interval

formal source

 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. -/
 207theorem timelike_iff_subluminal (v : Displacement) :
 208    interval v < 0 ↔ spatial_norm_sq v < temporal_sq v := by
 209  rw [interval_eq_spatial_minus_temporal]; constructor <;> intro h <;> linarith
 210
 211/-- **Spacelike condition**: ds² > 0 iff |Δx|² > (Δt)². -/
 212theorem spacelike_iff_superluminal (v : Displacement) :
 213    0 < interval v ↔ temporal_sq v < spatial_norm_sq v := by
 214  rw [interval_eq_spatial_minus_temporal]; constructor <;> intro h <;> linarith
 215