Physlib.Relativity.Tensors.RealTensor.Contraction.CrossToEnd

Let $d$ be the number of spatial dimensions in a $(1+d)$-dimensional spacetime. Let $t$ and $M$ be real Lorentz tensors with index color sequences $c_A$ (of length $n_A+1$) and $c_B$ (of length $n_B+1$) respectively. Let $i \in \{0, \dots, n_A\}$ and $j \in \{0, \dots, n_B\}$ be the indices of the slots to be contracted, such that the color of the $i$-th slot of $t$ is dual to the color of the $j$-th slot of $M$. For any multi-index $\phi$ of the resulting tensor of rank $n_A + n_B$, the component of the contraction $t \otimes_{i,j} M$ is given by the sum: $$[(t \otimes_{i,j} M)]_\phi = \sum_{x=0}^{d} t_{\text{ins}_i(x, \phi_{\text{left}})} \cdot M_{\text{ins}_j(x, \phi_{\text{right}})}$$ where $\text{ins}_k(x, \dots)$ denotes the operation of inserting the summation index $x$ into the $k$-th position of a multi-index, $\phi_{\text{left}}$ denotes the first $n_A$ indices of $\phi$, and $\phi_{\text{right}}$ denotes the remaining $n_B$ indices of $\phi$. This formula demonstrates that the contraction of real Lorentz tensors reduces to a single finite sum over the Lorentz indices.