Basic lemmas regarding metrics #
Symmetry properties #
The covariant metric is symmetric {η' | μ ν = η' | ν μ}ᵀ.
The contravariant metric is symmetric {η | μ ν = η | ν μ}ᵀ.
The left metric is antisymmetric {εL | α α' = - εL | α' α}ᵀ.
The right metric is antisymmetric {εR | β β' = - εR | β' β}ᵀ.
The alt-left metric is antisymmetric {εL' | α α' = - εL' | α' α}ᵀ.
The alt-right metric is antisymmetric {εR' | β β' = - εR' | β' β}ᵀ.
Contractions with each other #
The contraction of the covariant metric with the contravariant metric is the unit
{η' | μ ρ ⊗ η | ρ ν = δ' | μ ν}ᵀ.
The contraction of the contravariant metric with the covariant metric is the unit
{η | μ ρ ⊗ η' | ρ ν = δ | μ ν}ᵀ.
The contraction of the left metric with the alt-left metric is the unit
{εL | α β ⊗ εL' | β γ = δL | α γ}ᵀ.
The contraction of the right metric with the alt-right metric is the unit
{εR | α β ⊗ εR' | β γ = δR | α γ}ᵀ.
The contraction of the alt-left metric with the left metric is the unit
{εL' | α β ⊗ εL | β γ = δL' | α γ}ᵀ.
The contraction of the alt-right metric with the right metric is the unit
{εR' | α β ⊗ εR | β γ = δR' | α γ}ᵀ.