### Radiation reaction and the pitch angle changes for a charge undergoing synchrotron losses

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In synchrotron radiation formulas it is always assumed that the pitch angle of a charged particle remains constant during the radiation process. The argument employed is that as the radiation is beamed along the instantaneous direction of motion of the charge, the momentum loss will also be along the direction of motion. Accordingly radiation reaction should not cause any change in the direction of the velocity vector, and the pitch angle of the charge would therefore remain constant during the radiation process. However, it turns out that this picture is not relativistically covariant and that in the case of synchrotron losses, the pitch angle in general varies. While the component of the velocity vector perpendicular to the magnetic field does reduce in magnitude due to radiative losses, the parallel component does not undergo any change during radiation. Therefore there is a change in the ratio of the two components, implying a change in the pitch angle. This apparent paradox gets resolved and one gets a consistent picture only when effects on the charge motion are calculated from the Lorentz's radiation reaction formula. We derive the exact formula for life times of radiating electrons in a relativistically covariant way, by taking into account the change of the pitch angle due to radiative losses. We then compare it with the existing formula to examine if any revision in the life times of radiating charges, as computed in the erstwhile literature, is required.