The Super Collider: Round & Round It Goes. Where Does It Lead?

"Sir Ernest Rutherford is battering the atom by an electric attack under staggering voltage; and the negative electrons thus violently expelled from the domain of the positive nucleus are realizing the speculative guess made, centuries ago, by Hipparchus, that atoms could lose portions of themselves and thereby change their essence." — from a 1928 lecture given by the Scottish mathematician Andrew Russell Forsyth.

                                                                     By David Trulock

(published  in the Austin Chronicle on 13 January 1989)

Just after 6 p.m. on November 10^th I was waiting for my personal particle accelerator to warm up, so I could find out if Texas had been selected as the site for the much-ballyhooed Super Conducting Supercollider.

I was having a synchronization problem. My radiation detectors refused to synchronize with the target scanning rate of my accelerator. In other words, I was trying to watch the news on TV, but the stupid vertical hold wouldn’t hold.

Or maybe it’s the horizontal hold—I keep getting vertical and horizontal adjustments mixed up. I guess that’s one reason I majored in physics instead of becoming a TV repairman, although I did start studying physics while I was enrolled in a Vo-Tech electronics course. Anyway, when I saw the craggy features of Governor Bill Clements’ face flipping vertically across the screen (aha—vertical hold!), and heard him say how everyone pulled together to win the Super Collider, I knew: Waxahachie, Texas is the place.

Ooo coo coo. Golden cities. Golden towns

And great big signs and they all say:

Hallelujah. Yodel-lay hee hoo.
Every man for himself.

—Laurie Anderson, Big Science

A pretty accurate description of the real estate speculation going on in Waxahachie, wouldn’t you say? New shopping malls, a sports center, a freeway, and numerous drive-in banks—they’re all in the picture for the golden town. But beyond enriching landowners, real estate agents and construction contractors, is the Super Collider really worth building? What will it do that’s worth doing?

What it will do is convert energy into matter. The energy will come from accelerating protons to high speeds in opposite directions, then allowing them to smash into each other. For a very brief time after the collision, some of the energy of motion of the protons will be converted into unstable forms of matter. It is these unstable states of matter that coalesced into stable matter soon after the universe began—if the universe began as physicists and astronomers think it did. That “if” is by far the most compelling reason to collide particles at an energy 20 times greater than is now available.              

It also points to something that is being left out of the Super Collider discussion. From the way Texas politicians are talking, the Super Collider is the be-all and end-all of elementary particle physics. But particle physics, high-energy physics, will not end with the Super Collider, which will very likely generate more questions than answers. And then what? A higher energy machine will be needed.

So the Super Collider is a step up the high-energy physics ladder, not a complete stairway to heaven. The question therefore is simply this: Is it the right step?

In July 1983, the High Energy Advisory Panel at the Department of Energy decided the Super Collider was the right step. According to a 1985 article in Physics Today, the debate that led to that decision was “long and agonizing,” partly because the decision meant that proposals to expand existing accelerator facilities would not be approved. The Advisory Panel envisioned the Super Collider as a monolithic move forward for high-energy physics in the United States, a move that would “reassert American scientific supremacy.” President Reagan gave his enthusiastic stamp of approval to the project, and presumably President Bush will continue to support it when push comes to shove over funding in the upcoming session of Congress.

Some physicists, however, would prefer federal research money to be spread around a bit rather than concentrated in one project. A report last May in Physics Today noted that the Super Collider project “deeply divides the physics community, largely because of its cost and emphasis.” But the division seems to be more of a philosophical one rather than a technical one. “The polarization,” says the report, is mainly over priorities: big science or little science, crisis response or attention to long-term needs, national pre-eminence or international cooperation.” Every physicist for himself, in other words. What, then, is Congress going to think when it is faced with finding the funds to pay for the Super Collider? If the physicists aren’t sure about the project—well, hell, how are congressmen supposed to know what to do without solid support from a large block of lobbyists?

