Physlib.Relativity.Tensors.ComplexTensor.Metrics.Lemmas

The covariant metric tensor $\eta'_{\mu\nu}$ of the complex Lorentz species is symmetric with respect to its indices $\mu$ and $\nu$, satisfying $\eta'_{\mu\nu} = \eta'_{\nu\mu}$.

The contravariant metric tensor $\eta^{\mu\nu}$ is symmetric with respect to its indices, satisfying the relation $\eta^{\mu\nu} = \eta^{\nu\mu}$.

The left metric tensor $\varepsilon_L$ is antisymmetric with respect to its indices $\alpha$ and $\alpha'$. That is, the tensor satisfies the identity: $$\varepsilon_{L, \alpha \alpha'} = - \varepsilon_{L, \alpha' \alpha}$$ where $\alpha$ and $\alpha'$ denote the indices of the tensor.

The right metric tensor $\varepsilon_R$ for complex Lorentz tensors is antisymmetric with respect to its indices $\beta$ and $\beta'$, satisfying the identity $\varepsilon_R |_{\beta \beta'} = - \varepsilon_R |_{\beta' \beta}$.

The alt-left metric tensor, denoted as $\epsilon_{L'}$, is antisymmetric with respect to its indices $\alpha$ and $\alpha'$. Specifically, the tensor identity holds: $$\epsilon_{L' \alpha \alpha'} = -\epsilon_{L' \alpha' \alpha}$$ where $\epsilon_{L'}$ represents the alt-left metric in the context of complex Lorentz tensors.

The alt-right metric tensor $\epsilon_{R'}$ is antisymmetric with respect to its indices. That is, for any indices $\alpha$ and $\alpha'$, the components satisfy $\epsilon_{R' \alpha \alpha'} = -\epsilon_{R' \alpha' \alpha}$.

In the context of complex Lorentz tensors, the contraction of the covariant metric tensor $\eta_{\mu\rho}$ with the contravariant metric tensor $\eta^{\rho\nu}$ over the shared index $\rho$ results in the mixed Kronecker delta tensor $\delta_\mu^\nu$. This identity is expressed as: $$\eta_{\mu\rho} \eta^{\rho\nu} = \delta_\mu^\nu$$ where the contraction is performed on the second index of the covariant metric and the first index of the contravariant metric.

In the context of complex Lorentz tensors, the contraction of the contravariant metric tensor $\eta^{\mu\rho}$ with the covariant metric tensor $\eta_{\rho\nu}$ over the shared index $\rho$ is equal to the unit tensor (Kronecker delta) $\delta^\mu_\nu$.

For the complex Lorentz tensor species, the contraction of the left metric $\epsilon_L$ (with indices $\alpha, \beta$) and the alt-left metric $\epsilon_L'$ (with indices $\beta, \gamma$) over the shared index $\beta$ results in the left unit tensor $\delta_L$ (with indices $\alpha, \gamma$). In tensorial notation, this is expressed as: $$\{\epsilon_L |_{\alpha \beta} \otimes \epsilon_L' |_{\beta \gamma} = \delta_L |_{\alpha \gamma}\}^\text{T}$$ where the indices $\alpha, \beta, \gamma$ represent the color configurations of the respective tensor slots.

The contraction of the right metric tensor $\epsilon_R$ and the alt-right metric tensor $\epsilon'_R$ over their shared index $\beta$ is equal to the right unit tensor $\delta_R$. In index notation, this is expressed as $\{\epsilon_{R, \alpha \beta} \otimes \epsilon'_{R, \beta \gamma} = \delta_{R, \alpha \gamma}\}^T$.

Let $\varepsilon'_L$ denote the alt-left metric tensor and $\varepsilon_L$ denote the left metric tensor. The contraction of the tensor product $\varepsilon'_L \otimes \varepsilon_L$ over the second index of $\varepsilon'_L$ and the first index of $\varepsilon_L$ is equal to the unit tensor $\delta'_L$. This identity is expressed as $(\varepsilon'_L)_{\alpha \beta} (\varepsilon_L)_{\beta \gamma} = (\delta'_L)_{\alpha \gamma}$, where $\alpha, \beta, \gamma$ are indices.

In the context of complex Lorentz tensors, the contraction of the alt-right metric $\epsilon_{R'}$ and the right metric $\epsilon_R$ over a common index results in the unit tensor $\delta_{R'}$. Specifically, the contraction of the tensor product $\epsilon_{R' \alpha \beta} \otimes \epsilon_{R \beta \gamma}$ over the shared index $\beta$ is equal to the Kronecker delta (unit tensor) $\delta_{R' \alpha \gamma}$.