## 4/3 problem

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Модераторы: morozov, mike@in-russia, Editor

morozov
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### 4/3 problem

Electromagnetic mass

4/3 problem
Max von Laue in 1911 also used the Abraham–Lorentz equations of motion in his development of special relativistic dynamics, so that also in special relativity the 4/3-factor is present when the electromagnetic mass of a charged sphere is calculated. This contradicts the mass–energy equivalence formula, which requires the relation m_{em}=E_{em}/c^2 without the 4/3 factor, or in other words, four-momentum doesn't properly transform like a four-vector when the 4/3 factor is present. Laue found a solution equivalent to Poincaré's introduction of a non-electromagnetic potential (Poincaré stresses), but Laue showed its deeper, relativistic meaning by employing and advancing Hermann Minkowski's spacetime formalism. Laue's formalism required that there are additional components and forces, which guarantee that spatially extended systems (where both electromagnetic and non-electromagnetic energies are combined) are forming a stable or "closed system" and transform as a four-vector. That is, the 4/3 factor arises only with respect to electromagnetic mass, while the closed system has total rest mass and energy of m_{tot}=E_{tot}/c^2.
Another solution was found by authors such as Enrico Fermi (1922), Paul Dirac (1938) Fritz Rohrlich (1960), or Julian Schwinger (1983), who pointed out that the electron's stability and the 4/3-problem are two different things. They showed that the preceding definitions of four-momentum are non-relativistic per se, and by changing the definition into a relativistic form, the electromagnetic mass can simply written as m_{em}=E_{em}/c^2 and thus the 4/3 factor doesn't appear at all. So every part of the system, not only "closed" systems, properly transforms as a four-vector. However, binding forces like the Poincaré stresses are still necessary to prevent the electron from exploding due to Coulomb repulsion. But on the basis of the Fermi–Rohrlich definition, this is only a dynamical problem and has nothing to do with the transformation properties any more.

https://en.wikipedia.org/wiki/Electromagnetic_mass
Однако...
Пишут фигню. Задача решена и решение опубликовано в 2011

Натяжения Пуанкаре тут совершенно не причем, нет необходимости искать дополнительную массу все дело в неучтенном импульсе, который равен потоку энергии в в теле сферы и противоположен по направлению электромагнитному импульсу.
Однако задача пролежала в нерешенных сто лет.

morozov
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### Re: 4/3 problem

Обнаружилось, что ссылаются и неплохо на перевод моей статьи.
Я и не подозревал.
https://www.researchgate.net/publicatio ... arged_body

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