That is not to say, however, that Einstein’s general relativity was motivated by mathematical concerns alone, either the beauty of tensor algebra, which made his theory calculable, or that of Riemannian geometry, which Einstein had to master in order to ultimately describe curved space. Far from it. The origins of Einstein’s general theory of relativity stem from the same type of thought experiments involving physical phenomena that led to the special theory. In this case they came about as Einstein was pondering Newton’s law of gravity, electromagnetism, and special relativity in 1907, a year in which he later stated he had had “the happiest thought of his life.”
We have already seen how the relationship between electricity and magnetism implies that what one observer measures as a magnetic force, another could measure as an electric force. This “observer-dependence” of electromagnetism played a key role in the development of special relativity and the unification of space and time into space-time. Perhaps not surprisingly, a similar notion played a central role in Einstein’s thinking when, in 1907, while considering Newton’s gravity, he suddenly realized that it, too, was observer dependent.
He reasoned as follows: An observer who is free-falling in a gravitational field—like someone who jumps out of a plane—feels no gravitational forces at all. For this observer, the gravitational field is undetectable (at least until the rude awakening, followed by a quick demise, upon later hitting the ground). Ignoring any effects of air resistance, an object “dropped” from such an observer’s hand would fall at the same rate of acceleration as that of the observer, so it would remain at rest relative to the observer. For all intents and purposes, gravity wouldn’t exist for this individual. In this regard, as would be equally true for Galileo’s observer moving at a constant speed in the absence of gravity, such a free-falling observer would have every right to consider herself at rest, because all objects at rest in her frame would remain at rest if no other (nongravitational) force was applied to them.
In this sense gravity, like electricity or magnetism, seems to exist truly in the eye of the beholder. But this picture is true only if all objects fall at the same rate. If a single object accelerated at a different rate from all other objects in a gravitational field, the whole notion that gravity might be invisible would fall apart. A free-falling observer would see this object as accelerating relative to her, and thus would be able to conclude that some external force was acting upon it.
This idea—all objects fall at the same rate due to gravity, independent of their composition—Einstein labeled the Equivalence Principle, and it was central to his development of general relativity. Only if it remained true could gravity arise as an accident of one’s circumstances, just as the electric force that one might experience could actually be due to a distant, changing magnetic field.
While a violation of the equivalence principle would put an end to any chance of “replacing” gravity with something more fundamental and less observer-dependent, it is not obvious from this example what one might actually replace it with. Once again, Einstein provided a thought experiment that showed the way. If falling in a gravitational field can get rid of any observable effects of gravity, accelerating in the absence of one can create the appearance of a gravitational field. Consider the following famous example. Say, for some inexplicable reason, you are in an elevator deep in space. As everyone who has ever been in an elevator has experienced, when it first starts to accelerate upward, you feel slightly heavier; namely, you feel a greater force exerted by the floor on your feet. If you were in outer space, where you would otherwise feel weightless, and the elevator you were in started to accelerate upward, you would feel a similar force pushing you down against the floor.
Einstein reasoned that, if the equivalence principle was indeed true, then there is no experiment you could perform in the elevator that could distinguish between whether that elevator was accelerating upward in the absence of a gravitational field, or whether it was at rest in a gravitational field, where the force the observer would feel pushing her down against the floor would be due to gravity.
So far so good. Now, imagine what would happen if the observer in the accelerating elevator were to shine a laser beam from one side of the elevator to the other. Since, during the time the light beam was crossing the elevator, the elevator’s upward speed would have increased, this would mean that the light ray, which is traveling in a straight line relative to an observer at rest outside the elevator, would end up hitting the far side of the elevator somewhat below the height where it was emitted, relative to the floor of the elevator
.
Now, if gravity is to produce effects that are completely equivalent to those we would measure in an accelerating system this would mean that if I shined a laser beam in an elevator at rest in a gravitational field (say, on Earth), I would also see the light ray’s trajectory bend downward. (Of course, the effect would be very small, but since we are doing a thought experiment here, we are free to imagine an arbitrarily accurate measuring device.)
