Physlib.Relativity.LorentzGroup.Boosts.Basic

The Lorentz factor $\gamma$, also known as the gamma factor or Lorentz term, is a function that maps a real number $\beta$ (representing the velocity as a fraction of the speed of light) to the real value defined by $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$.

For any real number $\beta$ representing the velocity as a fraction of the speed of light, if $|\beta| < 1$, then the square of the Lorentz factor $\gamma(\beta)$ satisfies the identity $\gamma(\beta)^2 = \frac{1}{1 - \beta^2}$.

The Lorentz factor $\gamma$ evaluated at $0$ is equal to $1$, that is, $\gamma(0) = 1$.

For any real number $\beta$, the Lorentz factor $\gamma$ satisfies the identity $\gamma(-\beta) = \gamma(\beta)$.

For any real number $\beta$ such that $|\beta| < 1$, it holds that $1 - \beta^2 \neq 0$.

For a given spatial dimension $d$, a spatial direction $i \in \{1, \dots, d\}$, and a velocity parameter $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$, the Lorentz boost $\text{boost}(i, \beta)$ is the $(1+d) \times (1+d)$ matrix in the Lorentz group $\mathcal{L}$ whose components $\Lambda_{\mu \nu}$ are defined by: - $\Lambda_{0,0} = \gamma(\beta)$ - $\Lambda_{0,i} = \Lambda_{i,0} = -\beta \gamma(\beta)$ - $\Lambda_{i,i} = \gamma(\beta)$ - $\Lambda_{j,k} = \delta_{jk}$ for all other indices $j, k$, where $\gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}}$ is the Lorentz factor and $\delta$ denotes the Kronecker delta.

For a given number of spatial dimensions $d$, any spatial direction $i \in \{1, \dots, d\}$, and a velocity parameter $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$, the Lorentz boost matrix $\text{boost}(i, \beta)$ is equal to its transpose: $$\text{boost}(i, \beta)^T = \text{boost}(i, \beta)$$ In other words, Lorentz boost matrices are symmetric.

For a given number of spatial dimensions $d$, any spatial direction $i \in \{1, \dots, d\}$, and a velocity parameter $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$, the matrix representing the Lorentz boost in direction $i$ with speed $\beta$, denoted as $\text{boost}(i, \beta)$, is equal to its own transpose: $$\text{boost}(i, \beta)^T = \text{boost}(i, \beta)$$ In other words, the matrix of a Lorentz boost is symmetric.

For any spatial dimension $d$ and any spatial direction $i \in \{1, \dots, d\}$, the Lorentz boost in direction $i$ with velocity parameter $\beta = 0$ is equal to the identity element $I$ of the Lorentz group.

For any spatial dimension $d$, spatial direction $i \in \{1, \dots, d\}$, and velocity parameter $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$, the inverse of the Lorentz boost in direction $i$ with speed $\beta$ is equal to the Lorentz boost in the same direction with speed $-\beta$: $$\text{boost}(i, \beta)^{-1} = \text{boost}(i, -\beta)$$

For a given spatial dimension $d$, let $B$ be the Lorentz boost matrix in the direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$. The $(0,0)$-component (the time-time component) of this matrix is equal to the Lorentz factor $\gamma(\beta)$, where $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$.

For a given spatial dimension $d$, a spatial direction $i \in \{1, \dots, d\}$, and a velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$, the $(i, i)$-th component of the Lorentz boost matrix in direction $i$ is equal to the Lorentz factor $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$.

For a Lorentz boost $\Lambda$ in the spatial direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$, the matrix component $\Lambda_{0, i}$ (the entry at the temporal row and $i$-th spatial column) is given by: $$\Lambda_{0, i} = -\gamma(\beta) \beta$$ where $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$ is the Lorentz factor.

For a Lorentz boost $B$ in the spatial direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta \in \mathbb{R}$ (where $|\beta| < 1$), the matrix component at spatial row $i$ and temporal column $0$ is given by $B_{i,0} = -\gamma(\beta) \beta$, where $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$ is the Lorentz factor.

For a spatial dimension $d$, let $i$ and $j$ be distinct spatial indices ($j \neq i$). For any velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$, the $(0, j)$-th component of the Lorentz boost matrix in the direction $i$, denoted $(B_i(\beta))_{0,j}$, is equal to $0$. Here, the index $0$ represents the time-like dimension and $j$ represents a spatial dimension.

Consider a Lorentz boost $\Lambda$ in a spatial direction $i \in \{1, \dots, d\}$ with speed $\beta \in \mathbb{R}$ such that $|\beta| < 1$. For any spatial index $j \in \{1, \dots, d\}$ that is different from $i$ ($j \neq i$), the matrix component $\Lambda_{j, 0}$ (representing the coupling between the $j$-th spatial component and the temporal component) is equal to $0$.

In a spacetime with $d$ spatial dimensions, let $\Lambda$ be the Lorentz boost in the $i$-th spatial direction with speed $\beta$ (where $|\beta| < 1$). For any spatial index $j \in \{1, \dots, d\}$ that is not equal to $i$, the matrix component of the boost at row $i$ and column $j$ is zero, i.e., $\Lambda_{ij} = 0$.

Consider a Lorentz boost $\Lambda$ in the spatial direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta$ satisfying $|\beta| < 1$. For any spatial index $j \in \{1, \dots, d\}$ such that $j \neq i$, the matrix component $\Lambda_{ji}$ (the entry at the $j$-th spatial row and $i$-th spatial column) is $0$.

For a given number of spatial dimensions $d$, let $\Lambda$ be a Lorentz boost in the spatial direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta$ such that $|\beta| < 1$. For any spatial indices $j, k \in \{1, \dots, d\}$, if $j$ is not the boost direction ($j \neq i$), then the matrix element $\Lambda_{jk}$ is equal to the Kronecker delta $\delta_{jk}$ (where $\delta_{jk} = 1$ if $j = k$ and $0$ otherwise).

For a given number of spatial dimensions $d$, let $\Lambda$ be a Lorentz boost in the spatial direction $i \in \{1, \dots, d\}$ with velocity parameter $\beta$ such that $|\beta| < 1$. For any spatial indices $j, k \in \{1, \dots, d\}$, if the column index $j$ is not the boost direction ($j \neq i$), then the matrix element $\Lambda_{kj}$ (representing the entry at spatial row $k$ and spatial column $j$) is equal to the Kronecker delta $\delta_{kj}$ (where $\delta_{kj} = 1$ if $k = j$ and $0$ otherwise).

Consider a Minkowski space with $d+1$ spatial dimensions. For any velocity parameter $\beta \in \mathbb{R}$ such that $|\beta| < 1$, let $B_0(\beta)$ be the Lorentz boost matrix in the direction of the first spatial coordinate (index $0$). For any spatial index $j \in \{1, \dots, d\}$, the matrix element corresponding to the time-like dimension (row index $0$) and the $j$-th spatial dimension (column index $j$) is zero, i.e., $(B_0(\beta))_{0,j} = 0$.

For a Lorentz boost $\Lambda$ in the first spatial direction ($j=0$) with speed $\beta \in \mathbb{R}$ (where $|\beta| < 1$) in a spacetime of $1 + (d+1)$ dimensions, the matrix components $\Lambda_{j, 0}$ are equal to $0$ for all spatial indices $j \in \{1, \dots, d\}$. Here, the index $0$ in the column position refers to the temporal component, and the row index $j$ refers to the spatial components other than the direction of the boost.

Consider the Lorentz group in $d+1$ spatial dimensions. Let $B_0(\beta)$ be a Lorentz boost in the first spatial direction (indexed by $0$) with velocity $\beta$ such that $|\beta| < 1$. For any spatial index $j = i+1$ where $i \in \mathbb{N}$ and $j \leq d$, the matrix component corresponding to the time-like dimension and the $j$-th spatial dimension is zero, i.e., $(B_0(\beta))_{0, j} = 0$.

For a Lorentz boost $\Lambda$ in the first spatial direction (indexed by $0$) with speed $\beta \in \mathbb{R}$ satisfying $|\beta| < 1$, the matrix component $\Lambda_{j, 0}$ (the coupling between the $j$-th spatial component and the temporal component) is equal to $0$ for any spatial index $j = i + 1$.

In a spacetime with $d+1$ spatial dimensions, let $\Lambda$ be the Lorentz boost in the first spatial direction (indexed by $0$ in the spatial basis) with speed $\beta$ such that $|\beta| < 1$. For any spatial index $j \in \{2, \dots, d+1\}$, the matrix component of the boost in the first spatial row and the $j$-th spatial column is zero, i.e., $\Lambda_{1, j} = 0$.

Consider a Lorentz boost $\Lambda$ in the first spatial direction $0$ with velocity parameter $\beta$ satisfying $|\beta| < 1$. For any spatial index $j \in \{1, \dots, d\}$ (where $j > 0$), the matrix entry $\Lambda_{j, 0}$ (the component at the $j$-th spatial row and $0$-th spatial column) is equal to $0$.

In a spacetime with $d+1$ spatial dimensions, let $\Lambda$ be the Lorentz boost in the $0$-th spatial direction with speed $\beta$ (where $|\beta| < 1$). For any spatial index $j = i+1$ such that $1 \le j \le d$, the matrix component of the boost at spatial row $0$ and spatial column $j$ is zero, i.e., $\Lambda_{0,j} = 0$.

For a Lorentz boost $\Lambda$ in the $0$-th spatial direction with velocity $\beta$ satisfying $|\beta| < 1$, the matrix entry corresponding to the $(i+1)$-th spatial row and the $0$-th spatial column is $0$, where $i+1$ is a valid spatial index.

In a spacetime with $d+1$ spatial dimensions, let $\Lambda$ be the Lorentz boost in the $0$-th spatial direction with velocity parameter $\beta$ such that $|\beta| < 1$. For any spatial indices $j, k \in \{1, \dots, d\}$, the matrix element of the boost $\Lambda_{jk}$ is equal to the Kronecker delta $\delta_{jk}$ (i.e., $\Lambda_{jk} = 1$ if $j = k$ and $\Lambda_{jk} = 0$ if $j \neq k$).