Research proof index

The condition on an electromagnetic potential to be an extrema of the lagrangian. -/ def IsExtrema {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : Prop := gradLagrangian 𝓕 A J = 0 lemma isExtrema_iff_gradLagrangian {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : IsExtrema 𝓕 A J ↔ A....

This lemma expresses the component form of the transformed field strength tensor: when a Lorentz transformation Λ acts on the potential A, the resulting field strength tensor's components are given by the standard tensor transformation rule involving the Lorentz matrix elements Λ^μ_κ and Λ^ν_ρ applied to the original field components.

Index evaluation of the field strength tensor agrees with the corresponding component of the field strength matrix. -/ lemma toFieldStrength_eval_eq_fieldStrengthMatrix {d} (A : ElectromagneticPotential d) (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) : toField {A.toFieldStrength x | [μ] [ν]}ᵀ = A.fieldStrengthMatrix x (μ, ν) := by rw [toFieldStrength_eval_eq_b...

The field strength of the bare-gradient potential B^μ := ∂_μ χ for χ(x) = x⁰·xⁱ is nonzero (follows from fieldStrengthMatrix_bareGradient_inl_inr).

The statement that F = F^{μν} eᵤ ⊗ eᵥ written explicitly, with the components given by ∑ κ, (η μ κ * ∂_ κ A x ν - η ν κ * ∂_ κ A x μ).

The statement that F = F^{μν} eᵤ ⊗ eᵥ written explicitly, with with the components given by (η μ μ * ∂_ μ A x ν - η ν ν * ∂_ ν A x μ).