The Fundamental Hypotheses of the Theory of Electrons.
By M. Abraham.
The theory of electrons, the most promising continuation of Maxwell's electrodynamics, is based on the following general presuppositions:
A. The Maxwell-Hertz equations hold in space devoid of matter and electricity. They postulate a reference system, in which plane electromagnetic waves are propagating in all directions with the same velocity cm/sec; motions related to this reference frame are called absolute motions.
B. Electricity consists of discrete positive and negative particles, called "electrons". These are the ones, that convey the interaction of matter and aether.
C. Every electric current is a convection current of moving electrons. The density of the convection current is the product of density of electricity and its absolute velocity. The convection current excites the same magnetic field, as the equivalent conduction current of the Maxwell-Hertz theory.
From hypotheses A, B, C the field equations follow, specifying the velocity of electricity of the electromagnetic field at a given distribution. A statement is supplemented to them concerning the force (at a given field) acting upon a volume element filled by electricity:
D. The electromagnetic force is additively composed of forces, acting in the electric field upon resting electrons, and in the magnetic field upon moving electrons.
These four statements represent the general fundamental hypotheses of the theory of electrons. Every investigation based on them, and only such one, is to be denoted as belonging to the framework of the theory of electrons.
The phenomena observed at cathode rays, can be interpreted on the basis of D, when one considers the electromagnetic force of the external field as an external force, and one ascribes an inertial mass to the free electrons which are assumed to be in the cathode rays. On the other hand, the fundamental hypotheses A, B, C, D have led to the consequence, that this mass (at least partially) is the result of the field excited by the electron itself. The experiments of W. Kaufmann have shown, that at great velocities, the electromagnetic mass of the electron becomes of considerable influence.
In my investigations I have given a form to the dynamics of the electron, which suits to explain the experiments of Kaufmann on a purely electromagnetic basis. Thereby I have, besides the general fundamental hypotheses of the electron theory, introduced the following special hypotheses:
E. The electromagnetic forces of the external field and the field excited by the electron itself, are in equilibrium at the electron in the sense of the mechanics of rigid bodies.
F. The Electron is not at all capable of any change of its shape.
G. It is a sphere with uniform volume- or surface charge.
Hypothesis F is thereby to be understood as a conditional equation in the sense of Hertz's mechanics. It by no means obliges us to speak about forces holding together the volume elements of the electron; on the contrary, it says that such a force can never perform work and thus it renders its introduction superfluous.
On the basis of hypotheses A, B, C, D, E, F, the dynamics of the electron of arbitrary shape can be developed in a purely electromagnetic way. The details of the behavior of the electron, however, essentially depend on its form. I have extended the investigation also upon the ellipsoid electrons of invariable shape; it was given, that the translatory motion of such an electron is only stable in the direction of the major axis. An oblate rotational ellipsoid cannot move parallel to the rotation axis; the slightest push would cause it to turn.
On the basis of hypotheses A to G, I have calculated the electromagnetic momentum of the electron. I have shown in general, how to derive from it the electromagnetic masses, namely the longitudinal and transverse mass. The formula obtained for the latter, represents the deflection experiments by Kaufmann with a satisfying precision.
However, the theory of electron has given to itself another goal; it demands to totally enclose the electric and optical properties of the bodies. The optics of transparent bodies satisfying Maxwell's relation, is included in the electron theory by assuming quasi-elastic forces, which pull back the electrons into their equilibrium states. The dispersion of the bodies is interpreted by introduction of the inertial mass of the electrons, which together with those quasi-elastic force, cause the existence of proper oscillations. The oscillating electron represents the simplest image of an illuminating point; the Zeeman effect in its normal form shows, that this image agrees with reality for a great number of spectral lines. The velocity of the electron oscillations is thereby so low, that the variability of mass plays no role. Hypotheses E, F, G thus don't come into play as long as the body itself is at rest.
The matter is different in the optics of moving bodies. The aberration phenomena show, that the universal reference system (see A) doesn't share the orbital motion of Earth around the sun. How does it come, that the electric and optical processes happening on Earth's surface, nevertheless show no influence of Earth's motion? This question was studied by H. A. Lorentz. He has shown, that the absence of an influence of first order in the ratio of Earth's velocity and the speed of light, can very well be brought into agreement with the fundamental hypotheses A to D of electron theory.
