feat: some topological properties of lorentz group

HepLean/SpaceTime/LorentzGroup/Connectedness.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzGroup
+import Mathlib.GroupTheory.SpecificGroups.KleinFour
+import Mathlib.Topology.Constructions
+/-!
+# Connectedness of the restricted Lorentz group
+
+In this file we provide a proof that the restricted Lorentz group is connected.
+
+## References
+
+- The main argument follows: Guillem Cobos, The Lorentz Group, 2015:
+  https://diposit.ub.edu/dspace/bitstream/2445/68763/2/memoria.pdf
+
+-/
+noncomputable section
+namespace spaceTime
+
+def stepFunction : ℝ → ℝ := fun t =>
+  if t < -1 then -1 else
+    if t < 1 then t else 1
+
+lemma stepFunction_continuous : Continuous stepFunction := by
+  apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_id continuous_const)
+  all_goals
+    intro a ha
+    rw [@Set.Iio_def, @frontier_Iio, @Set.mem_singleton_iff] at ha
+    rw [ha]
+    simp  [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
+
+def lorentzDet (Λ : lorentzGroup) : ({-1, 1} : Set ℝ) :=
+  ⟨Λ.1.1.det, by
+    simp
+    exact Or.symm (lorentzGroup.det_eq_one_or_neg_one _)⟩
+
+lemma lorentzDet_continuous : Continuous lorentzDet := by
+  refine Continuous.subtype_mk ?_ fun x => lorentzDet.proof_1 x
+  exact Continuous.matrix_det $
+    Continuous.comp' Units.continuous_val continuous_subtype_val
+
+instance set_discrete : DiscreteTopology ({-1, 1} : Set ℝ) := discrete_of_t1_of_finite
+
+lemma lorentzDet_eq_if_joined {Λ Λ' : lorentzGroup} (h : Joined Λ Λ') :
+    Λ.1.1.det = Λ'.1.1.det := by
+  obtain ⟨f, hf, hf2⟩ := h
+  have h1 : Joined (lorentzDet Λ) (lorentzDet Λ') :=
+    ⟨ContinuousMap.comp ⟨lorentzDet, lorentzDet_continuous⟩ f, congrArg lorentzDet hf,
+    congrArg lorentzDet hf2⟩
+  obtain ⟨g, hg1, hg2⟩ := h1
+  have h2 := (@IsPreconnected.subsingleton ({-1, 1} : Set ℝ) _ _ _ (isPreconnected_range g.2))
+    (Set.mem_range_self 0) (Set.mem_range_self 1)
+  rw [hg1, hg2] at h2
+  simpa [lorentzDet] using h2
+
+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 ({-1, 1} : Set ℝ) :=
+    ContinuousMap.restrict s ⟨lorentzDet, lorentzDet_continuous⟩
+  haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
+  have hf := (@IsPreconnected.subsingleton ({-1, 1} : Set ℝ) _ _ _ (isPreconnected_range f.2))
+    (Set.mem_range_self ⟨Λ, hs.2⟩)  (Set.mem_range_self ⟨Λ', hΛ'⟩)
+  simpa [lorentzDet, f] using hf
+
+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
+
+namespace restrictedLorentzGroup
+
+/-- The set of spacetime points such that the time element is positive, and which are
+normalised to 1. -/
+def P : Set (spaceTime) := {x | x 0 > 0 ∧ ηLin x x = 1}
+
+lemma P_time_comp (x : P) : x.1 0 = √(1 + ‖x.1.space‖ ^ 2) := by
+  symm
+  rw [Real.sqrt_eq_cases]
+  apply Or.inl
+  refine And.intro ?_ (le_of_lt  x.2.1)
+  rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
+  simp only [Fin.isValue, space, Matrix.cons_val_zero, RCLike.inner_apply, conj_trivial,
+    Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
+    Matrix.tail_cons]
+  have h1 := x.2.2
+  rw [ηLin_expand] at h1
+  linear_combination h1
+
+def oneInP : P := ⟨![1, 0, 0, 0], by simp, by
+  rw [ηLin_expand]
+  simp⟩
+
+def pathToOneInP (u : P) : Path oneInP u where
+  toFun t := ⟨![√(1 + t ^ 2 * ‖u.1.space‖ ^ 2), t * u.1 1, t * u.1 2, t * u.1 3],
+    by
+      simp
+      refine Right.add_pos_of_pos_of_nonneg (Real.zero_lt_one) $
+          mul_nonneg (sq_nonneg _) (sq_nonneg _)
+    , by
+      rw [ηLin_expand]
+      simp
+      rw [Real.mul_self_sqrt, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three]
+      simp
+      ring
+      refine Right.add_nonneg (zero_le_one' ℝ) $
+            mul_nonneg (sq_nonneg _) (sq_nonneg _) ⟩
+  continuous_toFun := by
+    refine Continuous.subtype_mk ?_ _
+    apply (continuous_pi_iff).mpr
+    intro i
+    fin_cases i
+      <;> continuity
+  source' := by
+    simp
+    rfl
+  target' := by
+    simp
+    refine SetCoe.ext ?_
+    funext i
+    fin_cases i
+    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