The Cosmic Landscape (11 page)

Surrounding the space were about twenty offices. The entire school was housed in this one-time ballroom. It would have been very depressing except that several people were having a lively physics conversation at one of the boards. What’s more, I recognized some of them. I saw Dave Finkelstein, who had arranged my new job. Dave was a charismatic and brilliant theoretical physicist who had just written a paper on the use of topology in quantum field theory that was to become a classic of theoretical physics. I also saw P. A. M. Dirac, arguably the greatest theoretical physicist of the twentieth century after Einstein. Dave introduced me to Yakir Aharonov, whose discovery of the Aharonov-Bohm effect had made him famous. He was talking to Roger Penrose, who is now Sir Roger. Roger and Dave were two of the most important pioneers in the theory of black holes. I saw an open door with a sign that said Joel Lebowitz. Joel, a very well-known mathematical physicist, was arguing with Elliot Lieb, whose name was also familiar. It was the most brilliant collection of physicists that I had ever seen assembled in one place.

They were talking about vacuum energy. Dave was arguing that the vacuum was full of zero-point energy and that this energy ought to affect the gravitational field. Dirac didn’t like vacuum energy because whenever physicists tried to calculate its magnitude, the answer would come out infinite. He thought that if it came out infinite, the mathematics must be wrong and that the right answer is that there is no vacuum energy. Dave pulled me into the conversation, explaining as he went. For me this conversation was a fateful turning point—my introduction to a problem that would obsess me for almost forty years and that eventually led to
The Cosmic Landscape.

The part of the mind—I guess we call it the ego—that gets pleasure from being proved right is especially well developed in theoretical physicists. To make a theory of some phenomenon followed by a clever calculation and then finally to have the result confirmed by an experiment provides a tremendous source of satisfaction. In some instances the experiment takes place before the calculation, in which case it’s not predicting but, rather, explaining a result, and it’s almost as good. Even very good physicists now and then make wrong predictions. We tend to forget about them, but one wrong prediction just will not go away. It is by far the all-time worst calculation of a numerical result that any physicist ever made. It was not the work of any one person, and it was so wrong that no experiment was ever needed to prove it so. The problem is that the wrong result seems to be an inevitable consequence of our best theory of nature, quantum field theory.

Before I tell you what the quantity is, let me tell you how wrong the prediction is. If the result of a calculation disagrees with an experiment by being 10 times too large or too small, we say that it was off by one order of magnitude; if wrong by a factor of 100, then it’s two orders of magnitude off; a factor of 1,000, three orders; and so on. Being wrong by one order of magnitude is bad; two orders, a disaster; three, a disgrace. Well, the best efforts of the best physicists, using our best theories, predict Einstein’s cosmological constant incorrectly by 120 orders of magnitude! That’s so bad that it’s funny.

Einstein was the first to get burned by the cosmological constant. In 1917, one year after the completion of the General Theory of Relativity, Einstein wrote a paper that he later regretted as his worst mistake. The paper, titled “Cosmological Considerations on the General Theory of Relativity,” was written a few years before astronomers understood that the faint smudges of light called nebulae were actually distant galaxies. It was still twelve years till the American astronomer Edwin Hubble would revolutionize astronomy and cosmology, demonstrating that the galaxies are all receding away from us with a velocity that grows with distance. Einstein didn’t know in 1917 that the universe was expanding. As far as he or anyone else knew, the galaxies were stationary, occupying the same location for all eternity.

According to Einstein’s theory the universe is
closed and bounded,
which first of all means that space is finite in extent. But it doesn’t mean that it has an edge. The surface of the earth is an example of a closed-and-bounded space. No point on earth is more than twelve thousand miles from any other point. Moreover, there is no edge to the earth: no place that represents the boundary of the world. A sheet of paper is finite, but it has an edge: some people would say four edges. But on the earth’s surface, if you keep going in any direction, you never come to the end of space. Like Magellan you will eventually come back to the same place.
¹

We often say that the earth is a sphere, but to be precise, the term
sphere
refers only to the surface. The correct mathematical term for the solid earth is a ball. To understand the analogy between the surface of the earth and the universe of Einstein, you must learn to think
only
of the surface and not the solid ball. Let’s imagine creatures—call them flatbugs—that inhabit the surface of a sphere. Assume that they can never leave that surface: they can’t fly, and they can’t dig. Let’s also assume that the only signals they can receive or emit travel along the surface. For example, they might communicate with their environment by emitting and detecting surface waves of some kind. These creatures would have no concept of the third dimension and no use for it. They would truly inhabit a two-dimensional closed-and-bounded world. A mathematician would call it a
2-sphere,
because it is two-dimensional.

We are not flatbugs living in a two-dimensional world. But according to Einstein’s theory, we live in a three-dimensional analog of a sphere. A three-dimensional closed-and-bounded space is more difficult to picture, but it makes perfect sense. The mathematical term for such a space is a
3-sphere.
Just like the flatbugs, we would discover that we live in a 3-sphere by traveling out along any direction and eventually finding that we always return to the starting point. According to Einstein’s theory, space is a 3-sphere.

In fact spheres come in every dimension. An ordinary circle is the simplest example. A circle is one-dimensional like a line: if you lived on it, you could move only along a single direction. Another name for a circle is a
1-sphere.
Moving along the circle is much like moving along a line except that you come back to the same place after a while. To define a circle, start with a two-dimensional plane and draw a closed curve. If every point on the curve is the same distance from a central point (the center), the curve is a circle. Notice that we began with a two-dimensional plane in order to define the 1-sphere.

