Inertial reaction forces (discussed in "The Origin of Inertia") are a commonplace of everyday life. When we push on stuff, it pushes back because of its inertial mass. Less common in everyday life are pronounced recoil forces -- a special type of inertial reaction force -- like those experienced when shooting a gun or stepping out of a small boat onto a dock. But we know that they're quite real. The generalization of those experiences is the realization that whenever some massive object ejects part of itself, since the ejected part carries away energy and momentum, the original object must experience a recoil force. It changes the object's momentum in such a way that the momentum of the two parts after ejection is the same as the its momentum before ejection. That's just a convoluted way of saying momentum must be "conserved" in any "isolated system". When folks figured out that when accelerating electric charges launch electromagnetic waves that move away from the source charges at the speed of light carrying energy and momentum with them, they realized that the charges must experience a recoil force, that is, a force of "radiation reaction".
Normally, electromagnetic radiation reaction forces are ridiculously small. For example, if you rigged up a radio antenna to put out a kilowatt of power in one direction, the reaction force on the antenna would be a fraction of a dyne (the weight of several fleas roughly). So in almost all circumstances you can just ignore radiative reaction effects, pretend that they don't exist. But not always. In high energy elementary particle accelerators (like the ones at Fermilab or CERN) radiation reaction is an obvious fact of life. As the particles traveling at nearly the speed of light are bent into their circular paths by magnets, they are accelerated. And they radiate. The reaction force produced by the radiation slows the particles down, unless power is applied to replace the radiated energy and momentum.
Radiation reaction is rarely a major part of a formal course of study in physics. In part this is because it's only important in rather unusual circumstances. And in part it's a consequence of the fact that radiation reaction has some peculiar features. The problems with radiation reaction have been known for almost a century. They were already old when Feynman put an outstanding summary of them into chapter 28 of volume two of his Lectures on Physics, now nearly forty years ago. A fair amount has been written on radiation reaction since then, but the difficulties he described remain. Chief among those difficulties are problems with "causality" (causes always preceding effects) and seeming transient violations of the conservation of energy and momentum. Much of what I'll say here follows Feynman's discussion, so you may want to take a look at what he has to say for yourself.
The specific problem with radiation reaction that we're going to be concerned with is: What happens to the mass of something as it's radiating? Eventually we'll be looking at this question in the context of gravity, but it turns out to be a problem in electrodynamics too. And since electrodynamics is well-studied, it's instructive to see how this all works in that case.
The masses of things nowadays are known not to be due chiefly to their electromagnetic properties. But early in the century some folks thought it might be possible to explain mass and inertia electromagnetically, especially after Einstein showed that energy and mass are equivalent (yes, E = mc2). Two lines of argument led to this belief. First, if you view electrons (the simplest of all the "elementary particles") as being little extended spheres made up of some electrical dust, since each particle of dust must repel all of the other particles of dust (like charges repel), a lot of work (energy) must have been invested in assembling them. By Einstein's relationship, the assembly energy must have mass. The assembly energy depends on how big the sphere of charged dust is (as one over the radius in fact). So the energy goes up as the radius goes down. If the radius is exactly the "classical electron radius", about 10-13 centimeters, then the assembly energy is computed to be the observed electron mass.
The other line of argument that suggests that the mass of electrons might be electromagnetic in origin is the fact that the electromagnetic field of a moving electron has momentum in it. When you calculate the momentum (see Feynman for details) it turns out that the coefficient of the velocity, the mass that is, is essentially the same as the expression for the mass you get in the assembly energy calculation. This looked fairly promising. But it didn't work out. The reasons aren't germane to our purpose. (The history of the subsequent developments in this business, however, is one of the heroic tales of the 20th century. As such, it has been recounted many times at all levels of sophistication. Of the popularizations I've seen, I especially like Crease and Mann's, The Second Creation.) Although the masses of elementary particles are now known not to be attributable exclusively, or even chiefly, to electromagnetism, electromagnetism does make a contribution. And that turns out to be important in considering radiation reaction.
We now ask: How does the launching of electromagnetic waves, radiation that is, produce the reaction force on electrons we know must be present if the conservation of energy and momentum are to be preserved? Well, if electrons are extended spheres of charged dust, then the electrical forces (and those that balance them) that act between the particles of the dust must take time to get across the distances separating them according to special relativity theory. So when we push on one part of the electron, only later do the other parts detect the changes our push has produced and adjust for them. Because of these time delays, during accelerations the forces between the dust particles are unbalanced. The net force due to the imbalance turns out to be the force of radiation reaction. The unbalanced force disappears when no accelerating force is present, even if the electron is moving.