But let’s leave the quark-barrel politics behind, and get back to the central issue—whether the Super Collider is the best thing for elementary particle physics. Particle physics is necessarily big science, but particle physicists need not put all their ergs in one 53-mile in circumference basket. (An erg is a unit of energy, and energy in physics is the payment for doing work, so energy is a lot like money. And just as there are different units of money—dollars, quarters, dimes, nickels, pennies—there are different units of energy—joules, ergs, kilowatt-hours, calories and electron volts, for instance. Protons in the Super Collider will each have 32 ergs of energy when they collide. That’s equivalent to 20 trillion electron volts, or 20 TeV, each—meager when compared with the 251,000 TeV contained in a can of light beer. And they call it high-energy physics!)

While most particle physicists agree with the advisory panel’s decision, there are some supporters of particle physics who do not. One of them is Freeman Dyson, who has earned respect as both a physicist and a writer, or, more generally, as a thinker. Dyson, a professor of physics at the Institute for Advanced Study in Princeton, New Jersey, expressed his viewpoint in a recent opinion column in Physics Today.

“Every new machine is a gamble,” says Dyson. “If we build the SSC, it might turn out to be a glorious success or it might turn out to a flop. In either case, we will want to build other machines to carry on from where the SSC stops. Unfortunately, the SSC is an end rather than a beginning.”

How would the Super Collider be a flop? It would be a flop if it didn’t generate new questions—if, for instance, its energy wasn’t high enough to adequately produce new phenomena not seen at presently available energies. It would be a flop if it merely added details to what we already know, although such details could be fodder for hundreds of scientific papers (each co-authored by fifty or so physicists). And it would be a flop if laser acceleration of particles became feasible in the near future, because laser acceleration could probably do the work of the Super Collider at a fraction of its cost.

In fact, one of the alternatives to the Super Collider suggested by Dyson is an investment in laser acceleration research. Right now, nobody knows how to accelerate particles to high energy with laser radiation. But if it could be done, laser accelerators would be a great deal smaller than today’s electromagnet-based accelerators. Being much smaller would presumably make them much cheaper to build.

If the Super Collider does turn out to be a less useful machine than its backers anticipate, there is no practical or inexpensive way to increase its energy. that is why Dyson calls it an end rather than a beginning. For circular accelerators like the Super Collider, an energy increase can be achieved either by having a larger circular path or by increasing the efficiency of the electromagnets used for acceleration. Either case means rebuilding the machine.

Linear colliders, which accelerate particles in opposite directions along a straight line, can be “pursued incrementally,” says Dyson. That is, the available energy can be increased without rebuilding the machine. Since laser acceleration is uniquely suited for use in a linear collider, Dyson suggests that the alternative to the Super Collider is a concurrent investment in laser acceleration research linear collider technology.

In May, three months after Dyson’s opinion column appeared, the letters section of Physics Today was heavily weighted with responses to Dyson’s challenge. The responses brim with practical, hard-headed reasoning. Several letters noted that a subpanel of the High Energy Physics Advisory Panel had considered linear colliders and found them to be less scientifically attractive than the Super Collider. Others noted that important particle physics research would be put on hold if laser acceleration research were to be chosen as an alternative to the Super Collider. And one letter said that the Super Collider “cannot be a flop,” citing calculations any interested physicist can follow through.”

Practical, hard-headed reasoning may be a necessary component of physics, but it is not sufficient for making great discoveries. Indeed, the greatest discoveries in physics have generally come from ideas that seemed crazy when subjected to the practical, hardheaded wisdom of a particular era. But such ideas have come from individuals, not from panels or subpanels. Collectively, people are more prone to do whatever is necessary to maintain some kind of status quo, which can lead to such things as bad investments by savings and loan institutions or continued use of an unsafe booster rocket for the space shuttle.

Not that Dyson’s ideas sound crazy. In his letter of rebuttal to his critics, he says that higher-energy particle accelerators are needed, and should be built “when we can do so cost effectively.” But he also says that higher energy is only one of several relevant criteria for continued research in particle physics. Then, after making his technical points, Dyson points to the philosophical ground on which he stands:

“In conclusion, I urge my critics to remember that the universe is, as the biologist J. B. S. Haldane said, not only queerer than we suppose, but queerer than we can suppose. There is no illusion more dangerous than the belief that the progress of science is predictable. If you look for nature’s secrets in only one direction, you are likely to miss the most important secrets—those that you did not have enough imagination to predict.”

