lorentzian_signature

formal source

 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
 177  rw [η_det]; norm_num
 178
 179/-! ## §5  The Spacetime Interval and Causal Classification -/
 180
 181/-- A spacetime displacement: 4-vector (Δt, Δx₁, Δx₂, Δx₃). -/
 182abbrev Displacement := Fin 4 → ℝ
 183
 184/-- The spacetime interval for a displacement vector. -/
 185def interval (v : Displacement) : ℝ := ∑ i : Fin 4, η i i * v i ^ 2
 186
 187/-- The spatial norm squared. -/
 188def spatial_norm_sq (v : Displacement) : ℝ :=
 189  v (1 : Fin 4) ^ 2 + v (2 : Fin 4) ^ 2 + v (3 : Fin 4) ^ 2
 190
 191/-- The temporal component squared. -/
 192def temporal_sq (v : Displacement) : ℝ := v (0 : Fin 4) ^ 2