# Special relativity

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The classical laws of physics were formulated by Newton in the *Principia* in 1687. According to this theory the motion of a particle has to be described relative to an inertial frame in which the particle, not subjected to external forces, will move at a constant velocity in a straight line. Two inertial frames are related in that they move in a fixed direction at a constant speed with respect to each other. Time in the frames differs by a constant and all times can be described relative to an absolute time. This 17^{th} Century theory was not challenged until the 19^{th} Century when electric and magnetic phenomena were studied theoretically.

It had long been known that sound required a medium to travel through and it was quite natural to postulate a medium for the transmission of light. Such a medium was called the ether and many 19^{th} Century scientists postulated an ether with various properties. Cauchy, Stokes, Thomson and Planck all postulated ethers with differing properties and by the end of the 19^{th} Century light, heat, electricity and magnetism all had their respective ethers.

A knowledge that the electromagnetic field was spread with a velocity essentially the same as the speed of light caused Maxwell to postulate that light itself was an electromagnetic phenomenon. Maxwell wrote an article on *Ether* for the 1878 edition of *Encyclopaedia Britannica*. He proposed the existence of a single ether and the article tells of a failed attempt by Maxwell to measure the effect of the ether drag on the earth's motion. He also proposed an astronomical determination of the ether drag by measuring the velocity of light using Jupiter's moons at different positions relative to the earth.

Prompted by Maxwell's ideas, Michelson began his own terrestrial experiments and in 1881 he reported

The result of the hypothesis of a stationary ether is shown to be incorrect, and the necessary conclusion follows that the hypothesis is erroneous.

Lorentz wrote a paper in 1886 where he criticised Michelson's experiment and really was not worried by the experimental result which he dismissed being doubtful of its accuracy. Michelson was persuaded by Thomson and others to repeat the experiment and he did so with Morley, again reporting that no effect had been found in 1887. It appeared that the velocity of light was independent of the velocity of the observer. [Michelson and Morley were to refine their experiment and repeat it many times up to 1929.]

Also in 1887 Voigt first wrote down the transformations

x' =x-vt,y' =y/g,z' =z/g,t' =t-vx/c^{2}

and showed that certain equations were invariant under these transformations. These transformations, with a different scale factor, are now known as the Lorentz equations and the group of Lorentz transformations gives the geometry of special relativity. All this was unknown to Voigt who was writing on the Doppler shift when he wrote down the transformations.

Voigt corresponded with Lorentz about the Michelson-Morley experiment in 1887 and 1888 but Lorentz does not seem to have learnt of the transformations at that stage. Lorentz however was now greatly worried by the new Michelson-Morley experiment of 1887.

In 1889 a short paper was published by the Irish physicist George FitzGerald in *Science*. The paper *The ether and the earth's atmosphere* takes up less than half a page and is non-technical. FitzGerald pointed out that the results of the Michelson-Morley experiment could be explained only if

... the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocities to that of light.

Lorentz was unaware of FitzGerald's paper and in 1892 he proposed an almost identical contraction in a paper which now took the Michelson-Morley experiment very seriously. When it was pointed out to Lorentz in 1894 that FitzGerald had published a similar theory he wrote to FitzGerald who replied that he had sent an article to *Science* but *I do not know if they ever published it .* He was glad to know that Lorentz agreed with him *for I have been rather laughed at for my view over here .* Lorentz took every opportunity after this to acknowledge that FitzGerald had proposed the idea first. Only FitzGerald, who did not know if his paper had been published, believed that Lorentz had published first!

Larmor wrote an article in 1898 *Ether and matter* in which he wrote down the Lorentz transformations (still not written down by Lorentz) and showed that the FitzGerald-Lorentz contraction was a consequence.

Lorentz wrote down the transformations, now named after him, in a paper of 1899, being the third person to write them down. He, like Larmor, showed that the FitzGerald-Lorentz contraction was a consequence of the Lorentz transformations.

The most amazing article relating to special relativity to be published before 1900 was a paper of Poincaré *La mesure du temps* which appeared in 1898. In this paper Poincaré says

... we have no direct intuition about the equality of two time intervals.

The simultaneity of two events or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.

By 1900 the concept of the ether as a material substance was being questioned. Paul Drude wrote

The conception of an ether absolutely at rest is the most simple and the most natural - at least if the ether is conceived to be not a substance but merely space endowed with certain physical properties.

Poincaré, in his opening address to the Paris Congress in 1900, asked *Does the ether really exist?* In 1904 Poincaré came very close to the theory of special relativity in an address to the International Congress of Arts and Science in St Louis. He pointed out that observers in different frames will have clocks which will

... mark what on may call the local time. ... as demanded by the relativity principle the observer cannot know whether he is at rest or in absolute motion.