It seems, however, that most particle physicists are not as concerned with imagination as they are with insurance: the Super Collider insures that experimental particle physics research will not be interrupted. But maybe it should be interrupted. Although I was hopping around my house in excitement on November 10^th when I heard the word on the Waxahachie site, I also know something about the risks of investing huge amounts of money in scientific research—that there is no insurance when it comes to making discoveries. The lack of desirable and expensive laboratory instruments has been the key to some very important discoveries, and the awarding of large research grants has often simply increased the number of unread and uninteresting scientific papers. Lord knows there are too many of those these days.

So, while physicists are still “battering the atom by an electric attack under staggering voltage,” the costs of doing so can be more staggering than the voltages involved, and the number of research papers and journals is battering the budgets of university libraries. I hesitate to complain—an official inquiry into the matter might result in a new scientific journal called The Journal of Particle Physics Economics.

Instead, I’ll celebrate the coming of the Super Collider to Texas with a six pack of Quark, the breakfast of Nobel Prize winners. Or, as Laurie Anderson put it: Hey Professor! Could you turn out the lights? Let’s roll the film.

--------------end of article--------------------

21st Century Update: the Super Collider project was cancelled by Congress in late 1993, five years after the Texas site was selected. About $2 billion had been spent; physicists who were on the payroll were left without jobs (including some foreign physicists who’d moved to the U.S. to work on the project); and, probably the worst side effect, numerous people had lost houses and family farms in the Waxahachie area that were not recoverable. The only good thing to come out of the project that I’m aware of is the lighthearted 2003 novel “A Hole In Texas,” by Herman Wouk. But, like the whole Super Collider foofaraw, it’s not particularly memorable.

The Large Hadron Collider (LHC) near Geneva, Switzerland, also collides protons, but has less energy than the Super Collider would have had, only 7 TeV. It nevertheless found the Holy Grail of Standard Model particle physics, the Higgs boson, in 2012.  Nothing else of consequence has been discovered since then, but the collider will come back on-line soon, with more power.

Classical Electron: Intro

(This is my master's thesis in physics, Southwest Texas State University, 1998.  SWT is now called Texas State University-San Marcos.   At the moment, this uploaded version doesn't have all the equations that are supposed to go with it, but I'm working on that.)

(And now, 15 October 2025, I've found my thesis is available online. So that's where you'll find the equations, and more.)

 

 

 

 

 

 

 

 

 

 

 

 

FOR DEBORAH

 

 

 

 

 

 

 

 

 

 

 

 

INTRODUCTION

          In the 100 years since the discovery of the electron, no one has produced a satisfactory answer to the most elementary questions about this first inhabitant of what has been called[i] the subatomic particle zoo.  Although it has been defined by precise measurements of its charge, charge-to-mass ratio, and magnetic moment, and even though the experimental value of magnetic moment fits very snugly into the calculational confines of quantum electrodynamics, the electron is still a mystery.  Why does it have the charge and mass it has?  Why is the charge precisely the same as the proton’s charge (but opposite in sign), and why do the electron and proton have the charge-to-mass ratios they have?  Until these conceptual questions are answered and the mass and charge of the electron are predicted by a theory, rather than inserted as parameters in quantum electrodynamics as they are now, we cannot say we understand what an electron is.

           The discovery of the electron in 1897 was accomplished by J. J. Thomson’s experimental determination of the charge-to-mass ratio for the formerly unexplained cathode rays, showing that the “rays” are actually streams of charged particles.  Several kinematical or dynamical similarities between charge and mass indicate the charge-to-mass ratio is of fundamental importance in the development of a theory of the electron:   Point charges and point masses produce inverse square fields, charge and mass are both relativistic invariants (scalars)--although mass is often improperly described as varying with velocity,[ii] and charge and mass both resist being accelerated.  Mass resists acceleration because of its mysterious property called inertia.  Charge resists acceleration because it creates a time-dependent electromagnetic field when accelerated and, in conserving momentum, the field reacts back on the charge with a force called the radiation reaction force. 

          The radiation reaction phenomenon has been chosen as the subject of this thesis because the difficulties which arose in classical models of the electron’s radiation reaction have not been solved by quantum electrodynamics. The phenomenon of radiation reaction also represents a partial unification of the ideas of charge and mass, because the radiation reaction force causes an electron to act as if it has an electromagnetic mass in addition to its bare or mechanical mass.  This apparent combination of two kinds of inertial mass--and their apparent inseparability--is a problem “we still don’t understand,” as Pais says.[iii]

           In classical electrodynamics, the Maxwell equations are used to calculate fields when charges and currents are specified, or fields can be given and the distributions of charge and current calculated.  Forces on charge and current distributions can also be calculated from the Lorentz force equation.  As Jackson[iv] notes, however, both regimes usually ignore the radiation reaction effect, partly because there is “negligible error” when acceleration of charges is not extremely rapid, and partly because “a completely satisfactory treatment of the reactive effects of radiation does not exist.”

          Although quantum electrodynamics has been successful in giving precise calculations of radiative effects, it fails to put such calculations on a sound theoretical foundation.  As Jackson[v] says:

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of an elementary particle....One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties....It is one of the triumphs of comparatively recent years (1948-1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment.  From a fundamental point of view, however, the difficulties still remain.

          One very significant domain where accelerations of a charge are rapid enough that the radiation reaction force cannot be neglected is in classical and quantum models of light emission and absorption by atoms.  Because of the oscillatory nature of the electron’s motion in this case, radiation reaction is called radiation damping.  In the classical model of light emission and absorption, the electron is bound to the nucleus by a linear (Hooke’s law) restoring force, and there is a velocity-dependent radiation damping term in the equation of motion of the electron.  However, this velocity-dependent term is just the simple harmonic motion approximation to the more general case where radiation damping is proportional to  , the rate of change of acceleration. 

          The need for a rate of change of acceleration in a single-particle force law is the distinguishing characteristic of radiation reaction. The presence of   in the electron’s equation of motion--the classical Abraham-Lorentz equation--leads to physically senseless solutions.  In one case, exponentially increasing “runaway” solutions are predicted.  Choosing reasonable asymptotic conditions on the electron’s acceleration and avoiding the runaway solutions leads to “pre-acceleration,” the electron accelerating before an external force is applied. These problems are discussed in Chapter 2 of this thesis.

          Another difficulty of both classical and quantum electrodynamic models of radiation reaction is that the electromagnetic mass of a point electron is infinite.  This subject is discussed in Chapter 3, along with the so-called “4/3 problem”--a disagreement between relativistic and purely electrodynamic predictions of the electromagnetic mass of a spherical shell of charge.  In Chapter 4, classical models of the point charge electron and the extended spherical shell electron are discussed, and the spherical shell model is shown to be free of the difficulties of runaway solutions, pre-acceleration, and infinite electromagnetic mass.

          In Chapter 5, Discussion and Conclusion, the 4/3 problem is  considered as logical consequence of the classical electron requiring a relativistic binding energy, just as a nucleus has a binding energy holding together its positively charged protons.  The problem of predicting the observed stability of the electron is also discussed.

          This thesis seeks to put into perspective the classical ideas of radiation reaction and electromagnetic mass, and to discuss how the classical ideas might influence quantum electrodynamics and thereby improve our understanding of the electron as an elementary particle.

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[i] Cindy Schwarz, A Tour of the Subatomic Zoo:  A Guide to Particle Physics , 2nd ed. (American Institute of Physics, New York, 1997).

[ii] Lev B. Okun, “The Concept of Mass,”  Physics Today, June 1989; Carl G. Adler, “Does mass really depend on velocity, Dad?” Am. J. Phys. 55, 739-743 (1987).

