We clarify several issues involving the concepts of time-reversal invariance and time asymmetry in classical electrodynamics. Specifically, we consider three questions: (I) If electrodynamics is time-reversal invariant, why are the radiative processes that occur in nature time asymmetric? (II) Why doesn't the time-reversal invariance of electrodynamics contradict the fact that a charged particle in motion feels a radiation damping force? (III) Why don't the advanced solutions to Maxwell's equations occur in nature—is there some principle that forbids them? We argue that these questions are not specific to electrodynamics, but also arise in many other systems in which waves are radiated, and answer them in the context of a simple model describing a coupled particle-field system in dimensions.
REFERENCES
The model was originally proposed in Ref. 9, which describes its use as a toy model of classical electrodynamics and provides detailed derivations of many of its properties; topics discussed include the general solution to the initial value problem for the model, derivations of the retarded and advanced fields, conservation laws, and the similarities and differences between the model and ordinary electrodynamics. In Sec. IV we describe the retarded and advanced formulations of the model; further discussion of these formulations is given in Ref. 10. The model can be easily simulated on a computer; a simulation algorithm and the results of simulations for several example radiation processes can be found in Ref. 11.
The tensor is the Levi-Civita tensor in dimensions, defined such that and .
In what follows we will often use the notation as a convenient way of grouping the E and B fields into a single quantity.
This lack of spacetime symmetry is a generic feature of systems in which nonrelativistic radiators are coupled to wave fields. Such systems are not Lorentz invariant, because the radiators obey nonrelativistic dynamics, and are not Galilean invariant, because the wave equation is not invariant under Galilean transformations.
The damping force can be viewed as a self-force arising from the coupling of the particle to its own retarded field, as given by Eq. (13). Since the retarded field at time t is determined by the particle velocity at time t, the damping force also has this property.
From the form of the damping force we see explicitly that there is a preferred reference frame in which the equations of motion for the model are valid: A particle in motion feels a drag force that ultimately brings it to rest relative to this preferred frame.
We assume that the amplitude A lies in the range , so .
In general, one can show that the in and out fields for processes A and B are related by EB,in(x, t) = EA,out(x, −t) and BB,in(x, t) = −BA,out(x, −t), so if the in fields are simple for process A then the out fields are simple for process B (and vice-versa).