ingularities, but it would be very difficult even to formulate lawsat such badly behaved points and we would have no guide from observations asto what those laws might be. However, what the singularity theorems reallyindicate is that the gravitational field becomes so strong that quantum gravita-tional effects become important: Classical theory is no longer a good descrip-tion of the universe. So one has to use a quantum theory of gravity to discussthe very early stages of the universe. As we shall see, it is possible in the quan-tum theory for the ordinary laws of science to hold everywhere, including at thebeginning of time. It is not necessary to postulate new laws for singularities,because there need not be any singularities in the quantum theory.We don’t yet have a complete and consistent theory that combines quantummechanics and gravity. However, we are thoroughly certain of some featuresthat such a unified theory should have. One is that it should incorporateFeynman’s proposal to formulate quantum theory in terms of a sum over histo-ries. In this approach, a particle going from A to B does not have just a singlehistory as it would in a classical theory. Instead, it is supposed to follow everypossible path in space-time. With each of these histories, there are associateda couple of numbers, one representing the size of a wave and the other repre-senting its position in the cycle-its phase.The probability that the particle, say, passes through some particular point isfound by adding up the waves associated with every possible history thatpasses through that point. When one actually tries to perform these sums,however, one runs into severe technical problems. The only way around theseis the following peculiar prescription: One must add up the waves for particlehistories that are not in the real time that you and I experience but take placein imaginary time.Imaginary time may sound like science fiction, but it is in fact a well-definedmathematical concept. To avoid the technical difficulties with Feynman’s sumover histories, one must use imaginary time. This has an interesting effect onspace-time: The distinction between time and space disappears completely. Aspace-time in which events have imaginary values of the time coordinate issaid to be Euclidean because the metric is positive definite.In Euclidean space-time there is no difference between the time direction anddirections in space. On the other hand, in real space-time, in which events arelabeled by real values of the time coordinate, it is easy to tell the difference. Thetime direction lies within the light cone, and space directions lie outside. Onecan regard the use of imaginary time as merely a mathematical device-ortrick-to calculate answers about real space-time. However, there may be moreto it than that. It may be that Euclidean space-time is the fundamental conceptand what we think of as real space-time is just a figment of our imagination.When we apply Feynman’s sum over histories to the universe, the analogue ofthe history of a particle is now a complete curved space-time which representsthe history of the whole universe. For the technical reasons mentioned above,these curved space-times must be taken to be Euclidean. That is, time isimaginary and is indistinguishable from directions in space. To calculate theprobability of finding a real space-time with some certain property, one addsup the waves associated with all the histories in imaginary time that have thatproperty. One can then work out what the probable history of the universewould be in real time.THE NO BOUNDARY CONDITIONIn the classical theory of gravity, which is based on real space-time, there areonly two possible ways the universe can behave. Either it has existed for an infi-nite time, or else it had a beginning at a singularity at some finite time in thepast. In fact, the singularity theorems show it must be the second possibility. Inthe 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 onthe same footing as directions in space, it is possible for space-time to be finitein 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 dimen-sions. The surface of the Earth is finite in extent but it doesn’t have a boundaryor edge. If you sail off into the sunset, you don’t fall off the edge or run into asingularity. I know, because I have been around the world.If Euclidean space-times direct back to infinite imaginary time or else startedat a singularity, we would have the same problem as in the classical theory ofspecifying the initial state of the universe. God may know how the universebegan, but we cannot give any particular reason for thinking it began one wayrather than another. On the other hand, the quantum theory of gravity hasopened up a new possibility. In this, there would be no boundary tospace-time. Thus, there would be no need to specify the behavior at theboundary. There would be no singularities at which the laws of science brokedown and no edge of space-time at which one would have to appeal to God orsome 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 uni-verse would be completely self-contained and not affected by anything outsideitself. It would be neither created nor destroyed. It would just be.It was at the conference in the Vatican that I first put forward the suggestionthat maybe time and space together formed a surface that was finite in size butdid not have any boundary or edge. My paper was rather mathematical, how-ever, so its implications for the role of God in the creation of the universe werenot noticed at the time-just as well for me. At the time of the Vatican confer-ence, I did not know how to use a no boundary idea to make predictions aboutthe universe. However, I spent the following summer at the University ofCalifornia, Santa Barbara. There, a friend and colleague of mine, Jim Hartle,worked out with me what conditions the universe must satisfy if space-timehad no boundary.I should