Hendrik Lorentz

“ If the body is moving with velocity , then (when a conduction current is present) both kinds of electrons will obtain different total velocities, thus also the quantity will have different values for both kinds. Now, if an observer which is moving with the body, is calculating the average of charge density for both kinds of electrons, then he can obtain the sum zero, even when for an observer (in whose reference system the body is moving) the averages of the positive and negative electrons are not compensating themselves. ”

“ The necessity of such a great value of stems from the fact, that the magnitude arises in its end formulas in the first power, where is the planetary velocity. However, if the propagation speed of gravitation shall have the speed of light, as required by the relativity principle, then a contradiction with observations can only then be avoided, when only magnitudes of second (or higher) order in arise in the expression for the modified law of gravitation. ”

“ That is a special property, which all forces of nature must have, when the relativity principle shall hold. If we presuppose this, then one can calculate the forces acting on moving bodies, when one know them for the case of rest. ”

“ The electron is a charged and expandable shell, and an inner normal stress of invariable magnitude is resisting to the electric repulsions of the individual points. According to the above, such normal stresses indeed satisfy the relativity principle. ”

“ If one confines oneself to magnitudes of second order, then a condition can easily be given on the basis of an obvious electron-theoretical analogy, which defines the modified law in a definite manner.
If one namely considers the force acting upon an electron moving with velocity :
then the vectors and also depend on velocities of the electrons generating the field ”

“ Therefore, the formulas for the mass can only then be correct, when the charge distribution and thus also the shape of the electron are variable with velocity in a certain way. One has to assume, that in consequence of the translation of the electron, which is a sphere when at rest, the electron becomes an oblate ellipsoid in the direction of motion; the amount of oblateness is
If we assume, that shape and magnitude of the electron are regulated by inner forces, then they must (to be compatible with the relativity principle) have such properties by which this oblateness is arising by itself. ”

“ Now it is clear, that such a dependency of the phenomena from the motion through the aether contradicts the relativity principle.
Fig. 5 To solve this contradiction, we want to simplify the state of facts in a schematic way. We imagine, that sun shall have a mass which is infinitely great in relation to that of the planets. Let the velocity of the solar system coincide with the -axis, which we let go through the sun. The intersections of the orbit of the planets with the -axis, is denoted by us as the upper and lower transit and .
We place the observer upon the sun. ”

“ The last remark concerns the circumstance, that the motion of Earth cannot have an influence upon electromagnetic processes according to the relativity principle. However, Liénard alluded to a phenomenon, where such an influence (namely amounting to first order) shall be expected; also Poincaré has discussed this case in his book Electricité et Optique. It is about the ponderomotive force upon an conductor. In order to determine it, one will make the obvious assumption for the force acting upon the conduction electrons per unit force ”

“ The considered body (e.g. a steel magnet) shall contain only conduction electrons and such ones (when the body is at rest) which produce , yet not ; it shall have the shape of an infinitely extended even plate, bounded by two planes : the middle plane is made by us to the -plane (Fig. 6) . When it is at rest, a constant magnetization may exist in the -direction, while . If the body acquires the velocity in the -direction, then an observer not participating at the motion, is observing the electric polarization ”

“ The origin of system moves towards system in the -direction with velocity , which is always smaller than . Generally, any velocity has to be assumed as being smaller than .
All state variables of any phenomenon, measured in one or the other system, are connected by certain transformation formulas. They read, e.g. for the velocity of a point:
where
We furthermore consider a system of pints, whose velocity is a steady function of the coordinates. ”

“ We describe the motion of a point by the equations
where is a constant which we call the "Minkowski mass". Vector is denoted by us as "Minkowski force".
The transformation formulas for this acceleration and force can be easily derived; is left unchanged by us. Then one has
The essential thing is now as follows: The relativity principle requires, that (at an actual phenomenon) the Minkowski forces are in a certain way depending on the coordinates, velocities, etc. ”

“ If observer is in the first, in the second reference system, and if any of them is equipped with measuring rods and clocks at rest in his respective system, then will measure the values of , while measure the values of , where it is to be remarked, that and can use the same measuring rods and the same clocks as well. ”

“ The difference stems from the fact, that one usually doesn't operate with our Minkowski force, but with the "Newtonian force" , and we see, that (for an electron) these two forces are connected as follows ”

“ If we imagine an agent (matter, electricity etc.) as connected with these points, and if we assume that observer has reason to connect the same amount of that agent with every point as observer , then the space density must be inversely proportional to the volume elements, i.e. ”