Physlib.Relativity.Tensors.RealTensor.Matrix.Pre

For any spatial dimension \( d \in \mathbb{N} \), there exists a linear equivalence between the vector space of the tensor product of two contravariant Lorentz vectors, denoted \( (\text{Contr} \ d \otimes \text{Contr} \ d).V \), and the space of real square matrices of size \( (1+d) \times (1+d) \). This equivalence maps a tensor in the product space to its matrix representation relative to the basis formed by the tensor product of the contravariant bases.

Let $M$ be a real $(1+d) \times (1+d)$ matrix whose indices $i, j$ range over $\text{Fin } 1 \oplus \text{Fin } d$ (representing temporal and spatial components). The inverse of the linear equivalence `contrContrToMatrixRe`, which maps a matrix to an element of the tensor product of two contravariant Lorentz vector spaces $(\text{Contr } d \otimes \text{Contr } d).V$, is given by the sum: $$\text{contrContrToMatrixRe}^{-1}(M) = \sum_{i} \sum_{j} M_{ij} (e_i \otimes e_j),$$ where $e_i$ and $e_j$ are the $i$-th and $j$-th basis vectors of the contravariant Lorentz space $\text{Contr } d$, respectively.

For a given dimension $d \in \mathbb{N}$, the linear equivalence `coCoToMatrixRe` identifies the tensor product of two covariant Lorentz vector spaces, $V_{\text{Co } d} \otimes_{\mathbb{R}} V_{\text{Co } d}$, with the space of $(1+d) \times (1+d)$ real matrices $\text{Mat}_{(1+d) \times (1+d)}(\mathbb{R})$. The matrix indices are taken from the set $\text{Fin } 1 \oplus \text{Fin } d$, representing the temporal and spatial components of a $1+d$ dimensional spacetime. The mapping is constructed by taking the representation of a tensor in the product basis $\text{coBasis } d \otimes \text{coBasis } d$ and currying the resulting coefficients into a matrix format.

For a real matrix $M$ of size $(1+d) \times (1+d)$ indexed by $i, j \in \text{Fin } 1 \oplus \text{Fin } d$, the inverse of the linear equivalence $\text{coCoToMatrixRe}$ (denoted as $\text{coCoToMatrixRe}^{-1}$) maps $M$ to the tensor product of covariant basis vectors according to the expansion: $$\text{coCoToMatrixRe}^{-1}(M) = \sum_{i} \sum_{j} M_{ij} \cdot (e_i \otimes e_j)$$ where $\{e_k\}$ denotes the standard basis for the covariant Lorentz vector space (`coBasis d`).

For a given spatial dimension $d \in \mathbb{N}$, the linear isomorphism $\text{contrCoToMatrixRe}$ identifies the tensor product of the space of contravariant Lorentz vectors $V$ (denoted as `Contr d`) and covariant Lorentz vectors $V^*$ (denoted as `Co d`) with the space of $(1+d) \times (1+d)$ real matrices $\text{Mat}_{1+d}(\mathbb{R})$. It maps a tensor product of vectors to its representation as a matrix relative to the standard bases of $V$ and $V^*$.

For any real matrix $M$ of size $(1+d) \times (1+d)$, the inverse of the linear isomorphism $\text{contrCoToMatrixRe}$ (which identifies the tensor product of contravariant and covariant Lorentz vectors with the space of matrices) is given by the basis expansion: $$\text{contrCoToMatrixRe}^{-1}(M) = \sum_{i, j} M_{i,j} (e_i \otimes \epsilon^j)$$ where $e_i$ is the $i$-th basis vector of the contravariant Lorentz space (`contrBasis`) and $\epsilon^j$ is the $j$-th basis vector of the covariant Lorentz space (`coBasis`), with indices $i, j \in \{0, 1, \dots, d\}$.

The definition establishes a linear equivalence (isomorphism of vector spaces) between the tensor product of the covariant Lorentz vector space $\text{Co } d$ and the contravariant Lorentz vector space $\text{Contr } d$ and the space of $(1+d) \times (1+d)$ real matrices. The indices of the matrices are taken from the set $\{0, 1, \dots, d\}$, represented by the direct sum of finite types $\text{Fin } 1 \oplus \text{Fin } d$.

