IndisputableMonolith.Relativity.Geometry.Connection

   1import Mathlib
   2import IndisputableMonolith.Relativity.Geometry.Tensor
   3import IndisputableMonolith.Relativity.Geometry.Metric
   4
   5/-!
   6# SCAFFOLD MODULE — NOT PART OF CERTIFICATE CHAIN
   7
   8**Status**: Scaffold / Placeholder
   9
  10This file provides placeholder Christoffel symbol and covariant derivative definitions.
  11It is **not** part of the verified certificate chain.
  12
  13All Christoffel symbols default to zero, and covariant derivatives collapse to zero.
  14These are structural placeholders, not rigorous differential geometry.
  15
  16**Do not cite these definitions as proven mathematics.**
  17-/
  18
  19namespace IndisputableMonolith
  20namespace Relativity
  21namespace Geometry
  22
  23/-- **SCAFFOLD**: Stub Christoffel symbol bundle; entries default to zero. -/
  24structure ChristoffelSymbols where
  25  Γ : (Fin 4 → ℝ) → Fin 4 → Fin 4 → Fin 4 → ℝ := fun _ _ _ _ => 0
  26
  27/-- Christoffel symmetry predicate: Γ^ρ_μν = Γ^ρ_νμ. -/
  28def ChristoffelSymmetric (cs : ChristoffelSymbols) : Prop :=
  29  ∀ x ρ μ ν, cs.Γ x ρ μ ν = cs.Γ x ρ ν μ
  30
  31/-- Christoffel symbols associated to a metric; trivial in the sealed build. -/
  32noncomputable def christoffel_from_metric (_g : MetricTensor) : ChristoffelSymbols := {}
  33
  34@[simp] lemma christoffel_from_metric_symmetric (g : MetricTensor) :
  35    ChristoffelSymmetric (christoffel_from_metric g) := by
  36  unfold ChristoffelSymmetric christoffel_from_metric
  37  intro x ρ μ ν
  38  rfl
  39
  40
  41/-- Covariant derivative of a vector field; collapses to zero. -/
  42noncomputable def covariant_deriv_vector (_g : MetricTensor)
  43  (_V : VectorField) (_μ : Fin 4) : VectorField := fun _ _ _ => 0
  44
  45/-- Covariant derivative of a covector field; collapses to zero. -/
  46noncomputable def covariant_deriv_covector (_g : MetricTensor)
  47  (_ω : CovectorField) (_μ : Fin 4) : CovectorField := fun _ _ _ => 0
  48
  49/-- Covariant derivative of a bilinear form; collapses to zero. -/
  50noncomputable def covariant_deriv_bilinear (_g : MetricTensor)
  51  (_B : BilinearForm) (_ρ : Fin 4) : BilinearForm := fun _ _ _ => 0
  52
  53/-- Metric compatibility: ∇_ρ g_μν = 0. -/
  54theorem metric_compatibility (g : MetricTensor) :
  55    ∀ ρ x up low, covariant_deriv_bilinear g g.g ρ x up low = 0 := by
  56  intro ρ x up low
  57  unfold covariant_deriv_bilinear
  58  rfl
  59
  60
  61@[simp] theorem minkowski_christoffel_zero
  62    (x : Fin 4 → ℝ) (ρ μ ν : Fin 4) :
  63    (christoffel_from_metric minkowski_tensor).Γ x ρ μ ν = 0 := rfl
  64
  65end Geometry
  66end Relativity
  67end IndisputableMonolith
  68