Physlib.Electromagnetism.Distributional.Basic

Given the speed of light $c$, this linear map constructs an electromagnetic potential distribution $A^\mu$ from a scalar potential distribution $\phi$ defined on spacetime $\text{Time} \times \text{Space}^d$. The resulting distribution $A^\mu$ is a Lorentz vector-valued distribution whose temporal component is $\frac{1}{c} \phi$ and whose spatial components are zero.

Given the speed of light $c$, this linear map defines an electromagnetic potential distribution $A^\mu$ from a static scalar potential distribution $\phi$ on $\text{Space } d$. The static distribution $\phi$ is first extended to a time-independent distribution on spacetime $\text{Time} \times \text{Space } d$. The resulting electromagnetic potential $A^\mu$ is a Lorentz vector-valued distribution whose temporal component is $A^0 = \frac{1}{c} \phi$ and whose spatial components $A^i$ are zero.

Given the speed of light $c$, this function constructs an electromagnetic potential distribution $A^\mu$ from a static scalar potential function $\varphi: \text{Space } d \to \mathbb{R}$. If $\varphi$ satisfies the distribution-boundedness condition (`IsDistBounded`), it is promoted to a distribution and used to define $A^\mu$ such that its temporal component is $A^0 = \frac{1}{c} \varphi$ (extended as a time-independent distribution on spacetime) and its spatial components $A^i$ are zero. If the function $\varphi$ is not distribution-bounded, the result is the zero distribution.

For a spatial dimension $d$ and speed of light $c$, let $\varphi: \text{Space } d \to \mathbb{R}$ be a static scalar potential function. If $\varphi$ satisfies the distribution-boundedness condition (`Space.IsDistBounded`), then the electromagnetic potential distribution $A^\mu$ constructed directly from the function $\varphi$ is equal to the electromagnetic potential distribution constructed from the distribution associated with $\varphi$ (the distribution defined by the integral $\eta \mapsto \int \eta(x) \varphi(x) dx$).

Given the spatial dimension $d$ and the speed of light $c$, let $\varphi: \text{Space } d \to \mathbb{R}$ be a static scalar potential function. If $\varphi$ does not satisfy the distribution-boundedness condition (denoted as $\neg \text{IsDistBounded}(\varphi)$), then the electromagnetic potential distribution $A^\mu$ constructed from $\varphi$ is the zero distribution.

For a given spatial dimension $d$ and speed of light $c$, this defines a linear map that transforms a vector potential $\mathbf{A}$, represented as a distribution from spacetime $\text{Time} \times \text{Space } d$ to the $d$-dimensional Euclidean space $\mathbb{R}^d$, into an electromagnetic 4-potential distribution $A^\mu$.

For a given spatial dimension $d$ and speed of light $c$, this defines a linear map that transforms a static vector potential $\mathbf{A}$—represented as a distribution from the spatial manifold $\text{Space } d$ to the $d$-dimensional Euclidean space $\mathbb{R}^d$—into an electromagnetic 4-potential distribution $A^\mu$. The resulting 4-potential distribution is constant in time and corresponds to the case where the scalar potential component is zero, effectively embedding the spatial distribution into the contravariant Lorentz vector components of $A^\mu$.

For an electromagnetic potential $A$ (defined as a distribution mapping spacetime to contravariant Lorentz vectors) and a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the distributional tensor derivative of $A$ evaluated at $\varepsilon$ is equal to the double sum over spacetime indices $\mu$ and $\nu$: $$\nabla A (\varepsilon) = \sum_{\mu, \nu} (\partial_\mu A)(\varepsilon)_\nu \cdot (e^\mu \otimes e_\nu)$$ where $\partial_\mu A$ is the distributional partial derivative of $A$ in the direction of the $\mu$-th coordinate, $(\partial_\mu A)(\varepsilon)_\nu$ is the $\nu$-th component of the resulting Lorentz vector, and $e^\mu$ and $e_\nu$ are the standard basis elements for Lorentz covectors and vectors, respectively.

For a $(1+d)$-dimensional spacetime, let $A$ be an electromagnetic potential distribution and $\varepsilon$ be a test function in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$. For any pair of spacetime indices $\mu, \nu \in \text{Fin } 1 \oplus \text{Fin } d$, the $(\mu, \nu)$-th component of the distributional tensor derivative $\text{distTensorDeriv}(A)(\varepsilon)$—expressed in the basis formed by the tensor product of Lorentz covectors and Lorentz vectors—is equal to the $\nu$-th component of the distributional partial derivative $(\partial_\mu A)(\varepsilon)$. Mathematically, this is expressed as: $$[\text{distTensorDeriv}(A)(\varepsilon)]_{\mu}{}^{\nu} = (\partial_\mu A)(\varepsilon)^\nu$$

Let $d$ be the number of spatial dimensions. Let $A$ be an electromagnetic potential distribution on a $(1+d)$-dimensional spacetime, taking values in contravariant Lorentz vectors. For any test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$ and any tensor multi-index $b = (\mu, \nu)$ representing a covariant (down) and a contravariant (up) index, the $b$-th component of the distributional tensor derivative $\nabla A$ evaluated at $\varepsilon$ is given by: $$[(\nabla A)(\varepsilon)]_{\mu}{}^{\nu} = (\partial_\mu A)(\varepsilon)^\nu$$ where $\partial_\mu$ is the distributional partial derivative in the direction of the $\mu$-th coordinate, and $(\partial_\mu A)(\varepsilon)^\nu$ is the $\nu$-th component of the resulting vector.