INTRODUCTION.
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 aestheitc 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:
"In the classical theory of general relativity, there are many different possible curved space-time, each corresponding to a different initial state of the universe. If we knew the initial state of our universe, we would know its entire history. Similarly, in the quantum theory of gravity, there are many different possible quantum states of the universe. Again, if we knew how the Euclidean curved space-time in the sum over histories behaved at early times, we would know the quantum state of the universe."In the classical theory of gravity, which is based on real space-time, 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. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is using Euclidean space-times, in which the time direction is on the same footing as the direction in space, 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. The surface of the earth is finite in extent but it doesn't have a boundary or edge: if one sails off into the sunset, he will not fall off the edge or run into a singularity.
"If Euclidean space-time stretches back to infinite imaginary time, or else starts at a singularity in imaginary time, we have the same problem as in the classical theory of specifying the initial state of the universe: God may know how the universe began, but we cannot give any particular reason for thinking it began one way rather than another. On the other hand, the quantum theory of gravity has opened up a new possibility, in which there would be no boundary to space-time and so there would be no need to specify the behavior at the boundary. There would be no singularity at which the laws of science broke down and no edge of space-time at which one would have to appeal to God or some new law to set the boundary conditions for space-time. One 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 of itself. It would neither be created nor destroyed. It would just BE." [3]
Hawking also writes:
"If the universe really is in such a quantum state, there would be no singularities in the history of the universe in imaginary time. It might seem therefore that my more recent work had completely undone the results of my earlier work on singularities. But, as indicated above, the real importance of the singularity theorems was that they showed that the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turn led to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When one goes back to real time in which we live, however, there will still appear to be singularities. The poor astronaut who falls into a black hole still will come to a sticky end; only if he lived in imaginary time would he encounter no singularities.So science does not provide us with the truth, but only with that which is useful. Although Hawking believes that knowledge of the origin of universe is unattainable, he is not an atheist. He emphatically rejects the label "atheist." His view of God comes closer to the position that is called "deism"; that is, the view that after God created the universe, He left it to operate according to the laws that He established. In his A Brief History of Time, Hawking writes,"This might suggest that the so-called imaginary time is real time; and that what we call real time is just a figment of our imaginationa. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic; and what we call real is just an idea that we invent to help us describe what we think the universe like. But, according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: Which is real, 'real' or 'imaginary' time? It is simply a matter of which is the more useful description." [4]
"Science seems to have uncovered a set of laws that, within the limits set by the uncertainty principle, tells us how the universe will develop with time, if we know its state at any one time. These laws may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them and does not now intervene in it." [5]Hawking goes on to conclude at the end of chapter 8 of his book:
"The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break those laws. However, those laws do not tell us what the universe should have looked like when it started -- it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?" [6]Hawking rejects this conclusion and believes that the universe had a creator.
"It would be very difficult to explain why the universe should have begun in just this way, except as the act of a God who intended to create beings like us." [7]
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ENDNOTES.
[1] Stephen W. Hawkings; A Brief History of Time:
From the Big Bang to Black Holes
(New York: Bantam Books, April, 1988), p. 116.