refactor: Major refactor of Lorentz group
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6 changed files with 563 additions and 931 deletions
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@ -4,6 +4,7 @@ Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.Metric
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import HepLean.SpaceTime.FourVelocity
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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/-!
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# The Lorentz Group
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@ -112,7 +113,7 @@ def lorentzGroup : Subgroup (GL (Fin 4) ℝ) where
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instance : TopologicalGroup lorentzGroup :=
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Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
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@[simps!]
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def PreservesηLin.liftLor {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) :
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lorentzGroup := ⟨liftGL h, h⟩
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@ -143,62 +144,6 @@ theorem isClosed_of_GL4 : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
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rw [kernalMap_kernal_eq_lorentzGroup]
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exact continuous_iff_isClosed.mp kernalMap.2 {1} isClosed_singleton
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section Relations
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/-- The first column of a lorentz matrix. -/
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@[simp]
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def fstCol (Λ : lorentzGroup) : spaceTime := fun i => Λ.1 i 0
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lemma ηLin_fstCol (Λ : lorentzGroup) : ηLin (fstCol Λ) (fstCol Λ) = 1 := by
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rw [ηLin_expand]
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have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp ((mem_iff Λ.1).mp Λ.2)) 0) 0
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simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
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not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
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zero_add, one_apply_eq] at h00
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simp only [η, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
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vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two,
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Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
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rw [← h00]
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simp only [fstCol, Fin.isValue]
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ring
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lemma zero_component (x : { x : spaceTime // ηLin x x = 1}) :
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x.1 0 ^ 2 = 1 + ‖x.1.space‖ ^ 2 := by
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sorry
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/-- The space-like part of the first row of of a Lorentz matrix. -/
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@[simp]
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def fstSpaceRow (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 0 i.succ
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/-- The space-like part of the first column of of a Lorentz matrix. -/
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@[simp]
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def fstSpaceCol (Λ : lorentzGroup) : EuclideanSpace ℝ (Fin 3) := fun i => Λ.1 i.succ 0
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lemma fstSpaceRow_transpose (Λ : lorentzGroup) : fstSpaceRow (transpose Λ) = fstSpaceCol Λ := by
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rfl
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lemma fstSpaceCol_transpose (Λ : lorentzGroup) : fstSpaceCol (transpose Λ) = fstSpaceRow Λ := by
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rfl
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lemma fst_col_normalized (Λ : lorentzGroup) :
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(Λ.1 0 0) ^ 2 - ‖fstSpaceCol Λ‖ ^ 2 = 1 := by
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rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
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simp
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rw [show Fin.succ 2 = 3 by rfl]
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have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp ((mem_iff Λ.1).mp Λ.2)) 0) 0
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simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
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not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
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zero_add, one_apply_eq] at h00
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simp only [η, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
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vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two,
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Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
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rw [← h00]
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ring
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lemma fst_row_normalized
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end Relations
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end lorentzGroup
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end spaceTime
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170
HepLean/SpaceTime/LorentzGroup/Boosts.lean
Normal file
170
HepLean/SpaceTime/LorentzGroup/Boosts.lean
Normal file
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@ -0,0 +1,170 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzGroup.Orthochronous
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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import Mathlib.Topology.Constructions
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/-!
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# Boosts
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This file defines those Lorentz which are boosts.
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We first define generalised boosts, which are restricted lorentz transforamations taking
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a four velocity `u` to a four velcoity `v`.
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A boost is the speical case of a generalised boost when `u = stdBasis 0`.
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## TODO
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- Show that generalised boosts are in the restircted Lorentz group.
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- Define boosts.
