The Orthochronous Lorentz Group

We define the give a series of lemmas related to the orthochronous property of lorentz matrices.

def LorentzGroup.IsOrthochronous {d : ℕ} (Λ : ↑(LorentzGroup d)) : Prop

A Lorentz transformation is orthochronous if its 0 0 element is non-negative.

Equations

• LorentzGroup.IsOrthochronous Λ = (0 ≤ ↑Λ (Sum.inl 0) (Sum.inl 0))

theorem LorentzGroup.isOrthochronous_iff_toVector_timeComponet_nonneg {d : ℕ} (Λ : ↑(LorentzGroup d)) : IsOrthochronous Λ ↔ 0 ≤ (toVector Λ).timeComponent

A Lorentz transformation is orthochronous if and only if its first column is future pointing.

theorem LorentzGroup.isOrthochronous_iff_transpose {d : ℕ} (Λ : ↑(LorentzGroup d)) : IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ)

A Lorentz transformation is orthochronous if and only if its transpose is orthochronous.

@[simp]

theorem LorentzGroup.isOrthochronous_inv_iff {d : ℕ} {Λ : ↑(LorentzGroup d)} : IsOrthochronous Λ⁻¹ ↔ IsOrthochronous Λ

theorem LorentzGroup.isOrthochronous_iff_ge_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : IsOrthochronous Λ ↔ 1 ≤ ↑Λ (Sum.inl 0) (Sum.inl 0)

A Lorentz transformation is orthochronous if and only if its 0 0 element is greater or equal to one.

def LorentzGroup.orthochronoustoVelocity {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : IsOrthochronous Λ) : ↑(Lorentz.Velocity d)

The Lorentz.Velocity from Lorentz.toVector of a orthochronous lorentz transformation.

Equations

• LorentzGroup.orthochronoustoVelocity h = ⟨LorentzGroup.toVector Λ, ⋯⟩

theorem LorentzGroup.not_isOrthochronous_iff_le_neg_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : ¬IsOrthochronous Λ ↔ ↑Λ (Sum.inl 0) (Sum.inl 0) ≤ -1

A Lorentz transformation is not orthochronous if and only if its 0 0 element is less than or equal to minus one.

theorem LorentzGroup.isOrthochronous_iff_not_neg {d : ℕ} (Λ : ↑(LorentzGroup d)) : IsOrthochronous Λ ↔ ¬IsOrthochronous (-Λ)

theorem LorentzGroup.neg_isOrthochronous_iff_not {d : ℕ} {Λ : ↑(LorentzGroup d)} : IsOrthochronous (-Λ) ↔ ¬IsOrthochronous Λ

theorem LorentzGroup.not_isOrthochronous_iff_le_zero {d : ℕ} (Λ : ↑(LorentzGroup d)) : ¬IsOrthochronous Λ ↔ ↑Λ (Sum.inl 0) (Sum.inl 0) ≤ 0

A Lorentz transformation is not orthochronous if and only if its 0 0 element is non-positive.

theorem LorentzGroup.not_isOrthochronous_iff_toVector_timeComponet_nonpos {d : ℕ} (Λ : ↑(LorentzGroup d)) : ¬IsOrthochronous Λ ↔ (toVector Λ).timeComponent ≤ 0

theorem LorentzGroup.id_isOrthochronous {d : ℕ} : IsOrthochronous 1

The identity Lorentz transformation is orthochronous.

def LorentzGroup.timeCompCont {d : ℕ} : C(↑(LorentzGroup d), ℝ)

The continuous map taking a Lorentz transformation to its 0 0 element.

Equations

• LorentzGroup.timeCompCont = { toFun := fun (Λ : ↑(LorentzGroup d)) => ↑Λ (Sum.inl 0) (Sum.inl 0), continuous_toFun := ⋯ }

def LorentzGroup.stepFunction : ℝ → ℝ

An auxiliary function used in the definition of orthchroMapReal. This function takes all elements of ℝ less than -1 to -1, all elements of R greater than 1 to 1 and preserves all other elements.

Equations

• LorentzGroup.stepFunction t = if t ≤ -1 then -1 else if 1 ≤ t then 1 else t

theorem LorentzGroup.stepFunction_continuous : Continuous stepFunction

The stepFunction is continuous.

def LorentzGroup.orthchroMapReal {d : ℕ} : C(↑(LorentzGroup d), ℝ)

The continuous map from lorentzGroup to ℝ wh taking Orthochronous elements to 1 and non-orthochronous to -1.

