= Solution
Let $z^\mu(\tau)$ be the point <electric charge>'s <world line>, with <proper time> $\tau$ and <four-velocity> $U^\mu=dz^\mu/d\tau=\gamma(c,\mathbf u)$. For the field event $x^\mu$, choose the \b[retarded] source event: $R^\mu=x^\mu-z^\mu(\tau_{\rm ret})$ is future-directed and null. Write $R^\mu=(R,R\mathbf n)$, where $R=|\mathbf x-\mathbf z(t_{\rm ret})|=c(t-t_{\rm ret})$. In the instantaneous rest <inertial frame> of that source event,
$$
U'^\mu=(c,\mathbf0),\qquad -R'_\mu U'^\mu=cR',\qquad
A'^\mu=\left(\frac{q}{4\pi\epsilon_0cR'},\mathbf0\right).
$$
Consequently the covariant expression matching this <Coulomb potential> is
$$
\boxed{A^\mu(x)=\frac{q}{4\pi\epsilon_0}\frac{U^\mu/c}{-R_\nu U^\nu}\bigg|_{\rm ret}.}
$$
The denominator is a <Lorentz scalar>, so the right-hand side is a <four-vector> with the required instantaneous-rest components. This describes the <retarded potential> in <Lorenz gauge>, with no added free <electromagnetic wave>. <Lorentz covariance> alone does not establish the boundary conditions or exclude additional terms depending on <four-acceleration>. The physical input is the retarded point-source solution of <Maxwell equations>: the potential at the retarded event depends on the instantaneous <four-current>, while its <derivatives> generate the acceleration field. Equivalently, evaluating the retarded source integral gives a Jacobian $1/(1-\mathbf n\cdot\boldsymbol\beta)$ from the light-cone constraint; this fixes that input for arbitrary smooth subluminal motion.<Retarded Potentials by Richard Fitzpatrick>[§1.7, equations (71)–(76): causal source integrals, with source quantities evaluated at retarded time]{ref}
In any <inertial frame>, put $\boldsymbol\beta=\mathbf u/c$ and $\kappa=1-\mathbf n\cdot\boldsymbol\beta$. Then $-R\cdot U=\gamma cR\kappa$, and the <Liénard–Wiechert potential> is
$$
\phi=\frac{q}{4\pi\epsilon_0R\kappa},\qquad \mathbf A=\frac{\mathbf u}{c^2}\phi.
$$
Differentiate using $\mathbf E=-\nabla\phi-\partial_t\mathbf A$ and $\mathbf B=\nabla\times\mathbf A$, including the position and time dependence of the <retarded time>. For example, $\partial t_{\rm ret}/\partial t=1/\kappa$ and $\nabla t_{\rm ret}=-\mathbf n/(c\kappa)$. The resulting <Liénard-Wiechert field> is
$$
\boxed{\mathbf E=\frac{q}{4\pi\epsilon_0}\left[
\frac{(1-\beta^2)(\mathbf n-\boldsymbol\beta)}{\kappa^3R^2}
+\frac{\mathbf n\times[(\mathbf n-\boldsymbol\beta)\times\dot{\boldsymbol\beta}]}{c\kappa^3R}
\right]_{\rm ret},\qquad
\mathbf B=\frac1c[\mathbf n\times\mathbf E]_{\rm ret}.}
$$
Here the dot denotes a <derivative> with respect to source coordinate time. The first term is the velocity or non-radiative field: it falls as $R^{-2}$ and exists even without <acceleration>. The second is the acceleration or <radiation field>, falling as $R^{-1}$ and transverse to $\mathbf n$. Its <Poynting flux> falls as $R^{-2}$, so its outward <power> integrated over a large sphere stays finite as its radius tends to infinity. The first term's squared-field flux instead vanishes in that limit. This large-distance distinction defines <radiation>; a moving near field can carry local <energy> without constituting radiation escaping to infinity.
Back to article page