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## References
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- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
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https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
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-/
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noncomputable section
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namespace spaceTime
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namespace lorentzGroup
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open FourVelocity
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def genBoostAux₁ (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime where
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toFun x := (2 * ηLin x u) • v.1.1
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map_add' x y := by
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simp only [map_add, LinearMap.add_apply]
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rw [mul_add, add_smul]
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map_smul' c x := by
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simp only [LinearMapClass.map_smul, LinearMap.smul_apply, smul_eq_mul,
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RingHom.id_apply]
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rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
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def genBoostAux₂ (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime where
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toFun x := - (ηLin x (u + v) / (1 + ηLin u v)) • (u + v)
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map_add' x y := by
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simp only
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rw [ηLin.map_add, div_eq_mul_one_div]
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rw [show (ηLin x + ηLin y) (↑u + ↑v) = ηLin x (u+v) + ηLin y (u+v) from rfl]
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rw [add_mul, neg_add, add_smul, ← div_eq_mul_one_div, ← div_eq_mul_one_div]
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map_smul' c x := by
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simp only
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rw [ηLin.map_smul]
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rw [show (c • ηLin x) (↑u + ↑v) = c * ηLin x (u+v) from rfl]
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rw [mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
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rfl
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def genBoost (u v : FourVelocity) : spaceTime →ₗ[ℝ] spaceTime :=
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LinearMap.id + genBoostAux₁ u v + genBoostAux₂ u v
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namespace genBoost
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lemma self (u : FourVelocity) : genBoost u u = LinearMap.id := by
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ext x
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simp [genBoost]
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rw [add_assoc]
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rw [@add_right_eq_self]
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rw [@add_eq_zero_iff_eq_neg]
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rw [genBoostAux₁, genBoostAux₂]
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simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]
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rw [← add_smul]
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apply congr
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apply congrArg
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repeat rw [u.1.2]
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field_simp
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ring
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rfl
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def toMatrix (u v : FourVelocity) : Matrix (Fin 4) (Fin 4) ℝ :=
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LinearMap.toMatrix stdBasis stdBasis (genBoost u v)
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lemma toMatrix_mulVec (u v : FourVelocity) (x : spaceTime) :
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(toMatrix u v).mulVec x = genBoost u v x :=
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LinearMap.toMatrix_mulVec_repr stdBasis stdBasis (genBoost u v) x
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lemma toMatrix_apply (u v : FourVelocity) (i j : Fin 4) :
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(toMatrix u v) i j =
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η i i * (ηLin (stdBasis i) (stdBasis j) + 2 * ηLin (stdBasis j) u * ηLin (stdBasis i) v -
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ηLin (stdBasis i) (u + v) * ηLin (stdBasis j) (u + v) / (1 + ηLin u v)) := by
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rw [ηLin_matrix_stdBasis' j i (toMatrix u v), toMatrix_mulVec]
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simp only [genBoost, genBoostAux₁, genBoostAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply,
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LinearMap.id_apply, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, smul_eq_mul,
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map_neg, mul_eq_mul_left_iff]
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apply Or.inl
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ring
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lemma toMatrix_continuous (u : FourVelocity) : Continuous (toMatrix u) := by
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refine continuous_matrix ?_
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intro i j
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simp only [toMatrix_apply]
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refine Continuous.comp' (continuous_mul_left (η i i)) ?hf
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refine Continuous.sub ?_ ?_
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refine Continuous.comp' (continuous_add_left ((ηLin (stdBasis i)) (stdBasis j))) ?_
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refine Continuous.comp' (continuous_mul_left (2 * (ηLin (stdBasis j)) ↑u)) ?_
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exact η_continuous _
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refine Continuous.mul ?_ ?_
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refine Continuous.mul ?_ ?_
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simp only [(ηLin _).map_add]
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refine Continuous.comp' ?_ ?_
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exact continuous_add_left ((ηLin (stdBasis i)) ↑u)
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exact η_continuous _
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simp only [(ηLin _).map_add]
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refine Continuous.comp' ?_ ?_
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exact continuous_add_left _
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exact η_continuous _
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refine Continuous.inv₀ ?_ ?_
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refine Continuous.comp' (continuous_add_left 1) ?_
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exact η_continuous _
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exact fun x => one_plus_ne_zero u x
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lemma toMatrix_PreservesηLin (u v : FourVelocity) : PreservesηLin (toMatrix u v) := by
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intro x y
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rw [toMatrix_mulVec, toMatrix_mulVec, genBoost, genBoostAux₁, genBoostAux₂]
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have h1 : (1 + (ηLin ↑u) ↑v) ≠ 0 := one_plus_ne_zero u v
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simp only [map_add, smul_add, neg_smul, LinearMap.add_apply, LinearMap.id_coe,
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id_eq, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, map_neg, LinearMap.smul_apply,
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smul_eq_mul, LinearMap.neg_apply]
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field_simp
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rw [u.1.2, v.1.2, ηLin_symm v u, ηLin_symm u y, ηLin_symm v y]
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ring
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def toLorentz (u v : FourVelocity) : lorentzGroup :=
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PreservesηLin.liftLor (toMatrix_PreservesηLin u v)
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lemma toLorentz_continuous (u : FourVelocity) : Continuous (toLorentz u) := by
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refine Continuous.subtype_mk ?_ fun x => (PreservesηLin.liftLor (toMatrix_PreservesηLin u x)).2
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refine Units.continuous_iff.mpr (And.intro ?_ ?_)
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exact toMatrix_continuous u
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change Continuous fun x => (η * (toMatrix u x).transpose * η)
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refine Continuous.matrix_mul ?_ continuous_const
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refine Continuous.matrix_mul continuous_const ?_
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exact Continuous.matrix_transpose (toMatrix_continuous u)
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lemma toLorentz_joined_to_1 (u v : FourVelocity) : Joined 1 (toLorentz u v) := by
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obtain ⟨f, _⟩ := isPathConnected_FourVelocity.joinedIn u trivial v trivial
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use ContinuousMap.comp ⟨toLorentz u, toLorentz_continuous u⟩ f
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· simp only [ContinuousMap.toFun_eq_coe, ContinuousMap.comp_apply, ContinuousMap.coe_coe,
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Path.source, ContinuousMap.coe_mk]
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rw [@Subtype.ext_iff, toLorentz, PreservesηLin.liftLor]
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refine Units.val_eq_one.mp ?_
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simp [PreservesηLin.liftGL, toMatrix, self u]
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· simp
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end genBoost
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end lorentzGroup
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end spaceTime
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end
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@ -1,293 +0,0 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.SpaceTime.LorentzGroup
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import Mathlib.GroupTheory.SpecificGroups.KleinFour
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import Mathlib.Topology.Constructions
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/-!
