Timelike spatial lt time squared
Lorentz.Vector.timelike_spatial_lt_time_squared
Plain-language statement
For timelike vectors, the spatial norm squared is strictly less than the time component squared
Exact Lean statement
lemma timelike_spatial_lt_time_squared {d : ℕ} {v : Vector d}
(hv : causalCharacter v = .timeLike) :
⟪spatialPart v, spatialPart v⟫_ℝ < (timeComponent v)^2Formal artifact
Lean source
lemma timelike_spatial_lt_time_squared {d : ℕ} {v : Vector d} (hv : causalCharacter v = .timeLike) : ⟪spatialPart v, spatialPart v⟫_ℝ < (timeComponent v)^2 := by rw [timeLike_iff_norm_sq_pos, minkowskiProduct_toCoord] at hv simp only [PiLp.inner_apply] have h_time : timeComponent v = v (Sum.inl 0) := rfl simp [h_time, pow_two] have h_norm_pos : 0 < v (Sum.inl 0) * v (Sum.inl 0) - ∑ i, v (Sum.inr i) * v (Sum.inr i) := hv exact lt_of_sub_pos h_norm_pos- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean:110-119
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