by Lee Smolin
So here we have again a story of two competing attempts at unification. The mechanists had a beautiful idea that unified physics: Everything is matter. Einstein believed in another kind of unification, the unification of motion and rest. To support this, he had to invent a still deeper unification—of space and time. In each case, something previously thought to be absolutely distinct becomes distinct only relative to the motion of the observer.
In the end, the conflict between the two proposals for unification was settled by experiment. If you believed the mechanists, you believed that an observer could measure his speed through the aether. If you believed Einstein, you knew that he couldn’t, as all observers are equivalent.
Several attempts had been made to detect Earth’s motion through the aether before 1905, when Einstein proposed special relativity, and they had failed.1 Proponents of the aether theory had just adjusted their predictions so as to make it harder and harder to detect Earth’s motion. This was easy to do, because when they did calculations they used Maxwell’s theory, which, when correctly interpreted, agreed with Einstein’s expectations that motion was not detectable. That is, the mechanists already had the right equations, they just had the wrong interpretations.
As for Einstein himself, it’s not clear how much he knew about the early experiments, but they wouldn’t have mattered to him, as he was already convinced that the motion of the earth was not detectable. Einstein was in fact only getting started. As we shall see in the next chapter, his unification of space and time was about to deepen considerably. By the time most physicists had caught up with him and accepted the special theory of relativity, Einstein was already moving far beyond it.
3
The World As Geometry
THE EARLY DECADES of the twentieth century saw several attempts at unification. A few succeeded, the rest failed. By briefly telling their stories, we can draw lessons that will help us understand the crisis facing the current attempts at unification.
From Newton to Einstein, a single idea dominated: The world is made of nothing but matter. Even electricity and magnetism were aspects of matter—just stresses in the aether. But this beautiful picture was crushed when special relativity triumphed, for if the whole notion of being at rest or in motion is meaningless, the aether must be a fiction.
The quest for unification had to go somewhere, and there was really only one place to go. This was to reverse the aether theory: If fields are not made from matter, perhaps fields are the fundamental stuff. Matter must then be made from fields. There were already models of electrons and atoms as stresses in the fields, so this was not such a big step.
But even as this idea gained adherents, there were still mysteries. For example, there are two different kinds of fields, the gravitational field and the electromagnetic field. Why two fields and not a single field? Is this the end of the story? The yearning for unification compelled physicists to ask whether gravity and electromagnetism were aspects of a single phenomenon. Thus was born the search for what we now call a unified-field theory.
Since Einstein had just incorporated electromagnetism into his special theory of relativity, the most logical way to proceed was to modify Newton’s theory of gravity so as to make it consistent with relativity theory. This turned out to be easy to do. Not only that, this modification led to a wonderful discovery that would become the core of unified theories to this day. In 1914, a Finnish physicist named Gunnar Nordström found that all you had to do to unify gravity with electromagnetism was increase the dimensions of space by one. He wrote the equations that describe electromagnetism in a world with four dimensions of space (and one of time), and out popped gravity. Just by the extra dimension of space, you got a unification of gravity with electromagnetism that was also perfectly consistent with Einstein’s special theory of relativity.
But if this is true, shouldn’t we be able to look out in this new dimension, as we look out in the three dimensions of space? If not, isn’t this theory then obviously wrong? To avoid this troublesome issue, we can make the new dimension a circle, so that when we look out, we in effect travel around it and come back to the same place.1 Then we can make the diameter of the circle very small, so that it is hard to see that the extra dimension is there at all. To understand how shrinking something can make it impossible to see, recall that light is made up of waves and each light wave has a wavelength, which is the distance between peaks. The wavelength of a light wave limits how small a thing you can see, for you cannot resolve an object smaller than the wavelength of the light you use to see it. Hence, one cannot detect the existence of an extra dimension smaller than the wavelength of light one can perceive.
One might think that Einstein, of all people, would have embraced this new theory. But by that time (1914), he was already traveling down a different road. Unlike his contemporaries, Einstein had taken a route to the unification of gravity with relativity that brought him back to the very foundation of the principle of relativity: the unification of motion and rest discovered by Galileo several centuries earlier. That unification involved only uniform motion—that is, motion in a straight line at a constant speed. Beginning around 1907, Einstein started to ask himself about other types of motion, such as accelerated motion. This is motion whose speed or direction changes. Shouldn’t the distinction between accelerating and nonaccelerating motion somehow be erased?
At first this seems a misstep, for while we can’t feel the effects of uniform motion, we certainly do feel the effects of acceleration. When an airplane takes off, we feel pushed back into our seats. When an elevator begins to rise, we feel the acceleration in the form of additional pressure pushing us into the floor.
