Einstein’s gravitational insight
A discussion triggered by the movie Gravity prompts me to offer this to correct some misconceptions about the nature of “free fall”. As Einstein said, “For an observer falling freely from the roof of a house, the gravitational field does not exist”. He meant that literally, and later described it as “the happiest thought of my life”.
Conventional wisdom has it that although special relativity was a product of its time, and that had Einstein not come up with the theory in 1905 someone else soon would have, under the pressure of the need to explain the conflict between Newtonian mechanics and the behaviour of light, general relativity was a work of unique inspiration, which sprang from Einstein’s genius alone, and which might not have been discovered for another fifty years, had he fallen under the wheels of a tram in 1906. I have even been guilty of perpetuating this myth myself. But now, it seems to me that this case does not stand up to close inspection. It is a case made by physicists, looking back at how Einstein’s theory describes material objects. The conflict between Newton and Maxwell pointed to a need for a new theory, but once that theory was in place, the argument runs, there were no outstanding observational conflicts that had still to be explained. Maybe. But by the 1900s many mathematicians were already intrigued by the notion of curved space. Once Herman Minkowski had presented the special theory of relativity as a theory of mechanics in flat four-dimensional spacetime, it would, surely, not have been long before somebody wondered how those laws of mechanics would be altered if the spacetime were curved. From a mathematical point of view, the general theory is every bit as much a child of its time as the special theory was, and a logical development from the special theory. This is certainly born out by the fact that it took the prodding of a mathematician to get the physicist Einstein moving along the right lines after 1909.
What Einstein lacked in terms of top flight mathematical skill and knowledge, though, he more than made up for in terms of physical intuition — his “feel” for the way the Universe worked was second to none. His special theory of relativity, for example, developed from Einstein wondering what the Universe would look like if you could ride with a light ray as it hurtled through space at nearly 300,000 km a second; and the seed that grew into the general theory was an inspired piece of reasoning about the behaviour of a light ray crossing a falling elevator. The seed was sown within a couple of years of the completion of the special theory; but, partly because at that time Einstein knew nothing of Riemannian geometry, it took a further nine years to grow to fruition.
The special theory of relativity tells us how the world looks to observers moving with different velocities. But it deals only with constant velocities — steady motion at the same speed in the same direction. Even in 1905, it was obvious that the theory failed to describe how objects behave under two important sets of conditions that exist in the real world. It does not describe the
behaviour of accelerated objects (by which physicists mean, remember, objects that change their speed îorï their direction, or both); and it does not describe the behaviour of objects that are under the influence of gravity. Einstein’s insight, which he first presented in 1907, was that both these sets of conditions are the same — that acceleration is exactly equivalent to gravity. This is such a cornerstone of our modern understanding of the Universe that it is known as the “principle of equivalence”.
Anyone who has travelled in a high-speed elevator knows what Einstein meant by the principle. When the elevator starts moving upward, you are pressed to the floor, as if your weight has increased; when it slows at the top of its rise, you feel lighter, as if gravity has been partly cancelled out. Clearly, acceleration and gravity have something in common; but it is a dramatic step to go from this observation to say that gravity and acceleration are exactly the same. An implausible scenario demonstrates just how equivalent they are. If the cable of the elevator snapped and all the safety devices failed, while the lift was falling freely down its shaft you would fall at the same rate, weightless, floating about inside the falling “room”.
But what would happen to a beam of light shone across the falling elevator from one side to the other? In the weightless falling room, according to Einstein, Newton’s laws apply and the light must travel in a straight line from one side to the other. Then, however, he went on to consider how such a beam of light would look to anyone outside the falling elevator, if the lift had walls made of glass and the path of the light beam could be tracked. In fact, the “weightless” elevator and everything inside it is being accelerated by the gravitational pull of the Earth. In the time it takes the light beam to cross the elevator, the falling room has increased its speed, and yet the light beam still strikes the spot on the opposite wall level (according to an observer in the lift) with the spot from where it started. This can only happen if, from the point of view of the outside observer, the light beam has bent downward slightly while crossing the falling elevator. And the only thing that could be doing the bending is gravity.
So, said Einstein, if acceleration and gravity are indeed precisely equivalent to one another, gravity must bend light. You can cancel out gravity while you are in free fall, constantly accelerating; and you can create an effect indistinguishable from gravity by providing an acceleration, which makes everything “fall” to the back of the accelerating vehicle. So objects or people in free fall, as in the movie Gravity, literally do not feel gravity! The title could not be more wrong, except in that it is an allusion to the gravity of the situation.