Formally, following Feynman's notation, the self reaction force on an electron in the case of a simple acceleration in the x direction is:
+ higher order terms. (1)
is a numerical factor of order one, e the electric charge of an electron, a the radius of the dust sphere, and each dot over the xs means differentiation with respect to time once. The higher order terms in this "series expansion" scale with positive powers of the radius. For any reasonable electron radius they are so small that they can be safely ignored.
The first term on the right hand side of Equation (1) is just the electromagnetic mass times -- that is, the normal inertial reaction force if a is taken to be the classical electron radius. The second term is the one that accounts for radiation reaction forces. It has two noteworthy properties. First, it doesn't depend on the size or shape of the electrical dust, telling us that it is independent of the self-energy business that presumably accounts for the inertial mass of the electron. Second, it depends on the third time derivative of position. This causes all sorts of trouble. It opens the way to "pre-accelerations" (accelerations that start before the force that causes them is applied) and "runaway solutions" (accelerations that continue after the applied force has been removed). Folks have developed some clever, if not entirely convincing, ways of trying to deal with these problems. We're going to ignore them.
To investigate energy and momentum conservation in radiative processes we have to let the force act to produce a change in the energy. The rate at which the energy changes, the power that is, at any instant is just the force times the velocity . So, ignoring the higher order terms:
Now is proportional to , so we see that the first term on the right hand side represents the change in the kinetic energy of the electron. To see the physical meaning of the radiation reaction term we rewrite it as:
When the minus sign is multiplied through, the term on the right hand side of Equation (3) is always positive. It represents the energy carried away by the radiation. The term, however, is different. It can be either positive or negative. And for periodic motion of electrons, it averages to zero over time. As Feynman remarked, this term -- Equation (3) -- must be included in any account that hopes to conserve energy and momentum since we know with certainty that radiation carries energy and momentum away from accelerated charges.
Radiation reaction is attended by some very thorny problems. I have already alluded to several of them, and Feynman discussed others. Arguably the nastiest problem associated with radiation reaction -- the problem that is directly related to transient mass fluctuations -- was not mentioned by Feynman. It is quite simple. It is known as a matter of fact that electric charges subjected to constant accelerations radiate electromagnetic waves, and the energy they carry away from their source charges is proportional to the square of the charges' acceleration. But when the acceleration of the radiating charges is constant, the time derivative of the acceleration vanishes, and with it the radiation reaction term in Equation (2) [i.e., Equation (3)] disappears too. So, our mathematical formalism tells us that during constant accelerations all of the work being done by the accelerating force goes into change of the kinetic energy of the charges. Nonetheless, they radiate energy too. We seem to be faced with an obvious violation of the conservation of energy here.
This problem has been known since the early part of the century. A small, but substantial literature has grown up around it. [Two rather old, but very good papers that explore this problem are: Candelas and Sciama, "Is There a Quantum Equivalence Principle," in: Essays in honor of Bryce deWitt (Adam Higler, 1984), pp. 78-88, and Fulton and Rohrlich, "Classical Radiation from a Uniformly Accelerated Charge," Annals of Physics, 9, 499-547 (1960).] In addition to the apparent violation of the conservation of energy, this problem has attracted attention because it figures into discussions of the "equivalence principle" (the proposition that all things fall with the same acceleration in a gravity field, which is thus equivalent to an accelerated frame of reference), one of the cornerstones of general relativity theory. We're going to leave general relativity theory out of our considerations too. Energy (and momentum) conservation is problem enough by itself.
The easiest way to see the details of this problem is to consider a sequence of accelerations of a charged particle and display graphically what's happening to all of the relevant quantities in time. The sequence of accelerations we examine is:
We take the rates of change of the acceleration all to be the same and the intervals of the accelerations to result in causing the charge, starting from zero velocity, first to speed up, then slow down to zero velocity. During the periods of increasing and decreasing acceleration will be non-zero and constant. All of this is shown in Figure 1. (The scales for , , and are arbitrary.)