It was at this point that Einstein had his most extraordinary insight. He realized that the effects of acceleration were indistinguishable from the effects of gravity. Think of a woman standing in an elevator waiting for it to move. She already feels a force pulling her to the floor. What happens when the elevator starts to ascend is not different in kind, only in degree: She feels the same force increase. Suppose that the elevator stays still but the strength of gravity increases momentarily? Einstein realized that she would feel exactly the same as if the elevator had accelerated upward.
There is a converse to this. Suppose that the cable holding the elevator is cut and the car, with its occupants, begins to fall. In free fall, the occupants of the elevator will feel weightless. They will feel exactly as astronauts do in orbit. That is to say, the acceleration of the falling elevator can exactly cancel the effects of gravity.
Einstein recalled realizing that a person falling from the roof of a building would feel no effects of gravity as he fell. He called this “the most fortunate thought of my life,” and he made it into a principle, which he called the principle of equivalence. It says that the effects of acceleration are indistinguishable from the effects of gravity.2
So Einstein succeeded in unifying all kinds of motion. Uniform motion is indistinguishable from rest. And acceleration is no different from being at rest but with a gravitational field turned on.
The unification of acceleration with gravity was a unification with great consequences. Even before the conceptual implications were worked out, there were huge implications for experiment. Some predictions could even be derived with high school algebra—for example, that clocks would slow down in a gravitational field, which was eventually confirmed. Another prediction—first made by Einstein, in 1911—was that light is bent when it passes through a gravitational field.
Notice here that, as in the successful unifications discussed earlier, more than one unification is happening at once. Two different kinds of motion are being unified; there is no longer a need to distinguish uniform from accelerated motion. And the effects of acceleration are being unified with the effects of gravity.
Even if Einstein could reason from the equivalence principle to a few predictions, the new principle was not a complete theory. The formulation of a complete theory was the greatest challeng
e of his life and took nearly a decade to accomplish. To see why, let us try to understand what it means to say that gravity bends light rays. Before this particular insight of Einstein’s, there had always been two different kinds of things in the world: the things that live in space and space itself.
We are not accustomed to thinking of space as an entity with properties of its own, but it certainly is. Space has three dimensions, and it also has a particular geometry, which we learn in school. Called Euclidean geometry—after Euclid, who worked out its postulates and axioms more than two thousand years ago—it is the study of the properties of space itself. The theorems of Euclidean geometry tell us what happens to triangles, circles, and lines drawn in space. But they hold for all objects, material or imagined.
A consequence of Maxwell’s theory of electromagnetism is that light rays move in straight lines. Thus it makes sense to use light rays when tracing the geometry of space. But if we adopt this idea, we see immediately that Einstein’s theory has great implications. For light rays are bent by gravitational fields, which, in turn, respond to the presence of matter. The only conclusion to draw is that the presence of matter affects the geometry of space.
In Euclidean geometry, if two straight lines are initially parallel, they can never meet. But two light rays that are initially parallel can meet in the real world, because if they pass on each side of a star, they will be bent toward each other. So Euclidean geometry is not true in the real world. Moreover, the geometry is constantly changing, because matter is constantly moving. The geometry of space is not like a flat, infinite plane. It is like the surface of the ocean—incredibly dynamic, with great waves and small ripples in it.
Thus, the geometry of space was revealed to be just another field. Indeed, the geometry of space is almost the same as the gravitational field. To explain why, we have to recall the partial unification of space and time that Einstein achieved in special relativity. In this unification, space and time together make up a four-dimensional entity called spacetime. This has a geometry analogous to Euclidean geometry, in the following precise way.
Consider a straight line in space. Two particles can travel along it, but one travels at a uniform speed, while the other is constantly accelerating. As far as space is concerned, the two particles travel on the same path. But they travel on different paths in spacetime. The particle with a constant speed travels on a straight line, not only in space but also in spacetime. The accelerating particle travels on a curved path in spacetime (see Fig. 3).
Fig. 3. A car decelerating along a straight line in space travels on a curved path in spacetime.
Hence, just as the geometry of space can distinguish a straight line from a curved path, the geometry of spacetime can distinguish a particle moving at a constant speed from one that is accelerating.
But Einstein’s equivalence principle tells us that the effects of gravity cannot be distinguished, over small distances, from the effects of acceleration.3 Hence, by telling which trajectories are accelerated and which are not, the geometry of spacetime describes the effects of gravity. The geometry of spacetime is therefore the gravitational field.
Thus, the double unification given by the equivalence principle becomes a triple unification: All motions are equivalent once the effects of gravity are taken into account, gravity is indistinguishable from acceleration, and the gravitational field is unified with the geometry of space and time. When worked out in detail, this became Einstein’s general theory of relativity, which he published in full form in 1915.
