<br><h3> Chapter One </h3> <b>Classical Theory</b> <p> <i>S. W. Hawking</i> <p> <p> In these lectures, Roger Penrose and I will put forward our related but rather different viewpoints on the nature of space and time. We shall speak alternately and shall give three lectures each, followed by a discussion on our different approaches. I should emphasize that these will be technical lectures. We shall assume a basic knowledge of general relativity and quantum theory. <p> There is a short article by Richard Feynman describing his experiences at a conference on general relativity. I think it was the Warsaw conference in 1962. It commented very unfavorably on the general competence of the people there and the relevance of what they were doing. That general relativity soon acquired a much better reputation, and more interest, is in considerable measure due to Roger's work. Up to then, general relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance. However, Roger brought in modern concepts like spinors and global methods. He was the first to show that one could discover general properties without solving the equations exactly. It was his first singularity theorem that introduced me to the study of causal structure and inspired my classical work on singularities and black holes. <p> I think Roger and I pretty much agree on the classical work. However, we differ in our approach to quantum gravity and indeed to quantum theory itself. Although I'm regarded as a dangerous radical by particle physicists for proposing that there maybe loss of quantum coherence, I'm definitely a conservative compared to Roger. I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation. I think Roger is a Platonist at heart but he must answer for himself. <p> Although there have been suggestions that spacetime may have a discrete structure, I see no reason to abandon the continuum theories that have been so successful. General relativity is a beautiful theory that agrees with every observation that has been made. It may require modifications on the Planck scale, but I don't think that will affect many of the predictions that can be obtained from it. It may be only a low energy approximation to some more fundamental theory, like string theory, but I think string theory has been oversold. First of all, it is not clear that general relativity, when combined with various other fields in a supergravity theory, cannot give a sensible quantum theory. Reports of the death of supergravity are exaggerations. One year everyone believed that supergravity was finite. The next year the fashion changed and everyone said that supergravity was bound to have divergences even though none had actually been found. My second reason for not discussing string theory is that it has not made any testable predictions. By contrast, the straightforward application of quantum theory to general relativity, which I will be talking about, has already made two testable predictions. One of these predictions, the development of small perturbations during inflation, seems to be confirmed by recent observations of fluctuations in the microwave background. The other prediction, that black holes should radiate thermally, is testable in principle. All we have to do is find a primordial black hole. Unfortunately, there don't seem to be many around in this neck of the woods. If there had been, we would know how to quantize gravity. <p> Neither of these predictions will be changed even if string theory is the ultimate theory of nature. But string theory, at least at its current state of development, is quite incapable of making these predictions except by appealing to general relativity as the low energy effective theory. I suspect this may always be the case and that there may not be any observable predictions of string theory that cannot also be predicted from general relativity or supergravity. If this is true, it raises the question of whether string theory is a genuine scientific theory. Is mathematical beauty and completeness enough in the absence of distinctive observationally tested predictions? Not that string theory in its present form is either beautiful or complete. <p> For these reasons, I shall talk about general relativity in these lectures. I shall concentrate on two areas where gravity seems to lead to features that are completely different from other field theories. The first is the idea that gravity should cause spacetime to have a beginning and maybe an end. The second is the discovery that there seems to be intrinsic gravitational entropy that is not the result of coarse graining. Some people have claimed that these predictions are only artifacts of the semiclassical approximation. They say that string theory, the true quantum theory of gravity, will smear out the singularities and will introduce correlations in the radiation from black holes so that it is only approximately thermal in the coarse-grained sense. It would be rather boring if this were the case. Gravity would be just like any other field. But I believe it is distinctively different, because it shapes the arena in which it acts, unlike other fields which act in a fixed spacetime background. It is this that leads to the possibility of time having a beginning. It also leads to regions of the universe that one can't observe, which in turn gives rise to the concept of gravitational entropy as a measure of what we can't know. <p> In this lecture I shall review the work in classical general relativity that leads to these ideas. In my second and third lectures (Chapters 3 and 5) I shall show how they are changed and extended when one goes to quantum theory. My second lecture will be about black holes, and the third will be on quantum cosmology. <p> The crucial technique for investigating singularities