Many Worlds in One: The Search for Other Universes

by Vilenkin, Alex

  To have a specific picture in mind, suppose there is an observer in every galaxy of our local region. Since the universe is expanding, each of these observers will see the others moving away. Galaxies may not exist in some regions of space and time, but we can still imagine the entire universe “sprinkled” with observers in such a way that all of them are moving away from one another.bj To give these observers some name, we shall call them “spectators.”

  Let us now introduce another observer who is moving relative to the spectators. We shall call him the space traveler. He is moving by inertia, with the engines of his spaceship turned off, and has been doing so for all eternity. As he passes the spectators, they register his velocity.

  Since the spectators are flying apart, the space traveler’s velocity relative to each successive spectator will be smaller than his velocity relative to the preceding one. Suppose, for example, the space traveler has just zoomed by the Earth at the speed of 100,000 kilometers per second and is now headed toward a distant galaxy, about a billion light-years away. That galaxy is moving away from us at 20,000 kilometers per second, so when the space traveler catches up with it, the observers there will see him moving at 80,000 kilometers per second.

  If the velocity of the space traveler relative to the spectators gets smaller and smaller into the future, it follows that his velocity should get larger and larger as we follow his history into the past. In the limit, his velocity should get arbitrarily close to the speed of light.

  The key insight of my paper with Borde and Guth is that as we go into the past and approach past infinity by the clocks of the spectators, the time elapsed by the clock of the space traveler is still finite. The reason is that according to Einstein’s theory of relativity, a moving clock ticks slower, and the closer you get to the speed of light, the more slowly it ticks. As we go backward in time, the speed of the space traveler approaches the speed of light and his clock essentially comes to a halt. This is from the spectator’s point of view. But the space traveler himself does not notice anything unusual. For him, what spectators perceive as a frozen moment, stretched into eternity, is a moment like any other, which has to be preceded by earlier moments. Like the histories of the spectators, the space traveler’s history should extend into the infinite past.

  The fact that the time elapsed by the space traveler’s clock is finite indicates that we do not have his full history. This means that some part of the past history of the universe is missing; it is not included in the model. Thus, the assumption that the entire spacetime can be covered by an expanding “dust” of observers has led to a contradiction, and therefore it cannot be true.6

  A remarkable thing about this theorem is its sweeping generality. We made no assumptions about the material content of the universe. We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modification, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe never gets below some nonzero value, no matter how small.7 This assumption should certainly be satisfied in the inflating false vacuum. The conclusion is that past-eternal inflation without a beginning is impossible.

  What about a cyclic universe? It has alternating periods of expansion and contraction. Can this help the universe to escape from the clutches of the theorem? The answer turns out to be no. An essential feature of the cyclic scenario, which allows it to avoid the heat-death problem, is that the volume of the universe increases in every cycle, so on average the universe is expanding. In my paper with Borde and Guth, we show that as a result of this expansion, the space traveler’s velocity increases on average as we go back in time and still approaches the speed of light in the limit. Hence, the same conclusions apply.8

  It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape: they have to face the problem of a cosmic beginning.

  Working with Alan Guth on a paper was a memorable experience. The idea of the proof emerged in e-mail exchanges between me, Alan, and Arvind, and the details were nailed down in two hours at the blackboard, when the three of us met in my Tufts office in August 2001. In about a month, we wrote a paper and submitted it to Physical Review Letters. I was amazed. What happened to Alan and his legendary procrastination? But I was not to be disappointed. In a few months, the editor sent us the report of a referee, asking us to clarify some points in the proof. And that is when the good old Alan returned in full glory. His e-mails started arriving at longer and longer intervals, with headings like “swamped at the moment” and “nothing done yet.” When he did find some time to work on the paper, he seems to have spent a fair fraction of it on issues such as whether we should thank “an anonymous referee” or “the anonymous referee” for his or her comments. He gave a detailed discussion of pros and cons for either version. Alan might have suspected that his editing of the paper was taking a bit too long, and at some point he wrote “I need to thank you guys for not shooting me.” In fairness, I should add that he spent some time on more substantive issues as well and that the drawn-out process of revising the paper resulted in its substantial improvement. The paper was finally published in April 2003.9

  A PROOF OF GOD?

  Theologians have often welcomed any evidence for the beginning of the universe, regarding it as evidence for the existence of God. In the 1950s the accumulating evidence for the big bang inspired enthusiasm in theological circles and among some religiously inclined scientists. “As to the first cause of the universe,” wrote the British physicist Edward Milne, “in the context of expansion, this is left for the reader to insert, but our picture is incomplete without Him.”10 The big bang theory even received an official endorsement from the Roman Catholic Church. In his 1951 address to the Pontifical Academy of Sciences, Pope Pius XII said that it “has confirmed … the well-founded deduction as to the epoch when the cosmos came forth from the hands of the Creator. Hence, Creation took place. Therefore there is a Creator. Therefore God exists!”11

  For the same reasons that made the pope so exuberant, the natural instinct of most scientists has been to reject the idea of a cosmic beginning. “To deny the infinite duration of time,” asserted the Nobel Prize-winning German chemist Walter Nernst, “would be to betray the very foundations of science.”12 The beginning of the universe looked too much like a divine intervention; there seemed to be no possibility to describe it scientifically. This was one thing scientists and theologians seemed to agree upon.

