`covariant_deriv_vector`

definition

show as: view math explainer →

module: IndisputableMonolith.Relativity.Geometry.Connection
domain: Relativity
line: 42 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.Relativity.Geometry.Connection on GitHub at line 42.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

formal source

  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

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.