by Lee Smolin
Each planetary orbit has a radius. Each planet also has an orbital speed. In addition, the speeds are not uniform; the planets speed up and slow down as they move around the sun on their orbits. All of these numbers seem arbitrary. Kepler had been seeking his whole life for a principle that would unify the motions of the planets and, by doing so, explain the data of the planetary orbits.
Kepler’s first try at a unification of the planets was in line with an ancient tradition that cosmological theory must employ only the simplest figures. One reason the Greeks had believed in circles moving on circles is that the circle is the simplest and hence, to them, the most beautiful of closed curves. Kepler searched for equally beautiful geometrical figures that might explain the sizes of the orbits of the planets. And he had a very elegant idea, illustrated in Fig. 1.
Fig. 1. Kepler’s first theory of the solar system, based on the Platonic solids.
Let us take the orbit of Earth as given. There are, then, five numbers to explain: the ratio of the diameters of the orbits of the other five planets to the diameter of Earth’s orbit. If they are to be explained, there must be some beautiful geometric construction that yields exactly five numbers. No more and no less. So is there a problem in geometry to which there are exactly five answers?
Yes. The cube is a perfect kind of solid, for each side is the same as every other side, and each edge is the same length as all the other edges. Such solids are called Platonic solids. How many are there? Exactly five: besides the cube, there is the tetrahedron, the octahedron, the dodecahedron, and the icosahedron.
It didn’t take Kepler long to make an amazing discovery. Embed the orbit of Earth in a sphere. Fit a dodecahedron around the sphere. Put a sphere over that. The orbit of Mars fits on that sphere. Put the tetrahedron around that sphere, and another sphere around the tetrahedron. Jupiter fits on that sphere. Around Jupiter’s orbit is the cube, with Saturn beyond. Inside Earth’s orbit, Kepler placed the icosahedron, about which Venus orbited, and within Venus’s orbit was the octahedron, for Mercury.
This unified theory explained the diameters of the orbits of the planets, something no theory had done before. It was mathematically beautiful. So why wasn’t it believed? As compelling as it was, it didn’t lead anywhere. No new phenomena were predicted on its basis. It didn’t even lead to an understanding of the planets’ orbital speeds. The idea was too static; it unified, but it didn’t take science anywhere interesting.
Kepler thought about this for a long time. Since the diameters of the orbits were explained, he just needed to explain the speeds of the different planets. Finally he proposed that as the planets travel they “sing,” and the frequencies of the notes are proportional to their speeds. The pitches sung by the different planets as they travel in their orbits make a harmony in six voices, which he called the harmony of the spheres.
This idea also has ancient roots, harking back to Pythagoras’s discovery that the roots of musical harmony are in ratios of numbers. But it suffers from clear problems. It is not unique: There are many beautiful harmonizations of six voices. Even worse, there turned out to be more than six planets. And Galileo, a contemporary of Kepler, discovered four moons orbiting Jupiter. So there was yet another system of orbits in the sky. If Kepler’s theories were right, they should apply to the newly discovered system. But they didn’t.
Apart from those two proposals for the mathematical structure of the cosmos, Kepler made three discoveries that did lead to real progress in science. These were the three laws he is now famous for, proposed after years spent painstakingly analyzing the data he stole from Tycho. They have none of the beauty of his other proposals, but they do work. Moreover, one of them accomplishes something he could do no other way, which is to find a relationship between the speeds and the diameters of the orbits. Kepler’s three laws not only agreed with the data on all six planets, they agreed with observations of Jupiter’s moons.
Kepler discovered these laws because he took Copernicus’s unification to its logical conclusion. Copernicus had said that the sun was at (or actually, near) the center of the universe, but in his theory the planets would move the same way whether the sun was there or not. Its only role was to light up the scene. The success of Copernicus’s theory led Kepler to ask whether the sun’s being near the center of each planet’s orbit could really be a coincidence. He wondered whether the sun might instead play some role in causing the planetary orbits. Might the sun in some way exert a force on the planets, and might that force be the explanation for their motion?
To answer these questions, Kepler had to find a role for the precise position of the sun in each orbit. His first big breakthrough was to discover that the orbits were not circles, they were ellipses. And the sun had an exact role: It was exactly at the focus of the ellipse of each orbit. This was his first law. Shortly after this, he discovered his second law, which was that the speed of a planet in its orbit increased or decreased as it moved closer to or farther from the sun. He later discovered his third law, which governed how the speeds of the planets were related.
These laws point to some deep fact unifying the solar system, because the laws apply to all the planets. The payoff is that for the first time we had a theory that could make predictions. Suppose a new planet is discovered. Can we predict what its orbit will be? Before Kepler, no one could. But given Kepler’s laws, all we need is two observations of its position and we can predict its orbit.
These discoveries paved the way for Newton. It was Newton’s great insight to see that the force the sun exerted on the planets is the same as the force of gravity that holds us on Earth, and hence to unify physics in the heavens with physics on Earth.