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# Connectedness of the restricted Lorentz group
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In this file we provide a proof that the restricted Lorentz group is connected.
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## References
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- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
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https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
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-/
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noncomputable section
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namespace spaceTime
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def stepFunction : ℝ → ℝ := fun t =>
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if t < -1 then -1 else
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if t < 1 then t else 1
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lemma stepFunction_continuous : Continuous stepFunction := by
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apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_id continuous_const)
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all_goals
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intro a ha
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rw [@Set.Iio_def, @frontier_Iio, @Set.mem_singleton_iff] at ha
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rw [ha]
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simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
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def lorentzDet (Λ : lorentzGroup) : ({-1, 1} : Set ℝ) :=
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⟨Λ.1.1.det, by
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simp
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exact Or.symm (lorentzGroup.det_eq_one_or_neg_one _)⟩
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lemma lorentzDet_continuous : Continuous lorentzDet := by
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refine Continuous.subtype_mk ?_ fun x => lorentzDet.proof_1 x
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exact Continuous.matrix_det $
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Continuous.comp' Units.continuous_val continuous_subtype_val
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instance set_discrete : DiscreteTopology ({-1, 1} : Set ℝ) := discrete_of_t1_of_finite
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lemma lorentzDet_eq_if_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') :
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Λ.1.1.det = Λ'.1.1.det := by
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obtain ⟨f, hf, hf2⟩ := h
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have h1 : Joined (lorentzDet Λ) (lorentzDet Λ') :=
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⟨ContinuousMap.comp ⟨lorentzDet, lorentzDet_continuous⟩ f, congrArg lorentzDet hf,
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congrArg lorentzDet hf2⟩
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obtain ⟨g, hg1, hg2⟩ := h1
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have h2 := (@IsPreconnected.subsingleton ({-1, 1} : Set ℝ) _ _ _ (isPreconnected_range g.2))
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(Set.mem_range_self 0) (Set.mem_range_self 1)
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rw [hg1, hg2] at h2
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simpa [lorentzDet] using h2
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lemma det_on_connected_component {Λ Λ' : lorentzGroup} (h : Λ' ∈ connectedComponent Λ) :
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Λ.1.1.det = Λ'.1.1.det := by
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obtain ⟨s, hs, hΛ'⟩ := h
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let f : ContinuousMap s ({-1, 1} : Set ℝ) :=
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ContinuousMap.restrict s ⟨lorentzDet, lorentzDet_continuous⟩
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haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
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have hf := (@IsPreconnected.subsingleton ({-1, 1} : Set ℝ) _ _ _ (isPreconnected_range f.2))
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(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
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simpa [lorentzDet, f] using hf
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lemma det_of_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') : Λ.1.1.det = Λ'.1.1.det :=
