General relativity n' Special Relativity

General relativity

General relativity is a theory of gravitation and to understand the background to the theory we have to look at how theories of gravitation developed. Aristotle's notion of the motion of bodies impeded understanding of gravitation for a long time. He believed that force could only be applied by contact, force at a distance being impossible, and a constant force was required to maintain a body in uniform motion.
Copernicus's view of the solar system was important as it allowed sensible consideration of gravitation. Kepler's laws of planetary motion and Galileo's understanding of the motion and falling bodies set the scene for Newton's theory of gravity which was presented in the Principia in 1687. Newton's law of gravitation is expressed by

F = G M1M2/d2
where F is the force between the bodies of masses M1, M2 and d is the distance between them. G is the universal gravitational constant.
After receiving their definitive analytic form from Euler, Newton's axioms of motion were reworked by Lagrange, Hamilton, and Jacobi into very powerful and general methods, which employed new analytic quantities, such as potential, related to force but remote from everyday experience. Newton's universal gravitation was considered proved correct, thanks to the work of Clairaut and Laplace. Laplace looked at the stability of the solar system in Traité du Mécanique Céleste in 1799. In fact the so-called three-body problem was extensively studied in the 19th Century and was not properly understood until much later. The study of the gravitational potential allowed variations in gravitation caused by irregularities in the shape of the earth to be studied both practically and theoretically. Poisson used the gravitational potential approach to give an equation which, unlike Newton's, could be solved under rather general conditions.

Newton's theory of gravitation was highly successful. There was little reason to question it except for one weakness which was to explain how each of the two bodies knew the other was there. Some profound remarks about gravitation were made by Maxwell in 1864. His major work A dynamical theory of the electromagnetic field (1864) was written

... to explain the electromagnetic action between distant bodies without assuming the existence of forces capable of acting directly at sensible distances.
At the end of the work Maxwell comments on gravitation.
After tracing to the action of the surrounding medium both the magnetic and the electric attractions and repulsions, and finding them to depend on the inverse square of the distance, we are naturally led to inquire whether the attraction of gravitation, which follows the same law of the distance, is not also traceable to the action of a surrounding medium.

However Maxwell notes that there is a paradox caused by the attraction of like bodies. The energy of the medium must be decreased by the presence of the bodies and Maxwell said As I am unable to understand in what way a medium can possess such properties, I cannot go further in this direction in searching for the cause of gravitation. In 1900 Lorentz conjectured that gravitation could be attributed to actions which propagate with the velocity of light. Poincaré, in a paper in July 1905 (submitted days before Einstein's special relativity paper), suggested that all forces should transform according the Lorentz transformations. In this case he notes that Newton's law of gravitation is not valid and proposed gravitational waves which propagated with the velocity of light.

In 1907, two years after proposing the special theory of relativity, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence:-

... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.

After the major step of the equivalence principle in 1907, Einstein published nothing further on gravitation until 1911. Then he realised that the bending of light in a gravitational field, which he knew in 1907 was a consequence of the equivalence principle, could be checked with astronomical observations. He had only thought in 1907 in terms of terrestrial observations where there seemed little chance of experimental verification. Also discussed at this time is the gravitational redshift, light leaving a massive body will be shifted towards the red by the energy loss of escaping the gravitational field.

Einstein published further papers on gravitation in 1912. In these he realised that the Lorentz transformations will not apply in this more general setting. Einstein also realised that the gravitational field equations were bound to be non-linear and the equivalence principle appeared to only hold locally.

This work by Einstein prompted others to produce gravitational theories. Work by Nordström, Abraham and Mie was all a consequence of Einstein's, so far failed, attempts to find a satisfactory theory. However Einstein realised his problems.

If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.

Einstein then remembered that he had studied Gauss's theory of surfaces as a student and suddenly realised that the foundations of geometry have physical significance. He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci (Ricci-Curbastro) and Levi-Civita. Einstein wrote

... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.

In 1913 Einstein and Grossmann published a joint paper where the tensor calculus of Ricci and Levi-Civita is employed to make further advances. Grossmann gave Einstein the Riemann-Christoffel tensor which, together with the Ricci tensor which can be derived from it, were to become the major tools in the future theory. Progress was being made in that gravitation was described for the first time by the metric tensor but still the theory was not right. When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said
As an older friend I must advise you against it for in the first place you will not succeed, and even if you succeed no one will believe you.

