Thus far we have learned how to solve two kinds of problems. Either the fields were assumed to be given, in which case the relativistic Lorentz force law yielded covariant equations of motion for a point charged massive particle interacting with these fields or the trajectory of a charged, point particle was given and the fields radiated by this particle were determined.
This, however, was clearly not enough, or at least was not consistent. That is because (as a few simple mental problems will show) each of these processes is only half of an interaction -- a complete, consistent field theory would include the self-consistent interaction of a charged particle with the field in its vicinity, or better yet, the self-consistent interaction of all particles and fields. We need to be able to calculate the total field (including the radiated field) at the position of any given point charge. Some of that field is due to the charge itself and some is due to the field produced by the other charges. But we do not know how to do this, really, since the one will affect the other, and there are clearly infinities present.
This sort of problem can also lead to Newtonian paradoxes, paradoxes that smack of the resurrection of Aristotelian dynamics. To see this, let us assume (non-physically) that we have a Universe consisting of a single point charge orbiting around an uncharged gravitational mass (or some other force center that causes the charge to move in a bound orbit). In that case, the point charge must (according to the laws of electrodynamics that we have thus far deduced) radiate energy and momentum into the electromagnetic field.
As it accelerates, it must radiate. As it radiates, energy and momentum must be carried away from the point particle to ``infinity''. The particle must therefore decrease its total energy. If the particle is bound in an attractive, negative potential well, the only way that total energy can be conserved is if its total energy decreases. The particle must therefore spiral inwards the center, converting its potential energy into radiative energy in the field, until it reaches the potential minimum and comes to rest.
There is only one difficulty with this picture. There is only one charged particle in the Universe, and it is interacting with only one attractive center. What acts to slow the particle down?
This is a non-question, of course - a thought experiment designed to help us understand where our equations of motion and classical picture are incomplete or inconsistent. The real universe has many charged particles, and they are all constantly interacting with all the other charged particles that lie within the ``event horizon'' of an event relative to the time of the big bang, which is the set of the most distant events in space-time in the past and in the future that can interact with the current event on the world line of each particle. It is the edge of the ``black hole'' that surrounds us21.1.
However, in our simplied Universe this question is very real. We have systematically rid ourselves of the fields of all the other particles, so now we must find a field based on the particle itself that yields the necessary ``radiation reaction'' force to balance the energy-momentum conservation equations. This approach will have many unsatisfactory aspects, but it works.
First, when will radiation reaction become important? When the energy radiated by a particle is a reasonable fraction of the total relevant energy of the particle under consideration. That is E_rad &sim#sim;23c e^2a^2 T4&pi#pi;&epsi#epsilon;_0 c^2 where is the total (e.g. centripetal) acceleration and is the period of the orbit associated with or the time a uniform acceleration is applied. If then we can neglect radiation reaction.
As before, if a particle is uniformly (linearly) accelerated for a time , then we can neglect radiation reaction when E_0 &sim#sim;m (a &tau#tau;_r)^2 &Gt#gg;23c e^2a^2 &tau#tau;_r4&pi#pi;&epsi#epsilon;_0 c^2
Radiation reaction is thus only significant when the opposite is
&tau#tau;_r & &sim#sim;&23c e^24&pi#pi;&epsi#epsilon;_0 mc^2
& &sim#sim;& 23 r_e/c = 23&tau#tau;_e Only if and is large will radiation reaction be appreciable. For electrons this time is around seconds. This was the situation we examined before for linear accelerators and electron-positron anihillation. Only in the latter case is radiation reaction likely.
The second case to consider is where the acceleration is centripetal. Then the potential and kinetic energy are commensurate in magnitude (virial theorem) and E_0 &sim#sim;m &omega#omega;_0^2 d^2 where and . As before, we can neglect radiation reaction if m&omega#omega;_0^2 d^2 &Gt#gg;23c e^2&omega#omega;_0^4 d^2 4&pi#pi;&epsi#epsilon;_0 c^2&omega#omega;_0 = &omega#omega;_0 d^2 23c e^24&pi#pi;&epsi#epsilon;_0 c^2
Radiation reaction is thus again significant per cycle only if &omega#omega;_0 &tau#tau;_r &sim#sim;1 (ignoring factors of order one) where is given above - another way of saying the same thing. is (within irrelevant factor of and ) the time associated with the motion, so only if this timescale corresponds to will radiation reaction be significant.
So far, our results are just a restatement of those we obtained discussing Larmor radiation except that we are going to be more interested in electrons in atomic scale periodic orbits rather than accelerators. Electrons in an atomic orbit would be constantly accelerating, so any small loss per cycle is summed over many cycles. A bit of very simple order-of-magnitude arithmetic will show you that radiative power loss need not be negligible as a rate compared to human timescales when is very small (e.g. order of seconds for e.g. optical frequency radiation). Charged particles (especially electrons) that move in a circle at a high enough (angular) speed do indeed radiate a significant fraction of their energy per second when the loss is summed over many cycles. The loss per cycle may be small, but it adds up inexorably.
How do we evaluate this ``radiation reaction force'' that has no obvious physical source in the equations that remain? The easy way is: try to balance energy (and momentum etc) and add a radiation reaction force to account for the ``missing energy''. This was the approach taken by Abraham and Lorentz many moons ago.