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det_on_connected_component $ pathComponent_subset_component _ h
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namespace restrictedLorentzGroup
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/-- The set of spacetime points such that the time element is positive, and which are
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normalised to 1. -/
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def P : Set (spaceTime) := {x | x 0 > 0 ∧ ηLin x x = 1}
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lemma P_time_comp (x : P) : x.1 0 = √(1 + ‖x.1.space‖ ^ 2) := by
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symm
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rw [Real.sqrt_eq_cases]
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apply Or.inl
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refine And.intro ?_ (le_of_lt x.2.1)
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rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
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simp only [Fin.isValue, space, Matrix.cons_val_zero, RCLike.inner_apply, conj_trivial,
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Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
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Matrix.tail_cons]
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have h1 := x.2.2
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rw [ηLin_expand] at h1
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linear_combination h1
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def oneInP : P := ⟨![1, 0, 0, 0], by simp, by
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rw [ηLin_expand]
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simp⟩
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def pathToOneInP (u : P) : Path oneInP u where
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toFun t := ⟨![√(1 + t ^ 2 * ‖u.1.space‖ ^ 2), t * u.1 1, t * u.1 2, t * u.1 3],
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by
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simp
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refine Right.add_pos_of_pos_of_nonneg (Real.zero_lt_one) $
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mul_nonneg (sq_nonneg _) (sq_nonneg _)
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, by
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rw [ηLin_expand]
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simp
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rw [Real.mul_self_sqrt, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
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simp
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ring
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refine Right.add_nonneg (zero_le_one' ℝ) $
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mul_nonneg (sq_nonneg _) (sq_nonneg _) ⟩
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continuous_toFun := by
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refine Continuous.subtype_mk ?_ _
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apply (continuous_pi_iff).mpr
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intro i
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fin_cases i
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<;> continuity
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source' := by
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simp
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rfl
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target' := by
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simp
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refine SetCoe.ext ?_
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funext i
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fin_cases i
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simp
|
||||
exact (P_time_comp u).symm
|
||||
all_goals rfl
|
||||
|
||||
|
||||
|
||||
/-- P is a topological space with the subset topology. -/
|
||||
instance : TopologicalSpace P := instTopologicalSpaceSubtype
|
||||
|
||||
lemma P_path_connected : IsPathConnected (@Set.univ P) := by
|
||||
use oneInP
|
||||
apply And.intro trivial ?_
|
||||
intro y _
|
||||
use pathToOneInP y
|
||||
simp only [Set.mem_univ, implies_true]
|
||||
|
||||
|
||||
|
||||
lemma P_abs_space_lt_time_comp (x : P) : ‖x.1.space‖ < x.1 0 := by
|
||||
rw [P_time_comp, Real.lt_sqrt]
|
||||
exact lt_one_add (‖(x.1).space‖ ^ 2)
|
||||
exact norm_nonneg (x.1).space
|
||||
|
||||
