2024-05-14 08:25:03 -04:00
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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 Mathlib.Data.Complex.Exponential
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import Mathlib.Geometry.Manifold.VectorBundle.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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/-- 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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instance euclideanNormedAddCommGroup : NormedAddCommGroup spaceTime := Pi.normedAddCommGroup
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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 structure of a smooth manifold on spacetime. -/
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def asSmoothManifold : ModelWithCorners ℝ spaceTime spaceTime := 𝓘(ℝ, spaceTime)
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2024-05-14 08:27:14 -04:00
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instance : ChartedSpace spaceTime spaceTime := chartedSpaceSelf spaceTime
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2024-05-14 08:25:03 -04:00
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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, LinearMap.stdBasis_apply']
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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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lemma stdBasis_0 : stdBasis 0 = ![1, 0, 0, 0] := by
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funext i
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fin_cases i
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<;> simp [stdBasis_apply]
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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
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<;> simp [stdBasis_apply]
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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
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<;> simp [stdBasis_apply]
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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
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<;> 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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2024-05-14 08:25:03 -04:00
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intro x h
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rw [stdBasis_apply, if_neg (Ne.symm h)]
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simp
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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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end spaceTime
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end
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