The year that special relativity finally came into existence was 1905. June of 1905 was a good month for papers on relativity, on the 5^{th} June Poincaré communicated an important work *Sur la dynamique de l'electron* while Einstein's first paper on relativity was received on 30th June. Poincaré stated that *It seems that this impossibility of demonstrating absolute motion is a general law of nature.* After naming the Lorentz transformations after Lorentz, Poincaré shows that these transformations, together with the rotations, form a group.

Einstein's paper is remarkable for the different approach it takes. It is not presented as an attempt to explain experimental results, it is presented because of its beauty and simplicity. In the introduction Einstein says

... the introduction of a light-ether will prove to be superfluous since, according to the view to be developed here, neither will a space in absolute rest endowed with special properties be introduced nor will a velocity vector be associated with a point of empty space in which electromagnetic processes take place.

Inertial frames are introduced which, by definition, are in uniform motion with respect to each other. The whole theory is based on two postulates:-

1. The laws of physics take the same form in all inertial frames.

2. In any inertial frame, the velocity of lightcis the same whether the light is emitted by a body at rest or by a body in uniform motion.

Einstein now deduced the Lorentz transformations from his two postulates and, like Poincaré proves the group property. Then the FitzGerald-Lorentz contraction is deduced. Also in the paper Einstein mentions the clock paradox. Einstein called it a theorem that if two synchronous clocks *C*_{1} and *C*_{2} start at a point *A* and *C*_{2} leaves *A* moving along a closed curve to return to *A* then *C*_{2} will run slow compared with *C*_{1}. He notes that no paradox results since *C*_{2} experiences acceleration while *C*_{1} does not.

In September 1905 Einstein published a short but important paper in which he proved the famous formula

E=mc^{2}.

The first paper on special relativity, other than by Einstein, was written in 1908 by Planck. It was largely due to the fact that relativity was taken up by someone as important as Planck that it became so rapidly accepted. At the time Einstein wrote the 1905 paper he was still a technical expert third class at the Bern patent office. Also in 1908 Minkowski published an important paper on relativity, presenting the Maxwell-Lorentz equations in tensor form. He also showed that the Newtonian theory of gravitation was not consistent with relativity.

The main contributors to special relativity were undoubtedly Lorentz, Poincaré and, of course, the founder of the theory Einstein. It is therefore interesting to see their respective reactions to the final formulation of the theory. Einstein, although he spent many years thinking about how to formulate the theory, once he had found the two postulates they were immediately natural to him. Einstein was always reluctant to acknowledge that the steps which others were taking due to the Michelson-Morley experiment had any influence on his thinking.

Poincaré's reaction to Einstein's 1905 paper was rather strange. When Poincaré lectured in Göttingen in 1909 on relativity he did not mention Einstein at all. He presented relativity with three postulates, the third being the FitzGerald-Lorentz contraction. It is impossible to believe that someone as brilliant as Poincaré had failed to understand Einstein's paper. In fact Poincaré never wrote a paper on relativity in which he mentioned Einstein. Einstein himself behaved in a similar fashion and Poincaré is only mentioned once in Einstein's papers. Lorentz, however, was praised by both Einstein and Poincaré and often cited in their work.

Lorentz himself poses a puzzle. Although he clearly understood Einstein's papers, he did not ever seem to accept their conclusions. He gave a lecture in 1913 when he remarked how rapidly relativity had been accepted. He for one was less sure.

As far as this lecturer is concerned he finds a certain satisfaction in the older interpretation according to which the ether possesses at least some substantiality, space and time can be sharply separated, and simultaneity without further specification can be spoken of. Finally it should be noted that the daring assertion that one can never observe velocities larger than the velocity of light contains a hypothetical restriction of what is accessible to us, a restriction which cannot be accepted without some reservation.

Despite Lorentz's caution the special theory of relativity was quickly accepted. In 1912 Lorentz and Einstein were jointly proposed for a Nobel prize for their work on special relativity. The recommendation is by Wien, the 1911 winner, and states

...

While Lorentz must be considered as the first to have found the mathematical content of the relativity principle, Einstein succeeded in reducing it to a simple principle. One should therefore assess the merits of both investigators as being comparable...

Einstein never received a Nobel prize for relativity. The committee was at first cautious and waited for experimental confirmation. By the time such confirmation was available Einstein had moved on to further momentous work.

**References (13 books/articles)**

**Article by:** *J J O'Connor* and *E F Robertson*