The Danish scientist Niels Bohr found a partial solution to this problem in 1913. He suggested that perhaps the electrons were not able to orbit at just am distance from the central nucleus but rather could orbit only at certain specified distances. Supposing that only one or two electrons could orbit at any one of these specified distances would solve the problem of the collapse, because once the limited number of inner orbits was full, the electrons could not spiral in any farther. This model explained quite well the structure of the simplest atom, hydrogen, which has only one electron orbiting around the nucleus. But it was not clear how to extend this model to more complicated atoms. Moreover, the idea of a limited set of allowed orbits seemed like a mere Band-Aid. It was a trick that worked mathematically, but no one knew why nature should behave that way, or what deeper law—if any—it represented. The new theory of quantum mechanics resolved this difficulty. It revealed that an electron orbiting around the nucleus could be thought of as a wave, with a wavelength that depended on its velocity. Imagine the wave circling the nucleus at specified distances, as Bohr had postulated. For certain orbits, the circumference of the orbit would correspond to a whole number (as opposed to a fractional number) of wavelengths of the electron. For these orbits the wave crest would be in the same position each time round, so the waves would reinforce each other. These orbits would correspond to Bohr’s allowed orbits. However, for orbits whose lengths were not a whole number of wavelengths, each wave crest would eventually be canceled out by a trough as the electrons went round. These orbits would not be allowed. Bohr’s law of allowed and forbidden orbits now had an explanation.
A nice way of visualizing the wave/particle duality is the so-called sum over histories introduced by the American scientist Richard Feynman. In this approach a particle is not supposed to have a single history or path in space-time, as it would in a classical, nonquantum theory. Instead it is supposed to go from point A to point B by every possible path. With each path between A and B, Feynman associated a couple of numbers. One represents the amplitude, or size, of a wave. The other represents the phase, or position in the cycle (that is, whether it is at a crest or a trough or somewhere in between). The probability of a particle going from A to B is found by adding up the waves for all the paths connecting A and B. In general, if one compares a set of neighboring paths, the phases or positions in the cycle will differ greatly. This means that the waves associated with these paths will almost exactly cancel each other out. However, for some sets of neighboring paths the phase will not vary much between paths, and the waves for these paths will not cancel out. Such paths correspond to Bohr’s allowed orbits.
Waves in Atomic Orbits
Niels Bohr imagined the atom as consisting of electron waves endlessly circling atomic nuclei In his picture, only orbits with circumferences corresponding to a whole number of electron wavelengths could survive without destructive interference
With these ideas in concrete mathematical form, it was relatively straightforward to calculate the allowed orbits in more complicated atoms and even in molecules, which are made up of a number of atoms held together by electrons in orbits that go around more than one nucleus. Since the structure of molecules and their reactions with each other underlie all of chemistry and biology, quantum mechanics allows us in principle to predict nearly everything we see around us, within the limits set by the uncertainty principle. (In practice, however, we cannot solve the equations for any atom besides the simplest one, hydrogen, which has only one electron, and we use approximations and computers to analyze more complicated atoms and molecules.)
Many Electron Paths
In Richard Feynman’s formulation of quantum theory, a particle, such as this one moving from source to screen, takes every possible path.
Quantum theory has been an outstandingly successful theory and underlies nearly all of modern science and technology. It governs the behavior of transistors and integrated circuits, which are the essential components of electronic devices such as televisions and computers, and it is also the basis of modern chemistry and biology. The only areas of physical science into which quantum mechanics has not yet been properly incorporated are gravity and the large-scale structure of the universe: Einstein’s general theory of relativity, as noted earlier, does not take account of the uncertainty principle of quantum mechanics, as it should for consistency with other theories.
As we saw in the last chapter, we already know that general relativity must be altered. By predicting points of infinite density—singularities—classical (that is, nonquantum) general relativity predicts its own downfall, just as classical mechanics predicted its downfall by suggesting that blackbodies should radiate infinite energy or that atoms should collapse to infinite density. And as with classical mechanics, we hope to eliminate these unacceptable singularities by making classical general relativity into a quantum theory—that is, by creating a quantum theory of gravity.
If general relativity is wrong, why have all experiments thus far supported it? The reason that we haven’t yet noticed am discrepancy with observation is that all the gravitational fields that we normally experience are very weak. But as we have seen, the gravitational field should get very strong when all the matter and energy in the universe are squeezed into a small volume in the early universe. In the presence of such strong fields, the effects of quantum theory should be important.
