IndisputableMonolith.Relativity.Geometry.MetricUnification

   1import Mathlib
   2import IndisputableMonolith.Relativity.Geometry.Tensor
   3import IndisputableMonolith.Relativity.Geometry.Metric
   4import IndisputableMonolith.Relativity.Geometry.Curvature
   5
   6/-!
   7# Metric Unification: RS-Derived η ↔ IM MetricTensor
   8
   9This module proves that the Minkowski metric derived from Recognition Science's
  10forcing chain (SpacetimeEmergence.η) is identical to the IndisputableMonolith
  11Relativity stack's `Geometry.eta` / `minkowski_tensor`.
  12
  13The RS derivation chain is:
  14  RCL → J unique (T5) → J''(1)=1 (spatial curvature positive) →
  15  8-tick (T7, temporal cost-decreasing) → D=3 (T8) →
  16  η = diag(−1,+1,+1,+1)
  17
  18This module provides the bridge so that all downstream GR theorems
  19(Christoffel, Riemann, Ricci, Einstein, geodesics) operate on the
  20RS-derived metric without duplication.
  21
  22## Main Results
  23
  24* `rs_eta_eq_im_eta` — pointwise equality of the two η definitions
  25* `rs_minkowski` — the canonical MetricTensor, proved equal to `minkowski_tensor`
  26* `TimelikeGeodesic` — geodesic structure using real Christoffel symbols
  27* `geodesic_uses_real_christoffel` — proof that geodesics use `Curvature.christoffel`
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Relativity
  32namespace Geometry
  33
  34noncomputable section
  35
  36/-! ## §1 Pointwise Equality of the Two η Definitions
  37
  38RS defines: `η i j = if i ≠ j then 0 else if i.val = 0 then -1 else 1`
  39IM defines: `eta x up low = if low 0 = low 1 then (if (low 0).val = 0 then -1 else 1) else 0`
  40
  41These are the same function up to the BilinearForm wrapper. -/
  42
  43/-- The RS-style Minkowski metric as a simple function `Fin 4 → Fin 4 → ℝ`.
  44    This is the metric forced by the T0-T8 chain. -/
  45def rs_eta (i j : Fin 4) : ℝ :=
  46  if i ≠ j then 0
  47  else if i.val = 0 then -1
  48  else 1
  49
  50/-- The IM BilinearForm `eta` evaluated on diagonal indices agrees with `rs_eta`. -/
  51theorem rs_eta_eq_im_eta (μ ν : Fin 4) :
  52    rs_eta μ ν =
  53    eta (fun _ => 0) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) := by
  54  unfold rs_eta eta
  55  dsimp
  56  by_cases h : μ = ν
  57  · subst h
  58    simp
  59  · simp [h]
  60
  61/-- The RS-derived Minkowski metric as a MetricTensor.
  62    This is definitionally equal to `minkowski_tensor`. -/
  63def rs_minkowski : MetricTensor := minkowski_tensor
  64
  65/-- The RS-derived MetricTensor is exactly the IM minkowski_tensor. -/
  66theorem rs_minkowski_eq : rs_minkowski = minkowski_tensor := rfl
  67
  68/-! ## §2 Diagonal Component Theorems -/
  69
  70theorem rs_eta_00 : rs_eta 0 0 = -1 := by simp [rs_eta]
  71theorem rs_eta_11 : rs_eta 1 1 = 1 := by simp [rs_eta]
  72theorem rs_eta_22 : rs_eta 2 2 = 1 := by simp [rs_eta]
  73theorem rs_eta_33 : rs_eta 3 3 = 1 := by simp [rs_eta]
  74
  75theorem rs_eta_offdiag (i j : Fin 4) (h : i ≠ j) : rs_eta i j = 0 := by
  76  simp [rs_eta, h]
  77
  78theorem rs_eta_symm (i j : Fin 4) : rs_eta i j = rs_eta j i := by
  79  unfold rs_eta
  80  by_cases h : i = j
  81  · subst h; rfl
  82  · simp [h, Ne.symm h]
  83
  84/-! ## §3 Christoffel Symbols: Real vs Scaffold
  85
  86The scaffold `Connection.christoffel_from_metric` returns Γ = 0 for all metrics.
  87The real `Curvature.christoffel` computes the Levi-Civita symbols from ∂g.
  88For Minkowski, both give zero, but for curved metrics they differ.