The possibility of light bending was neither new nor startling. Newtonian mechanics and the corpuscular theory suggest that light should be bent, for example when it passes near the Sun. Indeed, Einstein’s first calculations of gravitational light bending, based on the principle of equivalence, suggested that
the amount of bending would be exactly the same as in the old Newtonian theory. Fortunately, though, before anyone could carry out a test to measure the predicted effect (not that anyone was very interested in it while the theory was incomplete), Einstein had developed a full theory of gravity and accelerations, the general theory of relativity. In the general theory, the predicted light bending is twice as much as in the Newtonian version, and it was the measurement of this non-Newtonian effect that made people sit up and take notice of the general theory. But that wasn’t until 1919.
For more than three years after he first stated the principle of equivalence, Einstein did very little work on trying to develop a proper theory of gravity based on the principle. There were many reasons for this. As Einstein’s reputation grew, he took up a series of increasingly prestigious academic posts, first as a
Privatdozent in Bern, then assistant professor in Zürich, then on to be a full professor in Prague. He had a growing family — his son Hans had been born in 1904, and Eduard arrived in 1910. But, most important of all, during that period Einstein’s scientific attention was focussed on his contributions to the exciting new developments in quantum physics, and he simply didn’t have time to struggle with a new theory of gravity as well. It was after he had reached a temporary impasse with his work on quantum theory that, in Prague in the summer of 1911, he returned to the gravitational fray.
It was in 1911, in fact, that Einstein first applied the idea of light bending to rays passing close by the Sun, and came up with a prediction essentially the same size as the Newtonian prediction. The Newtonian version of the calculation had been made back in 1801, by the German Johann von Soldner, acting on the assumption that light is a stream of particles; Einstein, completely unaware of von
Soldner’s calculation, calculated his own initial version of light bending by the Sun in 1911 by treating light as a wave (even though he had himself been instrumental in showing that light sometimes does behave like a stream of particles!). The two calculations give almost precisely the same value for the bending. The simplest way to understand the first Einsteinian version of the effect is that it results from the distortion of time caused by the Sun’s gravitational field. In 1911, Einstein was struggling with a horribly complex and unwieldy set of equations that in effect corresponded to a combination of warped time with flat space, and as a result he was literally only halfway to the full value of the light bending effect.
Things began to look up, however, as soon as Einstein returned to Zürich, after staying in Prague for only a year. His return to Switzerland was engineered by a friend whose lecture notes he had borrowed in his student days a dozen years before — Marcel Grossman, who had now risen to become Dean of the physics and mathematics department of the Polytechnic.
Grossman’s own career had followed a much more conventional pattern than Einstein’s, although he had reached this eminence very young. He was just one year older than Einstein, and after graduating with Einstein in 1900 he worked as a teacher while writing his doctoral thesis, also producing two geometry books for high school students and several papers on non-Euclidean geometry. On the strength of this work, he joined the faculty at the Polytechnic, becoming a full professor in 1907 and Dean in 1911 at the age of 33. One of his first acts as Dean was to entice Einstein back to Zürich. He arrived on 10 August 1912, knowing that he had the basis of a workable theory of gravity, but uncomfortably aware that he lacked the right mathematical tools to finish the job. Much later, he recalled a plea he made at this time to his old friend — “Grossman, you must help me or I’ll go crazy!” (quotes in this artucle are from Abraham Pais, Subtle is the
Einstein had realised that the method for describing curved surfaces developed by Gauss might help with his difficulties, but he knew nothing about Riemannian geometry. He did, however, know that Grossman was a wizz at non-Euclidean geometry, which is why he turned to him for help — “I asked my friend whether my problem could be solved by Riemann’s theory”. The answer, in a word, was “yes”. Although it took a long time to sort out the details, what Grossman was able to tell Einstein immediately opened the door for him, and by 16 August he was able to write to another colleague “it is going splendidly with gravitation. If it is not all deception, then I have found the most general equations.”
Einstein and Grossman investigated the significance of curved spacetime (warping both space and time) for a theory of gravity in a paper published in 1913. The collaboration ended when Einstein accepted an appointment as Director of the new Institute of Physics at the Kaiser Wilhelm Institute in Berlin in 1914 — a post so tempting, requiring no teaching duties but allowing him to devote all his time to research, that it tore him away from Switzerland and Grossman. But the two remained firm friends until Grossman’s death, from multiple sclerosis, in 1936. It was in Berlin that Einstein, alone, completed the long journey from the special theory of relativity to the general theory.