emphasize that this idea that time and space should be finite withoutboundary is just a proposal. It cannot be deduced from some other principle.Like any other scientific theory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makes predictions thatagree with observation. This, however, is difficult to determine in the case ofquantum gravity, for two reasons. First, we are not yet sure exactly which the-ory successfully combines general relativity and quantum mechanics, thoughwe know quite a lot about the form such a theory must have. Second, anymodel that described the whole universe in detail would be much too compli-cated mathematically for us to be able to calculate exact predictions. Onetherefore has to make approximations-and even then, the problem ofextracting predictions remains a difficult one.One finds, under the no boundary proposal, that the chance of the universebeing found to be following most of the possible histories is negligible. Butthere is a particular family of histories that are much more probable than theothers. These histories may be pictured as being like the surface of the Earth,with a distance from the North Pole representing imaginary time; the size of acircle of latitude would represent the spatial size of the universe. The universestarts at the North Pole as a single point. As one moves south, the circle of lat-itude get bigger, corresponding to the universe expanding with imaginary time.The universe would reach a maximum size at the equator and would contractagain to a single point at the South Pole. Even though the universe wouldhave zero size at the North and South poles, these points would not be singu-larities any more than the North and South poles on the Earth are singular.The laws of science will hold at the beginning of the universe, just as they doat the North and South poles on the Earth.The history of the universe in real time, however, would look very different. Itwould appear to start at some minimum size, equal to the maximum size of thehistory in imaginary time. The universe would then expand in real time likethe inflationary model. However, one would not now have to assume that theuniverse was created somehow in the right sort of state. The universe wouldexpand to a very large size, but eventually it would collapse again into whatlooks like a singularity in real time. Thus, in a sense, we are still all doomed,even if we keep away from black holes. Only if we could picture the universein terms of imaginary time would there be no singularities.The singularity theorems of classical general relativity showed that the uni-verse must have a beginning, and that this beginning must be described interms of quantum theory. This in turn led to the idea that the universe couldbe finite in imaginary time, but without boundaries or singularities. When onegoes back to the real time in which we live, however, there will still appear tobe singularities. The poor astronaut who falls into a black hole will still cometo a sticky end. It is only if he could live in imaginary time that he wouldencounter no singularities.This might suggest that the so-called imaginary time is really the fundamen-tal time, and that what we call real time is something we create just in ourminds. In real time, the universe has a beginning and an end at singularitiesthat form a boundary to space-time and at which the laws of science breakdown. But in imaginary time, there are no singularities or boundaries. Somaybe what we call imaginary time is really more basic, and what we call realtime is just an idea that we invent to help us describe what we think the uni-verse is like. But according to the approach I described in the first lecture, ascientific theory is just a mathematical model we make to describe our obser-vations. It exists only in our minds. So it does not have any meaning to ask:Which is real, “real” or “imaginary” time? It is simply a matter of which is amore useful description.The no boundary proposal seems to predict that, in real time, the universeshould behave like the inflationary models. A particularly interesting problemis the size of the small departures from uniform density in the early universe.These are thought to have led to the formation first of the galaxies, then ofstars, and finally of beings like us. The uncertainty principle implies that theearly universe cannot have been completely uniform. Instead, there must havebeen some uncertainties or fluctuations in the positions and velocities of theparticles. Using the no boundary condition, one finds that the universe musthave started off with just the minimum possible nonuniformity allowed by theuncertainty principle.The universe would have then undergone a period of rapid expansion, like inthe inflationary models. During this period, the initial nonuniformities wouldhave been amplified until they could have been big enough to explain the ori-gin of galaxies. Thus, all the complicated structures that we see in the universemight be explained by the no boundary condition for the universe and theuncertainty principle of quantum mechanics.The idea that space and time may form a closed surface without boundary alsohas profound implications for the role of God in the affairs of the universe.With the success of scientific theories in describing events, most people havecome to believe that God allows the universe to evolve according to a set oflaws. He does not seem to intervene in the universe to break these laws.However, the laws do not tell us what the universe should have looked likewhen it started. It would still be up to God to wind up the clockwork andchoose how to start it off. So long as the universe had a beginning that was asingularity, one could suppose that it was created by an outside agency. But ifthe universe is really completely self-contained, having no boundary or edge,it would be neither created nor destroyed. It would simply be. What place,then, for a creator?