negative_eigenvalue_count

formal source

 143  · right; exact η_33
 144
 145/-- Count of negative diagonal entries = 1 (temporal). -/
 146theorem negative_eigenvalue_count :
 147    (Finset.univ.filter (fun i : Fin 4 => η i i < 0)).card = 1 := by
 148  suffices h : Finset.univ.filter (fun i : Fin 4 => η i i < 0) = {(0 : Fin 4)} by
 149    rw [h]; simp
 150  ext i; fin_cases i <;> simp [Finset.mem_filter, η_00, η_11, η_22, η_33]
 151
 152/-- Count of positive diagonal entries = 3 (spatial). -/
 153theorem positive_eigenvalue_count :
 154    (Finset.univ.filter (fun i : Fin 4 => 0 < η i i)).card = 3 := by
 155  suffices h : Finset.univ.filter (fun i : Fin 4 => 0 < η i i) =
 156    {(1 : Fin 4), (2 : Fin 4), (3 : Fin 4)} by
 157    rw [h]; decide
 158  ext i; fin_cases i <;>
 159    simp [Finset.mem_filter, η_00, η_11, η_22, η_33, Fin.ext_iff]
 160
 161/-- **SE-004: The metric signature is (1, 3) — Lorentzian.** -/
 162theorem lorentzian_signature :
 163    (Finset.univ.filter (fun i : Fin 4 => η i i < 0)).card = temporal_dim ∧
 164    (Finset.univ.filter (fun i : Fin 4 => 0 < η i i)).card = spatial_dim :=
 165  ⟨negative_eigenvalue_count, positive_eigenvalue_count⟩
 166
 167/-- The trace of the metric: Tr(η) = −1 + 1 + 1 + 1 = 2. -/
 168theorem η_trace : ∑ i : Fin 4, η i i = 2 := by
 169  simp only [Fin.sum_univ_four]; rw [η_00, η_11, η_22, η_33]; norm_num
 170
 171/-- The determinant of the metric: det(η) = −1. -/
 172theorem η_det : ∏ i : Fin 4, η i i = -1 := by
 173  simp only [Fin.prod_univ_four]; rw [η_00, η_11, η_22, η_33]; norm_num
 174
 175/-- Negative determinant confirms Lorentzian signature. -/
 176theorem lorentzian_from_det : ∏ i : Fin 4, η i i < 0 := by