The full version of the general theory was presented at three consecutive meetings of the Prussian Academy of Sciences in Berlin in November 1915, and published in 1916. What matters here is the way in which Einstein used Riemannian geometry to describe curved space. A massive object, like the Sun, can be thought of as making a dent in three-dimensional space, in a way analogous to the way an object like a bowling ball would make a dent in the two-dimensional surface of a stretched rubber sheet, or a trampoline. The shortest distance between two points on such a curved surface will be a curved geodesic, not what we are used to thinking of as a straight line, and this applies in the three-dimensional case as well. Because space is bent, light rays are bent. But Einstein had already discovered, as we have seen, that light rays are bent near a massive object by a warp in the time part of spacetime, as well. And, as it happens, the space warping alone bends the light by the same amount as the time warping effect that Einstein had already calculated. Overall, the general‹d‹
theory of relativity predicts îtwiceï as much light bending as Newtonian theory does.
Indeed, it is the “new” space warping effect discussed by Einstein in 1916 that is actually the equivalent of the old Newtonian effect; it is the time warping that makes the relativistic prediction different from the Newtonian calculation.‚ That is why, when the light bending was measured during the eclipse of 1919 and found to agree with Einstein, not Newton, the newspapers proclaimed that Newton’s theory of gravity had been overthrown. But that is wrong.
What Einstein had actually done was to explain Newton’s law of gravity. There are some subtle differences, such as with the bending of light by the Sun, between simple Newtonian theory and the general theory of relativity. But what really matters is that if gravity is explained as the result of curvature in four-dimensional spacetime, then, because of the nature of this curvature itself, it is virtually impossible to come up with any version of gravity except an inverse square law. An inverse square law of gravity is far and away the most natural, and likely, consequence of curvature in four-dimensional spacetime. Unlike Newton, Einstein dod “frame hypotheses” about the nature of gravity. His hypothesis was that spacetime curvature causes what we perceive as gravitational attraction, with the implication of that hypothesis is that gravity must obey an inverse square law. Far from overturning Newton’s theory, Einstein’s work actually explains Newton’s theory, and puts it on a more secure footing than ever before.
The best way to picture this is as a kind of dialogue between matter and spacetime. Because the distribution of matter across the Universe is uneven, the curvature of spacetime is uneven — the very geometry of spacetime is relative, and the nature of the metric, defined in terms of tiny Pythagorean triangles, depends on where you are in the Universe. Lumps of matter distort spacetime, not so much making hills, as William Clifford conjectured in the 19th century, but valleys. Within that curved spacetime, moving objects travel along geodesics, which can be thought of as lines of least resistance. And you can calculate the length of even a curved geodesic in general relativity in terms of many tiny Pythagorean triangles which each “measure” a tiny portion of its length, added together using the integral calculus developed originally by Newton. But a falling rock, or a planet in its orbit, doesn’t have to make the calculation — it just does what comes naturally. In other words, matter tells spacetime how to bend, and spacetime tells matter how to move.
There is, however, one important point which often causes misunderstandings and confusion that I ought to get clear about all this. We are not just dealing with curved space. The orbit of the Earth around the Sun, for example, forms a closed loop in space. If you imagine that this represents the curvature of space caused by gravity, you would leap to the false conclusion that space itself is vlosed around the Sun — which it obviously is not, since light (not to mention the Voyager space probes) can escape from the Solar System. What you have to remember is that the Earth and the Sun are each following their own world lines through four-dimensional spacetime. Because the factor of the speed of light comes in to the time part of Minkowski’s metric for spacetime, and this carries over into the equivalent metric in general relativity, these world lines are enormously elongated in the time direction. So the actual path of the Earth “around” the Sun is not a closed loop, but a very shallow helix, like an enormously stretched spring. It takes light eight and one third minutes to reach the Earth from the Sun. So each circuit that the Earth makes around the Sun is a distance of about 52 light minutes. But it takes a year for the Earth to complete such a circuit, and in that time it has moved along the time direction of spacetime by the equivalent of a light year — more than ten thousand times further than the length of its annual journey through space, and more than 63,000 times the distance from the Earth to the Sun. In other words, the pitch of the helix representing the Earth’s journey through spacetime is more than 63,000 times bigger than its radius. In flat spacetime, the world line would be a straight line; the presence of the Sun’s mass actually distorts spacetime only slightly, just enough to cause a slight bending of the world line, so that it weaves to and fro, very gently, as the Earth moves through spacetime. You need to have much more mass, or a much higher density of mass, in order to close space around an object.
For more about these topics, see my book Companion to the Cosmos.