lemma η_P_P_pos (u v : P) : 0 < ηLin u v :=
|
||||
lt_of_lt_of_le (sub_pos.mpr (mul_lt_mul_of_nonneg_of_pos
|
||||
(P_abs_space_lt_time_comp u) ((P_abs_space_lt_time_comp v).le)
|
||||
(norm_nonneg (u.1).space) v.2.1)) (ηLin_ge_sub_norm u v)
|
||||
|
||||
lemma η_P_continuous (u : spaceTime) : Continuous (fun (a : P) => ηLin u a) := by
|
||||
simp only [ηLin_expand]
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.add ?_ ?_
|
||||
refine Continuous.comp' (continuous_mul_left _) ?_
|
||||
apply (continuous_pi_iff).mp
|
||||
exact continuous_subtype_val
|
||||
all_goals apply Continuous.neg
|
||||
all_goals apply Continuous.comp' (continuous_mul_left _)
|
||||
all_goals apply (continuous_pi_iff).mp
|
||||
all_goals exact continuous_subtype_val
|
||||
|
||||
|
||||
lemma one_plus_η_P_P (u v : P) : 1 + ηLin u v ≠ 0 := by
|
||||
linarith [η_P_P_pos u v]
|
||||
|
||||
def φAux₁ (u v : P) : spaceTime →ₗ[ℝ] spaceTime where
|
||||
toFun x := (2 * ηLin x u) • v
|
||||
map_add' x y := by
|
||||
simp only [map_add, LinearMap.add_apply]
|
||||
rw [mul_add, add_smul]
|
||||
map_smul' c x := by
|
||||
simp only [LinearMapClass.map_smul, LinearMap.smul_apply, smul_eq_mul,
|
||||
RingHom.id_apply]
|
||||
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
|
||||
|
||||
|
||||
def φAux₂ (u v : P) : spaceTime →ₗ[ℝ] spaceTime where
|
||||
toFun x := - (ηLin x (u + v) / (1 + ηLin u v)) • (u + v)
|
||||
map_add' x y := by
|
||||
simp only
|
||||
rw [ηLin.map_add, div_eq_mul_one_div]
|
||||
rw [show (ηLin x + ηLin y) (↑u + ↑v) = ηLin x (u+v) + ηLin y (u+v) from rfl]
|
||||
rw [add_mul, neg_add, add_smul, ← div_eq_mul_one_div, ← div_eq_mul_one_div]
|
||||
map_smul' c x := by
|
||||
simp only
|
||||
rw [ηLin.map_smul]
|
||||
rw [show (c • ηLin x) (↑u + ↑v) = c * ηLin x (u+v) from rfl]
|
||||
rw [mul_div_assoc, neg_mul_eq_mul_neg, smul_smul]
|
||||
rfl
|
||||
|
||||
def φ (u v : P) : spaceTime →ₗ[ℝ] spaceTime := LinearMap.id + φAux₁ u v + φAux₂ u v
|
||||
|
||||
lemma φ_u_u_eq_id (u : P) : φ u u = LinearMap.id := by
|
||||
ext x
|
||||
simp [φ]
|
||||
rw [add_assoc]
|
||||
rw [@add_right_eq_self]
|
||||
rw [@add_eq_zero_iff_eq_neg]
|
||||
rw [φAux₁, φAux₂]
|
||||
simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]
|
||||
rw [← add_smul]
|
||||
apply congr
|
||||
apply congrArg
|
||||
repeat rw [u.2.2]
|
||||
field_simp
|
||||
ring
|
||||
rfl
|
||||
|
||||
def φMat (u v : P) : Matrix (Fin 4) (Fin 4) ℝ := LinearMap.toMatrix stdBasis stdBasis (φ u v)
|
||||
|
||||
lemma φMat_mulVec (u v : P) (x : spaceTime) : (φMat u v).mulVec x = φ u v x :=
|
||||
LinearMap.toMatrix_mulVec_repr stdBasis stdBasis (φ u v) x
|
||||
|
||||
|
||||
lemma φMat_element (u v : P) (i j : Fin 4) : (φMat u v) i j =
|
||||
η i i * (ηLin (stdBasis i) (stdBasis j) + 2 * ηLin (stdBasis j) u * ηLin (stdBasis i) v -
|
||||
ηLin (stdBasis i) (u + v) * ηLin (stdBasis j) (u + v) / (1 + ηLin u v)) := by
|
||||
rw [ηLin_matrix_stdBasis' j i (φMat u v), φMat_mulVec]
|
||||
simp only [φ, φAux₁, φAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply, LinearMap.id_apply,
|
||||
LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, smul_eq_mul, map_neg,
|
||||
mul_eq_mul_left_iff]
|
||||
apply Or.inl
|
||||
ring
|
||||
|
||||
|
||||
lemma φMat_continuous (u : P) : Continuous (φMat u) := by
|
||||
refine continuous_matrix ?_
|
||||
intro i j
|
||||
simp only [φMat_element]
|
||||
refine Continuous.comp' (continuous_mul_left (η i i)) ?hf
|
||||
refine Continuous.sub ?_ ?_
|
||||
refine Continuous.comp' (continuous_add_left ((ηLin (stdBasis i)) (stdBasis j))) ?_
|
||||
refine Continuous.comp' (continuous_mul_left (2 * (ηLin (stdBasis j)) ↑u)) ?_
|
||||
exact η_P_continuous _
|
||||
refine Continuous.mul ?_ ?_
|
||||
refine Continuous.mul ?_ ?_
|
||||
simp only [(ηLin _).map_add]
|
||||
refine Continuous.comp' ?_ ?_
|
||||
exact continuous_add_left ((ηLin (stdBasis i)) ↑u)
|
||||
exact η_P_continuous _
|
||||
simp only [(ηLin _).map_add]
|
||||
refine Continuous.comp' ?_ ?_
|
||||
exact continuous_add_left _
|
||||
exact η_P_continuous _
|
||||
refine Continuous.inv₀ ?_ ?_
|
||||
refine Continuous.comp' (continuous_add_left 1) ?_
|
||||
exact η_P_continuous _
|
||||
exact fun x => one_plus_η_P_P u x
|
||||
|
||||
|
||||
lemma φMat_in_lorentz (u v : P) (x y : spaceTime) :
|
||||
ηLin ((φMat u v).mulVec x) ((φMat u v).mulVec y) = ηLin x y := by
|
||||
rw [φMat_mulVec, φMat_mulVec, φ, φAux₁, φAux₂]
|
||||
have h1 : (1 + (ηLin ↑u) ↑v) ≠ 0 := one_plus_η_P_P u v
|
||||
simp only [map_add, smul_add, neg_smul, LinearMap.add_apply, LinearMap.id_coe,
|
||||
id_eq, LinearMap.coe_mk, AddHom.coe_mk, LinearMapClass.map_smul, map_neg, LinearMap.smul_apply,
|
||||
