232 lines
9 KiB
Text
232 lines
9 KiB
Text
/-
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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.Basic
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import Mathlib.Analysis.InnerProductSpace.Adjoint
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import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
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/-!
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# Spacetime Metric
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This file introduces the metric on spacetime in the (+, -, -, -) signature.
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-/
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noncomputable section
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namespace spaceTime
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open Manifold
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open Matrix
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open Complex
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open ComplexConjugate
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/-- The metric as a `4×4` real matrix. -/
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def η : Matrix (Fin 4) (Fin 4) ℝ :=
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!![1, 0, 0, 0; 0, -1, 0, 0; 0, 0, -1, 0; 0, 0, 0, -1]
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lemma η_off_diagonal {μ ν : Fin 4} (h : μ ≠ ν) : η μ ν = 0 := by
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fin_cases μ <;>
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fin_cases ν <;>
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simp_all [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
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Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
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Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
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vecHead, vecTail, Function.comp_apply]
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lemma η_symmetric (μ ν : Fin 4) : η μ ν = η ν μ := by
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by_cases h : μ = ν
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rw [h]
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rw [η_off_diagonal h]
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refine (η_off_diagonal ?_).symm
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exact fun a => h (id a.symm)
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lemma η_transpose : η.transpose = η := by
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funext μ ν
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rw [transpose_apply, η_symmetric]
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lemma det_η : η.det = - 1 := by
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simp only [η, det_succ_row_zero, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, of_apply,
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cons_val', empty_val', cons_val_fin_one, cons_val_zero, submatrix_apply, Fin.succ_zero_eq_one,
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cons_val_one, head_cons, submatrix_submatrix, Function.comp_apply, Fin.succ_one_eq_two,
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cons_val_two, tail_cons, det_unique, Fin.default_eq_zero, cons_val_succ, head_fin_const,
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Fin.sum_univ_succ, Fin.val_zero, pow_zero, one_mul, Fin.zero_succAbove, Finset.univ_unique,
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Fin.val_succ, Fin.coe_fin_one, zero_add, pow_one, neg_mul, Fin.succ_succAbove_zero,
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Finset.sum_neg_distrib, Finset.sum_singleton, Fin.succ_succAbove_one, even_two, Even.neg_pow,
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one_pow, mul_one, mul_neg, neg_neg, mul_zero, neg_zero, add_zero, zero_mul,
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Finset.sum_const_zero]
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lemma η_sq : η * η = 1 := by
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funext μ ν
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rw [mul_apply, Fin.sum_univ_four]
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fin_cases μ <;> fin_cases ν <;>
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simp [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
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Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
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Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
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vecHead, vecTail, Function.comp_apply]
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lemma η_block : η = Matrix.reindex finSumFinEquiv finSumFinEquiv (
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Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin 3) (Fin 3) ℝ)) := by
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apply Matrix.ext
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intro i j
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fin_cases i <;> fin_cases j
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<;> simp_all only [Fin.zero_eta, reindex_apply, submatrix_apply]
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any_goals rfl
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all_goals simp [finSumFinEquiv, Fin.addCases, η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
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Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
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Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
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vecHead, vecTail, Function.comp_apply]
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any_goals rfl
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all_goals split
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all_goals simp
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all_goals rfl
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lemma η_diag_mul_self (μ : Fin 4) : η μ μ * η μ μ = 1 := by
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fin_cases μ
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<;> simp [η]
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lemma η_mulVec (x : spaceTime) : η *ᵥ x = ![x 0, -x 1, -x 2, -x 3] := by
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rw [explicit x]
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rw [η]
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funext i
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rw [mulVec, dotProduct, Fin.sum_univ_four]
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fin_cases i <;>
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simp [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
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Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
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Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
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vecHead, vecTail, Function.comp_apply]
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/-- Given a point in spaceTime `x` the linear map `y → x ⬝ᵥ (η *ᵥ y)`. -/
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@[simps!]
