Physlib.Relativity.LorentzGroup.Boosts.Generalized

For a given spatial dimension $d$ and two velocities $u, v \in \text{Velocity}_d$, the linear map $f: \text{Vector}_d \to \text{Vector}_d$ is defined by: $$f(x) = (2 \langle x, u \rangle_m) v$$ where $\langle x, u \rangle_m$ denotes the Minkowski product of the Lorentz vectors $x$ and $u$. This map serves as an auxiliary component in the construction of a generalized boost that maps velocity $u$ to velocity $v$.

For given velocities $u$ and $v$ in a $d$-dimensional spatial setting, this defines a linear map from the space of Lorentz vectors $\text{Vector}_d$ to itself. For any Lorentz vector $x \in \text{Vector}_d$, the map is defined as: $$x \mapsto - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski product with signature $(1, d)$, and $u$ and $v$ are the 4-vectors associated with the velocities.

For any velocity $u$ in a space with $d$ spatial dimensions, the two auxiliary linear maps $\text{genBoostAux}_1(u, u)$ and $\text{genBoostAux}_2(u, u)$ are negatives of each other: $$\text{genBoostAux}_2(u, u) = - \text{genBoostAux}_1(u, u)$$ Specifically, $\text{genBoostAux}_1(u, u)$ is the map $x \mapsto 2 \langle x, u \rangle_m u$, and $\text{genBoostAux}_2(u, u)$ is the map $x \mapsto - \frac{\langle x, u + u \rangle_m}{1 + \langle u, u \rangle_m} (u + u)$, where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski product.

For any spatial dimension $d \in \mathbb{N}$, let $u, v \in \text{Velocity}_d$ be velocities (represented as Lorentz vectors) and let $e_\mu$ be the $\mu$-th standard basis vector in the Lorentz vector space $\text{Vector}_d$ for an index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$. The first auxiliary linear map for generalized boosts, $\text{genBoostAux}_1(u, v)$, when applied to the basis vector $e_\mu$, is given by: $$(\text{genBoostAux}_1(u, v))(e_\mu) = (2 \eta_{\mu\mu} u^\mu) v$$ where $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$ and $u^\mu$ is the $\mu$-th component of the Lorentz vector $u$.

For any spatial dimension $d \in \mathbb{N}$, let $u, v \in \text{Velocity}_d$ be velocities (represented as Lorentz vectors) and let $e_\mu$ be the $\mu$-th standard basis vector in the Lorentz vector space $\text{Vector}_d$ for an index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$. The second auxiliary linear map for generalized boosts, $\text{genBoostAux}_2(u, v)$, when applied to the basis vector $e_\mu$, is given by: $$(\text{genBoostAux}_2(u, v))(e_\mu) = -\frac{\eta_{\mu\mu} (u^\mu + v^\mu)}{1 + \langle u, v \rangle_m} (u + v)$$ where $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, $u^\mu$ and $v^\mu$ are the $\mu$-th components of the Lorentz vectors $u$ and $v$ respectively, and $\langle u, v \rangle_m$ is the Minkowski product of $u$ and $v$.

In a $(d+1)$-dimensional Minkowski space, let $u$ and $v$ be two velocities and let $f(x) = 2 \langle x, u \rangle_m v$ be the auxiliary linear map used in the construction of generalized boosts. For any indices $\mu, \nu \in \{0, 1, \dots, d\}$, the Minkowski product of the images of the standard basis vectors $e_\mu$ and $e_\nu$ under $f$ is given by: $$\langle f(e_\mu), f(e_\nu) \rangle_m = 4 \eta_{\mu\mu} \eta_{\nu\nu} u^\mu u^\nu$$ where $\eta_{\mu\mu}$ and $\eta_{\nu\nu}$ are the diagonal entries of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $u^\mu, u^\nu$ are the components of the velocity vector $u$.

Let $d$ be the spatial dimension and $u, v \in \text{Velocity}_d$ be two velocities. Consider the auxiliary linear map $f: \text{Vector}_d \to \text{Vector}_d$ defined by $f(x) = (2 \langle x, u \rangle_m) v$, where $\langle \cdot, \cdot \rangle_m$ is the Minkowski product. The entry in the $\mu$-th row and $\nu$-th column of the matrix representation of $f$ with respect to the standard basis is given by: $$(M_f)_{\mu \nu} = 2 \eta_{\nu\nu} u^\nu v^\mu$$ where $u^\nu$ and $v^\mu$ are the $\nu$-th and $\mu$-th components of the vectors $u$ and $v$ respectively, and $\eta_{\nu\nu}$ is the $\nu$-th diagonal element of the Minkowski metric matrix $\mathrm{diag}(1, -1, \dots, -1)$.

