Acceleration covariance in nonrelativistic radiation ($d⟨P⟩/d\Omega =q^2[\mathrm{tr}C-{\mathbf{n}}^{T}C\mathbf{n}]/(16{\pi }^2{ϵ}_0{c}^3)$)

For a nonrelativistic point charge, let $C_{ij}=⟨a_i{a}_j⟩$ be the time-averaged second moment of acceleration, without subtracting its mean. The outgoing radiation zone fields satisfy $\mathbf{B}=\mathbf{n}\times \mathbf{E}/c$ and $\mathbf{E}=q\mathbf{n}\times (\mathbf{n}\times \mathbf{a})/(4\pi {ϵ}_0{c}^{2}r)$. Their Poynting vector gives the displayed power per solid angle. Its integral is $⟨P⟩=q^2\mathrm{tr}C/(6\pi {ϵ}_0{c}^3)$, the averaged Larmor formula.

For periodic motion the mean acceleration vanishes and $C$ is also its covariance matrix. More generally the second moment, rather than the centered covariance matrix, is the relevant quantity. The angular formula is valid to leading nonrelativistic order and evaluates source motion at retarded time.