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def linearMapForSpaceTime (x : spaceTime) : spaceTime →ₗ[ℝ] ℝ where
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toFun y := x ⬝ᵥ (η *ᵥ y)
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map_add' y z := by
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simp only
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rw [mulVec_add, dotProduct_add]
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map_smul' c y := by
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simp only
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rw [mulVec_smul, dotProduct_smul]
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rfl
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/-- The metric as a bilinear map from `spaceTime` to `ℝ`. -/
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def ηLin : LinearMap.BilinForm ℝ spaceTime where
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toFun x := linearMapForSpaceTime x
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map_add' x y := by
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apply LinearMap.ext
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intro z
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simp only [linearMapForSpaceTime_apply, LinearMap.add_apply]
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rw [add_dotProduct]
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map_smul' c x := by
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apply LinearMap.ext
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intro z
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simp only [linearMapForSpaceTime_apply, RingHom.id_apply, LinearMap.smul_apply, smul_eq_mul]
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rw [smul_dotProduct]
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rfl
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lemma ηLin_expand (x y : spaceTime) : ηLin x y = x 0 * y 0 - x 1 * y 1 - x 2 * y 2 - x 3 * y 3 := by
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rw [ηLin]
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simp only [LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply, Fin.isValue]
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erw [η_mulVec]
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nth_rewrite 1 [explicit x]
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simp only [dotProduct, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Fin.sum_univ_four,
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cons_val_zero, cons_val_one, head_cons, mul_neg, cons_val_two, tail_cons, cons_val_three]
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ring
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lemma ηLin_expand_self (x : spaceTime) : ηLin x x = x 0 ^ 2 - ‖x.space‖ ^ 2 := by
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rw [← @real_inner_self_eq_norm_sq, @PiLp.inner_apply, Fin.sum_univ_three, ηLin_expand]
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simp only [Fin.isValue, space, cons_val_zero, RCLike.inner_apply, conj_trivial, cons_val_one,
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head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons]
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ring
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lemma time_elm_sq_of_ηLin (x : spaceTime) : x 0 ^ 2 = ηLin x x + ‖x.space‖ ^ 2 := by
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rw [ηLin_expand_self]
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ring
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lemma ηLin_leq_time_sq (x : spaceTime) : ηLin x x ≤ x 0 ^ 2 := by
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rw [time_elm_sq_of_ηLin]
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apply (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖x.space‖
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lemma ηLin_space_inner_product (x y : spaceTime) :
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ηLin x y = x 0 * y 0 - ⟪x.space, y.space⟫_ℝ := by
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rw [ηLin_expand, @PiLp.inner_apply, Fin.sum_univ_three]
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simp only [Fin.isValue, space, cons_val_zero, RCLike.inner_apply, conj_trivial, cons_val_one,
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head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons]
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ring
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lemma ηLin_ge_abs_inner_product (x y : spaceTime) :
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x 0 * y 0 - ‖⟪x.space, y.space⟫_ℝ‖ ≤ (ηLin x y) := by
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rw [ηLin_space_inner_product, sub_le_sub_iff_left]
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exact Real.le_norm_self ⟪x.space, y.space⟫_ℝ
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lemma ηLin_ge_sub_norm (x y : spaceTime) :
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x 0 * y 0 - ‖x.space‖ * ‖y.space‖ ≤ (ηLin x y) := by
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apply le_trans ?_ (ηLin_ge_abs_inner_product x y)
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rw [sub_le_sub_iff_left]
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exact norm_inner_le_norm x.space y.space
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lemma ηLin_symm (x y : spaceTime) : ηLin x y = ηLin y x := by
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rw [ηLin_expand, ηLin_expand]
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ring
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lemma ηLin_stdBasis_apply (μ : Fin 4) (x : spaceTime) : ηLin (stdBasis μ) x = η μ μ * x μ := by
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rw [ηLin_expand]
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fin_cases μ
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<;> simp [stdBasis_0, stdBasis_1, stdBasis_2, stdBasis_3, η]
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lemma ηLin_η_stdBasis (μ ν : Fin 4) : ηLin (stdBasis μ) (stdBasis ν) = η μ ν := by
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rw [ηLin_stdBasis_apply]
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by_cases h : μ = ν
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rw [stdBasis_apply]
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subst h
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simp only [↓reduceIte, mul_one]
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rw [stdBasis_not_eq, η_off_diagonal h]
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simp only [mul_zero]
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exact fun a => h (id a.symm)
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lemma ηLin_mulVec_left (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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ηLin (Λ *ᵥ x) y = ηLin x ((η * Λᵀ * η) *ᵥ y) := by
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simp only [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply, mulVec_mulVec]
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rw [(vecMul_transpose Λ x).symm, ← dotProduct_mulVec, mulVec_mulVec]
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rw [← mul_assoc, ← mul_assoc, η_sq, one_mul]
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lemma ηLin_mulVec_right (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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ηLin x (Λ *ᵥ y) = ηLin ((η * Λᵀ * η) *ᵥ x) y := by
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rw [ηLin_symm, ηLin_symm ((η * Λᵀ * η) *ᵥ x) _ ]
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exact ηLin_mulVec_left y x Λ
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lemma ηLin_matrix_stdBasis (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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ηLin (stdBasis ν) (Λ *ᵥ stdBasis μ) = η ν ν * Λ ν μ := by
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rw [ηLin_stdBasis_apply, stdBasis_mulVec]
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lemma ηLin_matrix_stdBasis' (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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Λ ν μ = η ν ν * ηLin (stdBasis ν) (Λ *ᵥ stdBasis μ) := by
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rw [ηLin_matrix_stdBasis, ← mul_assoc, η_diag_mul_self, one_mul]
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lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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Λ = 1 ↔ ∀ (x y : spaceTime), ηLin x y = ηLin x (Λ *ᵥ y) := by
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apply Iff.intro
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intro h
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subst h
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simp only [ηLin, one_mulVec, implies_true]
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intro h
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funext μ ν
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have h1 := h (stdBasis μ) (stdBasis ν)
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rw [ηLin_matrix_stdBasis, ηLin_η_stdBasis] at h1
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fin_cases μ <;> fin_cases ν <;>
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simp_all [η, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
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Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
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Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
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vecHead, vecTail, Function.comp_apply]
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/-- The metric as a quadratic form on `spaceTime`. -/
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def quadraticForm : QuadraticForm ℝ spaceTime := ηLin.toQuadraticForm
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end spaceTime
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end
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