Physlib.Relativity.Tensors.ComplexTensor.Lemmas

1 declaration

theorem

$A^{\mu\nu} S_{\mu\nu} = -A^{\mu\nu} S_{\mu\nu}$ for antisymmetric $A$ and symmetric $S$

Let $A$ be a complex Lorentz tensor with two contravariant indices ( $A \in \mathbb{C}T^{\mu\nu}$ ) and $S$ be a complex Lorentz tensor with two covariant indices ( $S \in \mathbb{C}T_{\mu\nu}$ ). If $A$ is antisymmetric, such that $A^{\mu\nu} = -A^{\nu\mu}$ , and $S$ is symmetric, such that $S_{\mu\nu} = S_{\nu\mu}$ , then their full contraction satisfies the identity $A^{\mu\nu} S_{\mu\nu} = -A^{\mu\nu} S_{\mu\nu}$ .

Physlib.Relativity.Tensors.ComplexTensor.Lemmas

AμνSμν=−AμνSμνA^{\mu\nu} S_{\mu\nu} = -A^{\mu\nu} S_{\mu\nu}AμνSμν​=−AμνSμν​ for antisymmetric AAA and symmetric SSS

$A^{\mu\nu} S_{\mu\nu} = -A^{\mu\nu} S_{\mu\nu}$ for antisymmetric $A$ and symmetric $S$