Physlib.Relativity.Tensors.RealTensor.Vector.Pre.Basic

For a given natural number $d$ representing spatial dimensions, $\text{Lorentz.Contr } d$ defines the representation of the Lorentz group $\mathcal{L}$ on the space of real contravariant Lorentz vectors $V \cong \mathbb{R}^{1+d}$. In physical index notation, these vectors are characterized by an upper index $\psi^i$, and the representation corresponds to the standard action of the group matrices on the vector space.

For a given number of spatial dimensions $d$, let $M$ be an element of the Lorentz group $\mathcal{L}$ (a $(1+d) \times (1+d)$ matrix). Let $\rho(M)$ be the linear transformation corresponding to $M$ in the contravariant representation of the Lorentz group. The matrix of the linear map $\rho(M)$ with respect to the standard contravariant basis $\text{contrBasis } d$ is equal to the matrix $M$. Specifically, for any indices $i, j \in \{0, 1, \dots, d\}$, the $(i, j)$-th entry of the matrix of $\rho(M)$ is equal to the $(i, j)$-th entry of $M$.

Let $d$ be the number of spatial dimensions, such that the spacetime dimension is $1+d$. For any index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the coordinate representation of the $i$-th basis vector of the contravariant Lorentz vector space $\text{Contr } d$ is the standard basis vector $\mathbf{e}_i$ in $\mathbb{R}^{1+d}$, which is defined as having the value 1 at index $i$ and 0 elsewhere.

Let $d$ be the number of spatial dimensions, such that the total spacetime dimension is $1+d$. For any index $i \in \{0, 1, \dots, d\}$, the coordinate representation of the $i$-th basis vector of the contravariant Lorentz vector space (denoted $\text{contrBasisFin } d \ i$) is the standard basis vector $\mathbf{e}_j$ in the coordinate space $\mathbb{R}^{1+d} \cong (\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$. Here, $j$ is the index in $\text{Fin } 1 \oplus \text{Fin } d$ corresponding to $i$ via the canonical isomorphism $\text{finSumFinEquiv}^{-1}$, and $\mathbf{e}_j$ is the vector with the value 1 at index $j$ and 0 elsewhere.

Let $d$ be the number of spatial dimensions. For any contravariant Lorentz vector $p \in \text{Contr } d$ and any index $i \in \{0, 1, \dots, d\}$, the $i$-th coordinate of $p$ with respect to the basis $\text{contrBasisFin } d$ is equal to the value of the vector $p$ at the spacetime index corresponding to $i$ (via the isomorphism $\text{finSumFinEquiv}$ between $\{0, \dots, d\}$ and $\text{Fin } 1 \oplus \text{Fin } d$).

For a given natural number $d$ representing the number of spatial dimensions, the space of contravariant real Lorentz vectors $\text{Contr } d$ is equipped with a topological space structure. This topology is defined as the induced topology (pullback topology) from the standard product topology on the coordinate space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ via the canonical linear isomorphism $\text{ContrMod } d \cong \mathbb{R}^{1+d}$.

Let $T$ be a topological space and let $\text{Contr } d$ denote the space of real contravariant Lorentz vectors in $1+d$ dimensions. A function $f: T \to \text{Contr } d$ is continuous if its coordinate representation, defined by the mapping $t \mapsto \text{toFin1d}\mathbb{R}(f(t))$ from $T$ to $\mathbb{R}^{1+d}$, is a continuous function. This coordinate representation $\text{toFin1d}\mathbb{R}$ maps an abstract Lorentz vector to its $1+d$ real components $(v^0, v^1, \dots, v^d)$.

Let $d$ be a natural number representing the number of spatial dimensions. For any topological space $T$ and any function $f: \text{Contr } d \to T$ from the space of real contravariant Lorentz vectors to $T$, $f$ is continuous if the composition $f \circ \Phi^{-1}: \mathbb{R}^{1+d} \to T$ is continuous, where $\Phi^{-1}$ is the inverse of the canonical linear isomorphism $\Phi: \text{Contr } d \xrightarrow{\cong} \mathbb{R}^{1+d}$ that identifies a contravariant vector with its $1+d$ real components $(v^0, v^1, \dots, v^d)$.

For a given natural number $d$ representing the spatial dimensions, this definition specifies the representation of the Lorentz group $\mathcal{L} = \mathrm{O}(1, d)$ on the space of real covariant Lorentz vectors. The representation maps each Lorentz transformation matrix $\Lambda \in \mathcal{L}$ to the linear operator $(\Lambda^{-1})^\intercal$, which corresponds to the standard transformation law for covectors. In index notation, these covariant vectors are represented with components $\psi^i$.

For a given number of spatial dimensions $d$, let $\Lambda$ be a Lorentz transformation in the Lorentz group $\mathcal{L}$. The matrix representation of the action of $\Lambda$ on the space of covariant vectors (co-vectors), relative to the standard covariant basis, has components equal to the transpose of the inverse of $\Lambda$. Specifically, for indices $i, j \in \{0, 1, \dots, d\}$, the $(i, j)$-th entry of the matrix representing the covariant representation $\rho(\Lambda)$ is $(\Lambda^{-1})^\intercal_{ij}$.

For a natural number $d$ representing spatial dimensions, the component representation of the $i$-th basis vector of the covariant Lorentz representation, $\text{coBasis}(d)_i$, in the space of real-valued functions on spacetime indices $\text{Fin } 1 \oplus \text{Fin } d$ is the standard basis vector $\delta_{i, \cdot}$, which takes the value $1$ at index $i$ and $0$ otherwise.

For a natural number $d$ representing spatial dimensions, the $i$-th basis vector of the covariant Lorentz representation $\text{coBasisFin}(d)_i$, where $i \in \text{Fin}(1+d)$, has a component representation in the space of real-valued functions on spacetime indices $\text{Fin } 1 \oplus \text{Fin } d$ given by the standard unit vector $\delta_{j, \cdot}$. Here, $j$ is the index in $\text{Fin } 1 \oplus \text{Fin } d$ corresponding to $i$ via the canonical isomorphism $\text{finSumFinEquiv}^{-1}: \text{Fin}(1+d) \to \text{Fin } 1 \oplus \text{Fin } d$.

For a natural number $d$ representing spatial dimensions, let $p$ be a covariant Lorentz vector in the representation $\text{Co } d$. For any index $i \in \{0, 1, \dots, d\}$, the $i$-th coordinate of $p$ with respect to the basis $\text{coBasisFin}(d)$ is equal to the component value of $p$ at the spacetime index corresponding to $i$ under the canonical isomorphism between $\text{Fin}(1+d)$ and $\text{Fin } 1 \oplus \text{Fin } d$.

For a given natural number $d$ representing the spatial dimensions, this definition specifies the morphism of representations $f: \text{Contr } d \to \text{Co } d$ from the contravariant Lorentz representation to the covariant Lorentz representation. The map is defined by multiplying a contravariant vector $\psi \in \mathbb{R}^{1+d}$ by the Minkowski metric matrix $\eta = \text{diag}(1, -1, \dots, -1)$, such that $f(\psi) = \eta \psi$. In physical tensor notation, this corresponds to the operation of lowering an index, $\psi_\mu = \eta_{\mu\nu} \psi^\nu$. This map is an intertwining operator because it satisfies $\eta (\Lambda \psi) = (\Lambda^{-1})^\intercal (\eta \psi)$ for all $\Lambda$ in the Lorentz group.

For a natural number $d$ representing spatial dimensions, let $\text{Co } d$ and $\text{Contr } d$ be the covariant and contravariant representations of the Lorentz group $\mathcal{L}$ on $\mathbb{R}^{1+d}$, respectively. This definition defines a morphism of representations $\text{Co } d \to \text{Contr } d$ that maps a covariant vector $\psi$ to a contravariant vector by multiplying its component vector by the Minkowski metric $\eta = \text{diag}(1, -1, \dots, -1)$. In physical terms, this map represents the process of "raising an index" using the metric tensor, $\psi^\mu = \eta^{\mu\nu} \psi_\nu$.

For a natural number $d$ representing spatial dimensions, this definition specifies an isomorphism of representations $\text{Contr } d \cong \text{Co } d$ between the contravariant and covariant representations of the Lorentz group $\mathcal{L}$. This isomorphism is induced by the Minkowski metric $\eta = \text{diag}(1, -1, \dots, -1)$. The forward morphism $f: \text{Contr } d \to \text{Co } d$ is given by $f(\psi) = \eta \psi$, which corresponds to the physical operation of lowering an index ($\psi_\mu = \eta_{\mu\nu} \psi^\nu$). The inverse morphism $f^{-1}: \text{Co } d \to \text{Contr } d$ is also given by multiplication with the metric $\eta$, corresponding to raising an index ($\psi^\mu = \eta^{\mu\nu} \psi_\nu$), which is an inverse because $\eta^2 = I$.

In $(1+3)$-dimensional spacetime, let $e_\mu$ denote the $\mu$-th standard basis vector of the space of contravariant Lorentz vectors. For any Lorentz transformation $\Lambda \in \mathcal{L}$ and its associated representation $\rho$, the transformation of the basis vector $e_\mu$ is given by $\rho(\Lambda) e_\mu = \sum_j \Lambda_{j\mu} e_j$, where $\Lambda_{j\mu}$ is the entry of the matrix $\Lambda$ at row $j$ and column $\mu$.