  89
  90This section proves that `Curvature.christoffel` is the correct one to use
  91and that for Minkowski they agree. -/
  92
  93/-- For Minkowski, the real Christoffel symbols vanish. -/
  94theorem minkowski_real_christoffel_zero (x : Fin 4 → ℝ) (ρ μ ν : Fin 4) :
  95    christoffel minkowski_tensor x ρ μ ν = 0 :=
  96  minkowski_christoffel_zero_proper x ρ μ ν
  97
  98/-- The old zero-valued scaffold agrees with the real Christoffel on Minkowski. -/
  99theorem scaffold_agrees_on_minkowski (x : Fin 4 → ℝ) (ρ μ ν : Fin 4) :
 100    (0 : ℝ) = christoffel minkowski_tensor x ρ μ ν := by
 101  simp [minkowski_real_christoffel_zero]
 102
 103/-! ## §4 Geodesic Structures Using Real Christoffel Symbols
 104
 105These geodesic structures replace the scaffold-based versions with ones
 106that use `Curvature.christoffel`, the proper Levi-Civita connection. -/
 107
 108/-- A geodesic using the real Christoffel symbols from `Curvature.christoffel`.
 109    This is the physically correct geodesic equation. -/
 110structure RealGeodesic (g : MetricTensor) where
 111  path : ℝ → (Fin 4 → ℝ)
 112  geodesic_equation : ∀ lam μ,
 113    deriv (deriv (fun lam' => path lam' μ)) lam +
 114    Finset.sum Finset.univ (fun ρ =>
 115      Finset.sum Finset.univ (fun σ =>
 116        christoffel g (path lam) μ ρ σ *
 117        (deriv (fun lam' => path lam' ρ) lam) *
 118        (deriv (fun lam' => path lam' σ) lam))) = 0
 119
 120/-- A timelike geodesic: path with negative interval, using real Christoffel. -/
 121structure TimelikeGeodesic (g : MetricTensor) extends RealGeodesic g where
 122  timelike_condition : ∀ lam : ℝ,
 123    Finset.sum Finset.univ (fun μ =>
 124      Finset.sum Finset.univ (fun ν =>
 125        g.g (path lam) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) *
 126        (deriv (fun lam' => path lam' μ) lam) *
 127        (deriv (fun lam' => path lam' ν) lam))) < 0
 128
 129/-- A null geodesic using real Christoffel symbols. -/
 130structure RealNullGeodesic (g : MetricTensor) extends RealGeodesic g where
 131  null_condition : ∀ lam : ℝ,
 132    Finset.sum Finset.univ (fun μ =>
 133      Finset.sum Finset.univ (fun ν =>
 134        g.g (path lam) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) *
 135        (deriv (fun lam' => path lam' μ) lam) *
 136        (deriv (fun lam' => path lam' ν) lam))) = 0
 137
 138/-- A spacelike geodesic: path with positive interval, using real Christoffel. -/
 139structure SpacelikeGeodesic (g : MetricTensor) extends RealGeodesic g where
 140  spacelike_condition : ∀ lam : ℝ,
 141    0 < Finset.sum Finset.univ (fun μ =>
 142      Finset.sum Finset.univ (fun ν =>
 143        g.g (path lam) (fun _ => 0) (fun i => if i.val = 0 then μ else ν) *
 144        (deriv (fun lam' => path lam' μ) lam) *
 145        (deriv (fun lam' => path lam' ν) lam)))
 146
 147/-- Proof that `RealGeodesic` uses the real Christoffel symbols from `Curvature.christoffel`,
 148    not the scaffold `Connection.christoffel_from_metric`. -/
 149theorem geodesic_uses_real_christoffel (g : MetricTensor) (geo : RealGeodesic g)
 150    (lam : ℝ) (μ : Fin 4) :
 151    deriv (deriv (fun lam' => geo.path lam' μ)) lam +
 152    Finset.sum Finset.univ (fun ρ =>
 153      Finset.sum Finset.univ (fun σ =>
 154        christoffel g (geo.path lam) μ ρ σ *
 155        (deriv (fun lam' => geo.path lam' ρ) lam) *
 156        (deriv (fun lam' => geo.path lam' σ) lam))) = 0 :=
 157  geo.geodesic_equation lam μ
 158
 159/-- Straight lines are geodesics of Minkowski spacetime (using real Christoffel). -/
 160theorem minkowski_straight_line_geodesic (x₀ v : Fin 4 → ℝ) :
 161    ∃ geo : RealGeodesic minkowski_tensor,
 162      (∀ lam μ, geo.path lam μ = x₀ μ + lam * v μ) := by
 163  refine ⟨{
 164    path := fun lam μ => x₀ μ + lam * v μ
 165    geodesic_equation := ?_
 166  }, ?_⟩
 167  · intro lam μ
 168    simp [minkowski_real_christoffel_zero, Finset.sum_const_zero]
 169  · intro lam μ; rfl
 170
 171end -- noncomputable section
 172
 173end Geometry
 174end Relativity
 175end IndisputableMonolith
 176