Physlib.Relativity.Tensors.MetricTensor

The metric tensor associated with a color \( c \). It is a tensor of shape \( (c, c) \).

Let $S$ be a tensor species and let $g_c$ denote the metric tensor associated with a color $c$. For any colors $c$ and $c_1$ such that $c = c_1$, the metric tensor $g_c$ is equal to the metric tensor $g_{c_1}$ subject to the identity permutation $\text{perm}_{\text{id}}$ (which accounts for the change in index types).

For any color $c \in C$ and any element $g$ of the group $G$, the metric tensor associated with $c$ in the tensor species $S$ is invariant under the action of $g$. That is, $g \cdot \text{metricTensor}(c) = \text{metricTensor}(c)$.

For any color $c$ in a tensor species $S$ with color set $C$ and duality map $\tau$, let $g_c$ and $g_{\tau(c)}$ be the metric tensors associated with $c$ and its dual $\tau(c)$, respectively. Then, applying a permutation that swaps the two indices (0 and 1) to the tensor formed by the contraction of the pair $(g_c, g_{\tau(c)})$ results in the unit tensor $\mathbb{1}_c$ of shape $(\tau(c), c)$.

For any color $c$ in a tensor species $S$ with duality map $\tau$, let $g_c$ and $g_{\tau(c)}$ be the metric tensors associated with $c$ and its dual $\tau(c)$, respectively. The contraction of the pair of tensors $(g_c, g_{\tau(c)})$ is equal to the unit tensor $\mathbb{1}_c$ after its indices 0 and 1 have been permuted.

For any color $c$ in a tensor species $S$ with duality map $\tau$, let $g_c$ and $g_{\tau(c)}$ be the metric tensors associated with $c$ and its dual $\tau(c)$, respectively. The contraction of the tensor product $g_c \otimes g_{\tau(c)}$ over the second index of $g_c$ and the first index of $g_{\tau(c)}$ is equal to the unit tensor $\mathbb{1}_c$ after its indices 0 and 1 have been permuted.

For any color $c$ in a tensor species $S$ with duality map $\tau$, let $g_c$ and $g_{\tau(c)}$ be the metric tensors associated with $c$ and its dual $\tau(c)$, respectively. The contraction of the tensor product $g_c \otimes g_{\tau(c)}$ over the second index of $g_c$ and the first index of $g_{\tau(c)}$ is equal to the unit tensor associated with the dual color $\tau(c)$, denoted $\mathbb{1}_{\tau(c)}$.

For any color $c$ in a tensor species $S$ with duality map $\tau$, let $g_{\tau(c)}$ and $g_c$ be the metric tensors associated with the dual color $\tau(c)$ and the color $c$, respectively. The contraction of the tensor product $g_{\tau(c)} \otimes g_c$ over the second index of $g_{\tau(c)}$ and the first index of $g_c$ is equal to the unit tensor $\mathbb{1}_c$ (subject to the identity permutation of indices).