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Einstein’s general theory of relativity when applied to the whole universe, predicted that space-time began at the big bang singularity and would come to an end either as the big crunch singularity (if the whole universe collapsed), or at a singularity inside of a black hole (if a local region, such as the sun, were to collapse). Any matter that fell into the hole would be destroyed at the singulariy, and only the gravitational effect of its mass would continue to be felt outside. On the other hand, when quantum effects were taken into account, it seemed that the mass or energy of the matter eventually be returned to the rest of the universe, and that the black hole, along with any singularity inside of it, would evaporate away and finally disappear. Could quantum mechanics have an equally dramatic effect on the big bang and and big crunch singularities? What really happened during the very early stages of the universe, when gravitational fields are so strong that quantum effects cannot be ignored? Does the universe in fact have a beginning or an end? And if so, what are they like?

Throughout the 1970s, Stephen Hawkings had been mainly studying black holes, but in 1981 his interest in the questions about the origin and fate of the universe was reawakened when he attended a conference on cosmology organized by the Jesuits in the Vatican. The Roman Catholic Church had with Galileo tried to lay down the law on questions of science, declaring that the sun went around the earth. Now, centuries later, the Roman Church decided to invite a number of experts, including Hawking, to advise them on cosmology. The Roman Catholic Church had seized on the big bang model and in 1951 they had officially pronounced it to be in accordance with the Bible. At the end of the conference in 1981, the participants were granted an audience with the pope. The pope told them that it was all right for them to study the evolution of the universe after the big bang, but that they should not inquire into the big bang itself, because that was the moment of Creation and therefore the work of God. Hawking had given during the conference a lecture in which he presented the possibility that space-time was finite but had no boundary, which meant that it had no beginning, no moment of Creation. Hawking wrote,

“I had no desire to share the fate of Galileo, with whom I had a strong sense of identity, partly because of the coincidence of having been born exactly 300 years after his death!” [1]

His paper was rather mathematical so that its implications for the role of God in the Creation of the universe were not generally recognized at the time. At the time of the conference, Hawking did not know how to use the “no boudary” idea to make predictions about the universe. But when he returned from the conference, Hawkings spent the following summer at the University of California, Santa Barbara, where a friend and colleage of his, Jim Hartle, worked with him on what conditions the universe must satisfy if space-time had no boundary. When he returned to Cambridge, Hawkings continued this work with two of his research assistants, Julian Lutrel and Jonathan Halliwell. [2]

In his paper given at the Vatical conference, Hawking put forth the suggestion that maybe time and space together formed a surface of finite size but did not have any boundary or edge. Hawking put forward this concept just as a proposal because it cannot be deduced from some other principle. And like any other scientific theory, it may be put forward for aesthetic or metaphysical reasons, but the real test is whether it makes predictions that agree with experiment or observation. But this is difficult to determine in the case of quantum gravity, for two reasons. First, we are not sure which theory exactly successfully combines general relativity and quantum mechanics, though we know quite a lot about the form such a theory must have. Second, any model that describes the whole universe in detail would be complicated mathematically for us to be able to calculate exact predictions. Therefore, the scientist has to make simplifying assumptions and approximations; but even then the problem of extracting predictions remain a formidable one.

In order to predict how the universe should start off, one needs to know what laws hold at the beginning of time. If the accepted theory of general relativity is correct, then the singularity theorem that Roger Penrose and Stephen Hawking proved shows that the beginning of time would have been a point of infinite density and infinite curvature of space-time. All known laws of science would break down at such a point. What the singularity theorem really indicated is that the gravity field becomes so strong that quantum gravitational effects would become important: the general theory of relativity is no longer a good description of the universe. So one has to use the quantum theory of gravity to discuss the very early stages of the universe. But it is not necessary to postulate new laws for the singularities, because there need not be any singularities in quantum theory. In the quantum theory, the ordinary laws of science will hold everywhere, including at the beginning of time.

Scientists don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. But they are fairly certain of some features that such a unified theory should have. One feature is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does not have just a single history, as it would in classical theory. Instead, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a couple of numbers, one representing the size of wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. But when one actually performs these sums, one runs into severe technical problems. The only way around these is the following peculiar presecription: One must add up the waves for particle histories that are not in the “real” time that we experience but take place in what is called imaginary time.

Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. Consider ordinary (or “real”) numbers: if we multiply a real number by itself, the result is a positive number. For example, 2 times 2 is 4, but so is -2 times -2. But there are special numbers (called “imaginary”) that give negative numbers when multiplied by themselves. There is one special number called the imaginary unit and symbolized by letter i when multiplied by itself, gives -1; it is equal to the square root of -1, so that

√-1 · √-1 = -1.

Thus 2i multiplied by itself gives -4, and so on.   To avoid technical difficulties with Feynman’s sum over histories, one must use imaginary time. That is, for purposes of the calculation, one must measure time using imaginary numbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid, who founded the study of geometry of two-dimensional surfaces. What is being now called here Euclidean space-time is very similar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary real values of the time coordinate, it is easy to tell the difference: the time direction at all points lies within the light cone, and space directions lie outside. In any case, as far as everyday quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate answers about real-time.

A second feature that Hawking believes must be part of any ultimate theory is Einstein’s idea that the gravitationsl field is represented by curved space-time: particles try to follow the nearest thing to a straight path in curved space, but because space-time is not flat their paths appear to be bent, as if by a gravitational field. When Feynman’s sum over histories is applied 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. To avoid the technical difficulties in actually performing the sum over histories, these curved space-times must be taken to be Euclidean. That is, time is imaginary and is indistinguishable from directions in space. To calculate the probability of finding a real space-time with some certain property, such as looking the same at every point and in every direction, one adds up the waves associated with all the histories that have that property.  Hawking writes: