Many Worlds in One: The Search for Other Universes

by Vilenkin, Alex

  THE FABRIC OF SPACE AND TIME

  Einstein created two theories of stunning beauty that forever changed our concepts of space, time, and gravitation. The first of the two, called the special theory of relativity, was published in 1905, when Einstein was twenty-six and by most standards could be regarded a failure. His fierce independence and his casual class attendance did not make him popular among the professors at Zurich Polytechnic, where he got his diploma. When the time came to apply for jobs, all his fellow graduates were appointed assistants at the Polytechnic, while Einstein failed to get any academic position. He thought himself lucky to have a job as a clerk at the patent office in Berne, which he got with the help of a former classmate. On the positive side, the patent office work was not without some interest and left plenty of time for Einstein’s research and other intellectual pursuits. He spent evenings with friends, smoking a pipe, reading Spinoza and Plato, and discussing his ideas about physics. He also played string quintets in the unlikely company of a lawyer, a bookbinder, a schoolteacher, and a prison guard. None of them suspected that their second violin had something profound to say about the nature of space and time.

  Einstein completed the special theory of relativity in less than six weeks of frenzied work. The theory shows that space and time intervals do not by themselves have absolute meaning, but rather depend on the state of motion of the observer who measures them. If two observers move relative to one another, then each of them will find that the other’s clock ticks more slowly than his own. Simultaneity is also relative. Events that are simultaneous for one observer will generally occur at different times for the other. We do not notice these effects in our everyday life because they are completely negligible at ordinary velocities. But if the speed of the two observers relative to each other is close to the speed of light, the discrepancies between their measurements can be very large. There is one thing, though, that all observers will agree upon: light always travels at the same speed, approximately 300,000 kilometers per second.

  The speed of light is the absolute speed limit in the universe. As you apply a force to a physical object, the object accelerates. Its velocity grows, and if you keep up the force, the velocity of the object will eventually approach the speed of light. Einstein showed that it would take increasingly large amounts of energy to get closer and closer to the speed of light, so the limit can never be reached.

  Perhaps the best-known consequence of special relativity is the equivalence of energy and mass, expressed in Einstein’s formula E = mc2. If you heat an object, its thermal (heat) energy grows, so it should weigh more. This may give you the idea to take a cold shower before you step on the scale. But this trick is likely to decrease your weight by no more than a few billionths of a pound. In conventional units, like meters and seconds, the conversion factor c2 between energy and mass is very large, and it takes a huge amount of energy to noticeably change the mass of a macroscopic body. Physicists often use another system of units, where c = 1, so that energy is simply equal to mass and can be measured in kilograms.a I will mostly follow this tradition and make no distinction between energy and mass.

  The word “special” in “special relativity” refers to the fact that this theory applies only in special circumstances when the effects of gravity are unimportant. This limitation is removed in Einstein’s second theory, the general theory of relativity, which is essentially a theory of gravitation.

  The general theory of relativity grew out of a simple observation, that the motion of objects under the action of gravity is independent of their mass, shape, or any other properties, as long as all nongravitational forces can be neglected. This was first recognized by Galileo, who forcefully argued the point in his famous Dialogues. The accepted view at the time, that of Aristotle, was that heavier objects fall faster. Indeed, a watermelon does fall faster than a feather, but Galileo realized that the difference was due only to air resistance. Legend has it that Galileo dropped rocks of different weight from the Leaning Tower of Pisa, to make sure that they landed at the same time. We do know that he experimented with marbles rolling down an inclined plane and found that the motion was independent of the mass. He also offered a theoretical proof that Aristotle could not be right. Suppose, says Galileo, that a heavy rock falls faster than a light rock. Imagine then tying them together with a very light string. How will this affect the fall of the heavy rock? On the one hand, the slower-moving light rock should make the fall of the heavy rock somewhat slower than it was before. On the other hand, viewed together, the two rocks now constitute one object that is more massive than the heavy rock was initially, and thus the two rocks together should fall faster. This contradiction demonstrates that Aristotle’s theory is inconsistent.

  Einstein was pondering this peculiar kind of motion, which is completely independent of what is moving. It reminded him of inertial motion: in the absence of forces, an object moves along a straight line at a constant speed, regardless of what it is made of. In effect, the motion of the object in space and time is the property of space and time themselves.

  Here Einstein made use of the ideas of his former mathematics professor, Hermann Minkowski. As a student, Einstein did not think much of Minkowski’s lectures, while Minkowski remembered Einstein as a “lazy dog” and did not expect him to do anything worthwhile. To Minkowski’s credit, he changed his mind quickly after reading Einstein’s 1905 paper.

  Minkowski realized that the mathematics of special relativity becomes simpler and more elegant if space and time are not regarded as separate, but are united in a single entity called spacetime. A point in spacetime is an event. It can be specified by four numbers: three for its spatial location and one for its time. Hence, spacetime has four dimensions. If you had all of spacetime in front of you, then you would know all the past, present, and future of the universe. The history of each particle is represented by a line in spacetime, which gives the position of the particle at every moment of time. This line is called the world line of the particle. (George Gamow, one of the founders of the big bang cosmology, called his autobiography My World Line.)

  The uniform motion of particles in the absence of gravity is represented by straight lines in spacetime. But gravity makes particles deviate from this simple motion, so their world lines are no longer straight. This led Einstein to a truly astonishing hypothesis: even deviant particles with curved world lines might still be following the straightest possible paths in spacetime, but the spacetime itself must be curved around massive bodies. Gravity, then, is nothing but the curvature of spacetime!

  The distortion of spacetime geometry by a massive body can be illustrated by a heavy object resting on a horizontally stretched rubber sheet (see Figure 2.1). The rubber surface is warped near the object, just as spacetime is warped near a gravitating body. If you try playing billiards on this rubber sheet, you will discover that the billiard balls are deflected on the curved surface, especially when they pass near the heavy mass. This analogy is not perfect—it illustrates only the warping of space, not that of spacetime—but it does capture the essence of the idea.

  Figure 2.1. A massive body causes space to curve.

  It took Einstein more than three years of truly heroic effort to express these ideas in mathematical terms. The equations of the new theory, which he called the general theory of relativity, relate the geometry of spacetime to the matter content of the universe. In the regime of slow motion and not-too-strong gravitational fields, the theory reproduced Newton’s law, with the force of gravity being inversely proportional to the square of the distance. There was also a small correction to this law, which was utterly negligible for planetary motion, except in the case of Mercury, the planet closest to the Sun. The effect of the correction was to cause a slow precession, or advance, of Mercury’s orbit. Astronomical observations did in fact show a tiny precession, which remained unexplained in Newton’s theory, but was in perfect agreement with Einstein’s calculation. At this point Einstein was certain that the theory wa
s correct. “I was beside myself with ecstasy for days,” he wrote to his friend Paul Ehrenfest.1

  Figure 2.2. Einstein’s equations.

  Perhaps the most remarkable thing about the general theory of relativity is how little factual input it required. The essential fact that Einstein placed at the foundation of the theory—that the motion of objects under the action of gravity is independent of their mass—was known already to Galileo. With this minimal input, he created a theory that reproduced Newton’s law in the appropriate limit and explained a deviation from this law. If you think about it, Newton’s law is in some sense arbitrary. It states that the gravitational force between two bodies is inversely proportional to the second power of their distance, but it does not say why. It could equally well be the fourth power or the 2.03rd power. In contrast, Einstein’s theory allows no freedom. The picture of gravity as curvature of spacetime inevitably leads to Einstein’s equations, and the equations yield the inverse square law. In this sense the general theory of relativity not only describes gravity, it explains gravity. So compelling was the logic of the theory and so beautiful its mathematical structure that Einstein felt it simply had to be right. In a letter to a senior colleague, Arnold Sommerfeld, he wrote, “Of the general theory of relativity you will be convinced, once you have studied it. Therefore I am not going to defend it with a single word.”2

  THE GRAVITY OF EMPTY SPACE

  With his general theory of relativity now complete, Einstein wasted no time in applying it to the entire universe. He was not interested in trivial details, such as the position of this star or that planet. Rather, he wanted to find a solution of his equations that would describe, in broad brushstrokes, the structure of the universe as a whole.

  Little was known at the time about the distribution of matter in the universe, so Einstein had to make some guesses. He made the simplest assumption that, on average, matter is uniformly spread throughout the cosmos. There are, of course, local deviations from uniformity, with the density of stars being a little higher in this place and a little lower in that. What Einstein assumed was that if matter is smoothed over large enough distance scales, then, to a good approximation, the universe can be described as perfectly homogeneous. This assumption implies that our location in space is not in any way special: all places in the universe are more or less the same. Einstein also assumed that the universe is on average isotropic, which means that from any point it looks more or less the same in all directions.

  Finally, Einstein assumed that the average properties of the universe do not change with time. In other words, the universe is static. Although Einstein had little observational evidence to support this assumption, the picture of an eternal, unchanging universe seemed very compelling.

  Having specified the kind of universe he was looking for, Einstein could now try to find a solution of his equations that would describe a universe with the desired properties. It did not take him long, however, to discover that his theory admitted no such solutions. The reason was very simple: masses distributed throughout the universe refused to stay at rest and “wanted” instead to collapse onto one another, because of their gravitational attraction. Einstein was deeply puzzled and perplexed by this situation. After a year of struggle, he decided that the equations of general relativity had to be modified to allow for the existence of a static world.

  Einstein realized that it was possible to add an extra term to his equations without violating the physical principles of the theory. The effect of the new term was to endow empty space, or vacuum, with nonzero energy and tension. Each cubic centimeter of empty space has a fixed amount of energy (and therefore mass). Einstein called this constant energy density of the vacuum the cosmological constant.b The mathematics of Einstein’s equations dictates that the tension of the vacuum is exactly equal to its energy density and is therefore determined by the same constant. The vacuum tension is like the tension in a stretched rubber band that would cause the band to shrink if you let it go. Tension is opposite to pressure, which causes things to expand—as when a balloon expands under the pressure of compressed air. Thus, tension acts as negative pressure.

