In classical relativistic Electrodynamics, we are often told that any accelerating point charge inherently radiates (Bremstrallung). (This is the basis for the need for a QM conception of electrons.)
We are also told that the innate (not due to the fields) energy of a moving particle is based on relativity, and is written as $E = \gamma m c^2$. Thus, the only dynamic dependence that the relativistic energy has is speed (not velocity ... no direction to $\gamma$).
This puzzles me. If both of these are true, then the speed (and thus the energy) of a point charge accelerating in a direction perpendicular to its velocity (i.e. turning without speeding up/slowing down) would not change during the acceleration. How then can it radiate away energy?
Note: I do realize that this radiation is experimentally confirmed. It is the generally accepted theory behind such observations that is leaving me confused.