SELF-FORCE ON A CHARGE DUMBBELL WITH ARBITRARY ACCELERATION
 

Elsewhere on my website I mention a mistake in my self-force book, namely the claim that the leading order term in the electromagnetic self-force on a charge dumbbell with arbitrary rigid acceleration will always be either aligned or counteraligned with that acceleration. I even provide a detailed proof that it was a mistake! However, it has been brought to my attention by Thomas Banaszek, a PhD student in Lancaster, that I may have neglected to make a crucial relativistic correction. If this correction is justified, it would mean that my mistake was not a mistake. Since then, Andrew Steane in Oxford has thrown other light on this matter. The whole story can be found in pdf form by clicking the green button. Note that this was updated yet again on 27 July to include further generalisations of proofs.

It turns out that people have been doing calculations with charge dumbbells for years. An early paper was D.J. Griffiths, R.E. Owen: Mass renormalization in classical electrodynamics, Am. J. Phys. 51, no. 12, 1120 (1983) and a more recent one A. Ori, E. Rosenthal: Calculation of the self-force using the extended object approach, arXiv:gr-qc/0309102v1 (21 Sep 2003). I discuss the latter in detail in the attached paper as an appendix. In this very lucid and elegant paper, the authors raise the crucial question of how to add up forces acting at different spacetime events.

 

Steane fully exposes this issue in his paper A.M. Steane: The non-existence of the self-accelerating dipole, and related questions, arXiv:1311.5798v3 (19 Mar 2014), showing that there may in some instances be several possible ways to define such a sum. The basic problem here is that there is generally no covariant way to sum forces acting at different spacetime events. This is a very typical relativity problem. Any such sum will depend on a local foliation of spacetime. On the other hand, we require something of this sum physically. We wish to see how it will affect the motion of the spatially extended charge distribution as a whole. So if we make different definitions, we need careful consideration of the resulting equations of motion for our spatially extended object. Ori and Rosenthal do explicitly raise the question of the equation of motion and I have drawn attention to it in the more standard analysis.

 

Another thing Steane does is to show how the Poincaré stresses contribute to the total self-force, and in particular, the way they cancel the part of the EM self-force that is not aligned with the acceleration. He does this using a pressure model for the case of the uniformly accelerating charge dipole and I extend this to the arbitrarily accelerating charge dipole. We both assume the motion is rigid. This is a very heavy constraint on the motion that effectively allows one to deduce what the self-force due to internal stresses (pressure in this model) must actually be. When one thinks of a complex bound system like a proton, it is perhaps surprising to think that one could deduce the self-force due to all other fields for which the components of the object are a source without actually knowing anything about them. This just illustrates how restricting the rigidity assumption is. No object like a proton will move rigidly. It is a dynamic bound state.

 

Still it is striking that one would not be able to 'renormalise classical electromagnetism' if one considered only the electromagnetic effects. That is, in the point particle limit, one cannot renormalise the mass in Newton's second law to account for electromagnetic effects alone, precisely because the potentially divergent leading order term in the EM self-force is not aligned with the acceleration of the system. One must consider the system as a whole, i.e., all forces at work within the system. 

 

This discovery throws light on the three ways of deriving a mass for the EM fields: by energy, by momentum, and by self-force. It is notable that the momentum-derived and self-force-derived masses are always equal. This is because momentum gets into the fields by doing work against the leading order term in the self-force. However, the energy-derived and momentum-derived masses are often different, although not always (see my book Self-Force and Inertia). Poincaré introduced his internal stresses precisely in order to explain this difference and I also explain in my book where the discrepancy comes from, from a slightly different angle. But with these calculations (thanks to Steane), we see how Poincaré's stresses can also be used directly to explain the discrepancy between the energy-derived and self-force-derived masses. 

 

The second half of Steane's paper is concerned with the charge dipole in a static homogeneous gravitational field. In order to say anything about the EM fields of such a system, one must invoke the strong equivalence principle (SEP), which tells us how to ship our non-gravitational theories of physics into the general relativistic (GR) context. Then the fields are precisely the same when expressed relative to the global inertial frame that exists in this context as the fields of a rigidly and uniformly accelerating dipole in a flat spacetime without gravitational effects, not even non-tidal ones. In other words, the second half of his paper is, mathematically speaking, about exactly the same problem as the first. And yet he feels it necessary, like many, to redefine physical quantities 'for' the uniformly accelerating observer 'moving with' the dipole in some sense.

