THE FIRST scientific description of time was given in 1689 by Sir Isaac Newton, who held the Lucasian chair at Cambridge that I used to occupy (though it wasn’t electrically operated in his time). In Newton’s theory, time was absolute and marched on relentlessly. There was no turning back and returning to an earlier age. The situation changed, however, when Einstein formulated his general theory of relativity, in which space-time was curved and distorted by the matter and energy in the universe. Time still increased locally, but there was now the possibility that space-time could be warped so much that one could move on a path that would bring one back before one set out.
One possibility that would allow for this would be wormholes, hypothetical tubes of space-time that might connect different regions of space and time. The idea is that you step into one mouth of the wormhole and step out of the other in a different place and at a different time. Wormholes, if they exist, would be ideal for rapid space travel. You might go through a wormhole to the other side of the galaxy and be back in time for dinner. However, one can show that if wormholes exist, you could also use them to get back before you set out. One would then think that you could do something like blow up your own spaceship on its original launch pad to prevent you from setting out in the first place. This is a variation of the so-called grandfather paradox: What happens if you go back in time and kill your grandfather before your father was conceived? Would you then exist in the current present? If not, you wouldn’t exist to go back and kill your grandfather. Of course, this is a paradox only if you believe you have the free will to do what you like and change history when you go back in time.
The real question is whether the laws of physics allow wormholes and space-time to be so warped that a macroscopic body such as a spaceship can return to its own past. According to Einstein’s theory, a spaceship necessarily travels at less than the local speed of light, and follows what is called a “time-like path” through space-time. Thus one can formulate the question in technical terms: does space-time admit time-like curves that are closed—that is, time-like curves that return to their starting point again and again?
There are three levels on which we can try to answer this question. The first is Einstein’s general theory of relativity. This is what is called a classical theory, which is to say it assumes the universe has a well-defined history, without any uncertainty. For classical general relativity, we have a fairly complete picture of how time travel might work. We know, however, that classical theory can’t be quite right, because we observe that matter in the universe is subject to fluctuations, and its behavior cannot be predicted precisely.
In the 1920s a new paradigm called quantum theory was developed to describe these fluctuations and quantify the uncertainty. One can therefore ask the question about time travel on this second level, called the semi-classical theory. In this, one considers quantum matter fields against a classical space-time background. Here the picture is less complete, but at least we have some idea how to proceed.
Finally, one has the full quantum theory of gravity, whatever that may be. Here it is not clear even how to pose the question “Is time travel possible?” Maybe the best one can do is to ask how observers at infinity would interpret their measurements. Would they think that time travel had taken place in the interior of the space-time?
RETURNING TO the classical theory: flat space-time does not contain closed time-like curves. Nor do other solutions of the Einstein equations that were known early on. It was therefore a great shock to Einstein when in 1949 Kurt Gödel discovered a solution that represented a universe full of rotating matter, with closed time-like curves through every point. The Gödel solution required a cosmological constant, which is known to exist, though other solutions were subsequently found without one.
A particularly interesting case to illustrate this would be two cosmic strings moving at high speed past each other. As their name suggests, cosmic strings are objects with length but a tiny cross section. Some theories of elementary particles predict their occurrence. The gravitational field of a single cosmic string is flat space with a wedge cut out, with the string at its sharp end. Thus if one goes in a circle around a cosmic string, the distance in space is less than one would expect, but time is not affected. This means that the space-time around a single cosmic string does not contain any closed time-like curves.
However, if there is a second cosmic string moving with respect to the first, the wedge that is cut out for it will shorten both spatial distances and time intervals. If the cosmic strings are moving at nearly the speed of light relative to each other, the saving of time going around both strings can be so great that one arrives back before one set out. In other words, there are closed time-like curves that one can follow to travel into the past.
The cosmic string space-time contains matter that has positive energy density, and thus it is physically reasonable. However, the warping that produces the closed time-like curves extends all the way out to infinity and back to the infinite past. Thus these space-times were created with time travel in them. We have no reason to believe that our own universe was created in such a warped fashion, and we have no reliable evidence of visitors from the future. (Discounting, of course, the conspiracy theory that UFOs are from the future, which the government knows and is covering up. But governments’ record of cover-ups is not that good.) One should therefore assume that there are no closed time-like curves to the past of some surface of constant time S.
The question is then whether some advanced civilization could build a time machine. That is, could it modify the space-time to the future of S, so that closed time-like curves appeared in a finite region? I say “a finite region” because no matter how advanced the civilization becomes, it could presumably control only a finite part of the universe.