and black holes that was introduced by Roger, and which I helped develop, was the study of the global causal structure of spacetime. Define <i>I</i><sup>+</sup><i>(p)</i> to be the set of all points of the spacetime M that can be reached from pby future-directed timelike curves (see fig. 1.1). One can think of <i>I</i><sup>+</sup><i>(p)</i> as the set of all events that can be influenced by what happens at p. There are similar definitions in which plus is replaced by minus and future by past. I shall regard such definitions as self-evident. <p> On now consider the boundary <i>I</i><sup>+</sup> <i>(S)</i> of the future of a set <i>S</i>. t is fairly easy to see that this boundary cannot b timelike. For in that case, a point <i>q</i> just outside the boundary would be to the future of a point <i>p</i> just inside, or can the boundary of the future be spacelike, except at the set <i>S</i> it If. For in that case every past-directed curve from a point <i>q</i>, just to the future of the boundary, would cross the boundary and leave the future of <i>S</i>. That would be a contradiction with the fact that <i>q</i> is in the future of <i>S</i> (fig. 1.2). <p> One therefore concludes that the boundary of the future is null apart from the set <i>S</i> itself. More precisely, if <i>q</i> is in the boundary of the future but is not in the closure of <i>S</i>, there is a past-directed null geodesic segment through <i>q</i> lying in the boundary (see fig. 1.3). There may be more than one null geodesic segment through q lying in the boundary, but in that case <i>q</i> will be a future endpoint of the segments. In other words, the boundary of the future of <i>S</i> is generated by null geodesics that have a future endpoint in the boundary and pass into the interior of the future if they intersect another generator. On the other hand, the null geodesic generators can have past endpoints only on <i>S</i>. It is possible, however, to have spacetimes in which there are generators of the boundary of the future of a set <i>S</i> that never intersect <i>S</i>. Such generators can have no past endpoint. <p> A simple example of this is Minkowski space with a horizontal line segment removed (see fig. 1.4). If the set <i>S</i> lies to the past of the horizontal line, the line will cast a shadow and there will be points just to the future of the line that are not in the future of <i>S</i>. There will be a generator of the boundary of the future of <i>S</i> that goes back to the end of the horizontal line. However, as the endpoint of the horizontal line has been removed from spacetime, this generator of the boundary will have no past endpoint. This spacetime is incomplete, but one can cure this by multiplying metric by a suitable conformal factor near the end of the horizontal line. Although spaces like this are very artificial, they are important in showing how careful you have to be in the study of causal structure. In fact, Roger Penrose, who was one of my Ph.D. examiners, pointed out that a space like then one I just described was a counterexample to some of the claims I made in my thesis. <p> To show that each generator of the boundary of the future has a past endpoint on the set, one has to impose some global condition on the causal structure. The strongest and physically most important condition is that of global hyperbolicity. An open set <i>U</i> is said to be globally hyperbolic if <p> 1. For every pair of points <i>p</i> and <i>q</i> in <i>U</i> the intersection of the future of <i>p</i> and the past of <i>q</i> has compact closure. In other words, it is a bounded diamond shaped region (fig. 1.5). <p> 2. Strong causality holds on <i>U</i>. That is there are no closed or almost closed timelike curves contained in <i>U</i>. <p> <p> The physical significance of global hyperbolicity comes from the fact that it implies that there is a family of Cauchy surfaces Σ<i>(t)</i> for <i>U</i> (see fig. 1.6). A Cauchy surface for <i>U</i> is a spacelike or null surface that intersects every timelike curve in <i>U</i> once and once only. One can predict what will happen in <i>U</i> from data on the Cauchy surface, and one can formulate a well-behaved quantum field theory on a globally hyperbolic background. Whether one can formulate a sensible quantum field theory on a nonglobally hyperbolic background is less clear. So global hyperbolicity may be a physical necessity. But my viewpoint is that one shouldn't assume it because that may be ruling out something that gravity is trying to tell us. Rather, one should deduce that certain regions of spacetime are globally hyperbolic from other physically reasonable assumptions. <p> The significance of global hyperbolicity for singularity theorems stems from the following. Let <i>U</i> be globally hyperbolic and let <i>p</i> and <i>q</i> be points of <i>U</i> that can be joined by a timelike or null curve. Then there is a timelike or null geodesic between <i>p</i> and <i>q</i> which maximizes the length of timelike or null curves from <i>p</i> to <i>q</i> (fig. 1.7). The method of proof is to show that the space of all timelike or null curves from <i>p</i> to <i>q</i> is compact in a certain topology. One then shows that the length of the curve is an upper semicontinuous function on this space. It must therefore attain its maximum, and the curve of maximum length will be a geodesic because otherwise a small variation will give a longer curve. <p> One can now consider the second variation of the length of a geodesic γ. One can show that y can be varied to a longer curve if there is an infinitesimally neighboring geodesic from <i>p</i> which intersects γ again at a point <i>r</i> between <i>p</i> and <i>q</i>. The point <i>r</i> is said to be conjugate to <i>p</i> (fig. 1.8). Onecan illustrate