deriv_constant

formal source

  63  ring
  64
  65/-- Derivative of constant field is zero. -/
  66theorem deriv_constant (c : ℝ) (μ : Fin 4) (x : Fin 4 → ℝ) :
  67  directional_deriv (constant c) μ x = 0 := by
  68  simp only [directional_deriv, constant]
  69  norm_num
  70
  71/-- Gradient: collection of all directional derivatives ∂_μ ψ. -/
  72noncomputable def gradient (φ : ScalarField) (x : Fin 4 → ℝ) : Fin 4 → ℝ :=
  73  fun μ => directional_deriv φ μ x
  74
  75/-- Squared gradient g^{μν} (∂_μ ψ)(∂_ν ψ) with inverse metric. -/
  76noncomputable def gradient_squared (φ : ScalarField) (g : MetricTensor) (x : Fin 4 → ℝ) : ℝ :=
  77  Finset.sum (Finset.univ : Finset (Fin 4)) (fun μ =>
  78    Finset.sum (Finset.univ : Finset (Fin 4)) (fun ν =>
  79      (inverse_metric g) x (fun i => if (i : ℕ) = 0 then μ else ν) (fun _ => 0) *
  80      (gradient φ x μ) * (gradient φ x ν)))
  81
  82/-- Gradient squared for Minkowski metric. -/
  83theorem gradient_squared_minkowski (φ : ScalarField) (x : Fin 4 → ℝ) :
  84  gradient_squared φ minkowski_tensor x =
  85    -(gradient φ x 0)^2 + (gradient φ x 1)^2 + (gradient φ x 2)^2 + (gradient φ x 3)^2 := by
  86  unfold gradient_squared
  87  apply gradient_squared_minkowski_sum
  88
  89/-- Field squared. -/
  90noncomputable def field_squared (φ : ScalarField) (x : Fin 4 → ℝ) : ℝ :=
  91  (φ.ψ x) ^ 2
  92
  93theorem field_squared_nonneg (φ : ScalarField) (x : Fin 4 → ℝ) :
  94  field_squared φ x ≥ 0 := by
  95  simp [field_squared]
  96  exact sq_nonneg _