refactor: Remove double empty lines
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@ -17,7 +17,6 @@ We define
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-/
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namespace SpaceTime
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open Matrix
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open TensorProduct
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@ -34,7 +33,6 @@ lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
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erw [mem_skewAdjointMatricesLieSubalgebra] at h1
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simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
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lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
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(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
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erw [mem_skewAdjointMatricesLieSubalgebra]
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@ -58,7 +56,6 @@ lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
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rw [minkowskiMatrix.sq]
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all_goals noncomm_ring
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lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 := by
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have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2
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simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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@ -67,22 +64,18 @@ lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 :=
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simpa using h
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simpa using h
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lemma time_comps (Λ : lorentzAlgebra) (i : Fin 3) :
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Λ.1 (Sum.inr i) (Sum.inl 0) = Λ.1 (Sum.inl 0) (Sum.inr i) := by
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simpa only [Fin.isValue, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
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transpose_apply, Sum.elim_inr, mul_neg, mul_one, diagonal_neg, diagonal_mul, Sum.elim_inl,
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neg_mul, one_mul, neg_inj] using congrArg (fun M ↦ M (Sum.inl 0) (Sum.inr i)) $ mem_iff.mp Λ.2
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lemma space_comps (Λ : lorentzAlgebra) (i j : Fin 3) :
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Λ.1 (Sum.inr i) (Sum.inr j) = - Λ.1 (Sum.inr j) (Sum.inr i) := by
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simpa only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_neg, diagonal_mul,
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Sum.elim_inr, neg_neg, one_mul, mul_diagonal, transpose_apply, mul_neg, mul_one] using
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(congrArg (fun M ↦ M (Sum.inr i) (Sum.inr j)) $ mem_iff.mp Λ.2).symm
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end lorentzAlgebra
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@[simps!]
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@ -104,5 +97,4 @@ instance spaceTimeAsLieModule : LieModule ℝ lorentzAlgebra (LorentzVector 3) w
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simp [Bracket.bracket]
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rw [mulVec_smul]
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end SpaceTime
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