feat: Add properties of rotations

HepLean/GroupTheory/SO3/Basic.lean

@@ -7,6 +7,9 @@ import Mathlib.LinearAlgebra.UnitaryGroup
 import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
 import Mathlib.Data.Complex.Exponential
 import Mathlib.Geometry.Manifold.VectorBundle.Basic
+import Mathlib.LinearAlgebra.EigenSpace.Basic
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Analysis.InnerProductSpace.Adjoint
 /-!
 # the 3d special orthogonal group
 
@@ -154,8 +157,74 @@ lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
       _ = -  det (A.1 - 1)  := by simp [pow_three]
   simpa using h1
 
+@[simp]
+lemma det_id_minus (A : SO(3)) : det (1 - A.1) = 0 := by
+  have h1 : det (1 - A.1) = - det (A.1 - 1) := by
+    calc
+      det (1 - A.1) = det (- (A.1 - 1)) := by simp
+      _ = (- 1) ^ 3  *  det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
+      _ = -  det (A.1 - 1) := by simp [pow_three]
+  rw [h1, det_minus_id]
+  simp only [neg_zero]
+
+@[simp]
+lemma one_in_spectrum (A : SO(3)) : 1 ∈ spectrum ℝ (A.1) := by
+  rw [spectrum.mem_iff]
+  simp
+  rw [Matrix.isUnit_iff_isUnit_det]
+  simp
+
+noncomputable section action
+open Module
+
+@[simps!]
+def toEnd (A : SO(3)) : End ℝ (EuclideanSpace ℝ (Fin 3)) :=
+  Matrix.toLin (EuclideanSpace.basisFun (Fin 3) ℝ).toBasis
+  (EuclideanSpace.basisFun (Fin 3) ℝ).toBasis A.1
+
+lemma one_is_eigenvalue (A : SO(3)) : A.toEnd.HasEigenvalue 1 := by
+  rw [End.hasEigenvalue_iff_mem_spectrum]
+  erw [AlgEquiv.spectrum_eq (Matrix.toLinAlgEquiv (EuclideanSpace.basisFun (Fin 3) ℝ).toBasis ) A.1]
+  exact one_in_spectrum A
+
+lemma exists_stationary_vec (A : SO(3)) :
+    ∃ (v : EuclideanSpace ℝ (Fin 3)),
+    Orthonormal ℝ (({0} : Set (Fin 3)).restrict (fun _ => v ))
+    ∧ A.toEnd v = v := by
+  obtain ⟨v, hv⟩ := End.HasEigenvalue.exists_hasEigenvector $ one_is_eigenvalue A
+  have hvn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv.2
+  use (1/‖v‖) • v
+  apply And.intro
+  rw [@orthonormal_iff_ite]
+  intro v1 v2
+  have hv1 := v1.2
+  have hv2 := v2.2
+  simp_all only [one_div, Set.mem_singleton_iff]
+  have hveq : v1 = v2 := by
+    rw [@Subtype.ext_iff]
+    simp_all only
+  subst hveq
+  simp only [Set.restrict_apply, PiLp.smul_apply, smul_eq_mul,
+    _root_.map_mul, map_inv₀, conj_trivial, ↓reduceIte]
+  rw [inner_smul_right, inner_smul_left, real_inner_self_eq_norm_sq v]
+  field_simp
+  ring
+  rw [_root_.map_smul, End.mem_eigenspace_iff.mp hv.1 ]
+  simp
+
+lemma exists_basis_preserved (A : SO(3)) :
+    ∃ (b : OrthonormalBasis (Fin 3) ℝ (EuclideanSpace ℝ (Fin 3))), A.toEnd (b 0) = b 0 := by
+  obtain ⟨v, hv⟩ := exists_stationary_vec A
+  have h3 : FiniteDimensional.finrank ℝ (EuclideanSpace ℝ (Fin 3)) = Fintype.card (Fin 3) := by
+    simp_all only [toEnd_apply, finrank_euclideanSpace, Fintype.card_fin]
+  obtain ⟨b, hb⟩ :=  Orthonormal.exists_orthonormalBasis_extension_of_card_eq h3 hv.1
+  simp at hb
+  use b
+  rw [hb, hv.2]
 
 
+
+end action
 end SO3
 
 

HepLean/SpaceTime/LorentzGroup/Rotations.lean

@@ -18,7 +18,22 @@ namespace spaceTime
 namespace lorentzGroup
 open GroupTheory
 
+@[simp]
+def SO3ToMatrix (A : SO(3)) : Matrix (Fin 4) (Fin 4) ℝ :=
+  Matrix.reindex finSumFinEquiv finSumFinEquiv (Matrix.fromBlocks 1 0 0 A.1)
 
+@[simp]
+lemma SO3ToMatrix_det (A : SO(3)) : (SO3ToMatrix A).det = 1 := by
+  simp only [SO3ToMatrix, Nat.reduceAdd, Matrix.reindex_apply, Matrix.det_submatrix_equiv_self,
+    Matrix.det_fromBlocks_zero₂₁, Matrix.det_unique, Fin.default_eq_zero, Fin.isValue,
+    Matrix.one_apply_eq, A.2, mul_one]
+
+lemma SO3ToMatrix_lorentz (A : SO(3)) : η * (SO3ToMatrix A).transpose * η  * SO3ToMatrix A = 1 := by
+  simp only [η_block, Nat.reduceAdd, Matrix.reindex_apply, SO3ToMatrix, Matrix.transpose_submatrix,
+    Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,
+    Matrix.submatrix_mul_equiv, Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero,
+    Matrix.zero_mul, Matrix.mul_one, neg_mul, one_mul, zero_add, Matrix.mul_neg, neg_zero, mul_neg,
+    neg_neg, Matrix.mul_eq_one_comm.mpr A.2.2, Matrix.fromBlocks_one, Matrix.submatrix_one_equiv]
 
 
 end lorentzGroup

HepLean/SpaceTime/Metric.lean

@@ -66,6 +66,23 @@ lemma η_sq : η * η = 1 := by
       Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
       vecHead, vecTail, Function.comp_apply]
 
+lemma η_block : η = Matrix.reindex finSumFinEquiv finSumFinEquiv (
+    Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin 3) (Fin 3) ℝ)) := by
+  apply Matrix.ext
+  intro i j
+  fin_cases i <;> fin_cases j
+    <;> simp_all only [Fin.zero_eta, reindex_apply, submatrix_apply]
+  any_goals rfl
+  all_goals simp [finSumFinEquiv, Fin.addCases, η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
+      Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
+      Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
+      vecHead, vecTail, Function.comp_apply]
+  any_goals rfl
+  all_goals split
+  all_goals simp
+  all_goals rfl
+
+
 
 lemma η_diag_mul_self (μ : Fin 4) : η μ μ * η μ μ  = 1 := by
   fin_cases μ