Although we do not yet possess a quantum theory of gravity, w e do know a number of features we believe it should have. One is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over histories. A second feature that we believe must be part of any ultimate theory is Einstein’s idea that the gravitational field is represented by curved space-time: particles try to follow the nearest thing to a straight path in a curved space, but because space-time is not flat, their paths appear to be bent, as if by a gravitational field. When we apply Feynman’s sum over histories to Einstein’s view of gravity, the analogue of the history of a particle is now a complete curved space-time that represents the history of the whole universe.
In the classical theory of gravity, there are only two possible ways the universe can behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time in the past. For reasons we discussed earlier, we believe that the universe has not existed forever. Yet if it had a beginning, according to classical general relativity, in order to know which solution of Einstein’s equations describes our universe, we must know its initial state—that is, exactly how the universe began. God may have originally decreed the laws of nature, but it appears that He has since left the universe to evolve according to them and does not now intervene in it. How did He choose the initial state or configuration of the universe? What were the boundary conditions at the beginning of time? In classical general relativity this is a problem, because classical general relativity breaks down at the beginning of the universe.
In the quantum theory of gravity, on the other hand, a new possibility arises that, if true, would remedy this problem. In the quantum theory, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions. As was pointed out before, if you keep traveling in a certain direction on the surface of the earth, you never come up against an impassable barrier or fall over the edge, but eventually come back to where you started, without running into a singularity. So if this turns out to be the case, then the quantum theory of gravity has opened up a new possibility in w hich there would be no singularities at w hich the laws of science broke down.
If there is no boundary to space-time, there is no need to specify the behavior at the boundary—no need to know the initial state of the universe. There is no edge of space-time at which we would have to appeal to God or some new law to set the boundary conditions for space-time. We could say: “The boundary condition of the universe is
that it has no boundary.” The universe would be completely self-contained and not affected by anything outside itself. It would neither be created nor destroyed. It would just BE. As long as we believed the universe had a beginning, the role of a creator seemed clear. But if the universe is really completely self-contained, having no boundary or edge, having neither beginning nor end, then the answer is not so obvious: what is the role of a creator?
10
WORMHOLES AND TIME TRAVEL
IN PREVIOUS CHAPTERS WE HAVE SEEN how our views of the nature of time have changed over the years. Until the beginning of the twentieth century, people believed in an absolute time. That is, each event could be labeled by a number called “time” in a unique way, and all good clocks would agree on the time interval between two events. However, the discovery that the speed of light appeared the same to every observer, no matter how he was moving, led to the theory of relativity—and the abandoning of the idea that there was a unique absolute time. The time of events could not be labeled in a unique way. Instead, each observer would have his own measure of time as recorded by a clock that he carried, and clocks carried by different observers would not necessarily agree. Thus time became a more personal concept, relative to the observer who measured it. Still, time was treated as if it were a straight railway line on which you could go only one way or the other. But what if the railway line had loops and branches, so a train could keep going forward but come back to a station it had already passed? In other words, might it be possible for someone to travel into the future or the past? H. G. Wells in The Time Machine explored these possibilities, as have countless other writers of science fiction. Yet many of the ideas of science fiction, like submarines and travel to the moon, have become matters of science fact. So what are the prospects for time travel?
Time Machine
The authors in a time machine
It is possible to travel to the future. That is, relativity shows that it is possible to create a time machine that will jump you forward in time. You step into the time machine, wait, step out, and find that much more time has passed on the earth than has passed for you. We do not have the technology today to do this, but it is just a matter of engineering: we know it can be done. One method of building such a machine would be to exploit the situation we discussed in Chapter 6 regarding the twins paradox. In this method, while you are sitting in the time machine, it blasts off, accelerating to nearly the speed of light, continues for a while (depending upon how far forward in time you wish to travel), and then returns. It shouldn’t surprise you that the time machine is also a spaceship, because according to relativity, time and space are related. In any case, as far as you are concerned, the only “place” you will be during the whole process is inside the time machine. And when you step out, you will find that more time has passed on the earth than has gone by for you. You have traveled to the future. But can you go back? Can we create the conditions necessary to travel backward in time?
The first indication that the laws of physics might really allow people to travel backward in time came in 1949 when Kurt Gödel discovered a new solution to Einstein’s equations; that is, a new space-time allowed by the theory of general relativity. Many different mathematical models of the universe satisfy Einstein’s equations, but that doesn’t mean they correspond to the universe we live in. They differ, for instance, in their initial or boundary conditions. We must check the physical predictions of these models in order to decide whether or not they might correspond to our universe.
Gödel was a mathematician who was famous for proving that it is impossible to prove all true statements, even if you limit yourself to trying to prove all the true statements in a subject as apparently cut-and-dried as arithmetic. Like the uncertainty principle, Gödel’s incompleteness theorem may be a fundamental limitation on our ability to understand and predict the universe. Gödel got to know about general relativity when he and Einstein spent their later years at the Institute for Advanced Study in Princeton. Gödel’s space-time had the curious property that the whole universe was rotating.