  If the vacuum has energy and tension, how come they seem to have no effect on us? Why don’t we see empty space shrink because of its tension? The reason is that it is not so easy to notice constant energy and tension. If you increase the pressure inside a balloon, it will expand. But if you also increase the air pressure outside the balloon by the same amount, then there will be no effect. Similarly, if negative-pressure vacuum fills the entire universe, its overall effect is nil. The energy of the vacuum is elusive because it is impossible to extract this energy. You cannot burn the vacuum; you cannot use it to run a car or a hair dryer. Its energy is set by the cosmological constant and cannot be reduced. Thus, the energy and tension of the vacuum are undetectable—except for their gravitational effect.

  The gravity of the vacuum turned out to hold a big surprise. According to general relativity, pressure and tension contribute to the gravitational force of massive bodies. If you compress an object, its gravity is enhanced, and if you stretch it, gravity is reduced. This effect is normally very small, but if you could keep stretching the object without breaking it, you could in principle reduce gravity to the point of completely neutralizing it, or even making it repulsive. This is precisely what happens in the case of the vacuum. The repulsive gravity of vacuum tension is more than sufficient to overcome the attractive pull of its mass, so the net result is gravitational repulsion.

  This property was exactly what Einstein needed to solve his problem. He could now adjust the value of the cosmological constant so that the attractive gravitational force of matter is balanced by the repulsive gravity of the vacuum. The result is a static universe. He found from his equations that the balance is achieved when the cosmological constant is half the energy density of matter.

  A striking consequence of the modified equations was that the space of a static universe must be curved, so that it closes in upon itself like the surface of a sphere. A spaceship moving straight ahead in such a closed universe would eventually come back to its starting point. This closed space is called a three-dimensional sphere. Its volume is finite, although it has no boundary.

  Einstein described his closed-universe model in a paper published in 1917. He admitted that he had no observational evidence for a nonzero cosmological constant. His only reason for introducing it was to save the static picture of the world. More than a decade later, when the expansion of the universe was discovered, Einstein regretted he had ever proposed the idea and called it the greatest blunder of his life.3 After this unsuccessful debut, repulsive gravity disappeared from mainstream physics research for nearly half a century—but only to return later with a vengeance.

  3

  Creation and Its Discontents

  As a scientist I simply do not believe that the universe began with a bang.

  —SIR ARTHUR EDDINGTON

  FRIEDMANN’S UNIVERSES

  The cold and hungry Petrograd of the early 1920s was not on anyone’s list of places where the next breakthrough in cosmology was likely to occur. Classes at Petrograd University had just resumed, after six years of war and Russian revolution. A young, bespectacled professor was lecturing in a freezing classroom to an audience of students in overcoats and fur hats. His name was Alexander Friedmann. The lectures were meticulously prepared and emphasized mathematical rigor. The courses he taught ranged from mathematics and meteorology, his main areas of expertise, to his most recent passion, the general theory of relativity.

  Friedmann was fascinated by Einstein’s theory and threw himself into studying it with his usual intensity. “I am an ignoramus,” he used to say. “I don’t know anything. I have to sleep even less and not allow myself any distractions, because all this so-called ‘life’ is a complete waste of time.”1 It was as if he knew that he had only a few years left to live—and so much to accomplish.

  Having mastered the mathematics of
general relativity, Friedmann focused on what he thought was its central problem: the structure of the entire universe. He learned from Einstein’s paper that without a cosmological constant, the theory had no static solutions. He wanted to know, however, what kind of solutions it did have. Here, Friedmann made a radical step that would immortalize his name. Following Einstein, he assumed that the universe was homogeneous, isotropic, and closed, having the geometry of a three-dimensional sphere. But he broke away from the static paradigm and allowed the universe to move. The radius of the sphere and the density of matter could now change with time. With the requirement of a static universe lifted, Friedmann found that Einstein’s equations do have a solution. It describes a spherical universe that starts from a point, expands to some maximum size, and then recollapses back to a point. At the initial moment, which we now call the big bang, all matter in the universe is packed into a single point, so the density of matter is infinite. The density decreases as the universe expands and then grows as it recontracts, to become infinite again at the moment of the “big crunch,” when the universe shrinks back to a point.

  The big bang and the big crunch mark the beginning and the end of the universe. Because of the vanishing size and the infinite density of matter, the mathematical quantities appearing in Einstein’s equations become ill-defined, and spacetime cannot be extended beyond these points. Such points are called spacetime singularities.

  A two-dimensional spherical universe can be pictured as an expanding and recontracting balloon (see Figure 3.1). The squiggles on the surface of the balloon represent galaxies, and as the balloon expands, the distances between the galaxies grow. Hence, an observer in each galaxy sees other galaxies rush away. The expansion is gradually slowed down by gravity; it will eventually come to a halt and be followed by the contraction. In the contracting phase, the distances between the galaxies will decrease and all observers will see galaxies moving toward them.