Timelike time dominates space
Lorentz.Vector.timelike_time_dominates_space
Plain-language statement
For timeLike vectors in Minkowski space, the inner product of the spatial part is less than the square of the time component
Exact Lean statement
lemma timelike_time_dominates_space {d : ℕ} {v : Vector d}
(hv : causalCharacter v = .timeLike) :
⟪spatialPart v, spatialPart v⟫_ℝ < (timeComponent v) * (timeComponent v)Formal artifact
Lean source
lemma timelike_time_dominates_space {d : ℕ} {v : Vector d} (hv : causalCharacter v = .timeLike) : ⟪spatialPart v, spatialPart v⟫_ℝ < (timeComponent v) * (timeComponent v) := by rw [timeLike_iff_norm_sq_pos] at hv rw [minkowskiProduct_toCoord] at hv simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial] have h_spatial_sum : ∑ x, spatialPart v x * spatialPart v x = ∑ i, v (Sum.inr i) * v (Sum.inr i) := by simp only have h_time : timeComponent v = v (Sum.inl 0) := rfl rw [h_spatial_sum, h_time] have h_norm_pos : 0 < v (Sum.inl 0) * v (Sum.inl 0) - ∑ i, v (Sum.inr i) * v (Sum.inr i) := hv -- Rearrange the inequality have h : ∑ i, v (Sum.inr i) * v (Sum.inr i) < v (Sum.inl 0) * v (Sum.inl 0) := by exact lt_of_sub_pos h_norm_pos exact h- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean:48-65
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.