[iii] Abraham Pais, ‘Subtle is the Lord...’: The Science and the Life of Albert Einstein  (Oxford University Press, New York, 1982), p.155.

[iv] J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 781.

[v] Ibid.

Classical Electron: Radiation Reaction

RADIATION REACTION

2.1  PHYSICAL DESCRIPTION AND HISTORICAL ORIGIN

          A good introduction to the physical significance of the radiation reaction force is given by Robert March in Physics for Poets:[i]

...when electric charge is moved, an additional force is required above and beyond that needed to accelerate an uncharged mass.  This force is a consequence of the interaction of the charge with its own field, regardless of whether or not other charges are present.  In the process, forces are exerted on, and thus momentum is imparted to and work is done on, the field itself.

          This description alludes to the importance of the idea of the electromagnetic field, the necessary carrier of the radiation reaction force as well as the carrier of the outgoing radiation emitted by the accelerating charge. The radiation reaction problem has been subjected to investigation for nearly as long as Maxwell’s equations for the electric and magnetic fields have existed.  Thus, radiation reaction has been studied theoretically for over a hundred years,[ii] from the work of Larmor and Stewart in the 1870s, and Planck, Lorentz, and Abraham around the turn of the century, to Dirac in the 1930s, and the work of Bethe, Feynman, Schwinger, Tomonaga, and Dyson in the late 1940s that resulted in a renormalizable theory of quantum electrodynamics.  “Renormalizable” means, among other things, that a way was found around the difficulties presented by the infinite electromagnetic mass of a point electron interacting via radiation reaction with its own electromagnetic field.

          One of the recurrent questions is whether the electron is a point charge or an extended charge.  Partly because of this open question, there have been many papers written about the issue of classical radiation reaction since the advent of quantum electrodynamics.  The review article just cited is a comprehensive guide to research published from the 1870’s to 1961. The Quantum Vacuum: An Introduction to Quantum Electrodynamics, by Peter W. Milonni,[iii] provides a more recent discussion of both classical and quantum radiation reaction research.

2.2  THE ABRAHAM-LORENTZ EQUATION

     The most significant early work was done in the years 1903-1905, when the German physicist Max Abraham and the Dutch physicist H. A. Lorentz independently developed an equation of motion for a spherical electron.[iv]  Their equation includes a radiation reaction force.

          A derivation of the Abraham-Lorentz equation involves calculating the force of each part of an accelerating spherical shell of charge on all the other parts and summing the contributions, giving a net self-force due to the retardation time or propagation time of the electromagnetic field across the finite extent of the electron.  As described by Griffiths and Szeto:[v]  

Lorentz showed that the radiation reaction force is attributable to the breakdown of Newton's Third Law in classical electrodynamics.  When an extended charge accelerates, the force of one part on another is not equal and opposite to the force of the second part on the first.  When these imbalances are integrated over the entire configuration, the result is a net force of the charge on itself.

          Lorentz’s calculation is given in his book Theory of Electrons,[vi] and similar calculations appear in several recent texts.[vii]  The method uses a series expansion  of the vector potential in the retardation time 2a/c, where a is the electron radius and c  is the speed of light.  For a radius of 10^-15 meters the retardation time is on the order of 10^-24  seconds, and the electron is considered to remain approximately stationary during this very short time interval.  In this approximation “only terms which are linear in the particle’s velocity or its time derivatives”[viii] are kept.  Thus, magnetic effects are not considered, and the calculation is therefore nonrelativisitic.  It can be made fully relativistic by including terms of order (v/c)² and higher.

          The nonrelativistic result for the self-force, simplified to the case of one-dimensional motion, is given by Feynman:[ix]

F_self (x,t) = —(e²/6ε₀ac²) d²x/dt²  +  (e²/6ε₀c³) d³x/dt³  + (ae²/6ε₀c⁴) d⁴x/dt⁴  + …

The first term is a constant times acceleration, and thus the units of the constant are mass units (kilograms in the mks system used here).  This is the electromagnetic mass term.  The electron radius a appears in the denominator of this term, but does not appear in the second term and then appears linearly in the third term and to succeedingly higher powers in subsequent terms.  Thus if the electron radius approaches zero, as it does in the limit of a point charge, the higher order terms go to zero, the second term is unaffected, and the electromagnetic mass term approaches infinity.