smul_eq_mul, LinearMap.neg_apply]
|
||||
field_simp
|
||||
simp only [v.2.2, mul_one, u.2.2]
|
||||
rw [ηLin_symm v u, ηLin_symm u y, ηLin_symm v y]
|
||||
ring
|
||||
|
||||
def φLor (u v : P) : lorentzGroup :=
|
||||
preserveηLinLorentzLift (φMat_in_lorentz u v)
|
||||
|
||||
lemma φLor_continuous (u : P) : Continuous (φLor u) := by
|
||||
refine Continuous.subtype_mk ?_ fun x => (preserveηLinLorentzLift (φMat_in_lorentz u x)).2
|
||||
refine Units.continuous_iff.mpr (And.intro ?_ ?_)
|
||||
exact φMat_continuous u
|
||||
change Continuous fun x => (η * (φMat u x).transpose * η)
|
||||
refine Continuous.matrix_mul ?_ continuous_const
|
||||
refine Continuous.matrix_mul continuous_const ?_
|
||||
exact Continuous.matrix_transpose (φMat_continuous u)
|
||||
|
||||
|
||||
lemma φLor_joined_to_identity (u v : P) : Joined 1 (φLor u v) := by
|
||||
obtain ⟨f, _⟩ := P_path_connected.joinedIn u trivial v trivial
|
||||
use ContinuousMap.comp ⟨φLor u, φLor_continuous u⟩ f
|
||||
· simp only [ContinuousMap.toFun_eq_coe, ContinuousMap.comp_apply, ContinuousMap.coe_coe,
|
||||
Path.source, ContinuousMap.coe_mk]
|
||||
rw [@Subtype.ext_iff, φLor, preserveηLinLorentzLift]
|
||||
refine Units.val_eq_one.mp ?_
|
||||
simp [preserveηLinGLnLift, φMat, φ_u_u_eq_id u ]
|
||||
· simp
|
||||
|
||||
end restrictedLorentzGroup
|
||||
|
||||
|
||||
|
||||
end spaceTime
|
||||
end
|
||||
|
|
@ -3,6 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.SpaceTime.FourVelocity
|
||||
import HepLean.SpaceTime.LorentzGroup.Proper
|
||||
import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
||||
/-!
|
||||
|
|
@ -11,6 +12,10 @@ import Mathlib.GroupTheory.SpecificGroups.KleinFour
|
|||
We define the give a series of lemmas related to the orthochronous property of lorentz
|
||||
matrices.
|
||||
|
||||
## TODO
|
||||
|
||||
- Prove topological properties.
|
||||
|
||||
-/
|
||||
|
||||
|
||||
|
|
@ -24,100 +29,164 @@ open Complex
|
|||
open ComplexConjugate
|
||||
|
||||
namespace lorentzGroup
|
||||
open PreFourVelocity
|
||||
|
||||
/-- The determinant of a member of the lorentz group is `1` or `-1`. -/
|
||||
lemma det_eq_one_or_neg_one (Λ : lorentzGroup) : Λ.1.1.det = 1 ∨ Λ.1.1.det = -1 := by
|
||||
simpa [← sq, det_one, det_mul, det_mul, det_mul, det_transpose, det_η] using
|
||||
(congrArg det ((PreservesηLin.iff_matrix' Λ.1).mp ((mem_iff Λ.1).mp Λ.2)))
|
||||
/-- The first column of a lorentz matrix as a `PreFourVelocity`. -/
|
||||
@[simp]
|
||||
def fstCol (Λ : lorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
|
||||
rw [mem_PreFourVelocity_iff, ηLin_expand]
|
||||
simp only [Fin.isValue, stdBasis_mulVec]
|
||||
have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp ((mem_iff Λ.1).mp Λ.2)) 0) 0
|
||||
simp only [Fin.isValue, mul_apply, transpose_apply, Fin.sum_univ_four, ne_eq, zero_ne_one,
|
||||
not_false_eq_true, η_off_diagonal, zero_mul, add_zero, Fin.reduceEq, one_ne_zero, mul_zero,
|
||||
zero_add, one_apply_eq] at h00
|
||||
simp only [η, Fin.isValue, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
|
||||
vecCons_const, one_mul, mul_one, cons_val_one, head_cons, mul_neg, neg_mul, cons_val_two,
|
||||
Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons, cons_val_three, head_fin_const] at h00
|
||||
rw [← h00]
|
||||
ring⟩
|
||||
|
||||
def IsOrthochronous (Λ : lorentzGroup) : Prop := 0 ≤ Λ.1 0 0
|
||||
|
||||
lemma IsOrthochronous_iff_transpose (Λ : lorentzGroup) :
|
||||
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by
|
||||
simp [IsOrthochronous]
|
||||
simp [transpose]
|
||||
|
||||
lemma IsOrthochronous_iff_fstCol_IsFourVelocity (Λ : lorentzGroup) :
|
||||
IsOrthochronous Λ ↔ IsFourVelocity (fstCol Λ) := by
|
||||
simp [IsOrthochronous, IsFourVelocity]
|
||||
rw [stdBasis_mulVec]
|
||||
|
||||
def mapZeroZeroComp : C(lorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0, by
|
||||
refine Continuous.matrix_elem ?_ 0 0
|
||||
refine Continuous.comp' Units.continuous_val continuous_subtype_val⟩
|
||||
|
||||
|
||||
def stepFunction : ℝ → ℝ := fun t =>
|
||||
if t ≤ -1 then -1 else
|
||||
if 1 ≤ t then 1 else t
|
||||
|
||||
lemma stepFunction_continuous : Continuous stepFunction := by
|
||||
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
|
||||
<;> intro a ha
|
||||
rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
|
||||
rw [ha]
|
||||
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
|
||||
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
|
||||
rw [if_neg h1]
|
||||
rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
|
||||
simp [ha]
|
||||
|
||||
def orthchroMapReal : C(lorentzGroup, ℝ) := ContinuousMap.comp
|
||||
⟨stepFunction, stepFunction_continuous⟩ mapZeroZeroComp
