105 lines
3.4 KiB
Text
105 lines
3.4 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Geometry.Manifold.IsManifold.Basic
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import HepLean.Meta.TODO.Basic
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/-!
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# Space time
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This file introduce 4d Minkowski spacetime.
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-/
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noncomputable section
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TODO "SpaceTime should be refactored into a structure, or similar, to prevent casting."
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/-- The space-time -/
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def SpaceTime : Type := Fin 4 → ℝ
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/-- Give spacetime the structure of an additive commutative monoid. -/
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instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
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/-- Give spacetime the structure of a module over the reals. -/
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instance : Module ℝ SpaceTime := Pi.module _ _ _
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/-- The instance of a normed group on spacetime defined via the Euclidean norm. -/
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instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
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/-- The Euclidean norm-structure on space time. This is used to give it a smooth structure. -/
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instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
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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 space part of spacetime. -/
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@[simp]
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def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
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/-- The structure of a smooth manifold on spacetime. -/
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def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
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/-- The instance of a `ChartedSpace` on `SpaceTime`. -/
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instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
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/-- The standard basis for spacetime. -/
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def stdBasis : Basis (Fin 4) ℝ SpaceTime := Pi.basisFun ℝ (Fin 4)
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lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 := by
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erw [stdBasis, Pi.basisFun_apply, Pi.single_apply]
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refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
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exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
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lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 := by
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rw [stdBasis_apply]
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exact if_neg h
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/-- For space-time,`stdBasis 0` is equal to `![1, 0, 0, 0] `. -/
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lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
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funext i
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fin_cases i <;> simp [stdBasis_apply]
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/-- For space-time,`stdBasis 1` is equal to `![0, 1, 0, 0] `. -/
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lemma stdBasis_1 : stdBasis 1 = ![0, 1, 0, 0] := by
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funext i
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fin_cases i <;> simp [stdBasis_apply]
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/-- For space-time,`stdBasis 2` is equal to `![0, 0, 1, 0] `. -/
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lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] := by
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funext i
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fin_cases i <;> simp [stdBasis_apply]
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/-- For space-time,`stdBasis 3` is equal to `![0, 0, 0, 1] `. -/
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lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] := by
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funext i
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fin_cases i <;> simp [stdBasis_apply]
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lemma stdBasis_mulVec (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
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(Λ *ᵥ stdBasis μ) ν = Λ ν μ := by
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rw [mulVec, dotProduct, Fintype.sum_eq_single μ, stdBasis_apply]
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· simp only [↓reduceIte, mul_one]
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· intro x h
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rw [stdBasis_apply, if_neg (Ne.symm h)]
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exact CommMonoidWithZero.mul_zero (Λ ν x)
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/-- The explicit expansion of a point in spacetime as `![x 0, x 1, x 2, x 3]`. -/
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lemma explicit (x : SpaceTime) : x = ![x 0, x 1, x 2, x 3] := by
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funext i
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fin_cases i <;> rfl
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@[simp]
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lemma add_apply (x y : SpaceTime) (i : Fin 4) : (x + y) i = x i + y i := rfl
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@[simp]
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lemma smul_apply (x : SpaceTime) (a : ℝ) (i : Fin 4) : (a • x) i = a * x i := rfl
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end SpaceTime
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end
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