this by considering two points <i>p</i> and <i>q</i> on the surface of the Earth. Without loss of generality, one can take <i>p</i> to be at the north pole. Because the Earth has a positive definite metric rather than a Lorentzian one, there is a geodesic of minimal length, rather than a geodesic of maximum length. This minimal geodesic will be a line of longitude running from the north pole to the point <i>q</i>. But there will be another geodesic from <i>p</i> to <i>q</i> which runs down the back from the north pole to the south pole and then up to <i>q</i>. This geodesic contains a point conjugate to <i>p</i> at the south pole where all the geodesics from <i>p</i> intersect. Both geodesics from <i>p</i> to <i>q</i> are stationary points of the length under a small variation. But now in a positive definite metric the second variation of a geodesic containing a conjugate point can give a shorter curve from <i>p</i> to <i>q</i>. Thus, in the example of the Earth, we can deduce that the geodesic that goes down to the south pole and then comes up is not the shortest curve from <i>p</i> to <i>q</i>. This example is very obvious. However, in the case of spacetime one can show that under certain assumptions there ought to be a globally hyperbolic region in which there should be conjugate points on every geodesic between two points. This establishes a contradiction which shows that the assumption of geodesic completeness, which can be taken as a definition of a nonsingular spacetime, is false. <p> The reason one gets conjugate points in spacetime is that gravity is an attractive force. It therefore curves spacetime in such a way that neighboring geodesics are bent toward each other rather than away. One can see this from the Raychaudhuri or Newman-Penrose equation, which I will write in a unified form. <p> Here <i>v</i> is an affine parameter along a congruence of geodesics with tangent vector <i>l<sup>a</sup></i> which is hypersurface orthogonal. The quantity p is the average rate of convergence of the geodesics, while (J measures the shear. The term <i>R<sub>ab</sub>l<sup>a</sup>l<sup>b</sup></i> gives the direct gravitational effect of the matter on the convergence of the geodesics. <p> By the Einstein equations, it will be nonnegative for any null vector <i>l<sup>a</sup></i> if the matter obeys the so-called weak energy condition. This says that the energy density Too is nonnegative in any frame. The weak energy condition is obeyed by the classical energy momentum tensor of any reasonable matter, such as a scalar or electromagnetic field or a fluid with a reasonable equation of state. It may not, however, be satisfied locally by the quantum mechanical expectation value of the energy momentum tensor. This will be relevant in my second and third lectures (chapters 3 and 5). <p> Suppose the weak energy condition holds, and that the null geodesics from a point <i>p</i> begin to converge again and that ρ has the positive value ρ<sub>0</sub>. Then the Newman-Penrose equation would imply that the convergence ρ would become infinite at a point <i>q</i> within an affine parameter distance 1/ρ<sub>0</sub> if the null geodesic can be extended that far. <p> Infinitesimally neighboring null geodesics from <i>p</i> will intersect at <i>q</i>. This means the point <i>q</i> will be conjugate to <i>p</i> along the null geodesic γ joining them. For points on γ beyond the conjugate point <i>q</i> there will be a variation of γ that gives a timelike curve from <i>p</i>. Thus γ cannot lie in the boundary of the future of <i>p</i> beyond the conjugate point <i>q</i>. So γ will have a future endpoint as a generator of the boundary of the future of <i>p</i> (fig. 1.9). <p> The situation with timelike geodesics is similar, except that the strong energy condition that is required to make <i>R<sub>ab</sub>l<sup>a</sup>l<sup>b</sup></i> nonnegative for every timelike vector <i>l<sup>a</sup></i> is, as its name suggests, rather stronger. It is still, however, physically reasonable, at least in an averaged sense, in classical theory. If the strong energy condition holds, and the timelike geodesics from <i>p</i> begin converging again, then there will be a point <i>q</i> conjugate to <i>p</i>. <p> Finally, there is the generic energy condition. This says that first the strong energy condition holds. Second, every timelike or null geodesic encounters some point where there is some curvature that is not specially aligned with the geodesic. The generic energy condition is not satisfied by a number of known exact solutions. But these are rather special. One would expect it to be satisfied by a solution that was 1/generic" in an appropriate sense. If the generic energy condition holds, each geodesic will encounter a region of gravitational focussing. This will imply that there are pairs of conjugate points if one can extend the geodesic far enough in each direction. <p> One normally thinks of a spacetime singularity as a region in which the curvature becomes unboundedly large. However, the trouble with that as a definition is that one could simply leave out the singular points and say that the remaining manifold was the whole of spacetime. It is therefore better to define spacetime as the maximal manifold on which the metric is suitably smooth. One can then recognize the occurrence of singularities by the existence of incomplete geodesics that cannot be extended to infinite values of the affine parameter. <p> <i>(Continues...)</i> <p> <p> <!-- copyright notice --> <br></pre> <blockquote><hr noshade size='1'><font size='-2'> Excerpted from <b>The Nature of Space and Time</b> by <b>Stephen Hawking Roger Penrose</b> Copyright © 1996 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.<br>Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.