Abraham–Lorentz force

The Abraham - Lorentz equation is a result of classical electrodynamics. It represents an equation of motion for point charges, taking into account the reaction of the particle on itself, but results in their solution to fundamental problems in the classical theory. It is named after the physicists Max Abraham and Hendrik Antoon Lorentz.

The equation

As part of the formulation of the covariant electrodynamics the movement of a particle of mass and charge in a field ( field tensor ) will be described by the Einstein - Lorentz equation

( Four-dimensional trajectory, parametrized by arc length, the speed of light )

Taking into account that the particle is influenced not only by the external field described with, but also self radiates a field in its movement, which feeds back to itself, the result is the Abraham - Lorentz equation:

( with the Minkowski metric )

Resulting problems

A problem in the derivation of the bill is that on the left is actually a factor, where can be infinite. Therefore, one must carry out a so-called mass renormalization. This means first interpreted and re-write in the above form throughout this factor than the measurable mass.

However, even this (already questionable in the classical framework ) trick is not enough to save the physical significance of the equation: The right side contains a third derivative of the trajectory, which, according to the usual principles of mechanics can not happen in an equation of motion. In fact, the equation thus allows the so-called run-away solutions ( runaway solutions ) in which a particle is accelerated only short ( initial condition ), and then accelerates without further external influence to infinite speed, which apparently does not disclose a physical reality.

These paradoxes show the incompleteness of classical electrodynamics. The radiative effect of moving particles can only be accurately described by quantum electrodynamics.