Let $M$ be a real $(1+d) \times (1+d)$ matrix whose indices $i, j$ are taken from the set $\{0, 1, \dots, d\}$ (represented by the type $\text{Fin } 1 \oplus \text{Fin } d$). Let $\phi : \text{Co } d \otimes \text{Contr } d \simeq \text{Mat}_{(1+d) \times (1+d)}(\mathbb{R})$ be the linear equivalence `coContrToMatrixRe` between the tensor product of covariant and contravariant Lorentz vector spaces and the space of matrices. The inverse of this map, $\phi^{-1}$, applied to $M$ is equal to the sum of the tensor products of the basis vectors weighted by the matrix entries: $$\phi^{-1}(M) = \sum_{i} \sum_{j} M_{i,j} (\omega_i \otimes v^j)$$ where $\{\omega_i\}$ is the standard basis for the covariant Lorentz vector space $\text{Co } d$ and $\{v^j\}$ is the standard basis for the contravariant Lorentz vector space $\text{Contr } d$.

Let $d$ be a natural number representing the spatial dimension. Let $\text{Contr} \ d$ denote the space of contravariant Lorentz vectors, and let $\Phi$ be the linear equivalence that maps the tensor product $\text{Contr} \ d \otimes \text{Contr} \ d$ to the space of $(1+d) \times (1+d)$ real matrices. For any tensor $v$ in the tensor product space and any Lorentz transformation $M \in \text{LorentzGroup} \ d$, the matrix representation of the tensor transformed by the action of $M$ on both components is given by: $$\Phi((\rho(M) \otimes \rho(M))v) = M \Phi(v) M^T$$ where $\rho(M)$ is the representation of the Lorentz group on contravariant vectors, $M$ is the matrix of the Lorentz transformation, and $M^T$ is its transpose.

For any spatial dimension $d \in \mathbb{N}$, let $v \in V_{\text{Co } d} \otimes_{\mathbb{R}} V_{\text{Co } d}$ be a tensor in the product of two covariant Lorentz vector spaces. Let $M$ be an element of the Lorentz group with matrix representation $\Lambda$. The linear equivalence `coCoToMatrixRe`, which identifies tensors with $(1+d) \times (1+d)$ real matrices, satisfies the following transformation law under the Lorentz group action $\rho$: $$\text{coCoToMatrixRe}((\rho(M) \otimes \rho(M))v) = (\Lambda^{-1})^T (\text{coCoToMatrixRe } v) \Lambda^{-1}$$ where $\rho(M)$ is the representation of the Lorentz group acting on the covariant vector space.

For a spatial dimension $d$, let $V$ and $V^*$ denote the spaces of contravariant and covariant Lorentz vectors, respectively. Let $\Phi: V \otimes V^* \cong \text{Mat}_{1+d}(\mathbb{R})$ be the linear isomorphism (denoted as `contrCoToMatrixRe`) that identifies a $(1,1)$-tensor with its matrix representation relative to the standard bases. For any tensor $v \in V \otimes V^*$ and any Lorentz transformation $M \in \text{LorentzGroup}(d)$, the matrix representation of the tensor transformed by the action of $M$ is given by $$\Phi((\rho_V(M) \otimes \rho_{V^*}(M))v) = M \Phi(v) M^{-1}$$ where $\rho_V(M)$ and $\rho_{V^*}(M)$ are the representations of the Lorentz group acting on contravariant and covariant vectors, and $M$ on the right-hand side denotes the matrix representing the Lorentz transformation.

Let $d$ be a natural number. For any tensor $v$ in the tensor product of the covariant Lorentz vector space and the contravariant Lorentz vector space $(\text{Co } d \otimes \text{Contr } d)$, and for any Lorentz transformation $M$ in the Lorentz group $O(1, d)$, the matrix representation of the transformed tensor satisfies: $$\Phi((\rho_{\text{co}}(M) \otimes \rho_{\text{contr}}(M)) v) = (M^{-1})^T \Phi(v) M^T$$ where $\Phi$ denotes the linear isomorphism `coContrToMatrixRe` mapping the tensor to a $(1+d) \times (1+d)$ real matrix, $M^T$ is the transpose of the matrix $M$, $M^{-1}$ is its inverse, and $\rho_{\text{co}}, \rho_{\text{contr}}$ are the representation maps for the covariant and contravariant Lorentz spaces respectively.