Not bad for a guy who couldn’t initially get an academic job.
Thus, by 1916 there were two very different proposals for the future of physics, both based on a deep idea about unifying gravity with the rest of physics. There was Nordström’s elegant unification of gravity with electromagnetism by the simple postulation of an extra, hidden dimension of space. And there was Einstein’s general theory of relativity. Both seemed consistent theories and each did something unexpectedly elegant.
They could not both be true, so a choice had to be made. Fortunately, the two theories made different predictions for a doable experiment. Einstein’s general theory of relativity predicted that gravity must bend light rays—and by precisely how much. In Nordström’s theory, there was no such effect: Light always goes in straight lines, period.
In 1919, the great British astrophysicist Arthur Eddington led an expedition off the west coast of Africa to conduct an experiment, which ended by confirming that the gravitational field of the sun indeed bends light. This effect was observed during a total solar eclipse, which made it possible to see, near the rim of the occluded sun, the light from stars that were actually directly behind the sun. Had the sun’s gravity not bent their light, these stars would not have been visible. But they were. So the choice between the two profoundly different directions for unification was made in the only way it could have been—by experiment.
This is an important example, because it shows the limits of what can be accomplished by thought alone. Some physicists have argued that general relativity is a case in which pure thought sufficed to show the way forward. But the real story is the opposite. Without experiment, most theorists would probably have chosen Nordström’s unification; it is simpler and brought with it the powerful new idea of unification through extra dimensions.
Einstein’s unification of the gravitational field with the geometry of spacetime signaled a profound transformation in how we conceive of nature. Before Einstein, space and time were thought to have properties that were fixed for all eternity: The geometry of space is, was, and always would be as Euclid described. Time marched on independently of anything else. Things moved in space and evolved over time, but space and time themselves never altered.
For Newton, space and time constituted an absolute background. They provided a fixed stage on which a grand drama is played out. The geometry of space and time was needed to give meaning to the things that change, like the positions and motions of particles. But they themselves never changed. We have a name for theories of physics that rely on such an absolute, fixed framework: We call them background-dependent theories.
Einstein’s general theory of relativity is completely different. There is no fixed background. The geometry of space and time changes and evolves, as does everything else in nature. Different geometries of spacetime describe the histories of different universes. We no longer have fields moving in a fixed-background geometry. We have a bunch of fields all interacting with one another, all dynamical, all influencing one another, one of which is the geometry of spacetime. We call such a theory a background-independent theory.
Make note of the distinction between background-dependent and background-independent theories. The story that unfolds over the course of this book turns on the difference between them.
Einstein’s general theory of relativity satisfied all the tests we laid out in the last chapter for a successful unification. There were profound conceptual consequences, which were implied by the unifications involved. These quickly led to predictions of new phenomena, such as the expanding universe, the Big Bang, gravitational waves, and black holes, and there is good evidence for all of them. Our whole notion of cosmology was turned on its head. Proposals that once seemed radical, like the bending of light by matter, are now used as tools to trace the distribution of matter in the universe. And every time the predictions of the theory are tested in detail, they are beautifully borne out.4
But general relativity was just the start. Even before Einstein had published the final version of the theory, he and others were formulating new kinds of unified theories. They had in common a simple idea: If the gravitational force could be understood as a manifestation of the geometry of space, why couldn’t this also be true of electromagnetism? In 1915, Einstein had written to David Hilbert, perhaps the greatest mathematician then living, “I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism.”5
But it
took until 1918 for a really good idea about this particular unification to emerge. This theory, invented by the mathematician Herman Weyl, contained a beautiful mathematical idea that was to become the core of the standard model of particle physics. Yet it failed, because in Weyl’s original version it had big consequences that disagreed with experiment. One was that the length of an object would depend on the path it took. If you took two meter sticks, separated them, and then brought them back together and compared them, they would in general be different in length. This is more radical then special relativity, which holds that meter sticks can indeed appear to have different lengths, but only when they are moving relative to each other, not when they are compared at rest. It also, of course, disagrees with our experience of nature.
Einstein didn’t believe Weyl’s theory, but he admired it, writing to Weyl, “Apart from the [lack of] agreement with reality it is in any case a superb intellectual performance.”6 Weyl’s reply shows the power of mathematical beauty: “Your rejection of the theory for me is weighty, . . . But my own brain still keeps believing in it.”7
The tension between those caught in the allure of a beautiful theory they invented and more sober minds insisting on a connection to reality is a story we will see repeated in later attempts at unification. There is no easy resolution in these cases, because a theory can be fantastically beautiful, fruitful for the development of science, and yet at the same time completely wrong.