What does it mean to say that the whole universe is rotating? To rotate means to turn around and around, but doesn’t that imply the existence of a stationary point of reference? So you might ask, “Rotating with respect to what?” The answer is a bit technical, but it is basically that distant matter would be rotating with respect to directions that little tops or gyroscopes point in within the universe. In Gödel’s space-time, a mathematical side effect of this was that if you traveled a great distance away from earth and then returned, it would be possible to get back to earth before you set out.
That his equations might allow this possibility really upset Einstein, who had thought that general relativity wouldn’t allow time travel. But although it satisfies Einstein’s equations, the solution Gödel found doesn’t correspond to the universe we live in because our observations show that our universe is not rotating, at least not noticeably. Nor does Gödel’s universe expand, as ours does. However, since then, scientists studying Einstein’s equations have found other space-times allowed by general relativity that do permit travel into the past. Yet observations of the microwave background and of the abundances of elements such as hydrogen and helium indicate that the early universe did not have the kind of curvature these models require in order to allow time travel. The same conclusion follows on theoretical grounds if the no-boundary proposal is correct. So the question is this: if the universe starts out without the kind of curvature required for time travel, can we subsequently warp local regions of space-time sufficiently to allow it?
Again, since time and space are related, it might not surprise you that a problem closely related to the question of travel backward in time is the question of whether or not you can travel faster than light. That time travel implies faster-than-light travel is easy to see: by making the last phase of your trip a journey backward in time, you can make your overall trip in as little time as you wish, so you’d be able to travel with unlimited speed! But, as we’ll see, it also works the other way: if you can travel with unlimited speed, you can also travel backward in time. One cannot be possible without the other.
The issue of faster-than-light travel is a problem of much concern to writers of science fiction. Their problem is that, according to relativity, if we sent a spaceship to our nearest neighboring star, Proxima Centauri, which is about four light-years away, it would take at least eight years before we could expect the travelers to return and tell us what they had found. And if the expedition were to the center of our galaxy, it would be at least a hundred thousand years before it came back. Not a good situation if you want to write about intergalactic warfare! Still, the theory of relativity does allow one consolation, again along the lines of our discussion of the twins paradox in Chapter 6: it is possible for the journey to seem to be much shorter for the space travelers than for those who remain on earth. But there would not be much joy in returning from a space voyage a few years older to find that everyone you had left behind was dead and gone thousands of years ago. So in order to have any human interest in their stories, science fiction writers had to suppose that we would one day discover how to travel faster than light. Most of these authors don’t seem to have realized the fact that if you can travel faster than light, the theory of relativity implies you can also travel back in time, as the following limerick says:
There was a young lady of Wight
Who traveled much faster than light
She departed one day,
In a relative way,
And arrived on the previous night
The key to this connection is that the theory of relativity says not only that there is no unique measure of time on which all observers will agree but that, under certain circumstances, observers need not even agree on the order of events. In particular, if two events, A and B, are so far away in space that a rocket must travel faster than the speed of light to get from event A to event B, then two observers moving at differ
ent speeds can disagree on whether event A occurred before B, or event B occurred before event A. Suppose, for instance, that event A is the finish of the final hundred-meter race of the Olympic Games in 2012 and event B is the opening of the 100,004th meeting of the Congress of Proxima Centauri. Suppose that to an observer on Earth, event A happened first, and then event B. Let’s say that B happened a year later, in 2013 by Earth’s time. Since the earth and Proxima Centauri are some four light-years apart, these two events satisfy the above criterion: though A happens before B, to get from A to B you would have to travel faster than light. Then, to an observer on Proxima Centauri moving away from the earth at nearly the speed of light, it would appear that the order of the events is reversed: it appears that event B occurred before event A. This observer would say it is possible, if you could move faster than light, to get from event B to event A. In fact, if you went really fast, you could also get back from A to Proxima Centauri before the race and place a bet on it in the sure knowledge of who would win!
There is a problem with breaking the speed-of-light barrier. The theory of relativity says that the rocket power needed to accelerate a spaceship gets greater and greater the nearer it gets to the speed of light. We have experimental evidence for this, not with spaceships but with elementary particles in particle accelerators such as those at Fermilab or the European Centre for Nuclear Research (CERN). We can accelerate particles to 99.99 percent of the speed of light, but however much power we feed in, we can’t get them beyond the speed-of-light barrier. Similarly with spaceships: no matter how much rocket power they have, they can’t accelerate beyond the speed of light. And since travel backward in time is possible only if faster-than-light travel is possible, this might seem to rule out both rapid space travel and travel back in time.
A Briefer History of Time Page 9
