To describe these things mathematically for all smooth surfaces, Gauss found it convenient to imagine ‘painting’ curved coordinate lines on the surface. On a flat surface it is possible to introduce rectangular coordinate grids, but not if the surface is curved in an arbitrary way. So Gauss did the next best thing, which is to allow the coordinate lines to be curved, like the lines of latitude and longitude on the surface of the Earth. He showed how the distance between any two neighbouring points on a curved surface could be expressed by means of the distances along coordinate lines, and also how exactly the same distance relations could be expressed by means of a different system of coordinates on the same surface. About thirty years later, another great German mathematician, Bernhard Riemann, showed that not only two-dimensional surfaces but also three-dimensional and even higher-dimensional spaces can have intrinsic curvature. This is hard to visualize, but mathematically it is perfectly possible. Just as on the Earth, in a curved space of higher dimensions, you can, travelling always in the same direction, come back to the point you started from. These more general spaces with curvature are now called Riemannian spaces.
Einstein realized that he had to learn about all this work thoroughly, and it was very fortunate that he moved at that time to Zurich, where Marcel Grossmann, an old friend from student days, was working. Grossmann gave him a crash course in all the mathematics he needed. When he had fully familiarized himself with it, Einstein became extremely excited for two reasons.
First, Minkowski had shown that space-time could be regarded as a four-dimensional space with a ‘distance’ defined in it between any two points. Except that the ‘distance’ was sometimes positive and sometimes negative, whereas Riemann had assumed the distance to be always positive and had never envisaged time as a dimension, considered mathematically Minkowski’s space-time was just like one of Riemann’s spaces. But it was special in lacking curvature – it was like a sheet of paper rather than the Earth’s surface. Einstein had meanwhile become convinced that gravity curves space-time. This led to one of his most beautiful ideas: in special relativity, the world line (path) of a body moving inertially is a straight line in space-time. This is a special example of a ‘shortest curve’, or geodesic. The corresponding path in a space with curvature would be a geodesic, like a great circle on a sphere.
Einstein assumed that the world line of a body subject to inertia and gravity would be a geodesic. In this way he could achieve his dream of showing that inertia and gravity were simply different manifestations of the same thing – an innate tendency to follow a shortest path. This will be a straight line if no gravity is present, so that space-time has no curvature, but in general it will be a curved (but ‘straightest’) line in a genuinely curved space-time. Since matter causes gravity, Einstein assumed that matter must curve space-time in accordance with some law, for which he immediately started to look. Bodies moving in such a space-time would follow the geodesics corresponding to the curvature produced by the matter, so the gravitational effect of the matter would be expressed through the curvature it produces. Another important insight was that in small regions the effect of curvature would be barely noticeable, just as the Earth seems flat in a small region, so that in those small regions physical phenomena would appear to unfold just as in special relativity without gravity. This gave full expression to the equivalence principle.
The second reason why Einstein became so excited was that Gauss’s method matched his own idea of general relativity. He disliked the distinguished frames of special relativity because they corresponded to special ways of ‘painting’ coordinate systems onto space-time. He felt that this was the same as having absolute space and time. They would be eliminated only if the coordinate systems could be painted on space-time in an arbitrary way. But this was what Gauss’s method amounted to. In fact, in a curved space it is mathematically impossible to introduce rectangular coordinates. Mathematicians call the possibility of using completely arbitrary coordinate systems general covariance. Specifically, laws are said to be generally covariant if they take exactly the same form in all coordinate systems. Einstein identified this with his requirement of general relativity.
To summarize this part of the story, in 1912 Einstein became aware of the possibilities opened up by non-Euclidean geometry and the work initiated by Gauss. He had begun to suspect that gravitational fields would make geometry non-Euclidean. He was also almost desperate to find a formalism that did not presuppose distinguished frames of reference. He found that Gauss’s method of arbitrary coordinates was tailor-made for his ambitions. He also saw that, space and time having been so thoroughly fused by Minkowski, the only natural thing to do was to make space-time into a kind of Riemannian space. The ideas of Gauss and Riemann must be applied, not to space alone, but to space and time. This is the incredibly beautiful idea that Minkowski made possible: gravity was to be explained by curvature in space and time. Einstein thus conjectured that space-time is curved by gravity, and that bodies subject only to gravity and inertia follow geodesics determined by the distance properties of space-time, which encapsulate all its geometrical properties. Einstein’s conjecture has been brilliantly confirmed to great accuracy in recent decades.
Finding the law of motion of bodies in a gravitational field was only part of Einstein’s problem. He also had to find how matter created a gravitational field. He needed to find equations for the gravitational field somewhat like those that Maxwell had found for the electromagnetic field. They would establish how matter interacted with the gravitational field, and also how the field itself varied in regions of space-time free of matter (matching the way electromagnetic radiation propagated as light through space-time). This part of the problem created immense difficulties for Einstein, mostly through very bad luck.
Much as I would like to tell the complete story, which is fascinating and full of ironies, I shall have to content myself with saying that, after three nerve-wracking years, Einstein finally found a generally covariant law that described how matter determined the curvature of space-time. It involves mathematical structures called tensors, all the properties of which had already been studied by mathematicians. In particular, for space-time free of matter, Einstein was able to show that a tensor known as the Ricci tensor (because it had been studied by the Italian mathematician Gregorio Ricci-Curbastro) must be equal to zero. Ironically, Grossmann had already suggested to Einstein in 1912 that in empty space the vanishing of the Ricci tensor might be the generally covariant law he was seeking. However, some understandable mistakes prevented them from recognizing the truth at that time.
It is a striking fact that all the mathematics Einstein needed already existed. In fact, I believe it is significant that he did not have to invent any of it. In 1915, he was immediately able to show that, to the best accuracy astronomers could achieve at that time, his theory gave identical predictions to Newtonian gravity except for a very small correction to the motion of Mercury. All planetary orbits are ellipses. A planet’s elliptical orbit itself very slowly rotates, under the gravitational influence of the other planets. This is known as the advance of the perihelion, the perihelion being the point at which the planet is closest to the Sun, marking one end of the ellipse’s longest diameter. According to Einstein’s theory, Mercury’s perihelion should advance by 43 seconds of arc per century more than was predicted by Newtonian theory. This very small effect shows up for Mercury because it is closer to the Sun than the other planets, and also has a large orbital eccentricity. For many years, the sole discrepancy in the observed motions of the planets had been precisely such a perihelion advance for Mercury of exactly that magnitude. All attempts to explain it had hitherto failed. Einstein’s theory explained it straight off.