In science, finding the right formulation of a problem is often the key to solving it, and this was a good example. To define what was meant by a finite time machine, I went back to some early work of mine. I defined the future Cauchy development of S to be the set of points of space-time where events are determined completely by what happened on S. In other words, it is the region of space-time where every possible path that moves at less than the speed of light comes from S. However, if an advanced civilization managed to build a time machine, there would be a closed time-like curve, C, to the future of S. C will go round and round in the future of S, but it will not go back and intersect S. This means that points on C will not lie in the Cauchy development of S. Thus S will have a Cauchy horizon, a surface that is a future boundary to the Cauchy development of S.
Cauchy horizons occur inside some black hole solutions, or in anti–de Sitter space. However, in these cases, the light rays that form the Cauchy horizon start at infinity or at singularities. To create such a Cauchy horizon would require either warping space-time all the way out to infinity or the occurrence of a singularity in space-time. Warping space-time all the way to infinity would theoretically be beyond the powers of even the most advanced civilization, which could warp space-time only in a finite region. The advanced civilization could assemble enough matter to cause a gravitational collapse, which would produce a space-time singularity, at least according to classical general relativity. But the Einstein equations could not be defined at the singularity, so one could not predict what would happen beyond the Cauchy horizon, and in particular whether there would be any closed time-like curves.
One should therefore take as the criterion for a time machine what I call a finitely generated Cauchy horizon. That is a Cauchy horizon generated by light rays that all emerge from a compact region. In other words, they don’t come in from infinity, or from a singularity, but originate from a finite region containing closed time-like curves, the sort of region we have supposed our advanced civilization would create.
With Roger Penrose (top, middle) and Kip Thorne (bottom, far left), among others (above). With Roger and his wife, Vanessa (below). (illustration credit 11.1)
(illust
ration credit 11.2)
Adopting this definition as the footprint of a time machine has the advantage that one can use the machinery of causal structure that Roger Penrose and I developed to study singularities and black holes. Even without using the Einstein equations, I was able to show that, in general, a finitely generated Cauchy horizon will contain a closed light ray, or a light ray that keeps coming back to the same point over and over again. Moreover, each time the light comes around, it will be more and more blue-shifted, so the images will get bluer and bluer. The light rays may get defocused sufficiently each time round so that the energy of light doesn’t build up and become infinite. However, the blue shift will mean that a particle of light will have only a finite history, as defined by its own measure of time, even though it goes round and round in a finite region and does not hit a curvature singularity.
One might not care if a particle of light completes its history in a finite time. But I was also able to prove that there would be paths moving at less than the speed of light that had only finite duration. These could be the histories of observers who would be trapped in a finite region before the Cauchy horizon and would go round and round faster and faster until they reached the speed of light in a finite time.
So if a beautiful alien in a flying saucer invites you into her time machine, step with care. You might fall into one of these trapped repeating histories of only finite duration.
AS I said, these results depend not on the Einstein equations but only on the way space-time would have to warp to produce closed time-like curves in a finite region. However, one can now ask: What kind of matter would an advanced civilization need in order to warp space-time so as to build a finite-sized time machine? Can it have positive energy density everywhere, like in the cosmic string space-time? One might imagine that one could build a finite time machine using finite loops of cosmic string and have the energy density positive everywhere. I’m sorry to disappoint people wanting to return to the past, but it can’t be done with positive energy density everywhere. I proved that to build a finite time machine, you need negative energy.
In classical theory, all physically reasonable fields obey the weak energy condition, which says that the energy density for any observer is greater than or equal to zero. Thus time machines of finite size are ruled out in the purely classical theory. However, the situation is different in the semi-classical theory, in which one considers quantum fields on a classical space-time background. The uncertainty principle of quantum theory means that fields are always fluctuating up and down, even in apparently empty space. These quantum fluctuations make the energy density infinite. Thus one has to subtract an infinite quantity to get the finite energy density that is observed. Otherwise, the energy density would curve space-time up into a single point. This subtraction can leave the expectation value of the energy negative, at least locally. Even in flat space, one can find quantum states in which the expectation value of the energy density is negative locally, although the integrated total energy is positive.
One might wonder whether these negative expectation values actually cause space-time to warp in the appropriate way. But it seems they must. The uncertainty principle of quantum theory allows particles and radiation to leak out of a black hole. This causes the black hole to lose mass, thus evaporating slowly. For the horizon of the black hole to shrink in size, the energy density on the horizon must be negative and warp space-time to make light rays diverge from each other. If the energy density were always positive and warped space-time so as to bend light rays toward each other, the area of the horizon of a black hole could only increase with time.
The evaporation of black holes shows that the quantum energy momentum tensor of matter can sometimes warp space-time in the direction that would be needed to build a time machine. One might imagine, therefore, that some very advanced civilization could arrange that the expectation value of the energy density would be sufficiently negative to form a time machine that could be used by macroscopic objects.