But, special relativity tells us that light rays move at constant speed in straight lines. How can we reconcile this behavior with what you would measure in the elevator? Well, one way to go in a straight line and also travel in a curve is to travel on a straight line on a curved surface. This realization led Einstein on a long mental journey in the course of which he was drawn to the inescapable conclusion that space and time are not only coupled together, but are also themselves dramatically different than we perceive them to be. Space, and to some extent time, can be curved in the presence of mass or energy. The result was perhaps the most dramatic reformulation of our understanding of the underlying nature of the physical universe in the history of science.
Einstein’s journey was replete with false starts and dead ends, and the slowly dawning acceptance that mathematical concepts that he had vaguely been exposed to while a student might actually be useful for understanding the nature of gravity. In 1912 Einstein finally realized that the mathematics of Gauss, and then Riemann, which described the geometry of curved surfaces and ultimately curved spaces, held the key to unlocking the puzzle he had been wrestling with all those years. By November 1915, after almost having been scooped by the best mathematician of that generation, David Hilbert, Einstein unveiled the final form of his “gravitational field equations.” Einstein’s equations, as we usually call them, provide a relation between the energy and momentum of objects moving within space and the possible curvature of that space. There are at least two fascinating and unexpected facets of this relation. First, it turns out to be completely independent of whatever system of coordinates one might use to describe the position of objects within the curved space. Second, and true to the spirit of special relativity—which by tying together space and time also turned out to tie together mass and energy—energy becomes the source of gravity. In general relativity, however, such energy influences the very geometry of space itself—a fact that makes general relativity almost infinitely more complex and fascinating than Newton’s earlier law of gravitation. This is because the energy associated with a gravitational field, and hence with the curvature of space, in turn affects that curvature.
In the jargon of mathematicians, general relativity is a “nonlinear” theory. While technically speaking this means that it is difficult to solve the relevant equations, in physical terms it means that the distribution of mass and energy in space determines the strength of the gravitational field at any point, which in turn determines the curvature of space at any point, which in turn determines subsequent distribution of masses and energy, which in turn determines the curvature of space, and so on. Nevertheless, in spite of the difficulty of dealing with these equations, the single fact that affected Einstein during that fateful November in 1915 more deeply than perhaps any other discovery he had made in his lifetime was the realization that the mathematical theory he had just proposed explained an obscure but mysterious astronomical observation about the orbit of Mercury around the su
n. One of the most successful and stunning predictions of Newton’s law of gravity is that the orbit of planets around a central body such as the Sun should be described by mathematical curves called ellipses. That the planetary motions were not perfect circles had first been discovered, somewhat to his dismay, by Johannes Kepler, and in short order Newton proved that his universal law universally implied elliptical orbits. Nevertheless, in 1859 the French astronomer Urbain Jean Joseph Le Verrier discovered that the orbit of Mercury was anomalous. Instead of returning exactly to its initial position after each orbit, the planet advanced slightly, so that rather than forming perfect ellipses, the orbits traced a figure that was more like a spiral, with the axis of each successive orbit being slightly shifted compared to the one before it, as shown in an exaggerated view below:
This “precession” was extremely small, measuring only about 1/100 of a degree per century. Nevertheless, in physics, as in horseshoes, being merely close is not good enough; if Newton was correct, there should be no such precession. Barring the presence of some new, undiscovered massive body nearby exerting a gravitational pull on Mercury, the only way such a precession could be explained was to slightly alter the nature of Newtonian gravity.
Beyond its profound underlying physical implications, this is precisely what Einstein’s general theory does, making a small correction to Newton’s law. It turns out that when the force law is no longer precisely as Newton described it, then a precession is predicted. Einstein, to his credit, was able to derive an approximate solution to his equations that was accurate enough to predict the precession of Mercury’s orbit, and to his immense surprise and satisfaction, the prediction was precisely in agreement with this half-century-old puzzling result.
Years afterward Einstein recalled that, upon discovering this agreement between prediction and observation, he had the feeling that something had actually snapped within him. He suddenly realized that his journey of the mind had led him to more than mathematical fantasies. He said he was so excited that he had palpitations of the heart. Later in his career, Einstein would become more enamored with the simplicity of the mathematical principles that were the foundation for general relativity. But I think it is crucially important to recognize—and I shall have cause to return to this theme—that what distinguished Einstein the physicist from Hilbert the mathematician was that what Einstein wanted to do was explain the way nature worked, not merely derive beautiful equations. It was the excitement of seeing that, even by such a small effect, nature obeyed the laws he discovered in his mind that made Einstein weak with excitement.
In the same paper in which he derived the precession of Mercury’s orbit, written a week before the paper that presented the final form of general relativity, Einstein made another prediction. He calculated that light would indeed bend in a gravitational field, as he had realized almost a decade earlier, but that the actual magnitude of the bending would be twice as large as he had previously estimated, and twice as large as the value one might get by simply pretending that light had mass and then using Newton’s theory to calculate the effect of gravity on its trajectory. He thus predicted that light passing near the sun would be deflected by approximately 1/2000 of a degree. As small as this value was, its predicted effect would be measurable, as Einstein realized as early as 1911, when he was still in fact predicting the wrong value. If one observed stars near the sun during a solar eclipse, their position would be shifted by this very small amount compared to where one would otherwise predict them to lie. Fortunately for Einstein, war and other human idiocies prevented a successful eclipse expedition to test his ideas until three years after he had indeed made the correct prediction. In November 1919, two British expeditions reported on their observations of a May 1919 eclipse: Einstein, not Newton, was correct.
This discovery forever changed Einstein’s life and, with it, the world of physics. News of the eclipse observations spread across the headlines of papers throughout the world, and within weeks, Einstein attained a celebrity that would remain with him for the rest of his life. Special relativity had made him famous among physicists and perhaps even among educated intellectuals; general relativity made him a household name. His discovery that we are living in a possibly curved three-dimensional space had an immediate popular impact that might be akin to the revelation in Renaissance Europe that the earth wasn’t flat. In a single moment, everything changed, and Einstein’s fame would soon rival that of Columbus.
Part of the reason for his fame was surely the fact that he had now supplanted Newton as the father of gravity. But I think the general excitement that greeted his discovery was more deeply based, and for good reason. While special relativity had connected space and time in a new way that made separate measurements of length and time observer dependent, space-time itself nevertheless remained a fixed background in which the events of the universe played out. In Einstein’s general relativity, however, space and time become truly dynamic quantities. They are no longer mere backdrops in which the drama of life ensues, but respond to the presence of matter and energy, bending, contracting, or even expanding in the presence of appropriate forms of matter or energy. One of the predictions of general relativity that took almost half a century to verify empirically was that clocks tick more slowly in a gravitational field. Normally the effect is truly minuscule, and to measure it required careful optical techniques, unstable radioactive compounds, and ultimately the use of atomic clocks.
However, sometimes, if we take into account the fact that we live in a large universe, small effects can be magnified tremendously. One of my favorite examples of this (for reasons that will become obvious in a moment) involves some work a colleague of mine and I did shortly after the discovery, on February 23, 1987, of an exploding star on the outskirts of our galaxy, the first such event seen in almost four hundred years. Its demise was observed both via the light emitted by the star, which shined with a brightness approaching that of a billion stars for days, and also via the almost simultaneous detection of ghostlike elementary particles called neutrinos, which are in fact the dominant form of radiation emitted by exploding stars. Within a few weeks of the event, there were literally scores of scientific papers (including some by me) analyzing every aspect of these signals.
