# Translation:The Translation of Deformable Electrons and the Theorem of Conservation of Angular Momentum

The Translation of Deformable Electrons and the Theorem of Conservation of Angular Momentum.

by Paul Ehrenfest.

Abraham alluded to the fact, that for a (rigid) *non*-spherical electron, uniform translation cannot take place into all directions in a *force-free* manner. For example, if a rigid, homogeneously charged electron having the shape of an ellipsoid of three axes, shall execute an uniform translation inclined to its major axes, then a torque stemming from external forces must compensate the torque exerted by the field of the moving electron upon the electron itself.^{[1]}

Certain reservations against the ordinary definition and calculation of the apparent mass of deformable electrons are causing me to present the following remark, on whose solution the more accurate formulation of those reservations is depending.

Lorentzian relativity-electrodynamics, in the form as it was formulated by Einstein^{[2]}, is quite generally seen as a closed system.^{[3]} Accordingly, it must give in a pure deductive manner the answer to the question, which we obtain by transferring Abraham's problem from the rigid to the *deformable* electron: Provided that there exists a deformable electron, having any *non*-spherical and *non* -ellipsoid shape when at rest.^{[4]} When in uniform translation, then according to Einstein the electron undergoes the known Lorentz contraction. *Now, is an uniform translation into every direction in a force-free manner for this electron possible, or not*?

If it is *not possible*, then for the sake of the relativity principle, one has to exclude the existence of such electrons in favor of a *new* hypothesis; otherwise we indeed would possess by them an instrument to demonstrate absolute motion.

If it should be *possible*, then we would have to show as to how they can be derived from the Einsteinian system, without the use of totally new axioms.^{[5]}

(Received March 19, 1907.)

- ↑ M. Abraham, Ann. d. Phys. 10. p. 174. 1903; see also Theorie der Elektrizität 2. p. 170 — 173. In the case of motion, the forces exerted upon each other by the volume elements of the electron, are not directed any more in the direction of the connecting line. Thus every element-couple provides a torque. Only in the case of translation parallel to the major axes, the whole torque is zero due to symmetry.
- ↑ A. Einstein, Ann. d. Phys. 18. p. 639. 1905.
- ↑ See especially W. Kaufmann, Ann. d. Phys. 19. p. 487 and 20. p. 639. 1906.
- ↑ See e.g. M. Planck, Verhandl. d. Deutsch. Physik. Gesellsch. Berlin 1906. p. 137: "but instead there arises, on the other hand, the advantage that it's not necessary to ascribe to the electron neither a spherical form nor even any other form in order to arrive at a certain dependence of inertia on speed."
- ↑ If (in accordance with the relativity principle) a charged condenser shall have no torque when inclined with respect to Earth's motion, and if a discharging condenser moving with Earth shall have no recoil, then one must use the following hypothesis: that the molecular forces – produced by the charge of the condenser in the constituent parts holding the condenser plates away from each other – provide in both cases the corresponding back reactions. On the other hand, Abraham showed (Physik. Zeitschr. 5. p. 576. 1904; Theorie der Elektrizität 2. p. 205): When one calculates the longitudinal mass of the deformable electron in the usual manner, then one has to ascribe to the electron a non-electromagnetic energy of interior deformation forces, to maintain the energy theorem. Thus one could allow those non-electromagnetic forces (similarly to the molecular forces) to compensate the electromagnetic torque. So one would have given up the (pure electromagnetic) energy- and momentum conservations theorems, and would only maintain the center of gravity theorem. Yet it is only necessary to sacrifice the latter theorem too, as we already sacrificed the energy theorem –
*then we are able to define any desired apparent mass of the Lorentz electron, thus one can bring any measuring result in agreement with the relativity principle*. (However, the value of zero for the apparent mass offers itself as the most simple one; like a macroscopic condenser shall obtain no increase of mass by a charge according to relativity theory.)

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