Physlib.Electromagnetism.Dynamics.CurrentDensity

For a spatial dimension $d$, a speed of light $c$, and a Lorentz current density $J$, the associated charge density $\rho$ is equal to the time-slice of the function that maps each spacetime point $x$ to the temporal component of $J$ (the index $0$ component) scaled by $1/c$. That is, $$\rho = \text{timeSlice} \left( x \mapsto \frac{1}{c} J^0(x) \right)$$ where $J^0(x)$ denotes the temporal component of the Lorentz current density at point $x$.

For any spatial dimension $d$ and speed of light $c$, the charge density $\rho$ associated with the zero Lorentz current density $J = 0$ is itself zero.

For a given spatial dimension $d$, speed of light $c$, and Lorentz current density field $J$, if $J$ is differentiable over $\mathbb{R}$, then the associated charge density $\rho$ (considered as a function of both time and space) is also differentiable over $\mathbb{R}$.

For a given spatial dimension $d$, speed of light $c$, and Lorentz current density field $J$, if $J$ is differentiable over $\mathbb{R}$, then for any fixed time $t$, the associated charge density $\rho(t, \cdot)$ is differentiable with respect to the spatial coordinates $x$ over $\mathbb{R}$.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. If the Lorentz current density $J$ is a $C^n$ smooth function (i.e., of class $C^n$ over $\mathbb{R}$), then the associated charge density $\rho$ is also $C^n$ smooth.

Let $d$ be the number of spatial dimensions and $J$ be a Lorentz current density (a four-vector field on spacetime). The current density $\mathbf{j}$ associated with $J$ is equal to the time-sliced field of the spatial components of $J$. Mathematically, this means that for a given speed of light $c$, the current density is obtained by extracting the spatial components of the Lorentz four-vector (indexed by `Sum.inr`) at each spacetime point $x$ and organizing them as a vector field over time and space.

For any number of spatial dimensions $d$ and any speed of light $c$, the current density $\mathbf{j}$ associated with the zero Lorentz current density (the four-current field that is identically zero) is also identically zero.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. Let $J$ be a Lorentz current density (a four-vector field on spacetime). If $J$ is differentiable, then the associated current density vector field $\mathbf{j}$, viewed as a function of spacetime coordinates, is also differentiable.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. Let $J$ be a Lorentz current density, which is a four-vector field on spacetime. If $J$ is differentiable, then for any spatial index $i \in \{0, \dots, d-1\}$, the $i$-th component of the associated current density vector field $\mathbf{j}$, viewed as a function of spacetime $(t, \mathbf{x})$, is differentiable.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. Let $J$ be a Lorentz current density (a four-vector field on spacetime). If $J$ is differentiable, then for any fixed time $t$, the associated current density vector field $\mathbf{j}$ is differentiable as a function of the spatial coordinates $x$.

Let $d$ be the spatial dimension and $c$ be the speed of light. Let $J$ be a Lorentz current density (four-current). If $J$ is differentiable, then for any fixed time $t$ and spatial index $i \in \{1, \dots, d\}$, the $i$-th component of the associated current density vector field, $x \mapsto \mathbf{j}(t, x)_i$, is differentiable with respect to the spatial coordinates $x$.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. Let $J$ be a Lorentz current density (a four-vector field on spacetime). If $J$ is differentiable, then for any fixed spatial position $x$, the associated current density vector field $\mathbf{j}$ is differentiable as a function of time $t$.

Let $d$ be the number of spatial dimensions and $c$ be the speed of light. Let $J$ be a Lorentz current density (a four-vector field on spacetime). If $J$ is differentiable, then for any fixed spatial position $x \in \text{Space } d$ and any spatial component index $i \in \{1, \dots, d\}$, the $i$-th component of the associated spatial current density vector field, $t \mapsto \mathbf{j}(t, x)_i$, is differentiable with respect to time $t$.

For a given number of spatial dimensions $d$ and a speed of light $c$, let $J$ be a Lorentz current density (a four-vector field on spacetime). If $J$ is $n$-times continuously differentiable (of class $C^n$) over spacetime, then its associated spatial current density $\mathbf{j}$ is also $n$-times continuously differentiable (of class $C^n$).

For a given spatial dimension $d$ and speed of light $c$, this linear map assigns a distributional Lorentz current density $J$ to its corresponding distributional charge density $\rho$. The charge density is defined by extracting the temporal component $J^0$ of the Lorentz vector distribution and scaling it by the reciprocal of the speed of light, such that $\rho = \frac{1}{c} J^0$. The result is a scalar-valued distribution over spacetime $\mathbb{R} \times \mathbb{R}^d$.

Given the speed of light $c$, this is a linear map that associates a distributional Lorentz current density $J$ (a four-current distribution) with its corresponding spatial current density $\vec{j}$. The resulting distribution is defined over spacetime $\text{Time} \times \text{Space}_d$ and takes values in the $d$-dimensional Euclidean space $\mathbb{R}^d$. It is constructed by extracting the spatial components of the distributional four-vector.