About two months after this flurry, Scott Tremaine (who is now at Princeton, but at the time was at the Canadian Institute for Theoretical Astrophysics in Toronto) and I were at a meeting in Halifax, Canada, when we suddenly realized that one could calculate the extra time it would have taken for both the light and the neutrinos to travel from the distant star to Earth, due to the fact that both bursts were traveling in the gravitational field of our galaxy and hence not in a flat background. The result surprised both of us: The gravitational time delay was about six months. If it hadn’t been for the warping of both space and time as predicted in general relativity, Supernova 1987a, as it became known, would have been called Supernova 1986d, as it would have been observed sometime around the middle of the previous year!
It is virtually impossible for us, who are confined to live within a curved three-dimensional space, to physically picture what such a curvature implies. We can intuitively grasp a curved two-dimensional object, such as the surface of the earth, because we can embed it in a threedimensional background for viewing. But the possibility that a curved space can exist in any number of dimensions without being embedded in a higher-dimensional space is so foreign to our intuition that I am frequently asked, “If space is curved, what is it curving into?” There are, however, mathematical ways to define the geometry of a space without the existence of extrinsic quantities. The simplest example involves something with which we are all familiar. Consider a triangle drawn on this piece of paper.
As any European high school student could tell you, the sum of the angles inside this triangle is 180 degrees, independent of the shape or size of the triangle.
Now, however, consider the following figure:
All three angles of this triangle
are right angles, adding up to a sum of 270 degrees. Were we intelligent ants living on this curved surface, even if we could never circumnavigate it or view it from above, by drawing a large enough triangle and measuring the sum of its internal angles, we could nevertheless infer that we were living on a spherical surface. Another factor distinguishes a sphere from a flat piece of paper, which I alluded to earlier. Lines of longitude, extending from the North to the South Pole, are all parallel lines, yet all of these lines meet at both poles:
As obvious as all this might seem in retrospect, the notion that it might be possible to have a geometry where Euclid’s axioms about parallel lines or about the sum of angles in a triangle might not hold caused a revolution in philosophy. Euclid’s axioms had remained unchallenged for two thousand years when the mathematicians Gauss, Lobachevsky, and Bolyai independently discovered between the years 1824 and 1832 that one could build a consistent mathematical framework in which the axiom about parallel lines could be violated. So great was the resistance to these notions that the famous physicist Helmholtz felt it necessary to incorporate precisely the examples I have given here in his 1881 Popular Lectures on Scientific Subjects (published three years before Flatland ), in which he described a hypothetical world of two-dimensional beings living on the surface of a sphere, in order to convince people that the abstract mathematical notions of Gauss and others could be manifested in a consistent physical reality. Interestingly enough, both Gauss and Lobachevsky realized that if non-Euclidean geometry was possible in principle, it might also be possible in practice, and both conducted independent experiments to see if our three-dimensional space might be curved. Gauss was more modest in his attempts, merely measuring the sum of the angles in a large triangle formed by three distant mountain peaks. Lobachevsky, in contrast, performed a far more modern experiment. He observed the parallax of various distant stars, that is, the angle by which they shift compared to background objects when the earth is on one side of the sun, compared to when the earth is on the other side of the sun a half-year later. Plane geometry gives a straightforward prediction for what this shift should be for stars at a fixed distance, at least in a flat space. The shift would be different, however, if space was curved. Given the limited sensitivity of their observations, neither Gauss nor Lobachevsky was able to obtain any evidence whatsoever for the nonEuclidean nature of space. That evidence would have to wait for almost a century, until after Einstein had made it clear what to look for. While the British solar eclipse expeditions were able to detect the curvature of space in the vicinity of the sun, general relativity posed a much, much bigger challenge. This was a theory not merely of how objects might move throughout space and time, but of how space and time themselves might evolve. Einstein opened up the possibility of describing the dynamics of the universe itself, and since general relativity is a geometric theory, the central question of twentieth-century cosmology soon became: Is the geometry of the universe, on its largest scales, described by Euclid?
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 6
