Translation:Electrodynamic and Relativistic Theory of Electromagnetic Mass
Concerning a Contradiction between the Electrodynamic and Relativistic Theory of Electromagnetic Mass.
By Enrico Fermi.
1. It's known that simple electrodynamic considerations lead to the value for the electromagnetic mass of a spherical electricity-distribution of electrostatic energy , when denotes the speed of light. On the other hand, it's known that relativistic considerations for the mass of a system containing the energy give the value . Thus we stand before a contradiction between the two views, whose solution seems not unimportant to me, especially with respect to the great importance of the electromagnetic mass for general physics, as the foundation of the electron theory of matter.
Especially we will prove: The difference between the two values stems from the fact, that in ordinary electrodynamic theory of electromagnetic mass (though not explicitly) a relativistically forbidden concept of rigid bodies is applied. Contrary to that, the relativistically most natural and most appropriate concept of rigid bodies leads to the value for the electromagnetic mass.
We additionally notice, that relativistic dynamics of the electron was studied by M. Born, though from the standpoint only partially different from the ordinary electrodynamic one, so that the value for the Electron's mass was found of course.
In this paper, Hamilton's principle will serve as a basis, being most useful for the treatment of a problem subjected to very complicated conditions - conditions of a different nature than those considered in ordinary mechanics, because our system must contract in the direction of motion according to relativity theory. However, we notice that although this contraction is of order of magnitude , it changes the most important terms of electromagnetic mass, i.e, the rest mass.
2. Let us consider a system of electric charges connected by a rigid dielectric, being under the action of an electrostatic field, partially stemming from the system itself, and partially from external causes, describing a world-tube in the world. We further assume, that the motion is translatory, by which we understand: When we consider any Lorentz-Einstein reference frame and assume, in which at a certain moment, for example at time zero, a point of our conducting system is at rest, then all points of the conducting system must rest at time zero. From that it follows, that the world-lines of the points of our system are the orthogonal paths of a family of linear spaces; the rigidity is expressed by the condition, that the shape of the system in these spaces remains unchanged.
To be able to apply Hamilton's principle, we need a variation of motion of our system satisfying the condition of rigidity. Now we will prove, that we are led to or to as the electromagnetic mass, depending on whether one chooses the first or the second of both variation systems, which in the following will defined by us, and denoted with the letters and .
As we will immediately see, however, variation has to be excluded, because it is in contradiction with relativity theory.
Let be a world-tube described by the system. In the figure, we have indicated the space in a one-dimensional way by the -axis, and replaced time by , to reach a definite metric.
Variation : Consider as a variation satisfying the condition of rigidity, an infinitesimal displacement (parallel to space and rigid in the ordinary kinematic sense) of the parallel cross-sections of the world-tube parallel to the same space. We will obtain this variation in the figure, by displacing all cross-sections =const of the tube by arbitrary infinitesimal lines parallel to the -axis. If we confine ourselves to the consideration of translatory displacements, then are arbitrary functions of time, and .
Variation : Consider as a variation satisfying the condition of rigidity, an infinitesimal displacement (perpendicular to the world-tube and rigid in the ordinary kinematic sense) of the normal intersections of the world-tube. We will obtain such a variation in the figure, by displacing all normal-intersections of the tube parallel to itself, by arbitrary infinitesimal lines.
To be able to apply Hamilton's principle, we have to subject our variations to the additional condition, that it must vanish at the limits of the arbitrary integration field . Due to this additional condition, the integration field contracts to when the variation system is applied; so must vanish in the fields , because they must be constant for , and must vanish at the limits of , i.e., at the lines , . However, if we apply the variation system , the integration field contracts to due to the same reasons.
Now it can be immediately seen, that variation is in contradiction with relativity theory, because it has no invariant characteristics against the world-transformation, and is based on the arbitrary space . On the other hand, variation has the desired invariant characteristics, and is always based on the proper-space, i.e., the space perpendicular to the world-tube, thus it is without doubt to be preferred before the previous one.
3. Let us denote for the sake of convenience, and respectively as space-time coordinates, and as the electromagnetic field.
Hamilton's principle, that combines the laws of Maxwell-Lorentz, of Newton, and of mechanics, reads: The total action, i.e., the sum of actions of the electromagnetic field, of the electric charges, of the material masses, and in the case of general relativity, of the metric field, remains stationary when a variation (which satisfies the problem-conditions, and which vanishes at the limits of the arbitrary integration field "") of the following things takes place: of the components of the four-potential, of the coordinates of the points of the world-lines followed by the charges and the masses, and of the components of the metric tensor.
In our case, however, the metric field remains unchanged, because we assumed it to be Euclidean since the material masses are missing, and because the only magnitudes to be varied, are the coordinates of the points of the world-lines followed by the charges. Thus it suffices to set the variation of the charges , i.e.:
where the first integration is to be extended over the charge element of the system, the second over the distance contained in the integration field of the world-line followed by .
Now, we will separately derive the conclusions from the two variation systems and .
4. Conclusions from variation . In this case, the integration field contracts to . If and mean the times of and , and if we consider that only depend on the time, and that , then we can (1) write:
However, because are arbitrary functions of time, we obtain from it the three equations:
i.e. when and mean the electric and magnetic force:
and the two corresponding equations for and .
If the velocity in the reference system is equal to zero in the relevant instant, then the three previous equations contract to a single vectorial equation:
We would have arrived at these equations without further ado, when we (as it ordinarily happens in the derivation of the electromagnetic mass and as it was essentially done by M. Born as well) would have assumed from the outset, that the total force of the systems is equal to zero. However, we have derived eq. (2) from Hamilton's principle, to demonstrate the source of the error. From (2), immediately follows as electromagnetic mass.
Namely, if we notice that is the sum of a portion stemming from the system itself, and of a portion stemming from external causes, then we obtain from eq. (2):
On the other hand, either direct observation or in a familiar manner the consideration of electromagnetic momentum shows, that:
when means the acceleration. If we also notice, that represents the total external force , then we find:
A comparison of this equation with the basic law of point-dynamics eventually gives us:
5. Conclusions from variation : In this case, the integration field contracts to . Let us imagine, that it is cut by infinitely many spaces perpendicular to the world-tube, into infinitely thin layers. With respect to any layer we also assume, that is the rest-system. Then will be arbitrary constants for our layer.
It is also: , because the velocity relative to the rest-system vanishes at time , and the height of the layer . Here, means the vector whose origin is located at point , in which any arbitrary but fixed world-line hits space , and whose endpoint is located in point , in which the world-line followed by the charge element hits the space : and mean the curvature and the element of world-line contained in the layer, eventually means the scalar product. The contribution to integral (1) stemming from our layer, thus becomes:
Now, if is the acceleration, then it is:
If we also consider, that in the integration with respect to , and can be seen as constant, then we find for the previous integral:
This expression must vanish for all possible values of . Therefore we obtain three equations, that can be combined into a single vectorial equation:
This equation takes the place of (2), and gives the value for the electromagnetic mass. Namely, if we put in (3) instead of , then we find:
or as it was previously found:
It follows first, that is of order of magnitude . If we neglect terms containing , we can neglect the second integral and find, when we set as before:
In order to calculate the last integral, we consider that is equal to the sum of the Coulomb force
(where means the up-point, the source-point of charge , and the distance ), and a force of order . The last-mentioned one would only give terms of order , which are neglected by us. By that, the last integral (4) becomes:
If we interchange and , by which the integral is not changed, and take the arithmetic average of the two values thus obtained, then we find:
The -component of this expression is:
In the case of spherical symmetry, the integrals
still can be calculated without any ado, if one considers that the expressions
can be replaced by their mean value for all directions of ; these mean values are . By that, the three integrals become:
and the -component of (5):
thus the integral (5) becomes:
If one substitutes this value into (4), then one finds:
thus the electromagnetic mass .
- Pisa, January 1922.
(Received May 9, 1922.)
- M. Abraham, Theorie der Elektrizität; Lorentz, The theory of electrons, p. 37; Richardson, Electron Theory of matter, Chapter XI.
- M. Born, Ann. d. Phys. 30, 1, 1909.
- We consider this world as Euclidean, because we assume that the electromagnetic fields occurring in it, are not strong enough in order to change its metric structure.
- H. Weyl, Raum, Zeit, Materie. Berlin Springer, 1921.
- Richardson, Electron theory of matter.