Physlib.Relativity.Tensors.ComponentIdx.Pair

Given a sequence of two index types (or "colors") $c : \{0, 1\} \to C$, there is an equivalence between the type of component indices for a tensor with these colors, $\text{ComponentIdx}(c)$, and the Cartesian product of the individual basis index sets $\text{basisIdx}(c_0) \times \text{basisIdx}(c_1)$. This equivalence maps a functional index $b$ to the pair $(b_0, b_1)$.

For a tensor with two index types $c_0$ and $c_1$ (given by a mapping $cs : \{0, 1\} \to C$), let $b$ be a component index. The equivalence $\text{ComponentIdx.pair}$ maps the functional index $b$ to the ordered pair of its components $(b_0, b_1)$.

Given a sequence of two index types (or "colors") $c: \{0, 1\} \to C$, let $I_0 = \text{basisIdx}(c_0)$ and $I_1 = \text{basisIdx}(c_1)$ be the sets of basis indices for each color. Let $\text{pair} : \text{ComponentIdx}(c) \cong I_0 \times I_1$ be the equivalence mapping a multi-index $b$ to the pair $(b(0), b(1))$. For any pair of indices $x = (x_1, x_2) \in I_0 \times I_1$, the multi-index $b$ obtained from the inverse mapping $\text{pair}^{-1}(x)$ satisfies $b(0) = x_1$.

Let $cs: \{0, 1\} \to C$ be a sequence of two index types (colors). Let $\text{pair}$ be the equivalence between the type of component indices for a tensor with these colors, $\text{ComponentIdx}(cs)$, and the Cartesian product of the individual basis index sets $\text{basisIdx}(cs_0) \times \text{basisIdx}(cs_1)$. For any pair of basis indices $x = (x_0, x_1)$ in this product, applying the inverse equivalence $\text{pair}^{-1}$ to $x$ and evaluating the resulting component index function at index $1$ yields the second component of the pair, $x_1$.