Liénard radiated-power formula ($P={\mu }_0{q}^2{\gamma }^6(|a{|}^2-|v\times a{|}^2)/(6\pi )$, $c=1$)

The relativistic generalization of the Larmor formula uses the invariant proper-acceleration magnitude: $P=-{\mu }_0{q}^2{A}^a{A}_a/(6\pi )$ in signature $(+---)$. In coordinate time this is the displayed expression. For perpendicular magnetic circular motion, $\gamma m|a|=|q|vB$, giving $P={\mu }_0{q}^4({\gamma }^2-1)B^2/(6\pi m^2)$. The statement concerns total instantaneous emitted power; directional radiation flux is not itself invariant.

For one particle at a fixed Lorentz factor, the Liénard radiated-power formula gives $P_{\parallel }=C_q{\gamma }^6{a}_{\parallel }^2$ and $P_{\perp }=C_q{\gamma }^4{a}_{\perp }^2$, where $C_q=q^2/(6\pi {ϵ}_0{c}^3)$. Consequently equal coordinate acceleration magnitudes make transverse motion radiate less by ${\gamma }^2$. Equal applied laboratory force magnitudes instead give $a_{\parallel }=F/(m{\gamma }^3)$ and $a_{\perp }=F/(m\gamma )$, so $P_{\perp }/P_{\parallel }={\gamma }^2$. These opposite orderings concern distinct constraints and are consistent. An energy budget without a time, length, force, or acceleration constraint cannot by itself select a unique minimum-loss accelerator design.