The negative result of experiments, whose sensitivity is sufficient to discover an influence of second order, provides considerable difficulties for the electron theory. In two papers, Lorentz sought to overcome these difficulties. In the second of the cited papers, he states a system of hypotheses, which are sufficient to give account for all negative experimental results:
H. Due to Earth's motion, the bodies experience a certain contraction parallel to the direction of motion.
This hypothesis explains the negative result of the interference experiment of Michelson. It also explains the absence of a force-couple upon a condenser obliquely located to the Earth's direction of motion, which Trouton and Nobel tried to detect in vain.
One can render hypothesis H plausible, by interpreting the molecular forces as electrical forces.
I. The quasi-elastic force, which bind the electrons in their equilibrium positions, experience the same change due to Earth's motion, as the electric and the molecular forces.
Hypothesis I can also be made plausible, by considering the quasi-elastic forces themselves as electric forces.
To explain the absence of birefringence in the rest state of isotropic bodies due to Earth's motion, which has been shown by the experiments of Lord Rayleigh and D. B. Brace, it is sufficient to add hypothesis I to hypotheses A, B, C, D, H for bodies satisfying Maxwell's relation. For dispersing bodies, however, at which the inertia of the electrons come into play, birefringence due to Earth's motion is only then excluded when the longitudinal and transverse inertial forces are changed in the same way, as the quasi-elastic forces. This is the case according to H. A. Lorentz, when the dynamics of the electrons which oscillate in the interior of moving matter, is based on the following hypothesis:
F and G are replaced by:
K. The electron filled with uniform volume- or surface charge when at rest, oblates when in motion, by contracting its diameter parallel to the direction of motion in the ratio . It becomes a so-called Heaviside-Ellipsoid. H. A. Lorentz calculated the electromagnetic momentum for such an ellipsoid, from which both masses are given by my methods without further ado. He finds the longitudinal mass , the transverse mass . H. A. Lorentz shows, that his formula for the transverse mass is not substantially less in agreement with Kaufmann's experiments than the formulas of mine.
Due to K, on the other hand, the ratio of transverse and longitudinal mass is given equal to , yet because of F, G is is equal to , when terms of fourth and higher order are neglected; thus when F, G are introduced instead of K into Lorentz's system of hypotheses, a birefringence of order would be given for such bodies, for whose optical behavior the inertia of the electrons is decisive.
H. A. Lorentz finally remarks, that this influence of Earth's motion vanishes also for bodies with molecular motion, when the latter hypothesis is added.
L. The masses of the molecules are of electromagnetic nature.
We now want to examine hypothesis K more closely. H. A. Lorentz announces it with all restraint; he doesn't go so far as to state it as probable. Indeed, most serious objections can be raised against this hypothesis.
If such an electron is accelerated, its oblateness becomes increased; thus work must be performed against the electric forces. While (for the undeformable electron) the increase of energy is equal to the work performed by the external electric forces, this is not taken place here any more; the energy increase when the velocity increases, is greater as the work of the external forces.
The consequent development of hypothesis K forces us to assume (besides the inner electromagnetic forces) other inner forces which are non-electromagnetic, which determine the shape of the electrons together with the other ones. They would perform the necessary work during the contraction, being equivalent to the increase of electromagnetic energy of the electron together with the work of the external forces. The system of hypotheses A, B, C, D, E, K is incomplete, as long as one is not stating, according to which law these forces shall act.
The incompleteness of this system of hypotheses has the consequence, that one cannot be sure about the stability of an electron being subjected to those hypotheses. The motion of an oblate rotational ellipsoid of invariable form parallel to its rotation axis, is (as mentioned above) unstable. The confirmation is missing, that the non-electromagnetic supplementary forces are rendering stable the motion of the deformable electron.
The necessity to introduce non-electromagnetic forces shows, that the hypothesis of the deformable Heaviside ellipsoid, although it is mathematically simpler in a certain sense, is physically far more complicated though, as the hypothesis of the rigid spherical electron. The former fails indeed with respect to several equations, to which the latter gives a quite definite answer. I only mention the consequence drawn by P. Hertz from hypotheses A to G, that the electron can be brought by finite forces arbitrarily close to the speed of light, or even to the speed of light. The experiments of F. Paschen show, that negative electrons contained in the radiation of radium, possessing a much greater penetration capability and a much lower deflectability, as the quickest of the rays studied by Kaufmann. Here, the speed of light actually seems to be almost, if not entirely, reached. Here the paths followed (independently from one another) by the mathematical and experiments research, meet each other. – Hypothesis K, however, completely fails with respect to the question after the attainment of the speed of light.