Planck was wrong, but only just, for when Einstein was to succeed with his theory it was not readily accepted. It was the second half of 1915 that saw Einstein finally put the theory in place. Before that however he had written a paper in October 1914 nearly half of which is a treatise on tensor analysis and differential geometry. This paper led to a correspondence between Einstein and Levi-Civita in which Levi-Civita pointed out technical errors in Einstein's work on tensors. Einstein was delighted to be able to exchange ideas with Levi-Civita whom he found much more sympathetic to his ideas on relativity than his other colleagues.
At the end of June 1915 Einstein spent a week at Göttingen where he lectured for six 2 hour sessions on his (incorrect) October 1914 version of general relativity. Hilbert and Klein attended his lectures and Einstein commented after leaving Göttingen

To my great joy, I succeeded in convincing Hilbert and Klein completely.
The final steps to the theory of general relativity were taken by Einstein and Hilbert at almost the same time. Both had recognised flaws in Einstein's October 1914 work and a correspondence between the two men took place in November 1915. How much they learnt from each other is hard to measure but the fact that they both discovered the same final form of the gravitational field equations within days of each other must indicate that their exchange of ideas was helpful.
On the 18th November he made a discovery about which he wrote For a few days I was beside myself with joyous excitement . The problem involved the advance of the perihelion of the planet Mercury. Le Verrier, in 1859, had noted that the perihelion (the point where the planet is closest to the sun) advanced by 38" per century more than could be accounted for from other causes. Many possible solutions were proposed, Venus was 10% heavier than was thought, there was another planet inside Mercury's orbit, the sun was more oblate than observed, Mercury had a moon and, really the only one not ruled out by experiment, that Newton's inverse square law was incorrect. This last possibility would replace the 1/d2 by 1/dp, where p = 2+ for some very small number . By 1882 the advance was more accurately known, 43'' per century. From 1911 Einstein had realised the importance of astronomical observations to his theories and he had worked with Freundlich to make measurements of Mercury's orbit required to confirm the general theory of relativity. Freundlich confirmed 43" per century in a paper of 1913. Einstein applied his theory of gravitation and discovered that the advance of 43" per century was exactly accounted for without any need to postulate invisible moons or any other special hypothesis. Of course Einstein's 18 November paper still does not have the correct field equations but this did not affect the particular calculation regarding Mercury. Freundlich attempted other tests of general relativity based on gravitational redshift, but they were inconclusive.

Also in the 18 November paper Einstein discovered that the bending of light was out by a factor of 2 in his 1911 work, giving 1.74". In fact after many failed attempts (due to cloud, war, incompetence etc.) to measure the deflection, two British expeditions in 1919 were to confirm Einstein's prediction by obtaining 1.98" 0.30" and 1.61" 0.30".

On 25 November Einstein submitted his paper The field equations of gravitation which give the correct field equations for general relativity. The calculation of bending of light and the advance of Mercury's perihelion remained as he had calculated it one week earlier.

Five days before Einstein submitted his 25 November paper Hilbert had submitted a paper The foundations of physics which also contained the correct field equations for gravitation. Hilbert's paper contains some important contributions to relativity not found in Einstein's work. Hilbert applied the variational principle to gravitation and attributed one of the main theorem's concerning identities that arise to Emmy Noether who was in Göttingen in 1915. No proof of the theorem is given. Hilbert's paper contains the hope that his work will lead to the unification of gravitation and electromagnetism.

In fact Emmy Noether's theorem was published with a proof in 1918 in a paper which she wrote under her own name. This theorem has become a vital tool in theoretical physics. A special case of Emmy Noether's theorem was written down by Weyl in 1917 when he derived from it identities which, it was later realised, had been independently discovered by Ricci in 1889 and by Bianchi (a pupil of Klein) in 1902.

Immediately after Einstein's 1915 paper giving the correct field equations, Karl Schwarzschild found in 1916 a mathematical solution to the equations which corresponds to the gravitational field of a massive compact object. At the time this was purely theoretical work but, of course, work on neutron stars, pulsars and black holes relied entirely on Schwarzschild's solutions and has made this part of the most important work going on in astronomy today.

Einstein had reached the final version of general relativity after a slow road with progress but many errors along the way. In December 1915 he said of himself

That fellow Einstein suits his convenience. Every year he retracts what he wrote the year before.
Most of Einstein's colleagues were at a loss to understand the quick succession of papers, each correcting, modifying and extending what had been done earlier. In December 1915 Ehrenfest wrote to Lorentz referring to the theory of November 25, 1915. Ehrenfest and Lorentz corresponded about the general theory of relativity for two months as they tried to understand it. Eventually Lorentz understood the theory and wrote to Ehrenfest saying I have congratulated Einstein on his brilliant results . Ehrenfest responded
Your remark "I have congratulated Einstein on his brilliant results" has a similar meaning for me as when one Freemason recognises another by a secret sign.
In March 1916 Einstein completed an article explaining general relativity in terms more easily understood. The article was well received and he then wrote another article on relativity which was widely read and went through over 20 printings.
Today relativity plays a role in many areas, cosmology, the big bang theory etc. and now has been checked by experiment to a high degree of accuracy.

Special relativity

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 17th Century theory was not challenged until the 19th 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 19th 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 19th 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/c2
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 5th 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 light c is 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 C1 and C2 start at a point A and C2 leaves A moving along a closed curve to return to A then C2 will run slow compared with C1. He notes that no paradox results since C2 experiences acceleration while C1 does not.
In September 1905 Einstein published a short but important paper in which he proved the famous formula

E = mc2.
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.

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