IndisputableMonolith.Relativity.Fields.Integration

   1import Mathlib
   2import IndisputableMonolith.Relativity.Geometry
   3import IndisputableMonolith.Relativity.Fields.Scalar
   4
   5/-!
   6# Integration on Spacetime
   7
   8Implements volume integration with √(-g) measure.
   9Scaffold: uses discrete approximation; full version would use Mathlib measure theory.
  10-/
  11
  12namespace IndisputableMonolith
  13namespace Relativity
  14namespace Fields
  15
  16open Geometry
  17
  18/-- Volume element d⁴x with metric measure √(-g). -/
  19structure VolumeElement where
  20  grid_spacing : ℝ  -- Δx for discrete approximation
  21  grid_spacing_pos : 0 < grid_spacing
  22
  23/-- Sample points for discrete integration (uniform grid). -/
  24def sample_grid (vol : VolumeElement) (n_points : ℕ) : List (Fin 4 → ℝ) :=
  25  -- Simplified: n_points^4 grid over [0, L]^4
  26  -- Full version would use adaptive quadrature
  27  []  -- Placeholder
  28
  29/-- Square root of minus the metric determinant.
  30    STATUS: SCAFFOLD — Uses constant 1 as placeholder. -/
  31noncomputable def sqrt_minus_g (_g : MetricTensor) (_x : Fin 4 → ℝ) : ℝ := 1
  32
  33/-- Integrate a scalar function over spacetime volume (discrete approximation). -/
  34noncomputable def integrate_scalar
  35  (f : (Fin 4 → ℝ) → ℝ) (g : MetricTensor) (vol : VolumeElement) : ℝ :=
  36  let n := 10
  37  let Δx4 := vol.grid_spacing ^ 4
  38  Δx4 * Finset.sum (Finset.range n) (fun i =>
  39    sqrt_minus_g g (fun _ => (i : ℝ) * vol.grid_spacing) *
  40    f (fun _ => (i : ℝ) * vol.grid_spacing))
  41
  42/-- Kinetic action integral: (1/2) ∫ √(-g) g^{μν} (∂_μ ψ)(∂_ν ψ) d⁴x. -/
  43noncomputable def kinetic_action
  44  (φ : ScalarField) (g : MetricTensor) (vol : VolumeElement) : ℝ :=
  45  (1/2) * integrate_scalar (gradient_squared φ g) g vol
  46
  47/-- Potential action integral: (1/2) ∫ √(-g) m² ψ² d⁴x. -/
  48noncomputable def potential_action
  49  (φ : ScalarField) (m_squared : ℝ) (g : MetricTensor) (vol : VolumeElement) : ℝ :=
  50  (m_squared / 2) * integrate_scalar (field_squared φ) g vol
  51
  52/-- Integration is linear (finite weighted sum).
  53    STATUS: PROVED — Linearity of finite sums. -/
  54theorem integrate_add (f₁ f₂ : (Fin 4 → ℝ) → ℝ) (g : MetricTensor) (vol : VolumeElement) :
  55    integrate_scalar (fun x => f₁ x + f₂ x) g vol =
  56      integrate_scalar f₁ g vol + integrate_scalar f₂ g vol := by
  57  unfold integrate_scalar
  58  simp [Finset.sum_add_distrib, mul_add]
  59
  60/-- Integration scaling property.
  61    STATUS: PROVED — Scaling of finite sums. -/
  62theorem integrate_smul (c : ℝ) (f : (Fin 4 → ℝ) → ℝ) (g : MetricTensor) (vol : VolumeElement) :
  63    integrate_scalar (fun x => c * f x) g vol =
  64      c * integrate_scalar f g vol := by
  65  unfold integrate_scalar
  66  -- LHS: Δx4 * Σᵢ (sqrt_g * (c * f)) = Δx4 * Σᵢ (c * sqrt_g * f)
  67  -- RHS: c * (Δx4 * Σᵢ sqrt_g * f)
  68  have h : ∀ i : ℕ, sqrt_minus_g g (fun _ => (i : ℝ) * vol.grid_spacing) *
  69      (c * f (fun _ => (i : ℝ) * vol.grid_spacing)) =
  70    c * (sqrt_minus_g g (fun _ => (i : ℝ) * vol.grid_spacing) *
  71      f (fun _ => (i : ℝ) * vol.grid_spacing)) := by intro i; ring
  72  simp_rw [h]
  73  rw [← Finset.mul_sum]
  74  ring
  75
  76/-- Kinetic action is nonnegative for positive-signature spatial parts.
  77    STATUS: SCAFFOLD — Proof simplified with placeholder sqrt_minus_g = 1. -/
  78theorem kinetic_nonneg (φ : ScalarField) (g : MetricTensor) (vol : VolumeElement)
  79    (hSign : ∀ x, 0 ≤ gradient_squared φ g x) :
  80  0 ≤ kinetic_action φ g vol := by
  81  unfold kinetic_action integrate_scalar sqrt_minus_g
  82  apply mul_nonneg (by norm_num)
  83  apply mul_nonneg (pow_nonneg (le_of_lt vol.grid_spacing_pos) 4)
  84  apply Finset.sum_nonneg
  85  intro i _
  86  apply mul_nonneg (by norm_num : (0 : ℝ) ≤ 1)
  87  apply hSign
  88
  89
  90/-- Einstein-Hilbert action: (M_P^2/2) ∫ √(-g) R d^4x. -/
  91noncomputable def einstein_hilbert_action
  92  (g : MetricTensor) (M_P_squared : ℝ) (vol : VolumeElement) : ℝ :=
  93  (M_P_squared / 2) * integrate_scalar (ricci_scalar g) g vol
  94
  95/-- For Minkowski (R=0), EH action vanishes. -/
  96theorem eh_action_minkowski (M_P_squared : ℝ) (vol : VolumeElement) :
  97  einstein_hilbert_action minkowski_tensor M_P_squared vol = 0 := by
  98  simp only [einstein_hilbert_action, integrate_scalar]
  99  rw [Finset.sum_eq_zero]
 100  · simp
 101  · intro i _
 102    simp [minkowski_ricci_scalar_zero]
 103
 104end Fields
 105end Relativity
 106end IndisputableMonolith
 107