          For finite but small values of the retardation time (the assumption that leads to the series approximation), the third and higher order terms are negligible. The second term, which shows no dependence on the electron’s structure, is the radiation reaction term.  As described in Chapter 1, the effect of this term on the subsequent motion of the charged particle is often considered to be negligible.  The radiation reaction term, however, is necessary to precisely account for energy conservation, as will be shown in Section 2.3.

          Putting the remaining two self-interaction terms into the equation of motion for an electron with bare mass m₀ subjected to an external force gives

m₀ d²x/dt²  =  —(e²/6ε₀ac²) d²x/dt²  + (e²/6ε₀c³) d³x/dt³  + F_ext

or

(m₀ + e²/6ε₀ac²) d²x/dt²  =  (e²/6ε₀c³) d³x/dt³  + F_ext

The sum of mechanical mass and electromagnetic mass on the left-hand side of this equation is considered to be the observable or experimental  mass of the electron, m.  Writing the equation in terms of m gives the one-dimensional Abraham-Lorentz equation of motion for the electron,

m d²x/dt²  =  (e²/6ε₀c³) d³x/dt³  + F_ext

The coefficient of the d³x/dt³ term is often written as mτ ,  where τ is given by

τ = e²/6ε₀ mc³

which is of the same order of magnitude as the retardation time.  Then the Abraham-Lorentz equation is written

m(d²x/dt² - τd³x/dt³) = F_ext

As noted by Jiménez and Campos[x] (who use a for acceleration ), “the structure of the equation departs from the Newtonian canon because of the da/dt term, which involves an extension of the phase space and the specification of a at some time.”  The consequences of the departure from a standard Newton’s second law structure are discussed in Section 2.4. 

          As Erber[xi] points out, however, the A-L equation despite some of its pathological solutions has been accepted as valid because it accounts for the dynamical effects of radiation reaction “through an augmented form of Newton’s second law” so that one “universal form of [F_self] is sufficient for all cases,” and because the equation has a “natural place in a relativistic setting.”

          For a bound electron oscillating under the action of a linear restoring force,  the equation of motion including radiation reaction becomes

m(d²x/dt² )  =  mτ(d³x/dt³) - m(ω_o) ² x  +  F_ext

If the radiation reaction is considered to be small compared to the other terms, the simple harmonic motion (specified by assuming x(t) = x_ocos(wt)) allows replacement of  , giving

For an electric field E_x(t) as the external force, the equation becomes

which is the classical equation of motion for an electron in an atom interacting with light.  (The Appendix discusses the first derivation of these equations, published in 1897 by Max Planck.  Planck was considering the effects of electromagnetic damping on the entropy of abstract electromagnetic dipole oscillators, one aspect of the so-called blackbody problem.)

2.3  RADIATION REACTION AND ENERGY CONSERVATION

          When an uncharged particle of mass m is accelerated by a net force F, the work done in accelerating the particle is equal to the change in its kinetic energy.  However, an accelerating charged  particle, in addition to experiencing a change in kinetic energy, produces electromagnetic radiation.   As Griffiths and Szeto[xii] describe it, "radiation carries away energy, which, for a structureless particle, must come at the expense of kinetic energy."  Under this assumption--that translational kinetic energy alone is converted into electromagnetic energy--radiation reaction must somehow be included in a work-energy calculation in order to satisfy energy conservation.

          For the case of periodic motion, this will be accomplished if the rate at which the radiation reaction force does work on the charge is shown to be equal to the rate of energy loss to radiation, which is given by the Larmor formula.

