Does the 4/3 problem lie with the non vanishing of the divergence of the EM stress tensor?

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john mcandrew

Sep 6, 2020, 8:55:36 PM9/6/20
This quote comes from section 16.4 of Jackson's Classical Electrodynamics, where I've used @ = partial, a= alpha, B =beta, y = lambda:

So far our discussion of the Abraham-Lorentz model of a classical charged particle has been nonrelativistic, with apologies for the paradox of different electromagnetic "masses" from electrostatic and Lorentz force (dynamic) considerations -- the infamous 4/3 problem, first noted by J.J. Thomson (1881). The root of the difficulty lies in the nonvanishing of the 4-divergence of the electromagnetic stress tensor. In contrast to source-free fields, the stress tensor Theta^aB of any charged particle model has the divergence,

@_a Theta^aB =−F^By (J_y)/c=−f^B

Where f^α is the Lorentz force density. As stated in Problem 12.18, only if the 4-divergence of a stress tensor vanishes everywhere do the spatial integrals of Theta^a0 transform as a 4-vector.

I don't agree with the above. If an electromagnetic system is closed where all interactions are entirely via the sources with no additional forces such as Poincare stresses, then the total EM energy and momentum should still transform as a 4-vector. The root of the 4/3 difficulty lies with the 1/3 additional energy-momentum required to get the sources of the EM system to move at the same velocity as one another.

Any Comments appreciated in advance,


Jos Bergervoet

Sep 8, 2020, 8:41:24 AM9/8/20
It seems you can define "the 4/3 problem" in various ways. But always
by doing something wrong! Because there is no real 4/3 problem, it is
just a paradox.

In this discussion, for instance, entry 14 makes clear that the paradox
can be formulated by ignoring that a moving charged sphere becomes an
ellipsoid, which of course is an error, so no wonder a paradox results:


If there is no real problem then of course we can ignore loaded
questions like "does the problem remain in QM?" as in stackexchange's
discussion, or "does the problem lie with [insert some fact]" as in
this thread. We can just say that there is no problem if we avoid
errors and if we do not avoid errors we can create the problem in a
number of different ways..


john mcandrew

Nov 15, 2020, 11:15:58 PM11/15/20
Yes, I think it's important to look at all viewpoints; and this case has 'The Boyer-Rohrlich Controversy' history for it:
Philosophy of Physics Graduate Lunch Seminar (Monday - Week 3, HT19

"In the early 1980s a short argument concerning the proper definitions of field momentum and energy took place in the Physical Review journal. The argument was initiated by T. H. Boyer's criticism of the covariant definition of the field momentum and energy densities, suggested as a replacement of the classical definitions. A defense of the new definition was issued by F. Rohrlich, who was also one of its originators. Boyer argued against the new definition with the claim that the classical definition is a natural, conceptually clear way to define energy and momentum for the electromagnetic field; Boyer recapitulated the classical model of the electron as a body of finite-extension as an illustration of that claim. Rohrlich defended the covariant definition by arguing that it is improper for the energy and momentum of the electromagnetic field not to define a covariant 4-vector independently of the energy and momentum of any other auxiliary field - as is required in Boyer's approach. I will try to show that Boyer's approach coheres quite well with the dynamical interpretation of special relativity, while Rohrlich's approach can more adequately be based in the geometrical interpretation. I will also argue that consequently, Boyer's approach has some of the conceptual advantages of the former interpretation."

But back to where Jackson writes: @_a Theta^aB =−F^By (J_y)/c=−f^B, it's important to also note that integrating the RHS through out the closed electromagnetic system must equal zero from the total conservation of EM momentum and energy.

John McAndrew
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