## A3 Incompatibility of Maxwell's Theory of Electromagnetism

Einstein was always thorough in his thought and argumentation. In this section we investigate the contradictions of physics he wanted to redress with his special theory of relativity (STR).

Imagine the waiter in the dining car of a train, moving at 100 km/h on a long straight stretch of rails. The waiter moves at 5 km/h in the dining car both in and against the direction of travel of the train. At which speed does he actually travel?

We make two observations: First of all it is obvious that speeds are relative and not absolute. They always refer to a given coordinate system. We have the choice of fixing our coordinate system to the dining car or to the ties of the railway track. If we sit 'in peace' in the dining car, then the waiter moves forwards and back at ±5 km/h.

It was already clear to Galileo and Descartes, how fast the waiter moved in a reference system in which the rails are at rest and on which the train moves with 100 km/h. The velocities of the train and that of the waiter relative to the train are simply added: he moves with 105 km/h or with 95 km/h, always in the direction of the train's velocity. Thus our second observation is that in Newtonian mechanics velocities simply add. (If the speeds are not parallel as in our example, then not only the signs, but also the directions must be considered, i.e., the velocities must be added as vectors.)

In **D1** we will formally prove the correctness of this speed addition within Newtonian mechanics. The proof shows beautifully, how in particular the idea of absolute time is presupposed.

Where then is the problem?

In 1856 the physicist James Clerk Maxwell successfully condensed the rich research results of Michael Faraday and others in the areas of electricity and magnetism into four formulas. In 1862 he published these in his paper “On Physical Lines of Forces”. In 1873 his masterpiece “A Treatise on Electricity and Magnetism” appeared in two volumes. Maxwell demonstrated in pure mathematical form the fact that electrical and magnetic fields propagate as waves in space. In 1886 Heinrich Hertz proved the existence of such electromagnetic waves experimentally.

The propagation speed of these waves in a vacuum is given by the expression

where ε_{0} is the electrical field constant, which for example also arises in the force law of Coulomb, and μ_{0} is the appropriate magnetic field constant. Of course, Maxwell had already noticed that this value corresponds exactly to the speed of light in vacuum (which by the way differs from that in air only very slightly). This implies however that the speed of light must also be a universal constant, just as are the electrical and the magnetic field constants!

Thus this beautiful theory of Maxwell, which was distilled out of a large body of experimental work and which itself was afterwards splendidly confirmed, demonstrates that the speed of light in a vacuum is a universal constant! If we accept that the principle of relativity not only applies to mechanics, then it must also be true that Maxwell’s equations apply in any inertial frame, with the same values for the universal constants. The speed of light would be a constant, whose value would be the same in every inertial frame. The speed of the forward shining light of a forward moving locomotive must be exactly equal to that of one at rest or even one moving backwards! The speed of light is thus independent of the movement of its source. This however contradicts the vector addition of velocities, which we have presented as fact within Newtonian mechanics.

Seen as a package Newtonian mechanics, the principle of relativity and Maxwell's theory of electromagnetism are incompatible!

In order to describe the dilemma more clearly, we introduce the following abbreviations:

**N** mm Newtonian mechanics with absolute time and absolute space

**R** mm General principle of relativity: All inertial frames are equal

**M** mm Maxwell's theory of electromagnetism

One cannot have **N**, **R** and **M** at the same time. From **N** and **R** follows the addition of velocities which implies that the light of the forward moving locomotive moves in the rail track’s inertial frame at c + 100 km/h. This does not fit **M**. From **M** and **R** follows the constancy of the speed of light which implies that the light of the forward moving locomotive has in each inertial frame the speed of c, the measured value is independent of the movement of the source and the receiver!

** Which possibilities remain?**

One can keep **N** and limit **R** to the laws of **N**. In this case the beautiful equations of **M** apply unchanged only if the reference frame is at rest in Newton's absolute space, and they have to be adapted for other inertial frames. Ugly!

It amounts to almost the same, if one holds to **N** and **M** and abandons **R**. Now the experimenter has the additional task of determining his speed in absolute space, where the propagation medium of electromagnetic waves, the so-called ether, resides. This task was accepted by A. Michelson and E. Morley, following a suggestion by Maxwell.

If one tries to keep **R** and **M** then he must supply an 'improved' version of **N**! Before Einstein nobody had the courage to consistently pursue this path. Nevertheless Einstein could profit from many predecessors: FitzGerald and Lorentz suggested a formula for 'the contraction' of the measuring apparatus in the direction of motion of the earth through the ether. Also Lorentz had already around 1900 briefly introduced a 'local time', in order to explain the results of certain experiments. The great mathematician Poincaré coined in 1905 the expression 'Lorentz transformations' and (at the same time and independently of Einstein) showed that these transformations form a mathematical group and that **M** under such transformations is invariant. The fruit was thus ripe for the plucking (for details of the history of the STR refer to chapter 6 in [10]).

One could even claim that it was aesthetic reasons that actually induced Einstein to keep **R** and **M**. The introduction of his famous article of 1905 begins as follows:

"It is well known that Maxwell's electrodynamics - as usually understood at present - when applied to moving bodies, leads to asymmetries that do not seem to be inherent in the phenomena. Take, for example, the electrodynamic interaction between a magnet and a conductor. The observable phenomenon here depends only on the relative motion of conductor and magnet, whereas the customary view draws a sharp distinction between the two cases, in which either the one or the other of the two bodies is in motion.” [09-123]

This article does not presuppose (at least in the first part) any higher knowledge in mathematics and is very much recommended to the reader.