Next, using Figure 1, we plot in Figure 2 the rate at which energy is carried away by the radiation -- which is proportional to according to electrodynamics -- and the work done by the radiation reaction force -- which is proportional to .
Inspection of Figure 2 reveals the problem. If we consider the complete interval we know that the conservation of energy requires that the area under the curve be equal to minus the net area under the curve. Evidently, not only are the areas unequal, but instant-by-instant the energy flows don't balance anywhere during the process too. If we only consider the accelerating charge and the radiation, we have a sequence of transient energy conservation violations that some argue must balance out when averaged over time.
To give you a sense of the seriousness of this problem, let me relate some of Fulton and Rohrlich's comments on it.
"In the case of uniform acceleration, . . . , the total work done by the radiation reaction force vanishes. . . . The internal energy of the electron, . . . therefore, decreases while energy is being radiated. This result seems to lead to a very unphysical picture: The accelerated electron decreases its 'internal energy,' transforming it into radiation. Does this mean that the rest mass of the electron decreases? [They then do a little calculation.] Thus we obtain the comforting result that the change in internal energy of the particle does not affect its rest mass. Rather, the radiation energy is compensated by a decrease of that part of the field surrounding the charge, which does not escape to infinity (in the form of radiation) and which does not contribute to the (electromagnetic) mass of the particle."
This is a pretty remarkable statement. Energies of all forms have an equivalent mass. And the energy that resides in a non-radiative field coupled to the particle can be expected to contribute to the measured mass of the particle, just as the energy in the non-radiative part of the electromagnetic field does. (That's why they specifically exclude the electromagnetic field as the source of the internal energy that goes into the radiation when the reaction force is absent. The energy in the "static" or "inductive" electric and magnetic fields is proportional to the square of the field strengths. So if energy is being drawn from these fields to feed the radiation, their field strengths go down. And as they go down, the momentum in these fields goes down too. As a result, the mass of the electron goes down, contrary to expectation.) When you stop and think about this magical source of "acceleration energy", it doesn't sound very convincing. Without it, though, instantaneous violations of energy conservation would occur. Perhaps that's why Fulton and Rohrlich went on to say, "If the emerging physical picture seems unsatisfactory, one can reject the equations of motion. This is a possible alternative, but then the question of energy conservation simply cannot be answered, because the equations of motion are unknown."
How can this problem be dealt with? You might think that if we could let the rest mass change during accelerations we could get the energy we need for the radiation during constant accelerations. Alas, this doesn't help, because in reducing the mass to get the energy for the radiation, for a steadily applied force the acceleration increases, leading to yet more radiated energy. If Equation (2) is correct, this must be true, for the radiation reaction term vanishes. So there's no way to divert some of the energy that goes into kinetic energy into the radiation. Another possibility, taking a cue from quantum vacuum fluctuations, is to assume that transient violations of energy conservation actually occur. In light of the fluctuation-dissipation theorem that links vacuum fluctuations and radiation reaction, this might seem at least plausible. The problem with this approach, of course, is that we're not talking about quantum scale phenomena here. These transient violations can be made rather large and long. So large and so long that they could be produced in a well-equipped laboratory no doubt.
It's instructive to isolate the source of all the trouble here. That turns out to be the "little calculation" that led Fulton and Rohrlich to the conclusion that the rest mass of the accelerating electrically charged particle is constant. The way that they came to that conclusion was by noting that the "four-velocity" and the "four-acceleration" are "orthogonal". That is, the spacetime generalizations of the acceleration and velocity of an object are perpendicular to each other. (Although this, and much of what follows, is easily demonstrated, I will not spell out the details here. You can find them spelled out with crystal clarity in Wolfgang Rindler's outstanding little book, Introduction to Special Relativity [Oxford, 1991]. The relevant pages in the second edition are 58-60 and 90-93.) Since this is a simple kinematic relationship, it is always true. Now if the four-force (yes, the spacetime generalization of normal "three"-forces) points in the same direction as the four-acceleration, it turns out that restmass must be constant. This is how Fulton and Rohrlich came to the conclusion they did. The question one may pose here is: Do the four-force and four-acceleration have to always point in the same direction? Not necessarily.
