by John Gribbin
What Einstein should have known
The first person to go beyond Euclid and to appreciate the significance of what he was doing was the German Carl Friedrich Gauss, who was born in 1777 and had completed all of his great mathematical discoveries by 1799. But because he didn’t bother to publish many of his ideas, non-Euclidean geometry was independently discovered by the Russian Nikolai Ivanovich Lobachevsky, who was the first to publish a description of such geometry in 1829, and by a Hungarian, János Bolyai. They all hit on essentially the same kind of ‘new’ geometry, which applies on what is known as a ‘hyperbolic’ surface, which is shaped like a saddle, or a mountain pass. On such a curved surface, the angles of a triangle always add up to less than 180 degrees, and it is possible to draw a straight line and mark a point not on that line, through which you can draw many more lines, none of which crosses the first line and all of which are, therefore, parallel to it.
But it was Bernhard Riemann, a pupil of Gauss, who comprehensively put across the notion of non-Euclidean geometry in the 1850s, and who realised the possibility of yet another variation on the theme – the geometry that applies on the closed surface of a sphere (including the surface of the Earth). In spherical geometry, the angles of a triangle always add up to more than 180 degrees, and although all ‘lines of longitude’ cross the equator at right angles and must therefore all be parallel to one another, they all cross each other at the poles.
Riemann, who had been born in 1826, entered Göttingen University at the age of twenty and learned his mathematics initially from Gauss, who had turned 70 by the time Riemann moved on to Berlin in 1847, where he studied for two years before returning to Göttingen. He was awarded his doctorate in 1851, and worked for a time as an assistant to the physicist Wilhelm Weber, an electrical pioneer whose studies helped to establish the link between light and electrical phenomena, partially setting the scene for Maxwell’s theory of electromagnetism.
The accepted way for a young academic like Riemann to make his way in a German university in those days was to seek an appointment as a Privatdozent. In order to demonstrate his suitability for such an appointment, the applicant had to present a lecture to the faculty of the university, and the rules required the applicant to offer three possible topics for the lecture, from which the professors would choose the one they would like to hear. It was also a tradition, though, that although three topics had to be offered, the professors always chose one of the first two on the list. The story is that when Riemann presented his list for approval, it was headed by two topics which he had already thoroughly prepared, while the third, almost an afterthought, concerned the concepts that underpin geometry.
Riemann was certainly interested in geometry, but apparently he had not prepared anything along these lines at all, never expecting the topic to be chosen. But Gauss, still a dominating force in the University of Göttingen even in his seventies, found the third item on Riemann’s list irresistible, whatever convention might dictate, and the 27-year-old would-be Privatdozent learned to his surprise that he would have to lecture on it to win his spurs.
Perhaps partly under the strain of having to give a talk he had not prepared and on which his career depended, Riemann fell ill, missed the date set for the talk and did not recover until after Easter in 1854. He then prepared the lecture over a period of seven weeks, only for Gauss to call a postponement on the grounds of ill health. At last, the talk was delivered on 10 June 1854. The title, which had so intrigued Gauss, was ‘On the hypotheses which lie at the foundations of geometry’.
In that lecture – which was not published until 1867, the year after Riemann died – he covered an enormous variety of topics, including a workable definition of what is meant by the curvature of space and how it could be measured, the first description of spherical geometry (and even the speculation that the space in which we live might be gently curved, so that the entire Universe is closed up, like the surface of a sphere, but in three dimensions, not two), and, most important of all, the extension of geometry into many dimensions with the aid of algebra. Crucially, Riemann described mathematically the nature of a surface over which the geometry varies from place to place: flat in some places, hyperbolic in other places, spherical elsewhere.
Although Riemann’s extension of geometry into many dimensions was the most important feature of his lecture, the most astonishing, with hindsight, was his suggestion that space might be curved into a closed ball. More than half a century before Einstein came up with the General Theory of Relativity – indeed, a quarter of a century before Einstein was even born – Riemann was describing the possibility that the entire Universe might be contained within what we would now call a black hole. ‘Everybody knows’ that Einstein was the first person to describe the curvature of space in this way – and ‘everybody’ is wrong.d
Of course, Riemann got the job. Gauss died in 1855, just short of his 78th birthday, and less than a year after Riemann gave his classic exposition of the hypotheses on which geometry is based. In 1859, on the death of Gauss’s successor, Riemann himself took over as professor, just four years after the nerve-wracking experience of giving the lecture upon which his job as a humble Privatdozent had depended.