From all of these reasons, it would be highly premature to abandon hypotheses F, G without further ado in favor of hypothesis K. Of course, the dynamics of the electron is, as any physical theory, subject to the continuing tests by experiment. It is to be hoped, that the experiments which are now started again by W. Kaufmann with tireless endurance, give further insights.
The question, whether and why the influence of Earth's motion upon the electric and optical phenomena cannot be detected at Earth's surface, is for the time being by no means decided. H. A. Lorentz himself surely was hardly of the opinion, to definitely solve them by stating the system of hypotheses H, I, K, L. He probably wanted to show only, that the absence of a noticeable influence doesn't necessarily speak against the general fundamental hypotheses A, B, C, D of electron theory, but that these hypotheses can be combined with the other ones without contradiction, so that the influence of Earth's motion vanishes with respect to all observable phenomena.
If the dynamics based on hypotheses A to G should further be proven in the field of cathode and Becquerel rays, yet a birefringence of dispersing bodies of order (following from these hypotheses together with H, I) due to Earth's motion cannot be demonstrated, then there still remain several possibilities.
In the light of our incomplete knowledge concerning the molecular forces, it is near at hand to abandon or modify hypothesis H. Because it was by no means achieved for the time being, to electrically interpret the molecular forces of resting bodies in a satisfying way.
Also the nature of the assumed quasi-elastic forces, which shall pull the electrons into their equilibrium positions, is unknown to us. Its interpretation on the basis of electromagnetism would give the electromagnetic theory of spectral lines. Unfortunately, we don't posses such a theory; therefore we are far from completely understanding the optical properties of resting bodies on the basis of electron theory. Hypothesis I thus completely hangs in the air, it is very much capable of being changed.
In the course of assessing the probability of the various hypotheses, one has to care that ideas concerning the nature of the molecular forces and the quasi-elastic forces are far less clarified, and are far less accessible to experimental verification, than the ideas concerning the constitution of the free negative electrons. Thus one won't be willing to abandon a theory, which correctly describes the behavior of free negative electrons, but which doesn't agree in a satisfactory manner with an optics that is based on hypotheses H, I. One will rather try to modify hypotheses H, I, so that an agreement with the totality of observations is achieved.
In the ninth paragraph of my work concerning the dynamics of the electron I have stated the formulas for energy- and momentum radiation, emitted by a rapidly moving and simultaneously accelerated electron. Recently I have supplemented the extensive derivation of this formulas and discussed their meaning for the theory of a moving illuminating point. In this investigations, as I have explicitly emphasized several times, the electron is considered as point charge which is permitted in the calculation of radiation under certain circumstances. The results of this investigations are therefore independent from any hypothesis concerning the constitution of the electron; they are exclusively based on the fundamental hypotheses A to D of electron theory. The development of Lorentz's model and every other in agreement with it, must thus lead to exactly the same results with respect to radiation, unless reasoning errors occur, for example violations of Doppler's principle or erroneous applications of Pointing's theorem.
Thus it is to be wished urgently, that the authors writing in the field of electron theory, follow the example of H. A. Lorentz by giving account of their hypotheses (underlying their investigations) in a clear and unequivocal manner, instead of attempting to belatedly present their explanations as "free of hypotheses". A deviation in the end results of such theories that are "free of hypotheses", could occasionally be derived from a lack of care of the concerned author. Authors, who doesn't provide a clear representation of their fundamental hypotheses and a thorough development of the consequences derived from them, cannot demand to be noticed by serious consideration.
Edinburgh July 28, 1904.
(Received July 30, 1904)
- See H. A. Lorentz, Encykl. der mathem. Wissensch., Bd. V 2, Heft 1.
- M. Abraham, Ann. d. Phys. 10, 105, 1903.
- H. A. Lorentz, Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern. Leiden 1895.
- H. A. Lorentz, K. Akad. van Wetensch. te Amsterdam 1899, p. 507 and 1904, p. 809.
- P. Hertz, this journal, 6, 109, 1904. Untersuchungen über unstetige Bewegungen eines Elektrons. Inauguraldissertation. Göttingen 1904.
- F. Paschen, Ann. d. Phys. 14, 164 and 389, 1904.
- l. c. p. 153. The same formulas have been discovered, as I notice, by O. Heaviside, Nature 67, p. 6.
- Ann. d. Phys. 14, S. 273, 1904.