# With a little help from his friends

The people who put the geometry into relativity

Just how clever was Albert Einstein? The key feature of Einstein’s general theory of relativity is the idea of bent spacetime. But Einstein was neither the originator of the idea of spacetime geometry, nor the first to conceive of space being bent

ALBERT EINSTEIN first presented his general theory of relativity to the Prussian Academy of Sciences in Berlin in November 1915.  But he was about ten years later than he should have been in coming up with the idea. What took him so long?

The easy way to understand Einstein’s two theories of relativity is in terms of geometry. Space and time, we learn, are part of one four- dimensional entity, spacetime. The special theory of relativity, which deals with uniform motions at constant velocities, can be explained in terms of the geometry of a flat, four-dimensional surface. The equations of the special theory that, for example, describe such curious phenomena as time dilation and the way moving objects shrink are in essence the familiar equation of Pythagoras’ theorem, extended to four dimensions, and with the minor subtlety that the time dimension is measured in a negative direction. Once you have grasped this, it is easy to understand Einstein’s general theory of relativity, which is a theory of gravity and accelerations. What we are used to thinking of as forces caused by the presence of lumps of matter in the Universe (like the Sun) are due to distortions in the fabric of spacetime. The Sun, for example, makes a dent in the geometry of spacetime, and the orbit of the Earth around the Sun is a result of trying to follow the shortest possible path (a geodesic) through curved spacetime. Of course, you need a few equations if you want to work out details of the orbit. But that can be left to the mathematicians. The physics is disarmingly simple and straightforward, and this simplicity is often represented as an example of Einstein’s “unique genius”.

Only, none of this straightforward simplicity came from Einstein. Take the special theory first. When Einstein presented this to the world in 1905, it was a mathematical theory, based on equations. It didn’t make a huge impact at the time, and it was several years before the science community at large really began to sit up and take notice. They did so, in fact, only after Hermann Minkowski gave a lecture in Cologne in 1908. It was this lecture, published in 1909 shortly after Minkowski died, that first presented the ideas of the special theory in terms of spacetime geometry. His opening words indicate the power of the new insight:

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade into mere shadows, and only a kind of union of the two will preserve an independent reality.”

Minkowski’s enormous simplification of the special theory had a huge impact. It is no coincidence that Einstein received his first honorary doctorate, from the University of Geneva, in July 1909, nor that he was first proposed for the Nobel Prize in physics a year later. There is a delicious irony in all this. Minkowski had, in fact, been one of Einstein’s teachers at the Zrich polytechnic at the end of the nineteenth century. Just a few years before coming up with the special theory, Einstein had been described by Minkowski as a “lazy dog”, who “never bothered about mathematics at all”. The lazy dog himself was not, at first, impressed by the geometrization of relativity, and took some time to appreciate its significance. Never having bothered much with maths at the polytechnic, he was remarkably ignorant about one of the key mathematical developments of the nineteenth century, and he only began to move towards the notion of curved spacetime when prodded that way by his friend and colleague Marcel Grossman. This wasn’t the first time Einstein had enlisted Grossman’s help. Grossman had been an exact contemporary of Einstein at the polytechnic, but a much more assiduous student who not only attended the lectures (unlike Einstein) but kept detailed notes. It was those notes that Einstein used in a desperate bout of last-minute cramming which enabled him to scrape through his final examinations at the polytechnic in 1900.

What Grossman knew, but Einstein didn’t until Grossman told him, in 1912, was that there is more to geometry (even multi-dimensional geometry) than good old Euclidean “flat” geometry.

Euclidean geometry is the kind we encounter at school, where the angles of a triangle add up to exactly 180o, parallel lines never meet, and so on. The first person to go beyond Euclid and to appreciate the significance of what he was doing was the German Karl 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 Ivanovitch Lobachevsky, who was the first to publish a description of such geometry in 1829, and by a Hungarian, Janos 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 180o, 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 put the notion of non-Euclidean geometry on a comprehensive basis 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 180o, 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 Gottingen 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 Gottingen. 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 James Clerk 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 kind of lecturer known as a “Privatdozent”, whose income would come from the fees paid by students who voluntarily chose to take his course (an idea which it might be interesting to revive today). 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 Gottingen 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 that was what he would have to lecture on 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.

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.

Of course, Riemann got the job — though not because of his prescient ideas concerning the possible “closure” of the Universe. 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 (history does not record whether he ever succumbed to the temptation of asking later applicants for such posts to lecture on the third topic from their list).

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 (at the time, he was a Fellow of Newton’s old College, Trinity) 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 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 nineteenth 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.

When Einstein developed the special theory, he did so in blithe ignorance of all this nineteenth century mathematical work on the geometry of multi-dimensional and curved spaces. The great achievement of the special theory was that it reconciled the behaviour of light, described by Maxwell’s equations of electromagnetism (and in particular the fact that the speed of light is an absolute constant) with mechanics — albeit at the cost of discarding Newtonian mechanics and replacing them with something better.

Because the conflict between Newtonian mechanics and Maxwell’s equations was very apparent at the beginning of the twentieth century, it is often said that the special theory is very much a child of its time, and that if Einstein had not come up with it in 1905 then someone else would have, within a year or two.

On the other hand, Einstein’s great leap from the special theory to the general theory — a new, non-Newtonian theory of gravity — is generally regarded as a stroke of unique genius, decades ahead of its time, that sprang from Einstein alone, with no precursor in the problems faced by physicists of the day.

That may be true; but what this conventional story fails to acknowledge is that Einstein’s path from the special to the general theory (over more than ten tortuous years) was, in fact, more tortuous and complicated than it could, and should, have been. The general theory actually follows as naturally from the mathematics of the late nineteenth century as the special theory does from the physics of the late nineteenth century.

If Einstein had not been such a lazy dog, and had paid more attention to his maths lectures at the polytechnic, he could very well have come up with the general theory at about the same time that he developed the special theory, in 1905. And if Einstein had never been born, then it seems entirely likely that someone else, perhaps Grossman himself, would have been capable of jumping off from the work of Riemann and Clifford to come up with a geometrical theory of gravity during the second decade of the twentieth century.

If only Einstein had understood nineteenth century geometry, he would have got his two theories of relativity sorted out a lot quicker. It would have been obvious how they followed on from earlier work; and, perhaps, with less evidence of Einstein’s “unique insight” and a clearer view of how his ideas fitted in to mainstream mathematics, he might even have got the Nobel Prize for his general theory.

Einstein’s unique genius actually consisted of ignoring all the work that had gone before and stubbornly solving the problem his way, even if that meant ten years’ more work. He was adept at rediscovering the wheel, not just with his relativity theories but also in much of his other work. The lesson to be drawn is that it is, indeed, OK to skip your maths lectures — provided that you are clever enough, and patient enough, to work it all out from first principles yourself.