Of course, the idea of a force emanating from the sun to the planets was absurd to most scientists at the time. They believed that space was empty; there was no medium that could convey such a force. Furthermore, there was no visible manifestation of it—no arm reaching out from the sun to each planet—and nothing invisible could be real.
There are good lessons here for would-be unifiers. One is that mathematical beauty can be misleading. Simple observations made from the data are often more important. Another lesson is that correct unifications have consequences for phenomena unsuspected at the time a unification is invented, as in the case of the application of Kepler’s laws to Jupiter’s moons. Correct unifications also raise questions that may seem absurd at the time but lead to further unifications, as in Kepler’s postulation of a force from the sun to the planets.
Most important, we see that a real revolution often requires that several new proposals for unification come together to support one another. In the Newtonian revolution, there were several proposed unifications that triumphed at once: the unification of the earth with the planets, the unification of the sun with the stars, the unification of rest and uniform motion, and the unification of the gravitational force on Earth with the force by which the sun influences a planet’s motions. Singly, none of these ideas could have survived; together, they trounced their rivals. The result was a revolution that transformed every aspect of our understanding of nature.
In the history of physics, there is one unification that serves more than any other as a model for what physicists have been trying to do in the last thirty years. This is the unification of electricity and magnetism, achieved by James Clerk Maxwell in the 1860s. Maxwell made use of a powerful idea called a field, which had been invented by the British physicist Michael Faraday in the 1840s to explain how a force could be conveyed through empty space from one body to another. The idea is that a field is a quantity, like a number, one of which lives at each point in space. As you move through space, the value of the field changes continuously. The value of the field at a single point also evolves in time. The theory gives us laws that tell us how the field changes as you move in space and through time. These laws tell us that the value of the field at a particular point is influenced by the value of the field at nearby points. The field at a point can also be infl
uenced by a material body at the same point. Thus, a field can carry a force from one body to another. There is no need to believe in ghostly action at a distance.
One field Faraday studied was the electric field. This is not a number but a vector, which we may visualize as an arrow and which can vary its direction and length. Imagine such an arrow at each point of space. Imagine that the ends of the arrows at nearby points are attached to one another by rubber bands. If I pull on one, it pulls on the ones nearby. The arrows are also influenced by electric charges. The effect of the influence is that the arrows will arrange themselves so that they point to nearby negative charges and away from nearby positive charges.
Faraday also studied magnetism. He invented another field, another collection of arrows, which he called the magnetic field; these arrows like to point to poles of magnets (see Fig. 2).
Fig. 2. Lines of force trace the magnetic field arising from a bar magnet.
Faraday wrote down simple laws to describe how the electric and magnetized field arrows are influenced by nearby charges and magnetic poles and also by the arrows of nearby fields. He and others tested the laws and found they gave predictions that agreed with experiment.
Among the discoveries of the time were phenomena that mixed electric and magnetic effects. For example, a charge moving in a circle gave rise to magnetic fields. Maxwell realized that these discoveries pointed to a unification of electricity and magnetism. To fully unify them, he had to change the equations. When he did so, simply by adding one term, his unification became a unification with consequences.
The new equations allowed electric and magnetic fields to turn into each other. These transmutations give rise to waves of shifting patterns, in which first there is an electric field and then a magnetic field, and which move through space. Such moving patterns could be made by, among other things, waving an electric charge back and forth. The ensuing waves could carry energy from one place to another.
The most amazing thing was that Maxwell could compute the speed of these waves from this theory, and he found that they were the same as the speed of light. Then it must have hit him. The waves passing through the electric and magnetic fields are light. Maxwell did not set out to make a theory of light, he set out to unify electricity and magnetism. But in doing so, he achieved something greater. This is an example of how a good unification will have unexpected consequences for both theory and experiment.
New predictions immediately followed. Maxwell realized that there should be electromagnetic waves at all frequencies, not just those of visible light, and this led to the discovery of radio, infrared light, ultraviolet light, and so on. This illustrates another historical lesson: When someone proposes the right new unification, the implications become obvious very quickly. Many of these phenomena were observed in the first years after Maxwell published his theory.
This raises a point that will become important when we discuss other proposals for unification. All unifications have consequences because they lead to phenomena that arise because the things that were unified can transform into one another. In the good cases, these new phenomena are soon observed—the inventors have every right to celebrate the unification. But we will see that in other cases the predicted phenomena are already in conflict with observation. In this unhappy event, the proponents have to either give up their theory or constrain it unnaturally so as to hide the consequences of the unification.
But even as it triumphed, Maxwell’s unification of electricity and magnetism faced one formidable obstacle. In the mid-nineteenth century, most physicists believed that physics was unified because everything was made of matter (and had to be, in order to satisfy Newton’s laws). For these “mechanists,” the idea of a field just waving in space was hard to swallow. Maxwell’s theory made no sense to them without some stuff whose bending and stretching would constitute the true reality behind the electric and magnetic fields. Something material must be quivering when a light wave travels from a flower to one’s eye.