Riemann died of tuberculosis at the age of 39. If he had lived as long as Gauss, however, he would have seen his intriguing mathematical ideas about multi-dimensional space begin to find practical applications in Einstein’s new description of the way things move. But Einstein was not even the second person to think about the possibility of space in our Universe being curved, and he had to be set out along the path that was to lead to the General Theory of Relativity by mathematicians more familiar with the new geometry than he was. Chronologically, the gap between Riemann’s work and the birth of Einstein is nicely filled by the life and work of the English mathematician William Clifford, who lived from 1845 to 1879 and who, like Riemann, died of tuberculosis. Clifford translated Riemann’s work into English and played a major part in introducing the idea of curved space and the details of non-Euclidean geometry to the English-speaking world. He knew about the possibility that the three-dimensional Universe we live in might be closed and finite, in the same way that the two-dimensional surface of a sphere is closed and finite, but in a geometry involving at least four dimensions. This would mean, for example, that just as a traveller on Earth who sets off in any direction and keeps going in a straight line will eventually get back to their starting point, so a traveller in a closed universe could set off in any direction through space, keep moving straight ahead and eventually end up back at their starting point.
But Clifford realised that there might be more to space curvature than this gradual bending encompassing the whole Universe. In 1870, he presented a paper to the Cambridge Philosophical Society in which he described the possibility of ‘variation in the curvature of space’ from place to place, and suggested that: ‘Small portions of space are in fact of nature analogous to little hills on the surface [of the Earth] which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.’ In other words, still seven years before Einstein was born, Clifford was contemplating local distortions in the structure of space – although he had not got around to suggesting how such distortions might arise, nor what the observable consequences of their existence might be, and the General Theory of Relativity, as we shall see, actually portrays the Sun and stars as making dents, rather than hills, in spacetime, not just in space.
Clifford was just one of many researchers who studied non-Euclidean geometry in the second half of the 19th century – albeit one of the best, with some of the clearest insights into what this might mean for the real Universe. His insights were particularly profound, and it is tempting to speculate how far he might have gone in pre-empting Einstein, if he had not died eleven days before Einstein was born. So Einstein was following an established tradition when, with Grossmann’s help, he picked up the threads and completed his masterwork.
The masterwo
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In describing the Special Theory in geometric terms, a crucial property is the velocity of an object. Velocity involves both speed and direction, and in physics such directional entities are called ‘vectors’. In the curved spacetimes described by Riemannian geometry, the equivalents of vectors are called ‘tensors’, and they are more complicated in that they involve more numbers (there are more components to a tensor than there are to a vector). This is what makes calculations involving tensors tedious, although, once you know the rules, they are not difficult in the sense of requiring great brain power. Working out the rules is the clever bit.
The distances between points in Riemannian space are calculated in terms of a particular kind of tensor, known as the ‘metric tensor’. The metric tensor that applies to curved four-dimensional spacetime has been described as a ‘vector on steroids’ and has sixteen components, although only ten of them are completely independent of one another. It is usually written as gμ and pronounced ‘gee mu nu’. This expression means as much (or more!) in the General Theory as E = mc2 does in the Special Theory.
Its essence can be gleaned quite easily using Minkowski’s ideas about spacetime. When I mentioned earlier using coordinates to describe a triangle on a sheet of graph paper, I took it for granted you would be thinking of a flat sheet of paper. Einstein’s Special Theory describes the way things move about in what is called ‘flat spacetime’. Einstein’s General Theory of Relativity describes how things move in curved spacetime, and the curvature in spacetime is caused by the presence of matter. For lines drawn on the surface of a sphere, for example, the angles of a triangle don’t add up to 180 degrees, and lines that start out parallel to one another (like the lines of longitude crossing the equator) can end up crossing each other (in this case, at the North and South poles).
The usual example is to think of a stretched rubber sheet, like a trampoline. This is flat, and if you roll marbles across it they travel in straight lines. Now imagine dumping a heavy weight, like a bowling ball, on the stretched sheet. It makes a dent – it curves ‘spacetime’ – and if you roll marbles across the curved surface they will follow curved paths past the bowling ball. Einstein’s key insight was that gravity depends on the curvature of spacetime, telling material objects how to move, while the presence of material objects is what makes spacetime curved. He later explained his insight to his son Eduard: ‘When a blind beetle crawls across the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved. I was lucky enough to notice what the beetle didn’t notice’.9 But it took him three more years to turn that insight into a rigorous mathematical theory.
Einstein’s approach rested on two legs. One was physical. He knew that his theory of gravity must reproduce all the results of Newton’s theory in the case of weak gravitational fields, that it must contain the basic laws of ‘classical’ (that is, Newtonian) physics such as conservation of energy and momentum, and, his own Big Idea, that experiments carried out in an accelerating laboratory (or frame of reference) must give the same results as equivalent experiments carried out in a stationary laboratory in a gravitational field. The second leg was mathematical, using tensor theory to find an equation describing gravity (a field equation, or set of field equations) that is independent of arbitrary changes in the space and time coordinate systems (equivalent to changing the meridian we measure longitude from, from Greenwich to, say New York; wherever you measure from the distances between places stay the same). Such a theory is said to be covariant.
