lorentzian_from_det

formal source

 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
 193
 194/-- **Interval = spatial − temporal** (in signature −,+,+,+). -/
 195theorem interval_eq_spatial_minus_temporal (v : Displacement) :
 196    interval v = spatial_norm_sq v - temporal_sq v := by
 197  unfold interval spatial_norm_sq temporal_sq
 198  simp only [Fin.sum_univ_four]
 199  rw [η_00, η_11, η_22, η_33]; ring
 200
 201/-- **Light cone condition**: ds² = 0 iff |Δx|² = (Δt)². -/
 202theorem lightlike_iff_speed_c (v : Displacement) :
 203    interval v = 0 ↔ spatial_norm_sq v = temporal_sq v := by
 204  rw [interval_eq_spatial_minus_temporal]; constructor <;> intro h <;> linarith
 205
 206/-- **Timelike condition**: ds² < 0 iff |Δx|² < (Δt)². -/