 109  abel
 110
 111/-- Squared gradient helper for Minkowski metric. -/
 112lemma gradient_squared_minkowski_sum (grad : Fin 4 → ℝ) (x : Fin 4 → ℝ) :
 113    Finset.sum Finset.univ (fun μ => Finset.sum Finset.univ (fun ν =>
 114      inverse_metric minkowski_tensor x (fun i => if (i : ℕ) = 0 then μ else ν) (fun _ => 0) *
 115      grad μ * grad ν)) =
 116    -(grad 0)^2 + (grad 1)^2 + (grad 2)^2 + (grad 3)^2 := by
 117  rw [finset_sum_fin_4]
 118  simp only [finset_sum_fin_4, inverse_minkowski_apply]
 119  simp
 120  ring
 121
 122end Geometry
 123end Relativity
 124end IndisputableMonolith

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