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Time-reversal invariance and time asymmetry in classical electrodynamics
1.The Abraham-Lorentz formula is derived and discussed in D. J. Griffiths, Introduction to Electrodynamics, 2nd ed. (Prentice-Hall, New Jersey, 1989), Sec. 9.3.
2.The claim that the time asymmetry of radiative processes is due to initial conditions has been defended by many authors; see, for example, A. Einstein, A., The Collected Papers of Albert Einstein, Volume 2, The Swiss Years: Writings 1900–1909, translated by A. Beck (Princeton University Press, Princeton, 1989), p. 376;
2.and H. D. Zeh, The Physical Basis of the Direction of Time (Springer, Berlin, 2010), Sec. 2.
3.Although it is generally accepted that the equation of motion for a charged point particle including radiation damping is not time-reversal invariant, this claim has been disputed by F. Rohrlich; see, for example, F. Rohrlich, “The arrow of time in the equations of motion,” Found. Phys. 28 (7), 1045–1056 (1998);
3.and F. Rohrlich, Classical Charged Particles, 3rd ed. (World Scientific, New Jersey, 2007),
4.This example is motivated by the discussion of the time asymmetry of water waves presented in K. Popper, Ref. 2.
5.For a detailed discussion of the water wave-buoy system, see Y. H. Zhenga, Y. G. Youa, and Y. M. Shenb, “On the radiation and diffraction of water waves by a rectangular buoy,” Ocean Eng. 31, 1063–1082 (2004) and the references within.
6.Mass renormalization is discussed in A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980), pp. 190–195.
7.The runaway and preaccelerated solutions are discussed in J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 17.6, and Ref. 1, pp. 435–436.
8.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.
12.The tensor is the Levi-Civita tensor in dimensions, defined such that and .
13.In what follows we will often use the notation as a convenient way of grouping the E and B fields into a single quantity.
14.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.
15.The calculation is described in detail in Ref. 9; it proceeds analogously to the corresponding calculation for electrodynamics that is presented in J. D. Jackson, Ref. 7, Sec. 14.1.
16.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.
17.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.
18.Note that the time-reversal transformation of the E and B fields is the same as the time-reversal transformation of the electric and magnetic fields of electrodynamics; see, for example, J. D. Jackson, Ref. 7, p. 249.
19.We assume that the amplitude A lies in the range , so .
20.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).
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