Faraday and Maxwell were themselves mechanists, and they devoted a lot of time and trouble to addressing this problem. They were not alone; young gentlemen made good careers at renowned institutions by inventing elaborate constructions of the microscopic gears, pulleys, and belts that they posited underlay Maxwell’s equations. Prizes were given for those who could solve the convoluted equations that resulted.
There was one big and obvious manifestation of the problem, which is that light travels to us from the sun and stars, and outer space is empty of any matter. Were there any matter in space, it would retard the motion of the planets, which would thus have long since fallen into the sun. But how could electric and magnetic fields reside in a vacuum?
So the mechanists invented a new form of matter—the aether—and filled space with it. The aether had paradoxical properties: It had to be extremely dense and stiff, for light was to be essentially a sound wave through it. The huge ratio of the speed of light to that of sound had to be a consequence of the incredible density of the aether. At the same time, the aether had to offer absolutely no resistance to the passage of ordinary matter through it. This is harder to arrange than it looks. One can just say that the aether and ordinary matter don’t interact with each other—that is, that they exert no forces on each other. But then why should ordinary matter detect light—or electric or magnetic fields—if these are just stresses in the aether? No wonder professorships were given to those who cleverly worked it all out.
Could there have been a more beautiful unification than the aether theory? Not only were light, electricity, and magnetism unified, their unification was unified with matter.
However, while the aether theory was being developed, the physicists’ conception of matter was also changing. In the early nineteenth century, most physicists had thought of matter as continuous, but electrons were discovered late in the century, and the idea that matter is made of atoms was taken more seriously then—at least, by some physicists. But that raised another question: What were atoms and electrons in a world made of aether?
Picture field lines, like the lines of a magnetic field running from the north pole to the south pole of a magnet. The field lines can never end, unless they end on the pole of a magnet; this is one of Maxwell’s laws. But they can make closed circles, and those circles can tie themselves up in knots. So perhaps atoms are knots in magnetic field lines.
But as every sailor knows, there are different ways to tie a knot. Maybe that’s good, because there are different kinds of atoms. In 1867 Lord Kelvin proposed that the various atoms would correspond to different knots.
This may seem absurd, but recall that at the time we knew very little about atoms. We knew nothing about nuclei and had never heard of protons or neutrons. So this was not as crazy as it might seem.
At that time, we also knew very little about knots. No one knew how many ways there were to tie a knot or how to tell them apart. So, inspired by this idea, mathematicians began studying the problem of how to distinguish the various possible knots. This slowly turned into a whole field of mathematics called knot theory. It soon was proved that there are an infinite number of distinct ways to tie a knot, but it has taken a long time to learn how to tell them apart. Some progress was made in the 1980s, but there is still no known procedure for telling whether two complicated knots are the same or different.
Notice how a good idea of unification, even if it turns out to be wrong, can inspire new avenues of inquiry. We should keep in mind, though, that just because a unified theory is fruitful for mathematics does not mean that the physical theory is correct. Otherwise, the success of knot theory would require us to still believe that atoms are knots in a magnetic field.
There was a further problem: Maxwell’s theory appeared to contradict the principle of relativity from Newtonian physics. It turned out that by doing various experiments, including measuring the speed of light, observers studying an electromagnetic field could tell whether they were moving or not.
; Here is a conflict between two unifications, both central to Newton’s physics: the unification of everything as matter obeying Newton’s laws versus the unification of motion and rest. For many physicists, the answer was obvious: The idea of a material universe was more important than the perhaps accidental fact that it was hard to detect motion. But a few took the principle of relativity as more important. One of these was a young student studying in Zurich called Albert Einstein. He meditated on the puzzle for ten years, beginning at the age of 16, and finally, in 1905, realized that the resolution required a complete revision of our understanding of space and time.
Einstein solved the puzzle by playing the same great trick that Newton and Galileo had originally played to establish the relativity of motion. He realized that the distinction between electrical and magnetic effects depends on the motion of the observer. So Maxwell’s unification was deeper than even Maxwell had suspected. Not only were the electric and magnetic fields different aspects of a single phenomenon, but different observers would draw the distinction differently; that is, one observer might explain a particular phenomenon in terms of electricity, while another observer, moving relative to the first, would explain the same phenomenon in terms of magnetism. But the two would agree about what was happening. And so Einstein’s special theory of relativity was born, as a joining of Galileo’s unification of rest and motion with Maxwell’s unification of electricity and magnetism.
Much follows from this. One consequence is that light must have a universal speed, independent of the motion of the observer. Another is that there must be a unification of space and time. Previously, there had been a clear distinction: Time was universal, and everyone would agree on what it meant for two things to happen simultaneously. Einstein showed that observers in motion with respect to each other would disagree about whether two events at different places were happening at the same time or not. This unification was implicit in his 1905 paper titled “On the Electrodynamics of Moving Bodies,” and it was stated explicitly in 1907 by one of his teachers, Hermann Minkowski.
