5.0 ± 1.2

3.846 ± 0.012

the perihelia of the three inner planets compared with modern empirical and computational data. The level of agreement is impressive.

If general relativity is accepted as a successor to the universal gravitation of Newton, then the anomalous advance in the perihelion of Mercury is explained, and our understanding of the motions of the bodies in the Solar System essentially is complete. But in 1915 it was going to take more than one experimental success for people willingly to accept a theory that reinterprets the whole fabric of space and time. Moreover, as we have seen, there are a number of other possible influences on the motion of Mercury that are hard to quantify. The advance in technology in the second half of the twentieth century has led to numerous new tests for general relativity and the theory has passed each one (with many competing theories failing), and so the grounds for accepting Einstein's theory are now quite strong.83 But this was not the case when it was published.84

82 With the exception of the general relativistic value for Mercury, which is from Nobili and Will (1986), the numbers in the first two columns are from Whitrow and Morduch (1965). The value of 43.03", which is often quoted as the prediction of general relativity for the advance of Mercury, was derived by Clemence (l947) based on unconventional values for certain astronomical constants. The large error in the observational data for Venus is due to the fact that the eccentricity of Venus is very small and, hence, the determination of the perihelion is more difficult. The computations reported in the final column are from Taylor and Wheeler (2000) and were made by Myles Standish at the NASA Jet Propulsion Laboratory by computing the motions of the four inner planets over a 400-year period with and without relativistic effects. There is also a general relativistic effect due to the rotation of the Sun, but this turns out to be negligible (Whitrow and Morduch (l965)).

See Will (1979) for a detailed discussion. In the latter part of the twentieth century, general relativity has become an important tool in the construction of accurate ephemerides (see Brumberg (1991)) and plays an essential role in the operation of the Global Positioning System (Taylor and Wheeler (2000)).

There was considerable opposition within Germany, where Einstein's nationality (Swiss) and religion (Jewish) both worked against him. Ernst Gehrcke was particularly active in trying to discredit Einstein, and was responsible for the reprinting of a paper published originally in l902 by Paul Gerber. Gerber had developed (though his derivation was somewhat unclear) a velocity-dependent force law that gave exactly the same formula for the perihelion advance of Mercury as general relativity. A somewhat unsatisfactory debate ensued involving, among others, Seeliger and Einstein (see Roseveare (1982), Section 6.6). Note that the special relativistic advance of 7" should be added to Gerber's value, since his theory treats gravitation alone, whereas general relativity already contains the special theory.

In his first full-scale exposition of general relativity, Einstein wrote:

These equations, which proceed by the method of pure mathematics, from the requirement of the general theory of relativity, give us ... to a first approximation Newton's law of attraction, and to a second approximation the explanation of the motion of the perihelion of the planet Mercury discovered by Leverrier These facts must, in my opinion, be taken as convincing proof of the correctness of the theory.

Others disagreed. For example, Max von Laue objected on the grounds that the calculations assumed that the Sun and Mercury were point masses, and, unlike in the Newtonian theory, there was no reason to believe that one could treat extended bodies this way. Einstein recognized that more experimental confirmation should be sought, and at the end of the paper we find his calculations for the gravitational redshift, the bending of light (which is now double his original 1911 value), and the motion of Mercury.

For reasons described earlier (see p. 460), red-shift observations were insufficiently precise to be of much use. On the other hand, observations of light deflection made in 1919 made Einstein an international superstar. Given the negative reception that special relativity had in England and the fact that the First World War had made German journals virtually inaccessible to British scientists, it is perhaps surprising that it was a British team that provided the first new empirical justification for Einstein's theory. There was at least one significant champion of Einstein's inEngland - the secretary of the Royal Astronomical Society, Arthur Stanley Eddington - and he persuaded the Astronomer Royal, Frank Watson Dyson, to support two expeditions to measure the deflection of light during the solar eclipse of May 1919. One team set out to Sobral in Brazil, and another (led by Eddington) travelled to Principe, off the west coast of Africa. The results from these expeditions were far from conclusive. The Sobral group reported a deflection of 1" .98 ±0". 16 and those in Principe found 1" .61 ±0" .40, compared with the general relativistic value of 1" .75, but Dyson announced proudly that Einstein's prediction had been confirmed.89

85 Einstein Die Grundlage der allgemeinen Relativitatstheorie (1916). English translation in Lorentz, et al. (1923).

A history of the gravitational red shift problem from 1896, through the predictions of general relativity, and including a survey of subsequent attempts to quantify it, can be found in Forbes (1961) (see also Weinberg (1972), p. 81).

87 See Goldberg (1970).

These issues are explored in Earman and Glymour (1980) and Sponsel (2002). Details of the British expeditions and the many other attempts to measure light deflection during solar eclipses (both before and after 1919) can be found in von Kliiber (1960). There is a technical sense in which the bending of light is a less sensitive test of general relativity than perihelion shifts (see Duff (1974)).

On 7 November 1919, the London Times carried an article headed 'Revolution in Science/New Theory of the Universe/Newtonian Ideas Overthrown' and The New York Times carried some equally dramatic headlines 2 days later.

General relativity is a highly technical mathematical theory, yet it is grounded firmly on experiment. The guiding principles behind the theory - that all frames of reference are to be treated equally, special relativity should be recovered in the absence of gravitation, and Newtonian mechanics is appropriate when gravitational fields are weak - all have a sound empirical basis. Moreover, as more experimental tests have been performed, confidence in general relativity has grown. But the nonlinear nature of the theory makes it seriously challenging, and there is still plenty more to discover lying hidden within the deceptively simple looking Eqn (12.10).

A final thought

Some would say that, by reducing celestial motions to equations, science has robbed the heavens of their beauty and wonder.

When I heard the learn'd astronomer,

When the proofs, the figures, were ranged in columns before me, When I was shown the charts and the diagrams, to add, divide, and measure them,

When I, sitting, heard the astronomer, where he lectured with much applause in the lecture-room, How soon, unaccountable, I became tired and sick, Till rising and gliding out, I wander'd off by myself, In the mystical moist night-air, and from time to time, Look'd up in perfect silence at the stars.

'When I heard the Learn'd Astronomer' by Walt Whitman (1819-92)

However, I will leave the last word to Richard Feynman, a man who contributed perhaps more than anyone else to enhancing the public understanding of science in the twentieth century.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature It is too bad that it has to be mathematics, and that mathematics is hard for some people. It is reputed -1 do not know if it is true - that when one of the kings was trying to learn geometry from

Reproduced in Pais (1982), p. 307.

Euclid he complained that it was difficult. And Euclid said, "There is no royal road to geometry". And there is no royal road. Physicists cannot make a conversion to any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we

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