 

I would welcome any insights into the motivation for drawing up such pictures 'for' accelerating observers. The last few sections of the above paper discuss Steane's analysis in a critical way, particularly with regard to this question, but I have already discussed it at length in the summary of my talk in Bad Honnef. The green button below will summon this up. 

Steane considers that Ori and Rosenthal's approach is sometimes a better, or rather a more natural definition for the sum of forces acting at different spacetime events, but not in every case. He bases this assessment on what is more natural for this or that observer, but I feel that he does not justify his claims physically. The present view is that Ori and Rosenthal's account provides an alternative way of deducing the self-force effects in certain cases, but that those cases are not determined by what any observers are doing as suggested by Steane. Rather, their kind of analysis depends heavily on the equilibrium in the system being totally static in a certain sense, something that is assumed for all the calculations discussed here anyway. As mentioned, their reasoning is presented in appendix in the above paper.

 

In their paper, Ori and Rosenthal show that one may use the alternative definition of the EM self-force to obtain an equation of motion for the charge dumbbell in the point particle limit that is entirely consistent with the usual definition. Here we point out with Steane that the self-force due to internal forces (Poincaré stresses) must be zero when we use this same altered definition for sums of forces acting at different spacetime events, and we show by generalising a method due to Steane that this will indeed be the case, provided that the motion of the dumbbell is rigid. We also suggest that this whole method depends heavily on the rigidity of the motion and would not be generalisable to 'particles' whose internal structure involves a dynamic equilibrium, such as the proton.

 

We argue against Steane's proposal that Ori and Rosenthal's method is merely a natural choice for a certain kind of observer, namely one comoving with the dumbbell. What matters here is the equation of motion of the system and its resulting worldtube. In the present case, when the various assumptions are satisfied and the approximations justified, we obtain exactly the same worldline in the point particle limit as we would with the previous method, so this clearly constitutes a consistent reanalysis of the problem that has nothing to do with any observer. Outside the point particle limit, there are likely to be some small differences between the predictions due to the different approximations.

 

The point about equations of motion is that they give us something that is independent of any coordinate system, namely a worldline or a worldtube. It thus has an immediate physical significance. Pictures for observers are by definition coordinate dependent, and if there is one major lesson from the relativity theories, and in particular from the general theory of relativity, it is that coordinate particularities of things are physically irrelevant. Naturally, if one is doing a sky survey, it is important to be able to set up frames of reference, and for this one must make operational definitions and assumptions. However, the pictures here seem to be motivated largely by some theoretical urge and there is no mention of the actual practical problems, whence they appear worthless.

 

The title of Steane's paper refers to the non-existence of the self-accelerating dipole, which should seem rather surprising. It is a reference to a paper by Cornish who found a solution to the equation of motion of the dipole in which it would accelerate without the need for any external force! This was followed up by one by Griffiths, in which it was concluded that a dipole of the right dimensions might levitate in a gravitational field. This is all impossible of course, and Steane explains why. The two papers mentioned should probably never have been published because it is so obvious what is going wrong! Perhaps that is why nobody ever published a reply. Still, Steane's account is impeccable.

 

I should add, finally, that Steane never persevered with these discussions, at least with me.

 

Ori and Rosenthal generalise from the dumbbell to arbitrary assemblies of charged particles in rigid acceleration using their method. It turns out that one can use Steane's pressure model even here to show that the Poincaré stresses will not contribute a self-force under Ori and Rosenthal's definition of sums of forces acting at different spacetime events, a point that had to be assumed in Ori and Rosenthal's analysis (see the pdf). Furthermore, it seems likely that the very same pressure model can be used for the 'standard' definition of sums of forces, although it becomes somewhat strained in the sense that it would be hard to implement physically when there are many charged particles or a continuous charge distribution. It would thus be worthwhile finding a more elegant formulation of the Poincaré stresses. It might throw some light on the reason why the relativistic Navier-Stokes equation delivers such a convenient 'pressure' distribution for these purposes.

 

In the above paper (see the pdf), I have attempted to carry out a hidden momentum analysis, as suggested by Steane. The results are not beautiful and surely contain some errors, or misunderstanding of this method. Any comments would be welcome.