Equations

• LorentzGroup.orthchroMapReal = { toFun := LorentzGroup.stepFunction, continuous_toFun := LorentzGroup.stepFunction_continuous }.comp LorentzGroup.timeCompCont

theorem LorentzGroup.orthchroMapReal_on_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : IsOrthochronous Λ) : orthchroMapReal Λ = 1

A Lorentz transformation which is orthochronous maps under orthchroMapReal to 1.

theorem LorentzGroup.orthchroMapReal_on_not_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : ¬IsOrthochronous Λ) : orthchroMapReal Λ = -1

A Lorentz transformation which is not-orthochronous maps under orthchroMapReal to - 1.

theorem LorentzGroup.orthchroMapReal_minus_one_or_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1

Every Lorentz transformation maps under orthchroMapReal to either 1 or -1.

def LorentzGroup.orthchroMap {d : ℕ} : C(↑(LorentzGroup d), Multiplicative (ZMod 2))

A continuous map from lorentzGroup to ℤ₂ whose kernel are the Orthochronous elements.

Equations

• LorentzGroup.orthchroMap = LorentzGroup.coeForℤ₂.comp { toFun := fun (Λ : ↑(LorentzGroup d)) => ⟨LorentzGroup.orthchroMapReal Λ, ⋯⟩, continuous_toFun := ⋯ }

theorem LorentzGroup.orthchroMap_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : IsOrthochronous Λ) : orthchroMap Λ = 1

A Lorentz transformation which is orthochronous maps under orthchroMap to 1 in ℤ₂ (the identity element).

theorem LorentzGroup.orthchroMap_not_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : ¬IsOrthochronous Λ) : orthchroMap Λ = Additive.toMul 1

A Lorentz transformation which is not-orthochronous maps under orthchroMap to the non-identity element of ℤ₂.

theorem LorentzGroup.isOrthochronous_mul {d : ℕ} {Λ Λ' : ↑(LorentzGroup d)} (h : IsOrthochronous Λ) (h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ')

The product of two orthochronous Lorentz transformations is orthochronous.

theorem LorentzGroup.isOrthochronous_mul_iff {d : ℕ} {Λ Λ' : ↑(LorentzGroup d)} : IsOrthochronous (Λ * Λ') ↔ IsOrthochronous Λ = IsOrthochronous Λ'

def LorentzGroup.orthchroRep {d : ℕ} : ↑(LorentzGroup d) →* Multiplicative (ZMod 2)

The homomorphism from LorentzGroup to ℤ₂.

Equations

• LorentzGroup.orthchroRep = { toFun := ⇑LorentzGroup.orthchroMap, map_one' := ⋯, map_mul' := ⋯ }

theorem LorentzGroup.IsOrthochronous.iff_in_orthchroRep_ker {d : ℕ} (Λ : ↑(LorentzGroup d)) : IsOrthochronous Λ ↔ Λ ∈ orthchroRep.ker

The orthochronous Lorentz transformations form the kernel of the homomorphism from LorentzGroup to ℤ₂.

theorem LorentzGroup.orthchroRep_inv_eq_self {d : ℕ} (Λ : ↑(LorentzGroup d)) : orthchroRep Λ = orthchroRep Λ⁻¹

The homomorphism from LorentzGroup to ℤ₂ assigns the same value to any Lorentz transformation and its inverse.

theorem LorentzGroup.isOrthochronous_iff_of_orthchroMap_eq {d : ℕ} {Λ Λ' : ↑(LorentzGroup d)} (h : orthchroMap Λ = orthchroMap Λ') : IsOrthochronous Λ ↔ IsOrthochronous Λ'

Two Lorentz transformations are both orthochronous or both not orthochronous if they are mapped to the same element via the homomorphism from LorentzGroup to ℤ₂.

theorem LorentzGroup.isOrthochronous_on_connected_component {d : ℕ} {Λ Λ' : ↑(LorentzGroup d)} (h : Λ' ∈ connectedComponent Λ) : IsOrthochronous Λ ↔ IsOrthochronous Λ'

Two Lorentz transformations which are in the same connected component are either both orthochronous or both not orthochronous.