  So, what do we make of a proof that the beginning is unavoidable? Is it a proof of the existence of God? This view would be far too simplistic. Anyone who attempts to understand the origin of the universe should be prepared to address its logical paradoxes. In this regard, the theorem that I proved with my colleagues does not give much of an advantage to the theologian over the scientist. As evidenced by Jinasena’s remarks earlier in this chapter, religion is not immune to the paradoxes of Creation.

  Also, the scientists might have been too rash to admit that the cosmic beginning cannot be described in purely scientific terms. True, it is hard to see how this can be done. But things that seem to be impossible often reflect only the limitations of our imagination.

  17

  Creation of Universes from Nothing

  Nothing can be created from nothing.

  —LUCRETIUS

  INFLATION AT THE END OF THE TUNNEL

  Back in 1982, inflation was still a very new field, full of unexplored ideas and challenging problems—a gold mine for an aspiring young cosmologist. The most intriguing of these problems, and perhaps the least relevant for the present state of the universe, was the question of how inflation could have started. An inflating universe quickly “forgets” its initial conditions, so the state at the onset of inflation has little effect on what happens afterward. Thus, if you
want to find ways of testing inflation observationally, you should not waste your time worrying about how it began. But the puzzle of the beginning was still there and could not be avoided. It drew me like a magnet.

  At first sight, the problem looked relatively simple. We know that a small region of space filled with false vacuum is enough to drive inflation. So, all I had to figure out was how such a region could have arisen from some earlier state of the universe.

  The prevailing view at the time was based on the Friedmann model, where the universe expanded from a singular state of infinite curvature and infinite matter density. Assuming that the universe is filled with a high-energy false vacuum, any matter that was initially present is diluted, and the vacuum energy eventually dominates. At that point, the repulsive gravity of the vacuum takes over, and inflation begins.

  This would be fine, except, Why was the universe expanding to begin with? One of the achievements of inflation was to explain the expansion of the universe. Yet, it looked as if we needed to have expansion before inflation even started. The attractive gravity of matter is initially much stronger than the gravitational repulsion of the vacuum, so if we don’t postulate a strong initial blast of expansion, the universe would simply collapse and inflation would never begin.

  I pondered this argument for a while, but the logic was very simple and there seemed to be no escape. Then, suddenly, I realized that instead of collapsing, the universe could do something much more interesting and dramatic …

  Suppose we have a closed spherical universe, filled with a false vacuum and containing a certain amount of ordinary matter. Suppose also that this universe is momentarily at rest, neither expanding nor contracting. Its future will depend on its radius. If the radius is small, the matter is compressed to a high density and the universe will collapse to a point. If the radius is large, the vacuum energy dominates and the universe will inflate. Small and large radii are separated by an energy barrier, which cannot be crossed unless the universe is given a large expansion velocity.

  What I suddenly realized was that the collapse of a small universe was inevitable only in classical physics. In quantum theory, the universe could tunnel through the energy barrier and emerge on the other side—like a nuclear particle in Gamow’s theory of radioactive decay.

  This looked like a neat solution to the problem. The universe starts out extremely small and is most likely to collapse to a singularity. But there is a small chance that instead of collapsing, it will tunnel through the barrier to a bigger radius and start inflating (see Figure 17.1). So, in the grander scheme of things, there will be loads of failed universes that will exist only for a fleeting moment, but there will also be some that will make it big.

  I felt that I was making progress, so I pressed on. Is there any bound to how small the initial universe could be? What happens if we allow it to get smaller and smaller? To my surprise, I found that the tunneling probability did not vanish as the initial size approached zero. I also noticed that my calculations were greatly simplified when I allowed the initial radius of the universe to vanish. This was really crazy: what I had was a mathematical description of a universe tunneling from a zero size—from nothing!—to a finite radius and beginning to inflate. It looked as though there was no need for the initial universe!

  Figure 17.1. On the left, a spacetime diagram of a closed Friedmann universe expanding from a singularity, reaching a maximum radius and recollapsing. Time grows in the vertical direction, and horizontal circles give snapshots of the universe at different moments of time. On the right, a universe dominated by vacuum energy, which contracts and re-expands (de Sitter spacetime). Instead of recollapsing, the universe on the left can tunnel through the energy barrier to a larger radius and start expanding. The spacetime history of the universe will then include only the shaded parts of the two spacetimes.