Einstein’s notebooks show that by the end of 1912, when he was still in Zurich, he had found a field equation that satisfied the mathematical requirements, but did not seem to produce the same results as Newton for weak fields. Never a great mathematician, but a firm believer in physical intuition, this led Einstein to concentrate on the physical side of the problem in the months that followed, pushing the mathematics aside. This turned out to be a mistake; the equation he rejected in 1912 was very nearly the solution he had been looking for.
Starting out from the physical requirements, in the spring of 1913 Einstein and Grossmann produced what they called an ‘Outline of a Generalised Theory of Relativity and a Theory of Gravitation’, known now as the ‘Entwurf’, the German word for ‘outline’. The snag was, this theory was not covariant – people in different accelerated frames of reference would not see the universe in the same way. But it did make some interesting predictions. Among other things, it predicted that the orbit of Mercury would change over time in a regular way. There was an observed ‘advance in the perihelion of Mercury’ that could not be explained by Newtonian theory. The Entwurf did imply such a phenomenon, but Michele Besso showed that the predicted effect, although arguably better than nothing, was much too small to explain the observations. This echoes, with hindsight, the problem with Einstein’s light bending prediction of 1911: in the right direction, but too small. As I have mentioned, if Erwin Freundlich’s eclipse expedition of 1914 had been successful, Einstein would have been proved wrong!
Einstein struggled with the Entwurf idea during the months that saw turmoil in his domestic life and the move to Berlin in the spring of 1914. Living alone (but near to Elsa) in a seven-roomed apartment, he was free to work long and irregular hours, surrounded by piles of paper but the bare minimum of furniture, refining the theory and attempting to answer criticisms of it. By the spring of 1915, he was no longer referring in his notes to ‘a generalised theory’ but to ‘the general theory’, and he explained his progress so far in a series of lectures at the University of Göttingen in the summer of 1915. The lectures were a great success, and Einstein got on like a house on fire with the professor of mathematics there, 53-year-old David Hilbert, who enthusiastically espoused Einstein’s ideas. Or at least, one leg of them. As a mathematician, Hilbert was intrigued by the second leg of the theory, which Einstein had been neglecting for the past couple of years. He set out to find the required covariant field equation, without worrying too much about the physical requirements. And while he did so (making more progress than Einstein was aware of at the time), Einstein was finally forced to abandon the Entwurf approach.
In October 1915, Einstein finally convinced himself that the problems with covariance, with the orbit of Mercury and with the description of rotating systems made the Entwurf unworkable. Physical insight had, for once, let him down. So he went back to his Zurich notebooks and picked up the tensor leg of the theory from where he had left off. In a manner reminiscent of the way that problems can sometimes be solved by sleeping on them, when he looked again at the calculations from 1912 he saw almost immediately where he had gone wrong, and how with a relatively minor tweak he could come up with properly covariant field equations. But that still implied a great deal of calculation to get all his beans in a row. This led to a furious burst of work which he was able to present in instalments (lectures based on scientific papers), while still developing the theory, to four successive weekly meetings of the Prussian Academy of Sciences, starting on Thursday, 4 November 1915. Even when he gave the first lecture, though, he had not yet worked through to the fully covariant field equations; this was very much a work in progress.
Einstein knew that Hilbert was working on the problem, but not how much progress he had made. So he sent him a copy of the 4 November lecture, asking what Hilbert thought of this new approach. The lecture of 11 November, which still didn’t effectively solve the covariance problem, went off to Göttingen in the same way. Einstein was alarmed to receive a letter from Hilbert informing Einstein that he was on the brink of a solution to ‘your great problem’ and that if Einstein could come to Göttingen the following Tuesday, 16 November, he would be glad to ‘lay out my theory in very complete detail’. He even reminded Einstein of the times of the trains from Berlin to Göttingen. But what must have been the most alarming news from Hilbert was left for a postscript – ‘as far as I understand your new paper, the solution given by you is entirely different from mine’.
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Einstein had no intention of going to Göttingen with his own theory still incomplete and Hilbert possibly about to pre-empt him. On 15 November he wrote back declining the invitation, complaining of stomach pains and fatigue (he was genuinely ill as a result of overwork and a tendency not to bother with things like eating when he was busy), and asking for a copy of Hilbert’s theory ‘to mitigate my impatience’.
Flinging himself back into his calculations, Einstein made a great discovery, just in time for his third lecture on 18 November. Reworking the calculation of the orbit of Mercury with the revised version of his theory gave the right answer! The theory now predicted a shift in the perihelion of 43 seconds of arc per century, exactly matching observations. Einstein was so excited that he suffered heart palpitations and had to take a rest. But this wasn’t all. The same revision to the theory gave a new prediction for the bending of light by the Sun, not 0.85 seconds of arc as he had previously calculated, but exactly twice as much, 1.7 seconds of arc. This was less exciting, because in the absence of a convenient solar eclipse there were no observations to match the theory up with, and wouldn’t be for several years. But both results were included in the 18 November lecture.