|
||||
|
||||
lemma orthchroMapReal_on_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
|
||||
orthchroMapReal Λ = 1 := by
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
|
||||
simp only [IsFourVelocity] at h
|
||||
rw [zero_nonneg_iff] at h
|
||||
simp [stdBasis_mulVec] at h
|
||||
have h1 : ¬ Λ.1 0 0 ≤ (-1 : ℝ) := by linarith
|
||||
change stepFunction (Λ.1 0 0) = 1
|
||||
rw [stepFunction, if_neg h1, if_pos h]
|
||||
|
||||
|
||||
lemma orthchroMapReal_on_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
|
||||
orthchroMapReal Λ = - 1 := by
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
|
||||
rw [not_IsFourVelocity_iff, zero_nonpos_iff] at h
|
||||
simp only [fstCol, Fin.isValue, stdBasis_mulVec] at h
|
||||
change stepFunction (Λ.1 0 0) = - 1
|
||||
rw [stepFunction, if_pos h]
|
||||
|
||||
lemma orthchroMapReal_minus_one_or_one (Λ : lorentzGroup) :
|
||||
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
|
||||
by_cases h : IsOrthochronous Λ
|
||||
apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
|
||||
apply Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
|
||||
|
||||
local notation "ℤ₂" => Multiplicative (ZMod 2)
|
||||
|
||||
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
|
||||
def orthchroMap : C(lorentzGroup, ℤ₂) :=
|
||||
ContinuousMap.comp coeForℤ₂ {
|
||||
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
|
||||
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
|
||||
|
||||
instance : DiscreteTopology ℤ₂ := by
|
||||
exact forall_open_iff_discrete.mp fun _ => trivial
|
||||
lemma orthchroMap_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
|
||||
orthchroMap Λ = 1 := by
|
||||
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
|
||||
|
||||
instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
|
||||
lemma orthchroMap_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
|
||||
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
|
||||
simp [orthchroMap, orthchroMapReal_on_not_IsOrthochronous h]
|
||||
rfl
|
||||
|
||||
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
|
||||
@[simps!]
|
||||
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
|
||||
toFun x := if x = ⟨1, by simp⟩ then (Additive.toMul 0) else (Additive.toMul (1 : ZMod 2))
|
||||
continuous_toFun := by
|
||||
haveI : DiscreteTopology ({-1, 1} : Set ℝ) := discrete_of_t1_of_finite
|
||||
exact continuous_of_discreteTopology
|
||||
lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
|
||||
(Λ * Λ').1 0 0 = (fstCol (transpose Λ)).1 0 * (fstCol Λ').1 0 +
|
||||
⟪(fstCol (transpose Λ)).1.space, (fstCol Λ').1.space⟫_ℝ := by
|
||||
rw [@Subgroup.coe_mul, @GeneralLinearGroup.coe_mul, mul_apply, Fin.sum_univ_four]
|
||||
rw [@PiLp.inner_apply, Fin.sum_univ_three]
|
||||
simp [transpose, stdBasis_mulVec]
|
||||
ring
|
||||
|
||||
/-- The continuous map taking a lorentz matrix to its determinant. -/
|
||||
def detContinuous : C(lorentzGroup, ℤ₂) :=
|
||||
ContinuousMap.comp coeForℤ₂ {
|
||||
toFun := fun Λ => ⟨Λ.1.1.det, Or.symm (lorentzGroup.det_eq_one_or_neg_one _)⟩,
|
||||
continuous_toFun := by
|
||||
refine Continuous.subtype_mk ?_ _
|
||||
exact Continuous.matrix_det $
|
||||
Continuous.comp' Units.continuous_val continuous_subtype_val}
|
||||
lemma mul_othchron_of_othchron_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous, zero_zero_mul]
|
||||
exact euclid_norm_IsFourVelocity_IsFourVelocity h h'
|
||||
|
||||
lemma detContinuous_eq_iff_det_eq (Λ Λ' : lorentzGroup) :
|
||||
detContinuous Λ = detContinuous Λ' ↔ Λ.1.1.det = Λ'.1.1.det := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
simp [detContinuous] at h
|
||||
cases' det_eq_one_or_neg_one Λ with h1 h1
|
||||
<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
|
||||
<;> simp_all [h1, h2, h]
|
||||
rw [← toMul_zero, @Equiv.apply_eq_iff_eq] at h
|
||||
change (0 : Fin 2) = (1 : Fin 2) at h
|
||||
simp only [Fin.isValue, zero_ne_one] at h
|
||||
nth_rewrite 2 [← toMul_zero] at h
|
||||
rw [@Equiv.apply_eq_iff_eq] at h
|
||||
change (1 : Fin 2) = (0 : Fin 2) at h
|
||||
simp [Fin.isValue, zero_ne_one] at h
|
||||
intro h
|
||||
simp [detContinuous, h]
|
||||
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous, zero_zero_mul]
|
||||
exact euclid_norm_not_IsFourVelocity_not_IsFourVelocity h h'
|
||||
|
||||
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
|
||||
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
|
||||
simp [stdBasis_mulVec]
|
||||
change (Λ * Λ').1 0 0 ≤ _
|
||||
rw [zero_zero_mul]
|
||||
exact euclid_norm_IsFourVelocity_not_IsFourVelocity h h'
|
||||
|
||||
/-- The representation taking a lorentz matrix to its determinant. -/
|
||||
@[simps!]