  TUNNELING FROM NOTHING

  The concept of a universe materializing out of nothing boggles the mind. What exactly is meant by “nothing”? If this “nothing” could tunnel into something, what could have caused the primary tunneling event? And what about energy conservation? But as I kept thinking about it, the idea appeared to make more and more sense.

  The initial state prior to the tunneling is a universe of vanishing radius, that is, no universe at all. There is no matter and no space in this very peculiar state. Also, there is no time. Time has meaning only if something is happening in the universe. We measure time using periodic processes, like the rotation of the Earth about its axis, or its motion around the Sun. In the absence of space and matter, time is impossible to define.

  And yet, the state of “nothing” cannot be identified with absolute nothingness. The tunneling is described by the laws of quantum mechanics, and thus “nothing” should be subjected to these laws. The laws of physics must have existed, even though there was no universe. I will have more to say about this in Chapter 19.

  As a result of the tunneling event, a finite-sized universe, filled with a false vacuum, pops out of nowhere (“nucleates”) and immediately starts to inflate. The radius of the newborn universe is determined by the vacuum energy density: the higher the density, the smaller the radius. For a grand-unified vacuum, it is one hundred-trillionth of a centimeter. Because of inflation, this tiny universe grows at a staggering rate, and in a small fraction of a second it becomes much greater than the size of our observable region.

  If there was nothing before the universe popped out, then what could have caused the tunneling? Remarkably, the answer is that no cause is required. In classical physics, causality dictates what happens from one moment to the next, but in quantum mechanics the behavior of physical objects is inherently unpredictable and some quantum processes have no cause at all. Take, for example, a radioactive atom. It has some probability of decaying, which is the same from this minute to the next. Eventually, it will decay, but there will be nothing that causes it to decay at that particular moment. Nucleation of the universe is also a quantum process and does not require a cause.

  Most of our concepts are rooted in space and time, and it is not easy to create a mental picture of a universe popping out of nothing. You cannot imagine that you are sitting in “nothing” and waiting for a universe to materialize—because there is no space to sit in and there is no time.

  In some recently proposed models based on string theory, our space is a three-dimensional membrane (brane) floating in a higher-dimensional space. In such models, we can imagine a higher-dimensional observer watching small bubble universes—braneworlds—pop out here and there, like bubbles of vapor in a boiling pot of water. We live on one of the bubbles, which is an expanding three-dimensional spherical brane. For us, this brane is the only space there is. We cannot get out of it and are unaware of the extra dimensions. As we follow the history of our bubble universe back in time, we come to the moment of nucleation. Beyond that, our space and time disappear.

  From this picture, there is only a small step to the one that I originally proposed. Simply remove the higher-dimensional space. From our internal point of view, nothing will change. We live in a closed, three-dimensional space, but this space is not floating anywhere. As we go back in time, we discover that our universe had a beginning. There is no spacetime beyond that.

  An elegant mathematical description of quantum tunneling can be obtained using the so-called Euclidean time. This is not the kind of time you measure with your watch. It is expressed using imaginary numbers, like the square root of -1, and is introduced only for computational convenience. Making the time Euclidean has a peculiar effect on the character of spacetime : the distinction between time and the three spatial dimensions completely disappears, so instead of spacetime we have a four-dimensional space. If we could live in Euclidean time, we would measure it with a ruler, just as we measure length. Although it may appear rather odd, the Euclidean-time description is very useful: it provides a convenient way to determine the tunneling probability and the initial state of the universe as it emerges
into existence.

  The birth of the universe can be graphically represented by the spacetime diagram in Figure 17.2. The dark hemisphere at the bottom corresponds to quantum tunneling (time is Euclidean in this part of the spacetime). The light surface above it is the spacetime of the inflating universe. The boundary between the two spacetime regions is the universe at the moment of nucleation.

  A remarkable feature of this spacetime is that it has no singularities. A Friedmann spacetime has a singular point of infinite curvature at the beginning, where the mathematics of Einstein’s equations breaks down. It is represented by the sharp point (labeled “singularity”) at the bottom on the left-hand side of Figure 17.1. In contrast, the Euclidean spherical region has no such points; it has the same finite curvature everywhere. This was the first mathematically consistent description of how the universe could be born. The spacetime diagram of Figure 17.2, which looks a bit like a badminton shuttlecock is now on the logo of the Tufts Institute of Cosmology.

  I wrote all this up in a short paper entitled “Creation of Universes from Nothing.”1 Before submitting it to a journal, I made a day trip to Princeton University, to discuss these ideas with Malcolm Perry, a well-known expert on the quantum theory of gravitation. After an hour at the blackboard, Malcolm said, “Well, maybe this is not so crazy … How come I have not thought of it myself ?” What better compliment can you get from a fellow physicist!

  Figure 17.2. A spacetime diagram of the universe tunneling from nothing.

  THE UNIVERSE AS A QUANTUM FLUCTUATION