Physlib.Relativity.Tensors.RealTensor.Representation.Contraction

For a given natural number $d$ representing spatial dimensions, this defines the contraction of a Lorentz vector and a Lorentz covector. It is an intertwining linear map from the tensor product of the Lorentz vector representation $\text{Vector}_d$ and the Lorentz covector representation $\text{CoVector}_d$ to the trivial representation on $\mathbb{R}$. For a Lorentz vector $\phi$ and a Lorentz covector $\psi$, the contraction is defined by the sum of their components: $$\text{contract}(\phi \otimes \psi) = \sum_{i} \phi_i \psi_i$$ where the index $i$ ranges over the $(1+d)$-dimensional spacetime. As an intertwining map, this contraction is invariant under the action of the Lorentz group $\mathcal{L}$.

This definition provides the contraction map as an intertwining map between the tensor product of the space of covariant Lorentz vectors $\text{CoVector}(d)$ and contravariant Lorentz vectors $\text{Vector}(d)$ to the real numbers $\mathbb{R}$, where $\mathbb{R}$ is equipped with the trivial representation of the Lorentz group $\mathcal{L}$. For a covariant vector $\phi \in \text{CoVector}(d)$ and a contravariant vector $\psi \in \text{Vector}(d)$, the contraction is defined on pure tensors $\phi \otimes \psi$ as the sum $\sum_i \phi_i \psi_i$. The map is an intertwining map, meaning it is invariant under the action of the Lorentz group.

For any Lorentz vector $\phi \in \text{Vector}_d$ and Lorentz covector $\psi \in \text{CoVector}_d$ in a $(1+d)$-dimensional spacetime, the contraction of their tensor product $\phi \otimes \psi$ is given by the sum of the products of their components: $$\text{contract}(\phi \otimes \psi) = \sum_{i} \phi_i \psi_i$$ where the index $i$ ranges over the $(1+d)$ dimensions of the spacetime.

For a covariant Lorentz vector $\phi \in \text{CoVector}(d)$ and a contravariant Lorentz vector $\psi \in \text{Vector}(d)$, the contraction of their pure tensor product $\phi \otimes \psi$ is equal to the sum of the products of their components: $$\text{contract}(\phi \otimes \psi) = \sum_i \phi_i \psi_i$$ where the summation index $i$ ranges over the $(1+d)$ dimensions of spacetime.

For a natural number $d$ representing spatial dimensions, let $\mu \in \text{Fin } 1 \oplus \text{Fin } d$ be an index and $\psi \in \text{CoVector}(d)$ be a Lorentz covector. Let $e_\mu$ denote the $\mu$-th standard basis vector of the Lorentz vector space $\text{Vector}(d)$. The contraction of $e_\mu$ with $\psi$ is equal to the $\mu$-th component of $\psi$: $$\text{contract}(e_\mu \otimes \psi) = \psi_\mu$$

For a $(1+d)$-dimensional spacetime, let $\mu \in \{0\} \cup \{1, \dots, d\}$ be a spacetime index and let $e_\mu$ be the $\mu$-th standard basis element of the space of Lorentz covectors $\text{CoVector}(d)$. For any Lorentz vector $\varphi \in \text{Vector}(d)$, the contraction of the basis covector $e_\mu$ with $\varphi$ is equal to the $\mu$-th component of $\varphi$: $$\text{contract}(e_\mu \otimes \varphi) = \varphi_\mu$$

For any Lorentz vector $\phi$ in a $(1+d)$-dimensional spacetime and any spacetime index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the contraction of $\phi$ with the $\mu$-th basis covector $e^\mu$ of $\text{CoVector}_d$ is equal to the $\mu$-th component of the vector $\phi$. Mathematically, this is expressed as: $$\text{contract}(\phi \otimes e^\mu) = \phi_\mu$$ where $\otimes$ denotes the tensor product and $\phi_\mu$ denotes the component of $\phi$ at index $\mu$.

For a covariant Lorentz vector $\psi \in \text{CoVector}(d)$ and a spacetime index $\mu \in \{0\} \cup \{1, \dots, d\}$, the contraction of $\psi$ with the $\mu$-th standard basis vector $e_\mu$ of the contravariant Lorentz vector space $\text{Vector}(d)$ is equal to the $\mu$-th component of $\psi$. Mathematically, this is expressed as $\text{contract}(\psi \otimes e_\mu) = \psi_\mu$.

For any natural number $d$ representing spatial dimensions, given a contravariant Lorentz vector $\phi \in \text{Vector}(d)$ and a covariant Lorentz vector $\psi \in \text{CoVector}(d)$, the contraction of the vector $\phi$ with the covector $\psi$ is equal to the contraction of the covector $\psi$ with the vector $\phi$: $$\text{Vector.contract}(\phi \otimes \psi) = \text{CoVector.contract}(\psi \otimes \phi)$$ Both sides of the equation represent the scalar sum $\sum_i \phi_i \psi_i$ over the spacetime indices.

For any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$ of $d$ spatial dimensions, and for any Lorentz vector $\phi \in \text{Vector}_d$ and Lorentz covector $\psi \in \text{CoVector}_d$, the contraction of the transformed vector and transformed covector is equal to the contraction of the original vector and covector: $$\text{contract}(\rho_V(\Lambda)\phi \otimes \rho_{V^*}(\Lambda)\psi) = \text{contract}(\phi \otimes \psi)$$ where $\rho_V$ and $\rho_{V^*}$ are the representations of the Lorentz group acting on the space of vectors and covectors, respectively.

For any Lorentz transformation $\Lambda \in \mathcal{L}$ in $d$ spatial dimensions, any Lorentz covector $\phi \in \text{CoVector}(d)$, and any Lorentz vector $\psi \in \text{Vector}(d)$, the contraction of the transformed covector and vector is equal to the contraction of the original covector and vector: $$\text{contract}(\rho_{\text{co}}(\Lambda)\phi \otimes \rho_{\text{vec}}(\Lambda)\psi) = \text{contract}(\phi \otimes \psi)$$ where $\rho_{\text{co}}$ and $\rho_{\text{vec}}$ denote the representations of the Lorentz group on the space of covectors and vectors, respectively.