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Let $\mathcal{R}g(s,\omega )$ be the Radon transformTaking the large-radius limit in the Kirchhoff formula, with $\xi =t-r$ fixed for the outgoing limit and $\eta =t+r$ fixed for the incoming limit, gives the radiation fieldsIndeed, the expanding spheres converge after multiplication by $r/t$ to the planes $y\cdot \omega =-\xi$ and $y\cdot \omega =\eta$, respectively.

The Radon transforms of smooth compactly supported functions are smooth. If the data are supported in a ball of radius $R$, these transforms vanish for $|s|>R$. Hence both radiation fields are well-defined smooth functions of compact support in the null-time variable.

An accelerating point charge has a transverse radiation field in addition to its near field. At nonrelativistic speed and large distance $R$,The acceleration is evaluated at retarded time. Radiation fields decay as $1/R$, so their outward energy flux integrated over a sphere is independent of radius; the near electric field instead decays as $1/R^2$. The radiation vanishes along the acceleration and is strongest perpendicular to it.^[ref]

Using the Poynting vector givesThe integral uses $\int {\sin }^2\vartheta d\Omega =8\pi /3$. This is the Larmor formula. A harmonically accelerated charge with peak acceleration $a_0$ has mean power $q^2{a}_0^2/(12\pi {ϵ}_0{c}^3)$; external work supplies this radiated energy in steady operation. Uniform inertial motion does not radiate.

At relativistic speed the angular pattern becomes directionally concentrated and retarded-source factors matter. The total instantaneous radiated power generalizes to $q^2{\gamma }^6[a^2-|\mathbf{\beta }\times \mathbf{a}{|}^2]/(6\pi {ϵ}_0{c}^3)$, with $\mathbf{\beta }=\mathbf{v}/c$. The displayed simple ${\sin }^2\vartheta$ pattern is restricted to the nonrelativistic limit.

A blackbody absorbs all incident electromagnetic radiation, at every frequency, direction and polarization: its absorptivity is one. In thermal equilibrium, Kirchhoff's law of thermal radiation makes its emissivity one as well. Its emitted spectrum depends only on its absolute temperature, through Planck's law, rather than on the material of its surface.

A small aperture in a large isothermal cavity approximates a blackbody because radiation entering it undergoes many reflections and is very unlikely to escape before absorption. The equilibrium radiation field is isotropic and unpolarized. The total emitted radiant exitance is $\sigma T^4$ by the Stefan–Boltzmann law; the peak of the wavelength spectrum shifts to shorter wavelengths as temperature increases according to Wien's displacement law. An ideal absorber can therefore glow brightly when hot: “black” describes absorption, not the absence of thermal emission.

Motion relative to an isotropic radiation field produces a dipole: one broad hot hemisphere and an opposite cool hemisphere. To first order $T(\theta )=T_0[1+(v/c)\cos \theta ]$, where $\theta =0$ is the forward direction. This is not a narrow Galactic-plane feature and is not the intrinsic small-scale CMB pattern.

Let $z^{\mu }(\tau )$ be the point electric charge's world line, with proper time $\tau$ and four-velocity $U^{\mu }=d{z}^{\mu }/d\tau =\gamma (c,\mathbf{u})$. For the field event $x^{\mu }$, choose the retarded source event: $R^{\mu }=x^{\mu }-z^{\mu }({\tau }_{\mathrm{ret}})$ is future-directed and null. Write $R^{\mu }=(R,R\mathbf{n})$, where $R=|\mathbf{x}-\mathbf{z}(t_{\mathrm{ret}})|=c(t-t_{\mathrm{ret}})$. In the instantaneous rest inertial frame of that source event,Consequently the covariant expression matching this Coulomb potential isThe 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 $1/(1-\mathbf{n}\cdot \mathbf{\beta })$ from the light-cone constraint; this fixes that input for arbitrary smooth subluminal motion.^[ref]

In any inertial frame, put $\mathbf{\beta }=\mathbf{u}/c$ and $\kappa =1-\mathbf{n}\cdot \mathbf{\beta }$. Then $-R\cdot U=\gamma cR\kappa$, and the Liénard–Wiechert potential isDifferentiate using $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ and $\mathbf{B}=\nabla \times \mathbf{A}$, including the position and time dependence of the retarded time. For example, $\partial t_{\mathrm{ret}}/\partial t=1/\kappa$ and $\nabla t_{\mathrm{ret}}=-\mathbf{n}/(c\kappa )$. The resulting Liénard-Wiechert field isHere 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.