But there’s an important difference between a black hole horizon and the horizon in a time machine, which contains closed light rays that keep going round and round. This would make the energy density infinite, which would mean that a person or a spaceship that tried to cross the horizon to get into the time machine would get wiped out by a bolt of radiation. This might be a warning from nature not to meddle with the past.
So the future looks black for time travel, or should I say blindingly white? However, the expectation value of the energy momentum tensor depends on the quantum state of the fields on the background. One might speculate that there could be quantum states where the energy density was finite on the horizon, and there are examples where this is the case. How you achieve such a quantum state, or whether it would be stable against objects crossing the horizon, we don’t know. But it might be within the capabilities of an advanced civilization.
This is a question that physicists should be free to discuss without being laughed at or scorned. Even if it turns out that time travel is impossible, it is important that we understand why it is impossible.
We don’t know much about the fully quantized theory of gravity. However, one might expect it to differ from the semi-classical theory only on the Planck length, a million billion billion billionth part of a centimeter. Quantum fluctuations on the background of space-time may well create wormholes and time travel on a microscopic scale, but according to the general theory of relativity, macroscopic bodies will not be able to return to their past.
Even if some different theory is discovered in the future, I don’t think time travel will ever be possible. If it were, we would have been overrun by tourists from the future by now.
12
IMAGINARY TIME
WHILE WE WERE AT CALTECH, WE VISITED SANTA Barbara, which is a two-hour drive up the coast. There I worked with my friend and collaborator Jim Hartle on a new way of calculating how particles would be emitted by a black hole, adding up all the possible paths the particle could take to escape from the hole. We found that the probability that a particle would be emitted by a black hole was related to the probability that a particle would fall into the hole, in the same way that the probabilities for emission and absorption were related for a hot body. This again showed that black holes behave as if they have a temperature and an entropy proportional to their horizon area.
Our calculation made use of the concept of imaginary time, which can be regarded as a direction of time at right angles to ordinary real time. When I returned to Cambridge I developed this idea further with two of my former research students, Gary Gibbons and Malcolm Perry. We replaced ordinary time with imaginary time. This is called the Euclidean approach, because it makes time become a fourth direction of space. It met with a lot of resistance at first but is now generally accepted as the best way to study quantum gravity. The Euclidean space of black hole time is smooth and contains no singularity at which the equations of physics would break down. It solved the fundamental problem that the singularity theorems of Penrose and I had raised: that predictability would break down because of the singularity. Using the Euclidean approach, we were able to understand the deep reasons why black holes behaved like hot bodies and had entropy. Gary and I also showed that a universe that was expanding at an ever-increasing rate would behave as if it had an effective temperature like that of a black hole. At the time we thought this temperature could never be observed, but its significance became apparent fourteen years later.
With Don Page (top, far left), Kip Thorne (bottom, third from left), and Jim Hartle (bottom, far right), among others (illustration credit 12.1)
I HAD been working mainly on black holes, but my interest in cosmology was renewed by the suggestion that the early universe had gone through a period of inflationary expansion. Its size would have grown at an ever-increasing rate, just as prices go up in the shops. In 1982, using Euclidean methods, I showed that such a universe would become slightly non-uniform. Similar results were ob
tained by the Russian scientist Viatcheslav Mukhanov about the same time, but that only became known later in the West.
These non-uniformities can be regarded as arising from thermal fluctuations due to the effective temperature in an inflationary universe that Gary Gibbons and I had discovered eight years earlier. Several other people later made similar predictions. I held a workshop in Cambridge, attended by all the major players in the field, and at this meeting we established most of our present picture of inflation, including the all-important density fluctuations that give rise to galaxy formation, and so to our existence.
This was ten years before the Cosmic Background Explorer (COBE) satellite recorded differences in the microwave background in different directions produced by the density fluctuations. So again, in the study of gravity, theory was ahead of experiment. These fluctuations were later confirmed by the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite, and were found to agree exactly with predictions.
THE ORIGINAL scenario for inflation was that the universe began with a Big Bang singularity. As the universe expanded, it was supposed somehow to get into an inflationary state. I thought this was an unsatisfactory explanation, because all equations would break down at a singularity, as previously discussed. But unless one knew what came out of the initial singularity, one could not calculate how the universe would develop. Cosmology would not have any predictive power. What was needed was a space-time without a singularity, like in the Euclidean version of a black hole.
AFTER THE workshop in Cambridge, I spent the summer at the Institute for Theoretical Physics, Santa Barbara, which had just been set up. I talked to Jim Hartle about how to apply the Euclidean approach to cosmology. According to the Euclidean approach, the quantum behavior of the universe is given by a Feynman sum over a certain class of histories in imaginary time. Because imaginary time behaves like another direction in space, histories in imaginary time can be closed surfaces, like the surface of the Earth, with no beginning or end.
My Brief History Page 6
