Physlib.Electromagnetism.Kinematics.Boosts

In a spacetime with $d+1$ spatial dimensions, let $c$ be the speed of light and $\beta \in \mathbb{R}$ be a velocity parameter such that $|\beta| < 1$. Let $A$ be a differentiable electromagnetic potential. Let $\Lambda$ be the Lorentz boost in the $x$-direction (the first spatial coordinate, index $0$) with parameter $\beta$. Define the transformed time $t'$ and transformed spatial coordinates $\mathbf{x}'$ as: $$t' = \gamma(\beta) \left(t + \frac{\beta}{c} x_0\right)$$ $$x'_0 = \gamma(\beta) (x_0 + c \beta t)$$ $$x'_i = x_i \quad \text{for } i > 0$$ where $\gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}}$ is the Lorentz factor. Let $A'$ be the potential transformed under the Lorentz group action, given by $A'(y) = \Lambda \cdot A(\Lambda^{-1} y)$. Then the $x$-component (index $0$) of the electric field calculated from the transformed potential $A'$ at $(t, \mathbf{x})$ is equal to the $x$-component of the electric field calculated from the original potential $A$ at the boosted coordinates $(t', \mathbf{x}')$: $$E'_x(t, \mathbf{x}) = E_x(t', \mathbf{x}')$$

In a spacetime with $d+1$ spatial dimensions, let $c$ be the speed of light and $A$ be a differentiable electromagnetic potential. For a velocity parameter $\beta \in \mathbb{R}$ with $|\beta| < 1$, let $\gamma(\beta) = \frac{1}{\sqrt{1 - \beta^2}}$ be the Lorentz factor and $\Lambda$ be the Lorentz boost in the first spatial direction (index $0$). Given a spacetime point with time $t$ and spatial position $\mathbf{x} = (x_0, x_1, \dots, x_d)$, the transformed coordinates $(t', \mathbf{x}')$ are defined as: $$t' = \gamma(\beta) \left( t + \frac{\beta}{c} x_0 \right)$$ $$x'_0 = \gamma(\beta) \left( x_0 + c \beta t \right)$$ $$x'_j = x_j \quad \text{for } j > 0$$ For any spatial index $j \in \{1, \dots, d\}$ (represented as $i.\text{succ}$), the $j$-th component of the electric field $\mathbf{E}'$ in the boosted frame is related to the original electric field $\mathbf{E}$ and the magnetic field matrix $B$ by: $$E'_j(t, \mathbf{x}) = \gamma(\beta) \left( E_j(t', \mathbf{x}') + c \beta B_{0j}(t', \mathbf{x}') \right)$$

Let $c$ be the speed of light and $d+1$ be the number of spatial dimensions. Let $A$ be a differentiable electromagnetic potential field. Consider a Lorentz boost $\Lambda$ in the first spatial direction (index $0$) with a velocity parameter $\beta$ such that $|\beta| < 1$. Define the transformed coordinates $(t', \mathbf{x}')$ as: - $t' = \gamma(\beta) \left( t + \frac{\beta}{c} x_0 \right)$ - $x'_0 = \gamma(\beta) (x_0 + c \beta t)$ - $x'_j = x_j$ for $j > 0$ where $\gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}}$ is the Lorentz factor. Under the transformation of the potential $A'(x) = \Lambda A(\Lambda^{-1} x)$, the components of the magnetic field matrix $B$ involving the boost direction (the "longitudinal-transverse" components) transform as: $$B'_{0, j}(t, \mathbf{x}) = \gamma(\beta) \left( B_{0, j}(t', \mathbf{x'}) + \frac{\beta}{c} E_j(t', \mathbf{x'}) \right)$$ for any transverse spatial index $j \in \{1, \dots, d\}$, where $E_j$ is the $j$-th component of the electric field.

In a spacetime with $d+1$ spatial dimensions, let $A$ be a differentiable electromagnetic potential and $B$ be its associated magnetic field matrix. Consider a Lorentz boost $\Lambda$ in the first spatial direction (index $0$) with velocity parameter $\beta$ (where $|\beta| < 1$). Let $B'$ be the magnetic field matrix corresponding to the boosted potential $A'(x) = \Lambda A(\Lambda^{-1}x)$. For any spatial indices $i, j \in \{1, \dots, d\}$ representing directions transverse to the boost, the components of the magnetic field matrix satisfy: $$B'_{ij}(t, \mathbf{x}) = B_{ij}(t', \mathbf{x}')$$ where the boosted coordinates $(t', \mathbf{x}')$ are defined by $t' = \gamma(\beta)(t + \frac{\beta}{c} x_0)$, $x'_0 = \gamma(\beta)(x_0 + c\beta t)$, and $x'_k = x_k$ for all other spatial indices $k \in \{1, \dots, d\}$, with $\gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}}$ being the Lorentz factor.