It is, of course, obvious that restmass is not, in general, a constant during accelerations. Any deformable object stressed by an accelerating force stores part of the work done by the force, as the acceleration increases, in the form of elastic "internal" energy. That changes the restmass of the body. When the accelerating force is removed, the stresses relax, the added internal energy disappears, and the object recovers its original restmass. The increased restmass is present, quite clearly, even if the acceleration the object experiences is constant since the internal stress energy is stationary during constant accelerations. When the acceleration is changing, the internal energy (stresses) in the object will change so that the object does the requisite work on the accelerating agent. For example, as the accelerating force is removed, the transient stored internal energy present in the object during constant acceleration must be conveyed back to the agent. If something like this were going on during the acceleration of electrically charged particles, everything might be OK. The transient restmass increase, for a given applied force, will reduce the acceleration of the charge, decreasing both the amount of energy carried away by the radiation and the kinetic energy acquired by the charge during and after the acceleration. In effect, the increased restmass keeps all of the energy being delivered by the accelerating force from going into final-state kinetic energy so that some energy is available to feed the radiation field. When the charge is radiating, then, its restmass is greater than when it's in a state of inertial motion (i.e., unaccelerated).
Since the net reaction force on an accelerating electron, given by Equation (1), was recovered by taking into account the time-delays across a presumed finite size of the charged dust that makes up the electron, we might guess that if we let the cloud of dust be squished by the acceleration, the (carefully chosen) squish might alter the restmass so as to make things work out alright. Sad to say, this doesn't work. For one thing, electrons are known to be very much smaller than the classical electron radius, so the model of a deformable electron is questionable to start out with. For another, setting aside the issue of the electron's self-energy as a function of radius, we know that Equations (1) and (2) are basically correct. They can't be easily fudged to get the desired restmass behavior that might solve our problems. This is easily shown by substituting the equivalent expression for the radiation reaction term in Equation (3) into Equation (2):
If the sign of the term were reversed so that when is computed it didn't disappear, we'd have the sort of behavior that might get rid of the transient energy (and restmass) conservation violations we're stuck with. This would be true if instead of the term in Equation (2) we had . Then our radiation reaction term would be . And if we could assume the term to be small (ideally, completely negligible), the radiation reaction term would mimic the radiated power. Alas, there seems to be no justification for such fudging except for preserving the conservation of energy.
How, you may be wondering, have folks actually dealt with all this? Well, by simply asserting that energy and momentum must be conserved no matter what and making things work. Dirac was the person who did this. A modified version of Equation (4) is known as the Lorentz-Dirac equation. This has been much discussed, both in the journal literature and advanced textbooks on electrodynamics. Arguably the clearest presentation of how this is handled is to be found in sections 10 and 11 of of chapter 21 of Panofsky and Phillips' classic text, Classical Electricity and Magnetism. There one finds that when one creates the "covariant generalization" of the radiation reaction force term, some freedom in the equations allows you to stick in a term that ends up reducing the external force by just the amount needed to account for the radiation being emitted -- even during periods of constant acceleration -- without changing the restmass. Neat, Huh? But keep in mind Fulton and Rohrlich's remarks and "accelerational energy". They were made in the context of the Lorentz-Dirac treatment of radiation reaction.
I'm belaboring this business for a reason. When we consider gravity/inertia for accelerated stuff, as you might expect in view of the fact that in the lowest approximation the field equations are like those for electrodynamics, we'll find the same sort of higher order effects. (But instead of , we'll get which we see is just a classical radiative reaction effect plus a effect. It's worth noting that since is proportional to and thus the rate of change of kinetic energy, that , in general, will be proportional to the second time derivative of energy.) I'd like to be able to tell you that the gravity/inertia transient restmass effect could solve the instantaneous energy conservation problem here. But it doesn't. The reason why is the same as the one that louses up electromagnetic quantum vacuum fluctuation explanations of inertia: The charge to mass ratios for elementary particles aren't all the same.
There is an important message to take away from all of this. It's that whenever you encounter effects that involve stuff that looks like radiation reaction, you should be prepared for apparent transient violations of energy and momentum conservation. I say apparent because the Wheeler-Feynman "absorber" interpretation of radiation reaction makes plain that "non-local" (i.e., retarded/advanced) interactions with distant matter not normally considered to be part of an appropriate "isolated system" take place. So prepared for peculiar possibilities, we're now ready to look at gravitational/inertial transient mass fluctuations.
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Copyright © 1998, James F. Woodward. This work, whole or in part, may not be reproduced by any means for material or financial gain without the written permission of the author.