In a $(d+1)$-dimensional Minkowski spacetime, let $u$ and $v$ be two velocities (viewed as future-directed Lorentz vectors with Minkowski norm 1), and let $L$ be the second auxiliary linear map associated with the generalized boost from $u$ to $v$, defined by $L(x) = - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$. For any indices $\mu, \nu \in \{0, 1, \dots, d\}$, the Minkowski product of the images of the standard basis vectors $e_\mu$ and $e_\nu$ under $L$ is: $$\langle L(e_\mu), L(e_\nu) \rangle_m = \frac{2 \eta_{\mu\mu} \eta_{\nu\nu} (u^\mu + v^\mu) (u^\nu + v^\nu)}{1 + \langle u, v \rangle_m}$$ where $\eta_{\mu\mu}$ and $\eta_{\nu\nu}$ are the diagonal entries of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $u^\mu, v^\mu$ are the $\mu$-th components of the vectors $u$ and $v$ respectively.

For any spatial dimension $d \in \mathbb{N}$ and velocities $u, v \in \text{Velocity}_d$, let $e_\mu$ and $e_\nu$ be the standard basis vectors of the Lorentz vector space $\text{Vector}_d$ for indices $\mu, \nu \in \text{Fin } 1 \oplus \text{Fin } d$. Let $f_1 = \text{genBoostAux}_1(u, v)$ and $f_2 = \text{genBoostAux}_2(u, v)$ be the first and second auxiliary linear maps used in the construction of generalized boosts. The Minkowski product of $f_1(e_\mu)$ and $f_2(e_\nu)$ is given by: $$\langle f_1(e_\mu), f_2(e_\nu) \rangle_m = -2 \eta_{\mu\mu} \eta_{\nu\nu} u^\mu (u^\nu + v^\nu)$$ where $\eta_{\alpha\alpha}$ denotes the $\alpha$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $u^\mu, u^\nu, v^\nu$ are the components of the Lorentz vectors associated with the velocities $u$ and $v$.

For any two Lorentz velocities $u$ and $v$ in $(1+d)$-dimensional Minkowski spacetime, let $L$ be the second auxiliary linear map defined as $x \mapsto - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$. The entries of the matrix representing $L$ with respect to the standard basis are given by: $$(L)_{\mu\nu} = \eta_{\nu\nu} \frac{-(u^\mu + v^\mu)(u^\nu + v^\nu)}{1 + \langle u, v \rangle_m}$$ where $\mu$ and $\nu$ are spacetime indices, $u^\mu$ and $v^\nu$ are the components of the vectors associated with the velocities, $\eta_{\nu\nu}$ is the $\nu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product.

For any spatial dimension $d \in \mathbb{N}$ and velocities $u, v \in \text{Velocity}_d$, let $e_\mu$ and $e_\nu$ be the standard basis vectors of the Lorentz vector space for indices $\mu, \nu \in \text{Fin } 1 \oplus \text{Fin } d$. Let $f_1 = \text{genBoostAux}_1(u, v)$ and $f_2 = \text{genBoostAux}_2(u, v)$ be the first and second auxiliary linear maps used in the construction of generalized boosts. The Minkowski product of the sums $(f_1 + f_2)(e_\mu)$ and $(f_1 + f_2)(e_\nu)$ is given by: $$\langle f_1(e_\mu) + f_2(e_\mu), f_1(e_\nu) + f_2(e_\nu) \rangle_m = 2 \eta_{\mu\mu} \eta_{\nu\nu} \left( -u^\mu (u^\nu + v^\nu) - u^\nu (u^\mu + v^\mu) + \frac{(u^\mu + v^\mu)(u^\nu + v^\nu)}{1 + \langle u, v \rangle_m} + 2 u^\mu u^\nu \right)$$ where $\eta_{\alpha\alpha}$ denotes the $\alpha$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $u^\mu, u^\nu, v^\mu, v^\nu$ are the components of the Lorentz vectors associated with the velocities $u$ and $v$.

