= Acceleration covariance in nonrelativistic radiation
{title2=$d\langle P\rangle/d\Omega=q^2[\operatorname{tr}C-\mathbf n^TC\mathbf n]/(16\pi^2\epsilon_0c^3)$}
For a nonrelativistic point charge, let $C_{ij}=\langle a_i a_j\rangle$ 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 $\mathbf E=q\mathbf n\times(\mathbf n\times\mathbf a)/(4\pi\epsilon_0c^2r)$. Their <Poynting vector> gives the displayed <power> per <solid angle>. Its integral is $\langle P\rangle=q^2\operatorname{tr}C/(6\pi\epsilon_0c^3)$, 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>.
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