Let $d$ be a natural number representing the spatial dimension. Let $\Phi : (\text{Contr } d \otimes \text{Contr } d) \cong \text{Mat}_{1+d}(\mathbb{R})$ be the linear equivalence (denoted as `contrContrToMatrixRe`) that maps the tensor product of two contravariant Lorentz vector spaces to the space of $(1+d) \times (1+d)$ real matrices. For any matrix $v \in \text{Mat}_{1+d}(\mathbb{R})$ and any Lorentz transformation $M \in \text{LorentzGroup}(d)$, the action of the Lorentz group on the tensor corresponding to $v$ satisfies: $$(\rho(M) \otimes \rho(M))(\Phi^{-1}(v)) = \Phi^{-1}(M v M^T)$$ where $\rho(M)$ is the representation of the Lorentz group on contravariant vectors, $\Phi^{-1}$ is the inverse mapping from matrices to tensors, and $M^T$ is the transpose of the matrix $M$.

Let $d$ be a natural number representing the spatial dimension. Let $\Phi : (V_{\text{Co } d} \otimes_{\mathbb{R}} V_{\text{Co } d}) \cong \text{Mat}_{1+d}(\mathbb{R})$ be the linear equivalence (denoted as `coCoToMatrixRe`) that maps the tensor product of two covariant Lorentz vector spaces to the space of $(1+d) \times (1+d)$ real matrices. For any matrix $v \in \text{Mat}_{1+d}(\mathbb{R})$ and any Lorentz transformation $M \in \text{LorentzGroup}(d)$ with matrix representation $\Lambda$, the action of the Lorentz group on the tensor corresponding to $v$ satisfies: $$(\rho(M) \otimes \rho(M))(\Phi^{-1}(v)) = \Phi^{-1}((\Lambda^{-1})^T v \Lambda^{-1})$$ where $\rho(M)$ is the representation of the Lorentz group on covariant vectors, $\Phi^{-1}$ is the inverse mapping from matrices to tensors (denoted as `coCoToMatrixRe.symm`), and $(\Lambda^{-1})^T$ is the transpose of the inverse of the matrix $\Lambda$.

For a given spatial dimension $d \in \mathbb{N}$, let $V$ and $V^*$ denote the spaces of contravariant and covariant Lorentz vectors, respectively. Let $\Phi : V \otimes V^* \cong \text{Mat}_{1+d}(\mathbb{R})$ be the linear isomorphism (denoted as `contrCoToMatrixRe`) that identifies a $(1,1)$-tensor with its matrix representation relative to the standard bases. For any matrix $v \in \text{Mat}_{1+d}(\mathbb{R})$ and any Lorentz transformation $M \in \text{LorentzGroup}(d)$, the action of the Lorentz group on the tensor corresponding to $v$ satisfies: $$(\rho_V(M) \otimes \rho_{V^*}(M))(\Phi^{-1}(v)) = \Phi^{-1}(M v M^{-1})$$ where $\Phi^{-1}$ is the inverse mapping from matrices to tensors (denoted as `contrCoToMatrixRe.symm`), and $\rho_V(M)$ and $\rho_{V^*}(M)$ are the representations of the Lorentz group acting on contravariant and covariant vectors, respectively.

Let $d$ be a natural number representing the spatial dimension. Let $\Phi: (\text{Co } d \otimes \text{Contr } d) \to \text{Mat}_{(1+d) \times (1+d)}(\mathbb{R})$ be the linear isomorphism `coContrToMatrixRe` that maps a tensor from the product of covariant and contravariant Lorentz spaces to a real matrix. For any $(1+d) \times (1+d)$ matrix $v$ and any Lorentz transformation $M$ in the Lorentz group $O(1, d)$, the group action on the corresponding tensor satisfies: $$(\rho_{\text{co}}(M) \otimes \rho_{\text{contr}}(M)) (\Phi^{-1}(v)) = \Phi^{-1}((M^{-1})^T v M^T)$$ where $\Phi^{-1}$ denotes the inverse isomorphism (`coContrToMatrixRe.symm`), $\rho_{\text{co}}$ and $\rho_{\text{contr}}$ are the representation maps for the covariant and contravariant Lorentz spaces, $M^T$ is the transpose of the matrix $M$, and $M^{-1}$ is its inverse.