For any spatial dimension $d \in \mathbb{N}$, let $u, v \in \text{Velocity}_d$ be Lorentz velocities and let $e_\mu, e_\nu$ be the standard basis vectors for indices $\mu, \nu \in \text{Fin } 1 \oplus \text{Fin } d$. The Minkowski product of $e_\mu$ and the sum of the two auxiliary linear maps for generalized boosts applied to $e_\nu$ is given by: $$\langle e_\mu, \text{genBoostAux}_1(u, v, e_\nu) + \text{genBoostAux}_2(u, v, e_\nu) \rangle_m = \eta_{\mu\mu} \eta_{\nu\nu} \left( 2 u^\nu v^\mu - \frac{(u^\mu + v^\mu)(u^\nu + v^\nu)}{1 + \langle u, v \rangle_m} \right)$$ where $u^\mu, v^\mu$ are the $\mu$-th components of the vectors $u$ and $v$, $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski product.

For a given spatial dimension $d$ and two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ is the Lorentz transformation in $\mathcal{L}_d$ that maps the velocity $u$ to the velocity $v$. For any Lorentz vector $x \in \text{Vector}_d$, its action is defined by: $$\Lambda(u, v)x = x + 2\langle x, u \rangle_m v - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product with signature $(1, d)$.

For any velocities $u, v \in \text{Velocity}_d$ and any Lorentz vector $x \in \text{Vector}_d$, the action of the generalized boost $\Lambda(u, v)$ on $x$ is given by: $$\Lambda(u, v) \cdot x = x + \text{genBoostAux}_1(u, v, x) + \text{genBoostAux}_2(u, v, x)$$ where $\text{genBoostAux}_1(u, v, x) = 2\langle x, u \rangle_m v$ and $\text{genBoostAux}_2(u, v, x) = - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$, and $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product with signature $(1, d)$.

For any velocities $u, v \in \text{Velocity}_d$ and any Lorentz vector $x \in \text{Vector}_d$, the action of the generalized boost $\Lambda(u, v)$ on $x$ satisfies the following identity: $$(1 + \langle u, v \rangle_m) (\Lambda(u, v) \cdot x) = (1 + \langle u, v \rangle_m) x + 2 \langle x, u \rangle_m (1 + \langle u, v \rangle_m) v - \langle x, u + v \rangle_m (u + v)$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product with signature $(1, d)$.

For any velocities $u, v \in \text{Velocity}_d$ and any Lorentz vector $x \in \text{Vector}_d$, the action of the generalized boost $\Lambda(u, v)$ on $x$ expands to the explicit formula: $$\Lambda(u, v) \cdot x = x + 2\langle x, u \rangle_m v - \frac{\langle x, u + v \rangle_m}{1 + \langle u, v \rangle_m} (u + v)$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product.

For any two Lorentz velocities $u$ and $v$ in $d$ spatial dimensions, the generalized boost transformation $\Lambda(u, v)$ maps the four-velocity vector $u$ to the four-velocity vector $v$: $$\Lambda(u, v) \cdot u = v$$

For any two velocities $u, v \in \text{Velocity}_d$ in a space with $d$ spatial dimensions, the action of the generalized boost $\Lambda(u, v)$ on the velocity $v$ is given by: $$\Lambda(u, v) v = (2\langle u, v \rangle_m) v - u$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product with signature $(1, d)$.

For any four-velocity $u$ in a space with $d$ spatial dimensions, the generalized boost transformation $\Lambda(u, u)$, which maps the velocity $u$ to itself, is equal to the identity transformation $I$.

For any spatial dimension $d \in \mathbb{N}$, let $u, v \in \text{Velocity}_d$ be velocities (represented as Lorentz vectors) and let $e_\mu$ be the $\mu$-th standard basis vector in the Lorentz vector space $\text{Vector}_d$ for an index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$. The action of the generalized boost $\Lambda(u, v)$ on the basis vector $e_\mu$ is given by: $$\Lambda(u, v) \cdot e_\mu = e_\mu + (2 \eta_{\mu\mu} u^\mu) v - \frac{\eta_{\mu\mu} (u^\mu + v^\mu)}{1 + \langle u, v \rangle_m} (u + v)$$ where $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, $u^\mu$ and $v^\mu$ are the $\mu$-th components of the Lorentz vectors $u$ and $v$ respectively, and $\langle u, v \rangle_m$ is the Minkowski inner product of $u$ and $v$.

