Time Like iff time lt space
Lorentz.Vector.timeLike_iff_time_lt_space
Plain-language statement
A vector is timelike if and only if its time component squared is less than the sum of its spatial components squared
Exact Lean statement
lemma timeLike_iff_time_lt_space {d : ℕ} {v : Vector d} :
causalCharacter v = .timeLike ↔
⟪spatialPart v, spatialPart v⟫_ℝ < v (Sum.inl 0) * v (Sum.inl 0)Formal artifact
Lean source
lemma timeLike_iff_time_lt_space {d : ℕ} {v : Vector d} : causalCharacter v = .timeLike ↔ ⟪spatialPart v, spatialPart v⟫_ℝ < v (Sum.inl 0) * v (Sum.inl 0) := by constructor · intro h_timelike rw [timeLike_iff_norm_sq_pos, minkowskiProduct_toCoord] at h_timelike simp only [Fin.isValue, sub_pos] at h_timelike; exact h_timelike · intro h_time_lt_space rw [timeLike_iff_norm_sq_pos, minkowskiProduct_toCoord] simp only [Fin.isValue, sub_pos] exact h_time_lt_space- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean:89-99
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