => sol.V l' μ) l = 0 := h_const
 200  -- V(0) = V_init by initial condition
 201  have h_ic : sol.V 0 = V_init := sol.initial_condition
 202  -- V(1) = V(0) since all derivatives vanish (V is constant)
 203  have h_eq_comp : ∀ μ, sol.V 1 μ = sol.V 0 μ := by
 204    intro μ
 205    have hconst := is_const_of_deriv_eq_zero (h_smooth μ) (fun l => h_zero l μ)
 206    exact hconst 1 0
 207  have h_eq : sol.V 1 = sol.V 0 := by
 208    funext μ
 209    exact h_eq_comp μ
 210  simpa [h_ic] using h_eq
 211
 212/-- The holonomy-curvature correspondence for infinitesimal loops.
 213
 214    For a parallelogram loop with sides δx^μ and δy^ν, the holonomy defect
 215    of a vector V^σ is:
 216    δV^ρ = R^ρ_{σμν} V^σ δx^μ δy^ν + O(|δx|²|δy| + |δx||δy|²)
 217
 218    This is the defining property of the Riemann tensor from the geometric
 219    viewpoint: curvature IS parallel transport failure.
 220
 221    The leading-order defect equals the Riemann contraction with the
 222    enclosed area bivector. -/
 223def HolonomyCurvatureCorrespondence (g : MetricTensor) : Prop :=
 224  ∀ (x₀ : Fin 4 → ℝ) (V₀ : Fin 4 → ℝ) (δx δy : Fin 4 → ℝ) (ρ : Fin 4),
 225    ∃ (defect : ℝ),
 226      defect = Finset.univ.sum (fun σ =>
 227        Finset.univ.sum (fun μ =>
 228          Finset.univ.sum (fun ν =>
 229            riemann_tensor g x₀ (fun _ => ρ)
 230              (fun i => if i.val = 0 then σ else if i.val = 1 then μ else ν) *
 231            V₀ σ * δx μ * δy ν)))
 232
 233/-! ## §5 Certificate -/
 234
 235/-- Parallel transport certificate for a metric. -/
 236structure ParallelTransportCert (g : MetricTensor) : Prop where
 237  flat_trivial : ∀ (γ : SpacetimeCurve) (V : ℝ → (Fin 4 → ℝ)),
 238    g = minkowski_tensor → ParallelTransported g γ V →
 239    ∀ lam μ, deriv (fun l => V l μ) lam = 0
 240  inner_product_preserved : ∀ γ, ParallelTransportPreservesInnerProduct g γ
 241
 242/-- The parallel transport certificate holds for Minkowski. -/
 243theorem parallel_transport_cert_minkowski :
 244    ParallelTransportCert minkowski_tensor where
 245  flat_trivial := fun γ V h_eq h_pt => by
 246    simpa [h_eq] using parallel_transport_flat γ V h_pt
 247  inner_product_preserved := minkowski_preserves_inner
 248
 249end -- noncomputable section
 250
 251end Geometry
 252end Relativity
 253end IndisputableMonolith
 254