Disparities of Larmor’s/Liénard’s formulations with special relativity and energy-momentum conservation

It is shown that the familiar Larmor’s formula or its relativistic generalization, Liénard’s formula, widely believed to represent the instantaneous radiative losses from an accelerated charge, are not compatible with the special theory of relativity (STR). It is also shown that energy-momentum conservation is violated in all inertial frames when we compute the instantaneous rate of loss of kinetic energy and momentum of the radiating charge in accordance with Larmor’s/Liénard’s formulation. It is emphasized that one should clearly distinguish between the electromagnetic power going as radiation into space and the instantaneous loss of mechanical power by the charge. In literature both powers are treated as not only equal but almost synonymous; the two need not be the same, however. It is pointed out that a mathematical subtlety in the applicability of Poynting’s theorem seems to have been overlooked in the text-book derivation of Larmor’s formula, where a proper distinction between ‘real’ and ‘retarded’ times was not maintained. This has led to some century-old apparent anomalies, including the mysterious Schott energy, which happens to be nothing but merely a change in self-field energy of the charge between retarded and real times.