The 2-sphere is similar except that you begin with three-dimensional space. A surface is a 2-sphere if every point is the same distance from the center. Perhaps you can see how to generalize this to a 3-sphere or, for that matter, a sphere of any dimension. For the 3-sphere we begin with a four-dimensional space. You can think of it as a space described by four coordinates instead of the usual three. Now just pick out all the points that are at a common distance from the origin. Those points all lie on a 3-sphere.

Just as the flatbugs living on the two-sphere had no interest in anything but the surface of the sphere, the geometer studying a 3-sphere has no interest in the 4-dimensional space in which it is embedded. We can throw it away and concentrate only on the 3-sphere.

Einstein’s cosmology involved a space that has the overall shape of a 3-sphere, but like the earth’s surface, the spherical shape is not perfect. In the General Theory of Relativity, the properties of space are not rigidly fixed. Space is more like the deformable surface of a rubber balloon than the surface of a rigid steel ball. Picture the universe as the surface of such a giant, deformable balloon. Flatbugs live on the rubber surface, and the only signals they receive propagate along that surface. They know nothing of the other dimension of space. They have no concept of the interior or exterior of the balloon. But now their space is flexible, and the distance between points can change with time as the rubber stretches.

On the balloon are markings indicating the galaxies, which more or less cover the balloon uniformly. If the balloon expands, the galaxies move apart. If it shrinks, the galaxies move closer. All of this is fairly easy to understand. The hard part is the jump from two to three dimensions. Einstein’s theory describes a world in which space is flexible and stretchable but has the overall shape of a 3-sphere.

Now let’s add the element of gravitational attraction. According to both Newton’s and Einstein’s theories of gravity, every object in the universe attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. Unlike electric forces, which are sometimes attractive and sometimes repulsive, gravity is
always
attractive. The effect of gravitational attraction is to pull the galaxies together and shrink the universe. A similar effect exists on the surface of a real balloon, namely, the tension in the rubber that tries to shrink the balloon. If you want to see the effect of tension, just stick a pin in the rubber.

Unless some other force counteracts gravitational attraction, the galaxies should start to accelerate toward one another, collapsing the universe like a punctured balloon. But in 1917 the universe was thought to be static—unchanging. Astronomers, like ordinary people, looked at the sky and saw no motion of the distant stars (apart from that due to the earth’s motion). Einstein knew that a static universe was impossible if gravity was universally attractive. A static universe is like a stone, hovering above the surface of the earth, completely motionless. If the stone were thrown vertically upward, then a momentary glance might see it ascending or descending. You might even catch it at the precise instant when it was turning around. What the stone cannot do is to just eternally hover at a fixed height. That is, not unless some other force is acting on the stone opposing the gravitational attraction to the earth. In exactly the same way, a static universe defies the universal law of gravitational attraction.

What Einstein needed was a modification of his theory that would provide a compensating force. In the case of the balloon, the air pressure from inside is the force that counteracts the tension in the rubber. But the real universe doesn’t have an inside with air in it. There is only the surface. So Einstein reasoned that there must be some kind of repulsive force to counteract the gravitational pull. Could there be a hidden possibility of a repulsive force in the General Theory of Relativity?

Examining his equations, Einstein discovered an ambiguity. The equations could be modified without destroying their mathematical consistency by adding one more term. The meaning of the additional term was surprising: it represented an addition to the usual laws of gravity—a gravitational force that became increasingly strong with distance. The strength of this new force was proportional to a new constant of nature that Einstein denoted by the Greek letter λ (lambda). Ever since, the new constant has been called the cosmological constant, and it continues to be denoted by λ.

What had especially caught Einstein’s attention was that if λ were chosen to be a positive number, then the new term corresponded to a universal repulsion that increased in proportion to the distance. Einstein realized he could play off the new repulsive force against the usual gravitational attraction. The galaxies could be kept in equilibrium at a separation that could be controlled by choosing the magnitude of the new constant, λ. The way that it worked was simple. If the galaxies were closely spaced, their attraction would be strong and an equally strong repulsion would be needed to keep them in equilibrium. On the other hand, if the distance between galaxies were so large that they barely felt each other’s gravitational fields, then only a weak repulsion would be needed. Thus Einstein argued that the size of the cosmological constant should be closely connected to the average distance between the galaxies. Although from a mathematical perspective the cosmological constant could be anything, it could be easily determined if one knew the average distance separating galaxies. In fact at that time, Hubble was busy measuring the distance between galaxies. Einstein believed that he had the secret of the universe. It was a world kept in balance by competing attractive and repulsive forces.

Many things are wrong with this theory. From the theoretical point of view, the universe that Einstein had built was unstable. It was in equilibrium but
unstable equilibrium.
The difference between stable and unstable equilibrium is not hard to understand. Think of a pendulum. When the pendulum is vertical and the bob is at its low point, the pendulum is in stable equilibrium. This means that if you disturb it a little, for example, by giving it a slight push, it will return to its original position.

Now imagine turning the pendulum upside down so that the bob is delicately balanced in the straight-up position. If it is disturbed ever so slightly, perhaps by nothing more than the breeze from a butterfly’s wing, the disturbance will build up, and the pendulum will fall over. Moreover, the direction in which it falls is very unpredictable. Einstein’s static universe was like the unstable upside-down pendulum. The slightest perturbation would either cause it to explosively grow or implode it like a popped balloon. I don’t know whether Einstein missed this elementary point or if he just decided to ignore it.