Physlib.Relativity.Tensors.RealTensor.CoVector.Representation

For a $(1+d)$-dimensional spacetime, this defines the representation of the Lorentz group $\mathcal{L}$ on the vector space of Lorentz covectors $\text{CoVector}(d)$ over $\mathbb{R}$. The representation maps each Lorentz transformation $\Lambda \in \mathcal{L}$ to a linear automorphism of $\text{CoVector}(d)$ such that for a covector $v$, the transformed covector is given by the matrix-vector product $(\Lambda^{-1})^T v$, where $\Lambda^{-1}$ is the group inverse and $T$ denotes the matrix transpose.

For any natural number $d$ representing spatial dimensions, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ and $v$ be a Lorentz covector in $\text{CoVector}(d)$. The action of the Lorentz group representation on the covector $v$, denoted as $\text{rep}(\Lambda) v$, is equal to the matrix-vector product of the transpose of the inverse of $\Lambda$ and $v$: $$\text{rep}(\Lambda) v = (\Lambda^{-1})^T v$$

For a $(1+d)$-dimensional spacetime, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ and $v$ be a Lorentz covector. The $k$-th component of the transformed covector under the Lorentz representation is given by the sum $$(\text{rep}(\Lambda)v)_k = \sum_{j} (\Lambda^{-1})_{jk} v_j$$ where $(\Lambda^{-1})_{jk}$ is the $(j, k)$-th entry of the inverse matrix of $\Lambda$, and $v_j$ is the $j$-th component of the covector $v$.

For a $(1+d)$-dimensional spacetime, given a Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, a Lorentz covector $v \in \text{CoVector}(d)$, and a spacetime index $k$, the $k$-th component of the transformed covector $(\text{rep } \Lambda v)_k$ is equal to the sum $\sum_j (\Lambda^{-1})_{jk} v_j$, where $(\Lambda^{-1})_{jk}$ is the entry in the $j$-th row and $k$-th column of the inverse matrix of $\Lambda$.

For a $(1+d)$-dimensional spacetime, let $\{e_\mu\}$ be the standard basis for the space of Lorentz covectors $\text{CoVector}(d)$. For any Lorentz transformation $\Lambda$ and any index $\mu \in \{0, \dots, d\}$, the action of the representation of $\Lambda$ on the basis covector $e_\mu$ is given by $$\text{rep}(\Lambda)(e_\mu) = \sum_j (\Lambda^{-1})_{\mu j} e_j$$ where $(\Lambda^{-1})_{\mu j}$ denotes the $(\mu, j)$-th entry of the inverse matrix of $\Lambda$.

For a $(1+d)$-dimensional spacetime, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$. The matrix representation of the action of $\Lambda$ on the space of Lorentz covectors $\text{CoVector}(d)$ (the representation $\rho(\Lambda)$) with respect to the standard basis is equal to the transpose of the inverse of $\Lambda$, denoted as $(\Lambda^{-1})^T$.

For any natural number $d$ and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$ for $(1+d)$-dimensional spacetime, the linear map representing $\Lambda$ on the space of Lorentz covectors $\text{CoVector}(d)$, denoted as $\rho(\Lambda)$, is injective.

For any natural number $d$ representing spatial dimensions and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the linear map $\rho(\Lambda)$ representing the action of $\Lambda$ on the space of Lorentz covectors $\text{CoVector}(d)$ is surjective.

For any natural number $d$ representing the spatial dimensions of a $(1+d)$-dimensional spacetime, and for any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the linear map defined by the representation of $\Lambda$ on the space of Lorentz covectors $\text{CoVector}(d)$ is a bijective function.