Physlib.Relativity.Tensors.OfInt

A tensor with integer components with respect to the basis. Given a natural number \( n \) and an index configuration \( c : \{0, 1, \dots, n-1\} \to C \), it is defined as a function mapping the component indices associated with \( c \) to the integers \( \mathbb{Z} \).

Given a tensor with integer components $f \in \text{TensorInt}(S, c)$, where $f$ is a function mapping the set of component indices $\text{ComponentIdx}(c)$ to $\mathbb{Z}$, this function constructs the corresponding element in the tensor space $S.\text{Tensor}(c)$. It does so by casting each integer component $f(b)$ to the underlying field $k$ and using these as the coordinates for the basis of the tensor space.

Let $n$ be a natural number and $c: \{0, 1, \dots, n-1\} \to C$ be an index configuration. Let $f$ be a function mapping the component indices $\text{ComponentIdx}(c)$ to the integers $\mathbb{Z}$ (an element of $\text{TensorInt}(S, c)$). Let $\text{toTensor}(f)$ be the tensor in the space $S.\text{Tensor}(c)$ obtained by casting these integer components to the underlying field. Then, for any basis index $b \in \text{ComponentIdx}(c)$, the $b$-th coordinate of $\text{toTensor}(f)$ with respect to the basis $\text{Tensor.basis}(c)$ is equal to the integer $f(b)$ cast into the field.

Let $n$ be a natural number and $c: \{0, 1, \dots, n-1\} \to C$ be an index configuration. For any component index $b \in \text{ComponentIdx}(c)$, the $b$-th element of the basis for the tensor space $S.\text{Tensor}(c)$ is equal to the tensor $\text{toTensor}(f)$ constructed from the integer-valued function $f: \text{ComponentIdx}(c) \to \mathbb{Z}$ defined by $f(b') = 1$ if $b' = b$ and $f(b') = 0$ otherwise.