          The self-force equation, truncated to two terms, is

Since the first term is the electromagnetic mass term and is joined with mechanical or bare mass in the electron equation of motion, it will be included in the rate of change of kinetic energy of the electron caused by an external force.  Multiplying the self-force equation by  shows this and also shows how the rate of energy loss to radiation comes from the radiation reaction term.  Using  and multiplying the first term by  gives

,

showing how the electromagnetic mass contributes to the electron’s kinetic energy for the classical charged spherical shell model. The radiation reaction term multiplied by  is

,

which can be written as

The first term on the right-hand side averages to zero over any integral multiple of the period of motion.  The second term is the square of the acceleration and thus does not average to zero over a period of the motion.   The result is that the average rate of  energy loss due to the radiation reaction force is identical to the Larmor formula for energy radiated

The radiation reaction force, although it may be small compared to external forces, is required for the energy of an accelerating charge to be conserved.

2.4 RUNAWAY SOLUTIONS

The Abraham-Lorentz equation is abnormal as an equation of motion for a single particle.  Although it is linear, it contains a third time  derivative of position, which requires that an initial value of acceleration in addition to initial velocity and position.  Third order equations can show up in other single particle situations, for instance in solving simultaneous second order equations.  One example is the motion of a charged particle in a region with both a magnetic and an electric field.[xiii]  In this case the solutions make sense physically.

          The Abraham-Lorentz equation has been described as predicting “unphysical behavior because it is higher order in time differentiation than a mechanical equation of motion should be”[xiv] and this “involves an extension of the phase space.”[xv]  Erber[xvi] notes that the equation is actually a functional equation rather than a differential equation, and this “implicit x(t) dependence of F_self . . . endows the classical theory with its astonishing wealth of solutions.”

          The wealth of solutions includes a prediction of exponentially increasing acceleration even when no external force is applied.  This is the homogeneous equation

,

and solving for the acceleration gives

which implies that radiation reaction causes spontaneous acceleration of the electron.  The general solution--i.e., when an external force is included--necessarily contains the solution to the homogenous equation.  The general solution  can be written as[xvii]

which predicts that acceleration increases exponentially no matter what the form the external force takes.

          In Section 4.1, one method of avoiding the runaway solutions is discussed, but it leads to the problem of pre-acceleration.  In Section 4.2, where the Abraham-Lorentz equation is rewritten as a delay-differential equation, the problem of runaways and pre-acceleration is shown to be avoidable by avoiding the point charge limit and its associated infinite electromagnetic mass.  First the concept of electromagnetic mass needs to be discussed in more detail.

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[i] Robert March, Physics for Poets, 2nd ed. (McGraw-Hill, New York, 1978), p. 76.

[ii] Thomas Erber, “The Classical Theories of Radiation Reaction,” Fortschr. Phys. 9, 343-392 (1961).

[iii] Peter W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics  (Academic Press, San Diego, 1994), Chapter 5.

[iv] Jackson, p. 786.

[v] D. J. Griffiths and Ellen Szeto, “Dumbbell model for the classical radiation reaction” Am. J. Phys. 46, 244-248 (1978).

[vi] H. A. Lorentz, Theory of Electrons (B. G. Teubner, Leipzig 1909), Note 18.  This is a series of lectures Lorentz delivered at Columbia University in 1906, later reprinted in paperback (Dover Publications, New York, 1952).

[vii] See, for instance, Jack Vanderlinde, Classical Electromagnetic Theory (John Wiley, New York, 1993), Chapter 12.

[viii] D. H. Sharp, “Radiation Reaction in Nonrelativistic Quantum Theory,” in Foundations of Radiation Theory and Quantum Electrodynamics, ed. A.O. Barut (Plenum, New York, 1980), p. 130.

[ix] R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading MA, 1964),  Chapter 28.

[x] J. L. Jiménez and I. Campos, “A critical examination of the Abraham-Lorentz equation for a radiating charged particle,” Am. J. Phys. 55, 1017-1023 (1987).

[xi] Erber, p. 349.

[xii] Griffiths and Szeto, p. 244.

[xiii] K.R. Symon, Mechanics, 3rd edition (Addison-Wesley, Reading MA, 1971), pp. 144-148.

[xiv] Jackson, p. 798.

[xv] Jiménez and  Campos, p. 1019.

[xvi] Erber, p. 371.

[xvii] Milonni, p. 157.