For a given spatial dimension $d$, let $u$ and $v$ be two velocities in $\text{Velocity}_d$ and let $\mu, \nu$ be spacetime indices. The $(\mu, \nu)$-th component of the matrix representing the generalized boost $\Lambda(u, v)$ is given by: $$\Lambda(u, v)_{\mu\nu} = \eta_{\mu\mu} \left( \langle e_\mu, e_\nu \rangle_m + 2 \langle e_\nu, u \rangle_m \langle e_\mu, v \rangle_m - \frac{\langle e_\mu, u + v \rangle_m \langle e_\nu, u + v \rangle_m}{1 + \langle u, v \rangle_m} \right)$$ where $e_\mu$ and $e_\nu$ are the standard basis vectors for the space of Lorentz vectors $\text{Vector}_d$, $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix, and $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product with signature $(1, d)$.

For a given spatial dimension $d$ and two velocities $u, v \in \text{Velocity}_d$, the entry in the $\mu$-th row and $\nu$-th column of the matrix representation of the generalized boost $\Lambda(u, v)$ is: $$(\Lambda(u, v))_{\mu \nu} = \delta_{\mu \nu} + 2 \eta_{\nu \nu} u^\nu v^\mu - \eta_{\nu \nu} \frac{(u^\mu + v^\mu)(u^\nu + v^\nu)}{1 + \langle u, v \rangle_m}$$ where $\delta_{\mu \nu}$ is the Kronecker delta (the identity matrix entry), $\eta_{\nu \nu}$ is the $\nu$-th diagonal entry of the Minkowski metric matrix $\text{diag}(1, -1, \dots, -1)$, $u^\nu$ and $v^\mu$ are the components of the 4-vectors associated with the velocities, and $\langle u, v \rangle_m$ is the Minkowski inner product.

For a given spatial dimension $d$ and a fixed velocity $u \in \text{Velocity}_d$, the map $v \mapsto \Lambda(u, v)$, which sends a velocity $v$ to the generalized boost mapping $u$ to $v$, is continuous with respect to the topologies on $\text{Velocity}_d$ and the Lorentz group $\mathcal{L}_d$.

For a given spatial dimension $d$ and a fixed Lorentz velocity $u \in \text{Velocity}_d$, the function $v \mapsto \Lambda(v, u)$, which maps a Lorentz velocity $v$ to the generalized boost from $v$ to $u$, is continuous. Here, the set of velocities $\text{Velocity}_d$ and the Lorentz group $\mathcal{L}_d$ are equipped with their respective subspace topologies.

For any spatial dimension $d$ and any two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ is path-connected to the identity element $I$ in the Lorentz group $\mathcal{L}_d$. That is, there exists a continuous path $\gamma: [0, 1] \to \mathcal{L}_d$ such that $\gamma(0) = I$ and $\gamma(1) = \Lambda(u, v)$.

For any spatial dimension $d$ and any two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ belongs to the connected component of the identity element $I$ in the Lorentz group $\mathcal{L}_d$.

For any spatial dimension $d$ and any two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ is a proper Lorentz transformation, meaning that its determinant is equal to $1$.

For any spatial dimension $d$ and any two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ is orthochronous, meaning its $(0,0)$-component satisfies $\Lambda(u, v)_{00} \ge 0$.

For any spatial dimension $d$ and any two velocities $u, v \in \text{Velocity}_d$, the generalized boost $\Lambda(u, v)$ is an element of the restricted Lorentz group $\mathcal{L}_+^\uparrow$. That is, $\Lambda(u, v)$ is both a proper Lorentz transformation ($\det(\Lambda(u, v)) = 1$) and an orthochronous Lorentz transformation ($(\Lambda(u, v))_{00} \ge 0$).

For any two Lorentz velocities $u$ and $v$ in $(1+d)$-dimensional Minkowski spacetime, the inverse of the generalized boost $\Lambda(u, v)$ (which maps $u$ to $v$) is the generalized boost $\Lambda(v, u)$ (which maps $v$ to $u$): $$\Lambda(u, v)^{-1} = \Lambda(v, u)$$

For any spatial dimension $d \in \mathbb{N}$ and any two Lorentz velocities $u, v \in \text{Velocity}_d$, the time-time component (the entry at row index $0$ and column index $0$) of the matrix representation of the generalized boost $\Lambda(u, v)$ is given by: $$(\Lambda(u, v))_{00} = 1 + \frac{\|u^0 \mathbf{v} - v^0 \mathbf{u}\|^2}{1 + \langle u, v \rangle_m}$$ where $u^0$ and $v^0$ are the temporal components, $\mathbf{u}$ and $\mathbf{v}$ are the spatial parts of the velocities, $\|\cdot\|$ denotes the Euclidean norm, and $\langle \cdot, \cdot \rangle_m$ is the Minkowski inner product.