|
||||
def detRep : lorentzGroup →* ℤ₂ where
|
||||
toFun Λ := detContinuous Λ
|
||||
lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
|
||||
(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
|
||||
rw [IsOrthochronous_iff_transpose] at h
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
|
||||
rw [IsOrthochronous_iff_fstCol_IsFourVelocity, not_IsFourVelocity_iff]
|
||||
simp [stdBasis_mulVec]
|
||||
change (Λ * Λ').1 0 0 ≤ _
|
||||
rw [zero_zero_mul]
|
||||
exact euclid_norm_not_IsFourVelocity_IsFourVelocity h h'
|
||||
|
||||
def orthchroRep : lorentzGroup →* ℤ₂ where
|
||||
toFun := orthchroMap
|
||||
map_one' := by
|
||||
simp [detContinuous]
|
||||
map_mul' := by
|
||||
intro Λ1 Λ2
|
||||
simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul, det_mul, toMul_zero,
|
||||
mul_ite, mul_one, ite_mul, one_mul]
|
||||
cases' (det_eq_one_or_neg_one Λ1) with h1 h1
|
||||
<;> cases' (det_eq_one_or_neg_one Λ2) with h2 h2
|
||||
<;> simp [h1, h2, detContinuous]
|
||||
have h1 : IsOrthochronous 1 := by simp [IsOrthochronous]
|
||||
rw [orthchroMap_IsOrthochronous h1]
|
||||
map_mul' Λ Λ' := by
|
||||
simp
|
||||
by_cases h : IsOrthochronous Λ
|
||||
<;> by_cases h' : IsOrthochronous Λ'
|
||||
rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
|
||||
orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
|
||||
simp
|
||||
rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
|
||||
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
|
||||
simp
|
||||
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
|
||||
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
|
||||
simp
|
||||
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
|
||||
orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
|
||||
simp only [Nat.reduceAdd]
|
||||
rfl
|
||||
|
||||
lemma detRep_continuous : Continuous detRep := detContinuous.2
|
||||
|
||||
lemma det_on_connected_component {Λ Λ' : lorentzGroup} (h : Λ' ∈ connectedComponent Λ) :
|
||||
Λ.1.1.det = Λ'.1.1.det := by
|
||||
obtain ⟨s, hs, hΛ'⟩ := h
|
||||
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
|
||||
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
|
||||
simpa [f, detContinuous_eq_iff_det_eq] using
|
||||
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
|
||||
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
|
||||
|
||||
lemma det_of_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') : Λ.1.1.det = Λ'.1.1.det :=
|
||||
det_on_connected_component $ pathComponent_subset_component _ h
|
||||
|
||||
/-- A Lorentz Matrix is proper if its determinant is 1. -/
|
||||
@[simp]
|
||||
def IsProper (Λ : lorentzGroup) : Prop := Λ.1.1.det = 1
|
||||
|
||||
instance : DecidablePred IsProper := by
|
||||
intro Λ
|
||||
apply Real.decidableEq
|
||||
|
||||
lemma IsProper_iff (Λ : lorentzGroup) : IsProper Λ ↔ detRep Λ = 1 := by
|
||||
rw [show 1 = detRep 1 by simp]
|
||||
rw [detRep_apply, detRep_apply, detContinuous_eq_iff_det_eq]
|
||||
simp only [IsProper, OneMemClass.coe_one, Units.val_one, det_one]
|
||||
|
||||
|
||||
end lorentzGroup
|
||||
|
||||
end spaceTime
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue