Introduction
Introduction
The cosmological argument is a family of arguments that seek to demonstrate the existence of a Suffi cient Reason or First Cause of the existence of the cosmos. The roll of the defenders of this argument reads like a Who’s Who of Western philosophy: Plato, Aristotle, ibn Sina, al-Ghazali, Maimonides, Anselm, Aquinas, Scotus, Descartes, Spinoza, Leibniz, and Locke, to name but some. Cosmological arguments can be conveniently grouped into three basic types: the kalam cosmological argument for a First Cause of the beginning of the universe; the Thomist cosmological argument for a sustaining Ground of Being of the world; and the Leibnizian cosmological argument for a Suffi cient Reason why something exists rather than nothing.1
The kalam cosmological argument traces its roots to the efforts of early Christian theologians who, out of their commitment to the biblical teaching of creatio ex nihilo, sought to rebut the Aristotelian doctrine of the eternity of the universe. In his works Against Aristotle and On the Eternity of the World against Proclus, the Alexandrian Aristotelian commentator John Philoponus (d. 580?), the last great champion of creatio ex nihilo prior to the advent of Islam, initiated a tradition of argumentation in support of the doctrine of creation based on the impossibility of an infi nite temporal regress of events (Philoponus 1987; Philoponus & Simplicius 1991). Following the Muslim conquest of North Africa, this tradition was taken up and subsequently enriched by medieval Muslim and Jewish theologians before being transmitted back again into Christian scholastic theology.2
In light of the central role played by this form of the cosmological argument in medieval Islamic theology, as well as the substantive contribution to its development by its medieval Muslim proponents, we use the word “kalam” to denominate this version of the argument.
The Arabic word for speech, kalam was used by Muslim thinkers to denote a statement of 1. This typology has become somewhat standard ( Routledge Encyclopedia of Philosophy 1998; cf. Stanford Encyclopedia of Philosophy 2004a).
2. For an exposition of the argument in its historical context, see Craig (1980), Wolfson (1966), Wolfson (1976), Davidson (1987), and Dales (1990).
The Blackwell Companion to Natural Theology Edited William Lane Craig and J. P. Moreland
© 2009 Blackwell Publishing Ltd. ISBN: 978-1-405-17657-6
102 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
theological doctrine and eventually a statement of any intellectual position or an argument supporting such a position. According to the fourteenth-century Muslim theologian al-Idji, kalam is “the science which is concerned with fi rmly establishing religious beliefs by adduc-ing proofs and banishing doubts” (al-Idji 1971). Ultimately, kalam became the name of the whole movement within Muslim thought that might best be described as Islamic scholasticism. A practitioner of kalam is called a mutakallim (pl. mutakallimun). Jewish theologians in Muslim Spain, who rubbed shoulders both with the Arabic East and the Latin West, were the means by which the kalam cosmological argument found its way back into Christian thought. The subject of extended debate, the argument pitted al-Ghazali against ibn Rushd, Saadia ben Gaon against Maimonides, and Bonaventure against Aquinas. The debate was eventually enshrined during the modern era in the thesis and antithesis of Kant’s First Antinomy concerning time.
After suffering several centuries of eclipse, the argument has enjoyed a resurgence of interest in recent decades, doubtlessly spurred by the startling empirical evidence of contemporary astrophysical cosmology for a beginning of space and time. Kalam philosophical argumentation for the fi nitude of the past played a key role in the philosophy of time propounded by mathematician and cosmologist G. J. Whitrow (1980). As a piece of natural theology, the kalam argument was revived by Stuart Hackett in his little-noted The Resurrection of Theism (1957) and subsequently brought into philosophical prominence by his student William Lane Craig (1979). Noting the widespread debate over the argument today, Quentin Smith observes, “The fact that theists and atheists alike ‘cannot leave [the] Kalam argument alone’ suggests that it may be an argument of unusual philosophical interest or else has an attractive core of plausibility that keeps philosophers turning back to it and examining it once again” (Smith 2007, p. 183).
What is the kalam cosmological argument? In his Kitab al-Iqtisad, the medieval Muslim theologian al-Ghazali presented the following simple syllogism in support of the existence of a Creator: “Every being which begins has a cause for its beginning; now the world is a being which begins; therefore, it possesses a cause for its beginning”
(al-Ghazali 1962, pp. 15–6). In defense of the second premise, Ghazali offered various philosophical arguments to show the impossibility of an infi nite regress of temporal phenomena and, hence, of an infi nite past. The limit at which the fi nite past terminates Ghazali calls “the Eternal” (al-Ghazali 1963, p. 32), which he evidently takes to be a state of timelessness. Given the truth of the fi rst premise, the fi nite past must, therefore, “stop at an eternal being from which the fi rst temporal being should have originated” (al-Ghazali 1963, p. 33).
The argument, then, is extremely simple:3
1.0. Everything that begins to exist has a cause.
2.0. The universe began to exist.
3.0. Therefore, the universe has a cause.
3. In view of the bewildering variety of complex issues raised by cosmological arguments, Oppy cautions that even before looking at it, we should conclude that so simple an argument cannot plausibly be held to establish its conclusion (Oppy 2006b, p. 173). But the complexity of the issues involved in assessing the truth of an argument’s premises does not require, pace Oppy, that the argument itself have “dozens of complex premises.” The supporting arguments and responses to defeaters of the argument’s two basic premises can proliferate in an almost fractal-like fashion.
KA LA M C O S M O LO G I C A L A RG U M E N T 103
As a fi nal step one may explore the relevance of this conclusion for theism by means of a conceptual analysis of what it is to be a cause of the universe. In the sequel we shall examine each of the steps of the argument, beginning with Premise (2.0), since this is clearly the more controversial claim and since some attempts to subvert (1.0) are based upon cosmogonic theories – the discussion of which would be premature prior to their introduction in our treatment of (2.0).
2.0. Did the Universe Begin to Exist?
The crucial second premise of the kalam cosmological argument has been supported by both metaphysical and physical arguments. We shall examine two traditional philosophical arguments against the existence of an infi nite temporal regress of events, as well as scientifi c evidence in support of an absolute beginning of the universe.
2.1. Argument from the impossibility of an actual infi nite
One of the traditional arguments for the fi nitude of the past is based upon the impossibility of the existence of an actual infi nite. It may be formulated as follows: 2.11. An actual infi nite cannot exist.
2.12. An
infi nite temporal regress of events is an actual infi nite.
2.13. Therefore, an infi nite temporal regress of events cannot exist.
In order to assess this argument, we need to have a clear understanding of its key terms.
First and foremost among these is “actual infi nite.” Prior to the revolutionary work of mathematicians Bernard Bolzano (1781–1848), Richard Dedekind (1831–1916), and, especially, Georg Cantor (1845–1918), there was no clear mathematical understanding of the actual infi nite (Moore 1990, pt. I). Aristotle had argued at length that no actually infi nite magnitude can exist ( Physics 3.5.204b1–206a8). The only legitimate sense in which one can speak of the infi nite is in terms of potentiality: something may be infi nitely divisible or susceptible to infi nite addition, but this type of infi nity is potential only and can never be fully actualized ( Physics 8.8. 263a4–263b3). The concept of a potential infi nite is a dynamic notion, and strictly speaking, we must say that the potential infi nite is at any particular time fi nite.
This understanding of the infi nite prevailed all the way up to the nineteenth century.
But although the majority of philosophers and mathematicians adhered to the conception of the infi nite as an ideal limit, dissenting voices could also be heard. Bolzano argued vigorously against the then current defi nitions of the potential infi nite (Bolzano 1950, pp. 81–4).
He contended that infi nite multitudes can be of different sizes and observed the resultant paradox that although one infi nite might be larger than another, the individual elements of the two infi nites could nonetheless be matched against each other in a one-to-one correspondence (Bolzano 1950, pp. 95–6).4 It was precisely this paradoxical notion that Dedekind seized upon in his defi nition of the infi nite: a system is said to be infi nite if a part of 4. Despite the one-to-one correspondence, Bolzano insisted that two infi nites so matched might nevertheless be nonequivalent.
104 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
that system can be put into a one-to-one correspondence with the whole (Dedekind 1963, p. 63). According to Dedekind, the Euclidean maxim that the whole is greater than a part holds only for fi nite systems.
But it was undoubtedly Cantor who won for the actual infi nite the status of mathematical legitimacy that it enjoys today. Cantor called the potential infi nite a “variable fi nite” and attached the sign ∞ (called a lemniscate) to it; this signifi ed that it was an “improper infi -
nite” (Cantor 1915, pp. 55–6). The actual infi nite he pronounced the “true infi nite” and assigned the symbol ℵ0 (aleph zero) to it. This represented the number of all the numbers in the series 1, 2, 3, . . . and was the fi rst infi nite or transfi nite number, coming after all the fi nite numbers. According to Cantor, a collection or set is infi nite when a part of it is equivalent to the whole (Cantor 1915, p. 108). Utilizing this notion of the actual infi nite, Cantor was able to develop a whole system of transfi nite arithmetic. “Cantor’s . . . theory of transfi nite numbers . . . is, I think, the fi nest product of mathematical genius and one of the supreme achievements of purely intellectual human activity,” exclaimed the great German mathematician David Hilbert. “No one shall drive us out of the paradise which Cantor has created for us” (Hilbert 1964, pp. 139, 141).
Modern set theory, as a legacy of Cantor, is thus exclusively concerned with the actual as opposed to the potential infi nite. According to Cantor, a set is a collection into a whole of defi nite, distinct objects of our intuition or of our thought; these objects are called elements or members of the set. Fraenkel draws attention to the characteristics defi nite and distinct as particularly signifi cant (Fraenkel 1961, p. 10). That the members of a set are distinct means that each is different from the others. To say that they are defi nite means that given a set S, it should be intrinsically settled for any possible object x whether x is a member of S or not. This does not imply actual decidability with the present or even future resources of experience; rather a defi nition could settle the matter suffi ciently, such as the defi nition for “transcendental” in the set of all transcendental numbers.
Unfortunately, Cantor’s notion of a set as any logical collection was soon found to spawn various contradictions or antinomies within the naive set theory that threatened to bring down the whole structure. As a result, most mathematicians have renounced a defi nition of the general concept of set and chosen instead an axiomatic approach to set theory, by means of which the system is erected upon several given, undefi ned concepts formulated into axioms. An infi nite set in the Zermelo–Fraenkel axiomatic set theory is defi ned as any set R that has a proper subset that is equivalent to R. A proper subset is a subset that does not exhaust all the members of the original set, that is to say, at least one member of the original set is not also a member of the subset. Two sets are said to be equivalent if the members of one set can be related to the members of the other set in a one-to-one correspondence, that is, so related that a single member of the one set corresponds to a single member of the other set and vice versa. Equivalent sets are regarded as having the same number of members. This convention has recently been dubbed as Hume’s Principle5 (on the basis of Hume 1978, bk, I, pt. iii, sec. 1, p. 71). An infi nite set, then, is one that is such that the whole set has the same number of members as a proper subset. In contrast to this, a fi nite set is a set that is such that if n is a positive integer, the set has n members. Because set theory does not utilize the notion of potential infi nity, a set containing a potentially infi nite number of members is impossible. Such a collection would be one in which the 5. The appellation is due to Boolos (1986–7).
KA LA M C O S M O LO G I C A L A RG U M E N T 105
membership is not defi nite in number but may be increased without limit. It would best be described as indefi nite. The crucial difference between an infi nite set and an indefi nite collection would be that the former is conceived as a determinate whole actually possessing an infi nite number of members, while the latter never actually attains infi nity, although it increases perpetually. We have, then, three types of collection that we must keep conceptually distinct: fi nite, infi nite, and indefi nite.
When we use the word “exist,” we mean “be instantiated in the mind-independent
world.” We are inquiring whether there are extratheoretical correlates to the terms used in our mathematical theories. We thereby hope to differentiate the sense in which existence is denied to the actual infi nite in (2.11) from what is often called “mathematical existence.”
Kasner and Newman strongly differentiate the two when they assert, “ ‘Existence’ in the mathematical sense is wholly different from the existence of objects in the physical world”
(Kasner & Newman 1940, p. 61). “Mathematical existence” is frequently understood as roughly synonymous with “mathematical legitimacy.” Historically, certain mathematical concepts have been viewed with suspicion and, therefore, initially denied legitimacy in mathematics. Most famous of these are the complex numbers, which as multiples of √–1, were dubbed “imaginary” numbers. To say that complex numbers exist in the mathematical sense is simply to say that they are legitimate mathematical notions; they are in that sense as “real” as the real numbers. Even negative numbers and zero had to fi ght to win mathematical existence. The actual infi nite has, similarly, had to struggle for mathematical legitimacy. For many thinkers, a commitment to the mathematical legitimacy of some notion does not bring with it a commitment to the existence of the relevant entity in the nonmathematical sense. For formalist defenders of the actual infi nite such as Hilbert, mere logical consistency was suffi cient for existence in the mathematical sense. At the same time, Hilbert denied that the actual infi nite is anywhere instantiated in reality. Clearly, for such thinkers, there is a differentiation between mathematical existence and existence in the everyday sense of the word. We are not here endorsing two modes of existence but simply alerting readers to the equivocal way in which “existence” is often used in mathematical discussions, lest the denial of existence of the actual infi nite in (2.11) be misunderstood to be a denial of the mathematical legitimacy of the actual infi nite. A modern mutakallim might deny the mathematical legitimacy of the actual infi nite in favor of intuitionistic or constructivist views of mathematics, but he need not. When Kasner and Newman say, “the infi nite certainly does not exist in the same sense that we say, ‘There are fi sh in the sea’ ” (Kasner & Newman 1940, p. 61), that is the sense of existence that is at issue in (2.11).
These remarks make clear that when it is alleged that an actual infi nite “cannot” exist, the modality at issue is not strict logical possibility. Otherwise the presumed strict logical consistency of axiomatic set theory would be enough to guarantee that the existence of an actual infi nite is possible. Rather what is at issue here is so-called metaphysical possibility, which has to do with something’s being realizable or actualizable. This sort of modality, in terms of which popular possible worlds semantics is typically formulated, is often characterized as broadly logical possibility, but here a word of caution is in order. Insofar as by broadly logical possibility one means merely strict logical possibility augmented by the meaning of terms in the sentence within the scope of the modal operator, such a conception is still too narrow for the purposes of the present argument. Such a conception would enable us to see the necessity of analytic truths in virtue of logic and the meaning of sentential terms used in the expression of these truths (such as “All bachelors are unmarried”),
106 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
but it will not capture the metaphysical necessity or impossibility of synthetic truths, whether these be known a priori (such as “Everything that has a shape has a size”) or a posteriori (such as “This table could not have been made of ice”). Broad logical possibility, then, will not be broad enough for a proper understanding of the argument unless synthetic truths are among those truths that are classed as necessary.
The fact that the argument is framed in terms of metaphysical modality also has an important epistemic consequence. Since metaphysical modality is so much woollier a notion than strict logical modality, there may not be the sort of clean, decisive markers of what is possible or impossible that consistency in fi rst-order logic affords for strict logical modality. Arguments for metaphysical possibility or impossibility typically rely upon intuitions and conceivability arguments, which are obviously much less certain guides than strict logical consistency or inconsistency. The poorly defi ned nature of metaphysical modality cuts both ways dialectically: on the one hand, arguments for the metaphysical impossibility of some state of affairs will be much more subjective than arguments concerning strict logical impossibility; on the other hand, such arguments cannot be refuted by facile observations to the effect that such states of affairs have not been demonstrated to be strictly logically inconsistent.
Premise (2.12) speaks of a temporal regress of events. By an “event,” one means any change. Since any change takes time, there are no instantaneous events so defi ned.
Neither could there be an infi nitely slow event, since such an “event” would, in reality, be a changeless state. Therefore, any event will have a fi nite, nonzero duration. In order that all the events comprised by the temporal regress of past events be of equal duration, one arbitrarily stipulates some event as our standard and, taking as our point of departure the present standard event, we consider any series of such standard events ordered according to the relation earlier than. The question is whether this series of events comprises an actually infi nite number of events or not. If not, then since the universe cannot ever have existed in an absolutely quiescent state, the universe must have had a beginning.
It is therefore not relevant whether the temporal series had a beginning point (a fi rst temporal instant). The question is whether there was in the past an event occupying a nonzero, fi nite temporal interval which was absolutely fi rst, that is, not preceded by any equal interval.6
With these explications in mind, let us now turn to an examination of the argument’s two premises.
2.11. Existence of an actual infi nite
Premise (2.11) asserts that an actual infi nite cannot exist in the real world. It is frequently alleged that this sort of claim has been falsifi ed by Cantor’s work on the actual infi nite and by subsequent developments in set theory, which provide a convincing demonstration of the existence of actual infi nites. But this allegation is far too hasty. It not only begs the question against denials of the mathematical legitimacy of the actual infi nite on the part of certain mathematicians (such as intuitionists), but, more seriously, it begs the question against anti-Platonist views of mathematical objects. These are distinct questions, all too 6. This criterion allows that there may be events of shorter duration prior to the fi rst standard event. By stipulat-ing as one’s standard event a shorter interval, these can be made arbitrarily brief.
KA LA M C O S M O LO G I C A L A RG U M E N T 107
Mathematical objects
Exist
Meaningless question
Do not exist
(realism)
(anti-realism)
As abstract objects
As concrete objects
(Platonism)
Figuralism
Conventionalism
Constructibilism
Deductivism
Structuralism
Fictionalism
Physicalism
Conceptualism
Human
Divine
Figure 3.1 Some metaphysical options concerning the existence of abstract objects.
often confl ated by recent critics of the argument (Sobel 2004, pp. 181–9, 198;9; Oppy 2006a, pp. 291–3; cf. Craig, 2008). Most non-Platonists would not go to the intuitionistic extreme of denying mathematical legitimacy to the actual infi nite – hence, Hilbert’s defi ant declaration, “No one shall be able to drive us from the paradise that Cantor has created for us”
(Hilbert 1964, p. 141) – rather they would simply insist that acceptance of the mathematical legitimacy of certain notions does not imply an ontological commitment to the reality of various objects. Thus, in Hilbert’s view, “The infi nite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought. . . . The role that remains for the infi nite to play is solely that of an idea” (Hilbert 1964, p. 151). Cantor’s system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments. In view of the plethora of alternatives to Platonism (Figure 3.1), critics of the argument cannot justifi ably simply assume that the language of mathematics commits us ontologically to mind-independent entities, especially to such obscure objects as sets.
On antirealist views of mathematical objects such as Balaguer’s fi ctionalism (Balaguer 1998, pt. II; 2001, pp. 87–114; Stanford Encyclopedia of Philosophy 2004b), Yablo’s fi guralism (Yablo 2000, pp. 275–312; 2001, pp. 72–102; 2005, pp. 88–115), Chihara’s constructibilism (Chihara 1990, 2004; 2005, pp. 483–514), or Hellman’s Modal structuralism (Hellman 1989; 2001, pp. 129–57; 2005, pp. 536–62), mathematical discourse is not in any way abridged, but there are, notwithstanding, no mathematical objects at all, let alone an infi nite number of them. The abundance of nominalist (not to speak of conceptualist)
108 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
alternatives to Platonism renders the issue of the ontological status of mathematical entities at least a moot question. The Realist, then, if he is to maintain that mathematical objects furnish a decisive counterexample to the denial of the existence of the actual infi nite, must provide some overriding argument for the reality of mathematical objects, as well as rebutting defeaters of all the alternatives consistent with classical mathematics – a task whose prospects for success are dim, indeed. It is therefore open to the mutakallim to hold that while the actual infi nite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world.
The best way to support (2.11) is by way of thought experiments that illustrate the various absurdities that would result if an actual infi nite were to be instantiated in the real world.7 Benardete, who is especially creative and effective at concocting such thought experiments, puts it well: “Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face” (Benardete 1964, p. 238).8
Let us look at just one example: David Hilbert’s famous brainchild “Hilbert’s Hotel.”9
As a warm-up, let us fi rst imagine a hotel with a fi nite number of rooms. Suppose, 7. Ludwig Wittgenstein nicely enunciated this strategy when, in response to Hilbert’s solemn declaration, he quipped, “I wouldn’t dream of trying to drive anyone from this paradise. I would do something quite different: I would try to show you that it is not a paradise – so that you’ll leave of your own accord. I would say, ‘You’re welcome to this; just look about you’ . . .” (Wittgenstein 1976, p. 103). But here the strategy is employed on behalf of metaphysical, not mathematical, fi nitism. Oppy objects that such puzzles show, at most, that certain kinds of actual infi nities cannot exist, but that this conclusion cannot be generalized (Oppy 2006b, p. 140). The diffi culty with this attempt to blunt the force of the absurdities is twofold: (i) nothing in the various situations seems to be metaphysically impossible apart from the assumption of an actual infi nite and (ii) the absurdities are not tied to the particular kinds of objects involved.
8. He has in mind especially what he calls paradoxes of the serrated continuum, such as the following: Here is a book lying on the table. Open it. Look at the fi rst page. Measure its thickness. It is very thick indeed for a single sheet of paper – 1/2 inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? 1/4 inch thick. And the third page of the book, how thick is this third sheet of paper? 1/8 inch thick, &c. ad infi nitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one-half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a fi nite number of pages.
These two conditions are logically compatible: there is no certifi able contradiction in their joint assertion.
But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now – slowly – lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze. (Benardete 1964, pp. 236–7).
To our mind this conclusion itself is evidently metaphysically absurd. Although Oppy, following Hazen (1993), offers expansions of the story so that someone opening the book will have some sort of visual experience, rather than as it were, a blank (Oppy 2006a, pp. 83–5), that does not negate the conclusion that there is nothing there to see since there is no last page. Benardete imagines what would happen if we tried to touch the last page of the book. We cannot do it. Either there will be an impenetrable barrier at ω + 1, which seems like science fi ction, or else our fi ngers will penetrate through an infi nity of pages without fi rst penetrating a page, which recalls Zeno’s paradoxes in spades, since the pages are actual entities. What makes paradoxes such as these especially powerful, as Benardete points out, is that no process or supertask is involved here; each page is an actual entity having a fi nite thickness (none is a degenerate interval) which could be unbound from the others and all the pages scattered to the four winds, so that an actual infi nity of pages would exist throughout space. If such a book cannot exist, therefore, neither can an actual infi nite.
9. The story of Hilbert’s Hotel is related in Gamow (1946, p. 17).
KA LA M C O S M O LO G I C A L A RG U M E N T 109
furthermore, that all the rooms are occupied. When a new guest arrives asking for a room, the proprietor apologizes, “Sorry, all the rooms are full,” and that is the end of the story.
But now let us imagine a hotel with an infi nite number of rooms and suppose once more that all the rooms are occupied. There is not a single vacant room throughout the entire infi nite hotel. Now suppose a new guest shows up, asking for a room. “But of course!” says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4, and so on out to infi nity.
As a result of these room changes, room #1 now becomes vacant, and the new guest gratefully checks in. But remember, before he arrived, all the rooms were occupied! Equally curious, there are now no more persons in the hotel than there were before: the number is just infi nite. But how can this be? The proprietor just added the new guest’s name to the register and gave him his keys – how can there not be one more person in the hotel than before?
But the situation becomes even stranger. For suppose an infi nity of new guests show up at the desk, asking for a room. “Of course, of course!” says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infi nity, always putting each former occupant into the room number twice his own. Because any natural number multiplied by two always equals an even number, all the guests wind up in even-numbered rooms. As a result, all the odd-numbered rooms become vacant, and the infi nity of new guests is easily accommodated. And yet, before they came, all the rooms were occupied! And again, strangely enough, the number of guests in the hotel is the same after the infi nity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infi nitely many times and yet there would never be a single person more in the hotel than before.
But Hilbert’s Hotel is even stranger than the German mathematician made it out to be.
For suppose some of the guests start to check out. Suppose the guest in room #1 departs.
Is there not now one fewer person in the hotel? Not according to infi nite set theory!
Suppose the guests in rooms #1, 3, 5, . . . check out. In this case an infi nite number of people has left the hotel, but by Hume’s Principle, there are no fewer people in the hotel. In fact, we could have every other guest check out of the hotel and repeat this process infi nitely many times, and yet there would never be any fewer people in the hotel. Now suppose the proprietor does not like having a half-empty hotel (it looks bad for business). No matter!
By shifting guests in even-numbered rooms into rooms with numbers half their respective room numbers, he transforms his half-vacant hotel into one that is completely full. In fact, if the manager wanted double occupancy in each room, he would have no need of additional guests at all. Just carry out the dividing procedure when there is one guest in every room of the hotel, then do it again, and fi nally have one of the guests in each odd-numbered room walk next door to the higher even-numbered room, and one winds up with two people in every room!
One might think that by means of these maneuvers the proprietor could always
keep this strange hotel fully occupied. But one would be wrong. For suppose that the persons in rooms #4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infi nite converted to fi nitude.
And yet it would remain true that as many guests checked out this time as when the guests in rooms #1, 3, 5, . . . checked out! Can anyone believe that such a hotel could exist in reality?
110 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Hilbert’s Hotel is absurd. But if an actual infi nite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infi nite is not metaphysically possible.
Partisans of the actual infi nite might concede the absurdity of a Hilbert’s Hotel but maintain that this case is somehow peculiar and, therefore, its metaphysical impossibility warrants no inference that an actual infi nite is metaphysically impossible. This sort of response might seem appropriate with respect to certain absurdities involving actual infi nites; for example, those imagining the completion of a so-called supertask, the sequential execution of an actually infi nite number of defi nite and discrete operations in a fi nite time. But when it comes to situations involving the simultaneous existence of an actually infi nite number of familiar macroscopic objects, then this sort of response seems less plausible.10 If a (denumerably) actually infi nite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert’s Hotel. Since nothing hangs on the illustration’s involving a hotel, the metaphysical absurdity is plausibly attributed to the existence of an actual infi nite. Thus, thought experiments of this sort show, in general, that it is impossible for an actually infi nite number of things to exist in reality.
At this point, the actual infi nitist has little choice but, in Oppy’s words, simply to
“embrace the conclusion of one’s opponent’s reductio ad absurdum argument” (Oppy 2006a, p. 48). Oppy explains, “these allegedly absurd situations are just what one ought to expect if there were . . . physical infi nities” (Oppy 2006a, p. 48).
Oppy’s response, however, falls short: it does nothing to prove that the envisioned situations are not absurd but only serves to reiterate, in effect, that if an actual infi nite could exist in reality, then there could be a Hilbert’s Hotel, which is not in dispute. The problem cases would, after all, not be problematic if the alleged consequences would not ensue!
Rather the question is whether these consequences really are absurd.
Sobel similarly observes that such thought experiments bring into confl ict two “seemingly innocuous” principles, namely,
(i)
There are not more things in a multitude M than there are in a multitude M′ if there is a one-to-one correspondence of their members.
and
(ii) There are more things in M than there are in M′ if M′ is a proper submultitude of M.
We cannot have both of these principles along with
(iii) An
infi nite multitude exists.
10. Oppy, for example, makes the point that having a hotel with an infi nite number of occupied rooms does not commit one to the possibility of accommodating more guests by shifting guests about – maybe the hotel’s construction hinders the guests’ movements or the guests die off before their turn to move comes round. But as a Gedankenexperiment Hilbert’s Hotel can be confi gured as we please without regard to mere physical possibilities.
KA LA M C O S M O LO G I C A L A RG U M E N T 111
For Sobel, the choice to be taken is clear: “The choice we have taken from Cantor is to hold on to (i) while restricting the proper submultiplicity condition to fi nite multiplicities. In this way we can ‘have’ comparable infi nite multitudes” (Sobel 2004, pp. 186–7; cf. Mackie 1982, p. 93).
But the choice taken from Cantor of which Sobel speaks is a choice on the part of the mathematical community to reject intuitionism and fi nitism in favor of axiomatic infi nite set theory. Finitism would too radically truncate mathematics to be acceptable to most mathematicians. But, as already indicated, that choice does not validate metaphysical conclusions. The metaphysician wants to know why, in order to resolve the inconsistency among (i)–(iii), it is (ii) that should be jettisoned (or restricted). Why not instead reject or restrict to fi nite multiplicities (i), which is a mere set-theoretical convention? More to the point, why not reject (iii) instead of the apparently innocuous (i) or (ii)? It certainly lacks the innocuousness of those principles, and giving it up would enable us to affi rm both (i) and (ii). Remember: we can “have” comparable infi nite multiplicities in mathematics without admitting them into our ontology.
Sobel thus needs some argument for the falsity of (ii). Again, it is insuffi cient merely to point out that if (i) and (iii) are true, then (ii) is false, for that is merely to reiterate that if an actual infi nite were to exist, then the relevant situations would result, which is not in dispute.
Take Hilbert’s Hotel. Sobel says that the diffi culties with such a hotel are practical and physical; “they bring out the physical impossibility of this particular infi nity of concurrent real things, not its logical impossibility” (Sobel 2004, p. 187). But the claim is not that such a hotel is logically impossible but metaphysically impossible. As an illustrative embodiment of transfi nite arithmetic based on the axiomatic set theory, Hilbert’s Hotel will, of necessity, be as logically consistent as that system; otherwise it would be useless as an illustration.
But it also vividly illustrates the absurd situations to which the real existence of an infi nite multitude can lead. The absurdity is not merely practical and physical; it is ontologically absurd that a hotel exist which is completely full and yet can accommodate untold infi nities of new guests just by moving people around.
Oppy is prepared, if need be, simply to bite the bullet: “There can, after all, be a hotel in which infi nitely many new guests are accommodated, even though all the rooms are full, via the simple expedient of moving the guests in room N to room 2 N (for all N)”
(Oppy 2006a, p. 53). So asserting does nothing to alleviate one’s doubts that such a hotel is absurd. And would Oppy say something similar about what would happen
when an infi nite number of guests depart?11 In transfi nite arithmetic, inverse operations of subtraction and division with infi nite quantities are prohibited because they lead to contradictions; as Sobel says, “Of course, as operations and properties are extended from fi nite to transfi nite cardinals, some arithmetic principles are left confi ned to the fi nite” (Sobel 2007). But in reality, one cannot stop people from checking out of a hotel if they so desire! In this case, one does wind up with logically impossible situations, such 11. Oppy suggests using J. Conway’s recently developed constructions called surreal numbers to defi ne operations of subtraction and division of transfi nite numbers (Oppy 2006b, p. 140), but he explicitly denies that such non-canonical theories can be applied “to real-world problems, if one wishes to treat one’s models with full ontological seriousness” (Oppy 2006a, p. 272). Oppy does not show, nor does he think, that the results of operations on surreals would be any less counterintuitive when translated into the concrete realm.
112 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
as subtracting identical quantities from identical quantities and fi nding nonidentical differences.12
In response to the absurdities springing from performing inverse operations with infi -
nite quantities, David Yandell has insisted that subtraction of infi nite quantities does not yield contradictions . He writes,
Subtracting the even positive integers from the set of positive integers leaves an infi nite set, the odd positive integers. Subtracting all of the positive integers greater than 40 from the set of positive integers leaves a fi nite (forty-membered) set. Subtracting all of the positive integers from the set of positive integers leaves one with the null set. But none of these subtractions could possibly lead to any other conclusion than each leads to. This alleged contradictory feature of the infi nite seems not to generate any actual contradictions. (Yandell 2003, p. 132)
It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infi nite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infi nity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infi nity as the remainder. For this reason, subtraction and division of infi nite quantities are simply prohibited in transfi nite arithmetic – a mere stipulation which has no force in the nonmathematical realm.
Sometimes it is said that we can fi nd concrete counterexamples to the claim that an actually infi nite number of things cannot exist, so that Premise (2.11) must be false. For example, Walter Sinnott-Armstrong asserts that the continuity of space and time entails the existence of an actually infi nite number of points and instants (Craig & Sinnott-Armstrong 2003, p. 43). This familiar objection gratuitously assumes that space and time are composed of real points and instants, which has never been proven. Mathematically, the objection can be met by distinguishing a potential infi nite from an actual infi nite. While one can continue indefi nitely to divide conceptually any distance, the series of subintervals thereby generated is merely potentially infi nite, in that infi nity serves as a limit that one endlessly approaches but never reaches. This is the thoroughgoing Aristotelian position on the infi nite: only the potential infi nite exists. This position does not imply that minimal time atoms, or chronons, exist. Rather time, like space, is infi nitely divisible in the sense that division can proceed indefi nitely, but time is never actually infi nitely divided, neither does one arrive at an instantaneous point. If one thinks of a geometrical line as logically 12. It will not do, in order to avoid the contradiction, to assert that there is nothing in transfi nite arithmetic that forbids using set difference to form sets. Indeed, the thought experiment assumes that we can do such a thing.
Removing all the guests in the odd-numbered rooms always leaves an infi nite number of guests remaining, and removing all the guests in rooms numbered greater than four always leaves three guests remaining. That does not change the fact that in such cases identical quantities minus identical quantities yields nonidentical quantities, a contradiction.
KA LA M C O S M O LO G I C A L A RG U M E N T 113
prior to any points which one may care to specify on it rather than as a construction built up out of points (itself a paradoxical notion13), then one’s ability to specify certain points, like the halfway point along a certain distance, does not imply that such points actually exist independently of our specifi cation of them. As Grünbaum emphasizes, it is not infi -
nite divisibility as such which gives rise to Zeno’s paradoxes; the paradoxes presuppose the postulation of an actual infi nity of points ab initio. “. . . [A]ny attribution of (infi nite)
‘divisibility’ to a Cantorian line must be based on the fact that ab initio that line and the intervals are already ‘divided’ into an actual dense infi nity of point-elements of which the line (interval) is the aggregate. Accordingly, the Cantorian line can be said to be already actually infi nitely divided” (Grünbaum 1973, p. 169). By contrast, if we think of the line as logically prior to any points designated on it, then it is not an ordered aggregate of points nor actually infi nitely divided. Time as duration is then logically prior to the (potentially infi nite) divisions we make of it. Specifi ed instants are not temporal intervals but merely the boundary points of intervals, which are always nonzero in duration. If one simply assumes that any distance is already composed out of an actually infi nite number of points, then one is begging the question. The objector is assuming what he is supposed to prove, namely that there is a clear counterexample to the claim that an actually infi nite number of things cannot exist.
Some critics have charged that the Aristotelian position that only potential, but no actual, infi nites exist in reality is incoherent because a potential infi nite presupposes an actual infi nite. For example, Rudy Rucker claims that there must be a “defi nite class of possibilities,” which is actually infi nite in order for the mathematical intuitionist to regard the natural number series as potentially infi nite through repetition of certain mathematical operations (Rucker 1980, p. 66). Similarly, Richard Sorabji asserts that Aristotle’s view of the potentially infi nite divisibility of a line entails that there is an actually infi nite number of positions at which the line could be divided (Sorabji 1983, pp. 210–3, 322–4).
13. See Craig (1985). Consider, for example, the many variations on the Grim Reaper Paradox (Benardete 1964, pp. 259–61; Hawthorne 2000; Oppy 2006a, pp. 63–6, 81–3). There are denumerably infi nitely many Grim Reapers (whom we may identify as gods, so as to forestall any kinematic objections). You are alive at 12:00 p.m.
Grim Reaper 1 will strike you dead at 1:00 p.m. if you are still alive at that time. Grim Reaper 2 will strike you dead at 12:30 p.m. if you are still alive then. Grim Reaper 3 will strike you dead at 12:15 p.m., and so on. Such a situation seems clearly conceivable but leads to an impossibility: you cannot survive past 12:00 p.m. and yet you cannot be killed at any time past 12:00 p.m. Oppy’s solution to a similar paradox concerning infi nitely many deafening peals, viz. that there is no particular peal responsible for your deafness but that the collective effect of infi nitely many peals is to bring about deafness (Oppy 2006a, p. 83), not only involves a most bizarre form of retro-causation (Benardete 1964, p. 259) but is also in any case inapplicable to the Grim Reaper version since once you are dead no further Grim Reaper will swing his scythe, so that collective action is out of the question.
The most plausible way to avert such paradoxes is by denying that time and space are constructions out of an actually infi nite number of points. (My thanks to Alexander Pruss for drawing my attention to this version of the paradox.)
Moreover, on an A-Theory of time, according to which temporal becoming is an objective feature of reality, treating time as composed of instants (degenerate temporal intervals of zero duration) seems to land one in Zeno’s clutches since temporal becoming would require the actualization of consecutive instants, which is incoherent. For a good discussion, see Grünbaum (1950–1, pp. 143–86). Grünbaum succeeds in defending the continuity of time only at the expense of sacrifi cing temporal becoming, which his interlocutors James and Whitehead would not do. See further Craig (2000c).
114 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
If this line of argument were successful, it would, indeed, be a tour de force since it would show mathematical thought from Aristotle to Gauss to be not merely mistaken or incomplete but incoherent in this respect. But the objection is not successful. For the claim that a physical distance is, say, potentially infi nitely divisible does not entail that the distance is potentially divisible here and here and here and. . . . Potential infi nite divisibility (the property of being susceptible of division without end) does not entail actual infi nite divisibility (the property of being composed of an infi nite number of points where divisions can be made). The argument that it does is guilty of a modal operator shift, inferring from the true claim
(1) Possibly, there is some point at which x is divided
to the disputed claim
(2) There is some point at which x is possibly divided.
But it is coherent to deny the validity of such an inference. Hence, one can maintain that a physical distance is potentially infi nitely divisible without holding that there is an infi nite number of positions where it could be divided.
Rucker also argues that there are probably, in fact, physical infi nities (Rucker 1980, p.
69). If the mutakallim says, for example, that time is potentially infi nite, then Rucker will reply that the modern, scientifi c worldview sees the past, present, and future as merely different regions coexisting in space-time. If he says that any physical infi nity exists only as a temporal (potentially infi nite) process, Rucker will rejoin that it is artifi cial to make physical existence a by-product of human activity. If there are, for example, an infi nite number of bits of matter, this is a well-defi ned state of affairs which obtains right now regardless of our apprehension of it. Rucker concludes that it seems quite likely that there is some form of physical infi nity.
Rucker’s conclusion, however, clearly does not follow from his arguments. Time and space may well be fi nite. But could they be potentially infi nite? Concerning time, even if Rucker were correct that a tenseless four-dimensionalism is correct, that would provide no reason at all to think the space-time manifold to be temporally infi nite: there could well be fi nitely separated initial and fi nal singularities. In any case, Rucker is simply incorrect in saying that “the modern, scientifi c worldview” precludes a theory of time, according to which temporal becoming is a real and objective feature of reality. Following McTaggart, contemporary philosophers of space and time distinguish between the so-called A-Theory of time, according to which events are temporally ordered by tensed determinations of past, present, and future, and temporal becoming is an objective feature of physical reality, and the so-called B-Theory of time, according to which events are ordered by the tenseless relations of earlier than, simultaneous with, and later than, and temporal becoming is purely subjective. Although some thinkers have carelessly asserted that relativity theory has vindicated the B-Theory over against its rival, such claims are untenable. One could harmonize the A-Theory and relativity theory in at least three different ways: (1) distinguish metaphysical time from physical or clock time and maintain that while the former is A-Theoretic in nature, the latter is a bare abstraction therefrom, useful for scientifi c purposes and quite possibly B-Theoretic in character, the element of becoming having been abstracted out; (2) relativize becoming to reference frames, just as is done with simultaneity; and (3) select
KA LA M C O S M O LO G I C A L A RG U M E N T 115
a privileged reference frame to defi ne the time in which objective becoming occurs, most plausibly the cosmic time, which serves as the time parameter for hypersurfaces of homogeneity in space-time in the General Theory of Relativity. And concerning space, to say that space is potentially infi nite is not to say, with certain constructivists, that it depends on human activity (nor again, that there are actual places to which it can extend), but simply that space expands limitlessly as the distances between galaxies increase with time.
As for the number of bits of matter, there is no incoherence in saying that there is a fi nite number of bits or that matter is capable of only a fi nite number of physical subdivisions, although mathematically one could proceed to carve up matter potentially ad infi nitum.
The sober fact is that there is just no evidence that actual infi nities are anywhere instantiated in the physical world. It is therefore futile to seek to rebut (2.11) by appealing to clear counterexamples drawn from physical science.
2.12. An infi nite regress of events as an actual infi nite
The second premise states that an infi nite temporal regress of events is an actual infi nite. The point seems obvious enough, for if there has been a sequence composed of an infi nite number of events stretching back into the past, then the set of all events in the series would be an actually infi nite set.
But manifest as this may be to us, it was not always considered so. The point somehow eluded Aristotle himself, as well as his scholastic progeny, who regarded the series of past events as a potential infi nite. Aristotle contended that since things in time come to exist sequentially, an actual infi nite never exists at any one moment; only the present thing actually exists ( Physics 3.6.206a25–206b1). Similarly, Aquinas, after confessing the impossibility of the existence of an actual infi nite, nevertheless proceeded to assert that the existence of an infi nite regress of past events is possible ( Summa Theologiae 1.a.7.4.). This is because the series of past events does not exist in actuality. Past events do not now exist, and hence do not constitute an infi nite number of actually existing things. The series is only potentially infi nite, not actually infi nite, in that it is constantly increasing by the addition of new events.
These Aristotelian thinkers are clearly presupposing an A-Theory of time and an ontology of presentism, according to which the only temporal items which exist are those that presently exist. On a B-Theory of tenseless time, since there is ontological parity among all events, there can be no question that an infi nite temporal regress of events is composed of an actually infi nite number of events.14 Since all events are equally real, the fact that they exist (tenselessly) at different times loses any signifi cance. The question, then, is whether events’ temporal distribution over the past on a presentist ontology precludes our saying that the number of events in a beginningless series of events is actually infi nite.
Now we may take it as a datum that the presentist can accurately count things that have existed but no longer exist. He knows, for example, how many US presidents there have been up through the present incumbent, what day of the month it is, how many shots Oswald squeezed off, and so forth. He knows how old his children are and can reckon how many billion years have elapsed since the Big Bang, if there was such an event. The 14. Some philosophers of time, such as C. D. Broad and Michael Tooley, have defended a sort of hybrid A/B-Theory, according to which the past and present are on an ontological par, the past being a growing space-time block. On such a view, a beginningless series of past events is also, uncontroversially, actually infi nite.
116 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
nonexistence of such things or events is no hindrance to their being enumerated. Indeed, any obstacle here is merely epistemic, for aside from considerations of vagueness there must be a certain number of such things. So in a beginningless series of past events of equal duration, the number of past events must be infi nite, for it is larger than any natural number. But then the number of past events must be ℵ0, for ∞ is not a number but an ideal limit. Aquinas’ own example of a blacksmith working from eternity who uses one hammer after another as each one breaks furnishes a good example of an actual infi nite, for the collection of all the hammers employed by the smith is an actual infi nite. The fact that the broken hammers still exist is incidental to the story; even if they had all been destroyed after being broken, the number of hammers broken by the smith is the same.
Similarly, if we consider all the events in an infi nite temporal regress of events, they constitute an actual infi nite.
The question arises whether on the A-Theory the series of future events, if time will go on forever, is not also actually infi nite. Intuitively, it seems clear that the situation is not symmetrical, but this is notoriously diffi cult to express. It might rightly be pointed out that on presentism there are no future events and so no series of future events. Therefore, the number of future events is simply zero, not ℵ0. (By this statement, one means not that there are future events, and that their number is 0, but that there just are no future events.) But on presentism, the past is as unreal as the future and, therefore, the number of past events could, with equal justifi cation, be said to be zero. It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infi nite number of events, just as in a beginningless past there have been an actually infi nite number of events. But in a sense that assertion is false; for there never will be an actually infi nite number of events since it is impossible to count to infi nity. The only sense in which there will be an infi nite number of events is that the series of events will go toward infi nity as a limit. But that is the concept of a potential infi nite, not an actual infi nite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infi nite, that is to say, fi nite but indefi nitely increasing toward infi nity as a limit. The situation, signifi cantly, is not symmetrical: as we have seen, the series of events earlier than any arbitrarily selected future event cannot properly be regarded as potentially infi nite. So when we say that the number of past events is infi nite, we mean that prior to today, ℵ0 events have elapsed. But when we say that the number of future events is infi nite, we do not mean that ℵ0 events will elapse, for that is false. Ironically, then, it turns out that the series of future events cannot be actually infi nite regardless of the infi nity of the past or the metaphysical possibility of an actual infi nite, for it is the objectivity of temporal becoming that makes the future potentially infi nite only.
Because the series of past events is an actual infi nite, all the absurdities attending the existence of an actual infi nite apply to it. For example, if the series of past events is actually infi nite, then the number of events that have occurred up to the present is no greater than the number that have occurred at any point in the past. Or again, if we number the events beginning in the present, then there have occurred as many odd-numbered events as events.
If we mentally take away all the odd-numbered events, there are still an infi nite number of events left over; but if we take away all the events greater than three, there are only four events left, even though in both cases we took away the same number of events.
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2.13. Conclusion
Since an actual infi nite cannot exist and an infi nite temporal regress of events is an actual infi nite, we may conclude that an infi nite temporal regress of events cannot exist. Therefore, since the temporal regress of events is fi nite, the universe began to exist.
2.2. Argument from the impossibility of the formation of an actual infi nite by successive addition
We now turn to a second philosophical argument in support of the premise that the universe began to exist, the argument from the impossibility of the formation of an actual infi nite by successive addition. The argument may be simply formulated as follows: 2.21 A collection formed by successive addition cannot be an actual infi nite.
2.22 The temporal series of events is a collection formed by successive addition.
2.23 Therefore, the temporal series of events cannot be an actual infi nite.
This second argument is independent of the foregoing argument, for its conclusion is not incompatible with the existence of an actual infi nite. It rather denies that a collection containing an actually infi nite number of things can be formed by adding one member after another . If an actual infi nite cannot be formed by successive addition, then the series of past events must be fi nite since that series is formed by successive addition of one event after another in time.
2.21. Formation of an actual infi nite
Quite independent of the absurdities arising from the existence of an actually infi nite number of things are the further diffi culties arising as a result of the temporal formation of such a multitude through a process of successive addition. By “successive addition,” one means the accrual of one new element at a (later) time. The temporality of the process of accrual is critical here. For while it is true that 1 + 1 + 1 + . . . equals ℵ0, the operation of addition signifi ed by “+” is not applied successively but simultaneously or, better, timelessly.
One does not add the addenda in temporal succession: 1 + 1 = 2, then 2 + 1 = 3, then 3 +
1 = 4, . . . , but rather all together. By contrast, we are concerned here with a temporal process of successive addition of one element after another.
The impossibility of the formation of an actual infi nite by successive addition seems obvious in the case of beginning at some point and trying to reach infi nity.15 For given any fi nite number n, n + 1 equals a fi nite number. Hence, ℵ0 has no immediate predecessor; it is not the terminus of the natural number series but stands, as it were, outside it and is the 15. This despite the speculation concerning the possibility of supertasks, various thought experiments involving the completion of an infi nite number of tasks in a fi nite time by performing each successive task during half the time taken to perform its immediate predecessor. The fatal fl aw in all such scenarios is that the state at ω + 1 is causally unconnected to the successive states in the ω series of states. Since there is no last term in the ω series, the state of reality at ω + 1 appears mysteriously from nowhere. The absurdity of such supertasks underlines the metaphysical impossibility of trying to convert a potential into an actual infi nite.
118 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
number of all the members in the series. Notice that the impossibility of forming an actual infi nite by successive addition has nothing to do with the amount of time available. Sometimes it is wrongly alleged that the only reason an actual infi nite cannot be formed by successive addition is because there is not enough time.16 But this is mistaken. While we can imagine an actually infi nite series of events mapped onto a tenselessly existing infi nite series of temporal intervals, such that each consecutive event is correlated with a unique consecutive interval, the question remains whether such a sequence of intervals can be instantiated, not tenselessly, but one interval after another. The very nature of the actual infi nite precludes this. For regardless of the time available, a potential infi nite cannot be turned into an actual infi nite by any amount of successive addition since the result of every addition will always be fi nite. One sometimes, therefore, speaks of the impossibility of counting to infi nity, for no matter how many numbers one counts, one can always count one more number before arriving at infi nity. One sometimes speaks instead of the impossibility of traversing the infi nite. The diffi culty is the same: no matter how many steps one takes, the addition of one more step will not bring one to a point infi nitely distant from one’s starting point.
The question then arises whether, as a result of time’s asymmetry, an actually infi nite collection, although incapable of being formed by successive addition by beginning at a point and adding members, nevertheless could be formed by successive addition by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity. In this case, one is not engaged in the impossible task of trying to convert a potential into an actual infi nite by successive addition. Rather at every point the series already is actually infi nite, although allegedly successively formed.
Although the problems will be different, the formation of an actually infi nite collection by never beginning and ending at some point seems scarcely less diffi cult than the formation of such a collection by beginning at some point and never ending. If one cannot count to infi nity, how can one count down from infi nity? If one cannot traverse the infi nite by moving in one direction, how can one traverse it by moving in the opposite direction? In order for us to have “arrived” at today, temporal existence has, so to speak, traversed an infi nite number of prior events.17 But before the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infi nitum. One gets driven back and back into the infi nite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd.
16. For example, Oppy’s discussion of counting forward to infi nity is predicated upon Dretske’s assumption that if one never stops counting, then one does count to infi nity (Oppy 2006a, p. 61; cf. Dretske 1965). Oppy fails so much as to mention, much less take account, of the difference between an actual and a potential infi nite in this case. One who, having begun, never stops counting counts “to infi nity” only in the sense that one counts potentially infi nitely.
17. Richard Gale protests, “This argument depends on an anthropomorphic sense of ‘going through’ a set. The universe does not go through a set of events in the sense of planning which to go through fi rst, in order to get through the second, and so on” (Gale 2007, pp. 92–3). Of course not; but on an A-Theory of time, the universe does endure through successive intervals of time. It arrives at its present event-state only by enduring through a series of prior event-states. Gale’s framing the argument in terms of a “set of events” is maladroit since we are not talking about a set but about a series of events which elapse one after another.
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It is unavailing to say that an infi nite series of past events cannot be formed only in a fi nite time but that such formation is possible given infi nite time, for that riposte only pushes the question back a notch: how can an actually infi nite series of congruent temporal intervals successively elapse? Tenseless correlations are irrelevant here. Granted that the series of past events, if infi nite, can be mapped one-to-one onto an equally infi nite series of past temporal intervals, the question remains how such a temporal series can be lived through so as to arrive at the present.
The arguments against the formation of an actual infi nite by successive addition bear a clear resemblance to Zeno’s celebrated paradoxes of motion, in particular the Stadium and Dichotomy paradoxes, the Stadium in the case of beginning at some point and never ending and the Dichotomy in the case of never beginning and ending at some point. In the Dichotomy Paradox, Zeno argued that before Achilles could cross the stadium, he would have to cross halfway; but before he could cross halfway, he would have to cross a quarter of the way; but before he could cross a quarter of the way, he would have to cross an eighth of the way, and so on to infi nity. It is evident that Achilles could not arrive at any point. In the case of the infi nite past, we cannot speak meaningfully of halfway through the past or a quarter of the way, and so on since there is no beginning point, as there is in Achilles’ case. But the metrical distances traversed are not essential to the conundrum insofar as the series of past events is concerned since the essential point holds that before traversing any interval there will always be a prior interval to be traversed fi rst.
Now although Zeno’s paradoxes have proved very stubborn, scarely anybody has really believed that motion is impossible. Is the argument against the impossibility of traversing an infi nite past, as some critics allege, no more plausible than Zeno’s paradoxes? This cannot be said because the allgation fails to reckon with two crucial disanalogies of the case of an infi nite past to Zeno’s paradoxes: whereas in Zeno’s thought experimentsm the intervals traversed are potential and unequal, in the case of an infi nite past the intervals are actual and equal. The claim that Achilles must pass through an infi nite number of halfway points in order to cross the stadium already assumes that the whole interval is a composition of an infi nite number of points, whereas Zeno’s opponents, like Aristotle, take the line as a whole to be conceptually prior to any divisions which we might make in it. Moreover, Zeno’s intervals, being unequal, sum to a merely fi nite distance, whereas the intervals in an infi nite past sum to an infi nite distance. The question is not whether it is possible to traverse infi nitely many (progressively shorter) distances but whether it is possible to traverse an infi nite distance. Thus, the problem of traversing an infi nite distance comprising an infi nite number of equal, actual intervals to arrive at our present location cannot be dismissed on the basis of the argument’s resemblance in certain respects to Zeno’s puzzles.
It is surprising that a number of critics, such as Mackie and Sobel, have objected that the argument illicitly presupposes an infi nitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But if the past is infi nite, they say, then there would be no starting point whatever, not even an infi nitely distant one.
Nevertheless, from any given point in the past, there is only a fi nite distance to the present, which is easily “traversed” (Mackie 1982, p. 93; Sobel 2004, p. 182). But, in fact, no proponent of the kalam argument of whom we are aware has assumed that there was an infi nitely distant starting point in the past. The fact that there is no beginning at all, not even an infi nitely distant one, seems only to make the problem worse, not better. To say that the
120 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
infi nite past could have been formed by successive addition is like saying that someone has just succeeded in writing down all the negative numbers, ending at −1. And how is the claim that from any given moment in the past there is only a fi nite distance to the present even relevant to the issue? For the question is how the whole series can be formed, not a fi nite portion of it. Do Mackie and Sobel think that because every fi nite segment of the series can be formed by successive addition the whole infi nite series can be so formed? That is as logically fallacious as saying that because every part of an elephant is light in weight, the whole elephant is light in weight, or in other words, to commit the fallacy of composition. The claim that from any given moment in the past there is only a fi nite distance to the present is simply irrelevant.
Wholly apart from these Zenonian arguments, the notion that the series of past events could be actually infi nite is notoriously diffi cult. Consider, for example, al-Ghazali’s thought experiment involving two beginningless series of coordinated events. He envisions our solar system’s existing from eternity past, the orbital periods of the planets being so coordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many. If they have been orbiting from eternity, which planet has completed the most orbits?
The correct mathematical answer is that they have completed precisely the same number of orbits. But this seems absurd, for the longer they revolve, the greater becomes the disparity between them, so that they progressively approach a limit at which Jupiter has fallen infi nitely far behind Saturn. Yet, being now actually infi nite, their respective completed orbits are somehow magically identical. Indeed, they will have “attained” infi nity from eternity past: the number of completed orbits is always the same. Moreover, Ghazali asks, will the number of completed orbits be even or odd? Either answer seems absurd. We might be tempted to deny that the number of completed orbits is either even or odd. But post-Cantorian transfi nite arithmetic gives a quite different answer: the number of orbits completed is both even and odd! For a cardinal number n is even if there is a unique cardinal number m such that n = 2 m, and n is odd if there is a unique cardinal number m such that n = 2 m + 1. In the envisioned scenario, the number of completed orbits is (in both cases!) ℵ0, and ℵ0 = 2ℵ0 = 2ℵ0 + 1. So Jupiter and Saturn have each completed both an even and an odd number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any fi nite interval of time. This seems absurd.18
Or consider the case of Tristram Shandy, who, in the novel by Sterne, writes his autobiography so slowly that it takes him a whole year to record the events of a single day. Tristram Shandy laments that at this rate he can never come to an end.
According to Russell, if Tristram Shandy were immortal and did not weary of his task,
“no part of his biography would have remained unwritten,” since by Hume’s Principle to each day there would correspond 1 year, and both are infi nite (Russell, 1937, p. 358). Such an assertion is misleading, however. The fact that every part of the autobiography will be eventually written does not imply that the whole autobiography will be eventually written, which was, after all, Tristram Shandy’s concern. For every part of the autobiography there 18. Oppy’s discussion of al-Ghazali’s problem just fails to connect with the problem as we understand it (Oppy 2006a, pp. 49–51), probably because Oppy takes its point to be that there is a logical contradiction with respect to the number of orbits completed (Oppy 2006a, p. 8), so that he spends most of his space arguing that given Cantorian assumptions there is no unequivocal sense in which the number of orbits both is and is not same.
Temporal becoming is left wholly out of account.
KA LA M C O S M O LO G I C A L A RG U M E N T 121
is some time at which it will be completed, but there is not some time at which every part of the autobiography will be completed. Given an A-Theory of time, though he write forever, Tristram Shandy would only get farther and farther behind, so that instead of fi nishing his autobiography, he would progressively approach a state in which he would be infi nitely far behind.
But now turn the story about: suppose Tristram Shandy has been writing from eternity past at the rate of 1 day per year. Should not Tristram Shandy now be infi nitely far behind?
For if he has lived for an infi nite number of years, Tristram Shandy has recorded an equally infi nite number of past days. Given the thoroughness of his autobiography, these days are all consecutive days. At any point in the past or present, therefore, Tristram Shandy has recorded a beginningless, infi nite series of consecutive days. But now the question arises: Which days are these? Where in the temporal series of events are the days recorded by Tristram Shandy at any given point? The answer can only be that they are days infi nitely distant from the present. For there is no day on which Tristram Shandy is writing which is fi nitely distant from the last recorded day.
This may be seen through an incisive analysis of the Tristram Shandy Paradox given by Robin Small (1986, pp. 214–5). He points out that if Tristram Shandy has been writing for 1 year’s time, then the most recent day he could have recorded is 1 year ago. But if he has been writing for 2 years, then that same day could not have been recorded by him. For since his intention is to record consecutive days of his life, the most recent day he could have recorded is the day immediately after a day at least 2 years ago. This is because it takes a year to record a day, so that to record 2 days he must have 2 years. Similarly, if he has been writing 3 years, then the most recent day recorded could be no more recent than 3
years and 2 days ago. In other words, the longer he has written the further behind he has fallen. In fact, the recession into the past of the most recent recordable day can be plotted according to the formula (present date – n years of writing) + n – 1 days. But what happens if Tristram Shandy has, ex hypothesi, been writing for an infi nite number of years? The most recent day of his autobiography recedes to infi nity, that is to say, to a day infi nitely distant from the present. Nowhere in the past at a fi nite distance from the present can we fi nd a recorded day, for by now Tristram Shandy is infi nitely far behind. The beginningless, infi nite series of days which he has recorded are days which lie at an infi nite temporal distance from the present. This is not in itself a contradiction. The infi nite past must have in this case, not the order type of the negative numbers ω*, but the order type ω* + ω*, the order type of the series . . . , −3, −2, −1, . . . , −3, −2, −1. But there is no way to traverse the temporal interval from an infi nitely distant event to the present, or, more technically, for an event which was once present to recede to an infi nite temporal distance. Since the task of writing one’s autobiography at the rate of 1 year per day seems obviously coherent, what follows from the Tristram Shandy story is that an infi nite series of past events is absurd.19
But suppose that such an infi nite task could be completed by the present day. Suppose we meet a man who claims to have been counting down from infi nity and who is now fi nishing: . . . , −3, −2, −1, 0. We could ask, why did he not fi nish counting yesterday or the 19. Oppy rightly observes that it is the whole scenario that is impossible, which includes the requirement that consecutive days be recorded (Oppy 2006a, p. 57, n. 3). But given that the task of writing one’s autobiography at the rate of 1 day per year seems obviously coherent, it seem to us that the blame can be placed on the infi nity of the past.
122 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
day before or the year before? By then an infi nite time had already elapsed, so that he has had ample time to fi nish. Thus, at no point in the infi nite past should we ever fi nd the man fi nishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never fi nd the man counting at all, for at any point we reach he will already have fi nished. But if at no point in the past do we fi nd him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infi nite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infi nity.
Conway and Sorabji have responded that there is no reason to think that the man would at any point have already fi nished (Sorabji 1983 , pp. 219–22; Conway 1984). Sorabji thinks the argument confuses counting an infi nity of numbers with counting all the numbers. At any given point in the past, the man will have already counted an infi nity of negative numbers, but that does not entail that he will have counted all the negative numbers. Similarly, in Conway’s analysis, the nub of the argument lies in the conditional
(*)
If an infi nite number of numbers had been counted by yesterday, then the man will have fi nished by yesterday.
But Conway’s conditional is quite ambiguous, and the arguments that he suggests in support of it have no apparent relevance to the reasoning behind the paradox. The mutakallim is not making the obviously false claim that to count infi nitely many negative numbers is to count all the negative numbers! Rather, the conditional at the heart of the paradox is a counterfactual conditional like:
(**) If the man would have fi nished his countdown by today, then he would have fi nished it by yesterday,
and the truth of this conditional seems plausible in light of Hume’s Principle. It is on the basis of this principle that the defender of the infi nite past seeks to justify the intuitively impossible feat of someone’s counting down all the negative numbers and ending at 0.
Since the negative numbers can be put into a one-to-one correspondence with the series of, say, past hours, someone counting from eternity would have completed his countdown.
But by the same token, the man at any point in the past should have already completed his countdown, since by then a one-to-one correspondence exists between each negative number and a past hour. In this case, having infi nite time does seem to be a suffi cient condition of fi nishing the job. Having had infi nite time, the man should have already completed his task.
Such reasoning in support of the fi nitude of the past and the beginning of the universe is not mere armchair cosmology. P. C. W. Davies, for example, utilizes this reasoning in explaining two profound implications of the thermodynamic properties of the universe:
The fi rst is that the universe will eventually die, wallowing, as it were, in its own entropy. This is known among physicists as the ‘heat death’ of the universe. The second is that the universe cannot have existed for ever, otherwise it would have reached its equilibrium end state an infi nite time ago. Conclusion: the universe did not always exist. (Davies 1983, p. 11)
KA LA M C O S M O LO G I C A L A RG U M E N T 123
The second of these implications is a clear application of the reasoning that underlies the current paradox: even if the universe had infi nite energy, it would in infi nite time come to an equilibrium since at any point in the past infi nite time has elapsed, a beginningless universe would have already reached an equilibrium, or as Davies puts it, it would have reached an equilibrium an infi nite time ago. Therefore, the universe began to exist, quod erat demonstrandum.20
Oppy’s response to the problem at hand is to say that the man’s fi nishing his countdown when he does rather than earlier is just “a brute feature of the scenario, that is, a feature that has no explanation” (Oppy 2006a, p. 59; cf. p. 63; Oppy 2006b, pp. 141–2). It has always been the case that he will fi nish when he does, but why the man fi nishes when he does rather than at some other time is just inexplicable. Resting with inexplicability may seem unsatisfactory, however, especially in light of the respectable role such reasoning plays in scientifi c cosmological discussions. Oppy justifi es his response on the basis that principles of suffi cient reason requiring that there be an explanation in such a case are highly contentious. Oppy presents the typical objections to various versions of the Principle of Suffi cient Reason such as the impossibility of providing an explanation of what has been called the
“Big Contingent Conjunctive Fact” (BCCF), which is the conjunction of all the contingent facts there are, or of libertarian free choices (Oppy 2006a, pp. 279–80). The problem with this justifi cation, however, is twofold. First, plausible defenses of the Principle of Suffi cient Reason can be given.21 Second, and more to the point, there is no reason to think that requiring the need for an explanation in the present case demands for its acceptability or plausibility the enunciation and defense of some general Principle of Suffi cient Reason.
Indeed, any such principle is apt to be tested inductively for its adequacy by whether cases like this constitute plausible counterexamples. The exceptions offered by Oppy, such as the inexplicability of the BCCF and libertarian choices, are simply irrelevant to the present case, for the BCCF is not at stake nor can a person counting from eternity at a constant rate choose arbitrarily when to fi nish his countdown. In the case under discussion, we have a good reason to think that the man should have fi nished his countdown earlier than any 20. See the similar reasoning of Barrow and Tipler (1986, p. 601–8) against infl ationary steady-state cosmologies on the ground that any event which would have happened by now would have already happened before now if the past were infi nite.
21. See Pruss’ article in this volume. We shall leave to him the defense of principles of suffi cient reason. Oppy himself thinks that it is “very plausible” that there are acceptable instances of the following schema for a Principle of Suffi cient Reason:
O (for every FG of kind K, there is an F′G′ that partly explains why the GFs rather than Q possible alternatives),
where O is an operator like “necessarily,” “it is knowable a priori,” etc., G is an ontological category such as a proposition, state of affairs, etc., F is a restriction such as true, contingent, etc., and Q is a quantifi er like “any,”
“every,” etc. (Oppy 2006a, p. 285, cf. pp. 275–6). But he thinks that it is not at all clear that there are acceptable instances of this schema that can be used to rule out scenarios like counting down from infi nity. Although it is not clear what Oppy means by “GFs,” the following principle would seem to be an instance of his schema: Necessarily, for any contingent state of affairs involving concrete objects there is a contingent state of affairs that partly explains why that state of affairs obtains rather than any other. Such a principle would require that there be some partial explanation for why the man fi nishes his countdown today rather than at some other time. But not even a partial explanation can be given, for regardless of how we vary such factors as the rate of counting, they will be the same regardless of the time that he fi nishes and so do not furnish even a partial explanation of why he fi nishes today. So why is this instance of the schema not acceptable?
124 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
time that he does, namely, he has already had infi nite time to get the job done.22 If we deny that infi nite time is suffi cient for completing the task, then we shall wonder why he is fi nishing today rather than tomorrow or the day after tomorrow, or, indeed, at any time in the potentially infi nite future. It is not unreasonable to demand some sort of explanation for why, if he fi nishes today, he did not already fi nish yesterday. By contrast, if such a countdown is metaphysically impossible, then no such conundrum can arise. But clearly, there is no metaphysical impossibility in counting backward for all time, unless time is past eternal. It follows that the past cannot be infi nite.
For all of these reasons, the formation of an actual infi nite by successive addition is a notoriously diffi cult notion, even more so than the static existence of an actual infi nite.
2.22. Successive formation of the series of past events
Premise (2.22) may seem rather obvious. The past did not spring into being whole and entire but was formed sequentially, one event occurring after another. Notice, too, that the direction of this formation is “forward,” in the sense that the collection grows with time. Although we sometimes speak of an “infi nite regress” of events, in reality an infi nite past would be an “infi nite progress” of events with no beginning and its end in the present.
As obvious as this premise may seem at fi rst blush, it is, in fact, a matter of great controversy. It presupposes once again an A-Theory of time. On such a theory, the collection of all past events prior to any given event is not a collection whose members all tenselessly coexist. Rather it is a collection that is instantiated sequentially or successively in time, one event coming to pass on the heels of another. Since temporal becoming is an objective feature of the physical world, the series of past events is not a tenselessly existing manifold, all of whose members are equally real. Rather the members of the series come to be and pass away one after another.
Space does not permit a review of the arguments for and against the A- and B-Theories of time respectively. But on the basis of a case such as is presented by Craig (2000a,b), we take ourselves to be justifi ed in affi rming the objective reality of temporal becoming and, hence, the formation of the series of temporal events by successive addition. It is noteworthy that contemporary opponents of Zenonian arguments such as Grünbaum resolve those puzzles only by denying the objective reality of temporal becoming and treating time as a continuum of tenselessly existing point-instants. If moments of time and, hence, events really do come to be and elapse, then it remains mysterious how an infi nite number of such event-intervals can be traversed or manage successively to elapse.
2.23. Conclusion
It follows, then, that the temporal series of events cannot be actually infi nite. The only way a collection to which members are being successively added could be actually infi nite would 22. Notice, too, that if there is any probability of his fi nishing in infi nite time, then he will have already fi nished.
KA LA M C O S M O LO G I C A L A RG U M E N T 125
be for it to have an infi nite tenselessly existing “core” to which additions are being made.
But then, it would not be a collection formed by successive addition, for there would always exist a surd infi nite, itself not formed successively but simply given, to which a fi nite number of successive additions have been made. Clearly, the temporal series of events cannot be so characterized, for it is by nature successively formed throughout. Thus, prior to any arbitrarily designated point in the temporal series, one has a collection of past events up to that point which is successively formed and completed and cannot, therefore, be actually infi nite.
2.3. Scientifi c confi rmation
The sort of philosophical problems with the infi nity of the past, which have been the object of our discussion, are now being recognized in scientifi c papers by leading cosmologists and philosophers of science.23 For example, Ellis, Kirchner, and Stoeger ask, “Can there be an infi nite set of really existing universes? We suggest that, on the basis of well-known philosophical arguments, the answer is No” (Ellis, Kirchner, & Stoeger 2003, p. 14; emphasis added). Similarly, noting that an actual infi nite is not constructible and, therefore, not actualizable, they assert, “This is precisely why a realized past infi nity in time is not considered possible from this standpoint – since it involves an infi nite set of completed events or moments” (Ellis, Kirchner, & Stoeger 2003, p. 14). These misgivings represent endorse-ments of both the kalam arguments defended earlier. Ellis and his colleagues conclude,
“The arguments against an infi nite past time are strong – it’s simply not constructible in terms of events or instants of time, besides being conceptually indefi nite” (Ellis, Kirchner,
& Stoeger 2003, p. 14).
Apart from these philosophical arguments, there has emerged during the course of the twentieth century provocative empirical evidence that the universe is not past eternal. This physical evidence for the beginning of the universe comes from what is undoubtedly one of the most exciting and rapidly developing fi elds of science today: astronomy and astrophysics. Prior to the 1920s, scientists had always assumed that the universe was stationary and eternal. Tremors of the impending earthquake that would topple this traditional cosmology were fi rst felt in 1917, when Albert Einstein made a cosmological application of his newly discovered gravitational theory, the General Theory of Relativity (Einstein 1917, pp. 177–88). In so doing, he assumed that the universe is homogeneous and isotropic and that it exists in a steady state, with a constant mean mass density and a constant curvature of space. To his chagrin, however, he found that General Relativity (GR) would not permit such a model of the universe unless he introduced into his gravitational fi eld equations a certain “fudge factor” Λ in order to counterbalance the gravitational effect of matter and so ensure a static universe. Einstein’s universe was balanced on a razor’s edge, however, and the least perturbation – even the transport of matter from one part of the universe to another – would cause the universe either to implode or to expand. By taking this feature of Einstein’s model seriously, the Russian mathematician Alexander Friedmann and the Belgian astronomer Georges Lemaître were able to formulate independently in the 1920s solutions to the fi eld equations which predicted an expanding universe (Friedmann 1922; Lemaître 1927).
23. Besides the paper by Ellis et al., see Vaas (2004).
126 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
A “perfect” city
A
B
C
D
E
F
G
H
1st
2nd
3rd
4th
A
B C D
E F
G H
1st
5th
2nd
3rd
6th
4th
5th
7th
6th
7th
8th
8th
a = 1
a = ½
Figure 3.2 Analogy of the universe as a city laid out in a grid.
Friedmann’s fi rst equation is:
2
å ⎞
8π G
k
H 2 = ⎛
ρ Λ
,
2
⎝ a⎠ =
+
−
3
3
a
where H = Hubble parameter, a = scale factor, G = gravitational constant, r = mass density of universe, Λ = cosmological constant, k = curvature parameter.24 By way of explanation, the scale factor, “a,” of the universe is a global multiplier to universe size. Imagine the universe as a perfectly laid out city with streets that travel only north-south and east-west. Streets are spaced at equal-distance intervals. Street intersections then defi ne perfectly symmetric city blocks. One could go further and think of buildings in the city as analogous with galaxies in the universe.
The distance from one city block to another is a function of two values: the originally laid out distance (call that the “normalized” distance) and the scale factor multiplier “a.”
Note that, as in Figure 3.2, when one multiplies by a scale factor of 1/2, one still has precisely the same city with the same number of city blocks. The only thing that has changed is the distance interval between the city blocks.
Now consider buildings within the city. If the city block distance were reduced to the size of buildings, clearly something must give. The buildings would be squeezed together and destroyed. This is analogous to what happens with matter in the real universe. The sizes of nonelementary particle mass structures such as protons, neutrons, atomic nuclei, and so on are fi xed; they do not change with the scale factor. Other physical structures, such as massless particles, do adjust with the scale factor. The wavelength of radiation 24. This form of the equation assumes a unit system where the speed of light, “c,” is equal to one. Otherwise the cosmological constant term and the curvature parameter term would be seen as multipled by c 2.
KA LA M C O S M O LO G I C A L A RG U M E N T 127
adjusts and, hence, would gain (in the case of contraction) or lose (in the case of expansion) energy as a result25. When one crowds these particles in upon themselves, one will see a transition to different physics.
Recall that the full city is always present regardless of the value of the scale factor. So now consider two additional situations. First, suppose that the scale factor were to shrink to zero. Space (and time) would disappear. Any structure that could not transform to zero size would be destroyed. If there were no physical process that would allow such a thing to happen, we should seem to have a paradox. Either there must be an undiscovered physical process or the scale factor cannot, in reality, assume a null value.
Second, imagine that the city is of infi nite size. Conceptually, there is no problem with extending the streets north-south and east-west to infi nity in each direction. What does it mean to scale the universe’s size in such a situation? No matter what scale factor one adopts, the size of the universe remains infi nite. Nevertheless, the idea of scaling still retains coher-ency in that we can apply a multiplier to the fi nite distance between city blocks. Yet what would be the meaning of applying a zero scale factor in this situation? Now it would appear that the size of the full universe is “zero times infi nity,” which, in general, can be any fi nite number (Barrow 2005, p. 160). What does this mean, given that the distance between any spot in the universe to any other spot must still be zero? GR simply breaks down at zero scale factor.
Whether or not the full universe is of infi nite or fi nite size is given in the Friedmann equation by the curvature parameter “k.” A positive k indicates that the universe, much like the surface of the Earth, is unbounded yet of fi nite size. Going back to the analogy, imagine that the city is laid out over Earth’s entire curved surface. A traveler on 1st street would never come to the end; rather he would eventually come back to the location where he started. A positive k yields positive curvature and a closed universe. This is one type of
“compact metric” within GR.
A zero value for “k” yields a “fl at” universe. The 1st street is unbounded and of infi nite length (in both directions). A similar situation obtains for a negative k value. Here one has negative, or “saddle-shaped,” curvature. Two travelers moving east and side-by-side up 1st and 2nd streets would actually get laterally farther apart as the curvature of the surface causes the streets to diverge from each other. The latter case gives an infi nitely sized “open”
universe.
The components of the universe (all the energy, keeping in mind that E = mc 2) determine what type of curvature the universe possesses. The “strength” of gravity, included in the equation via the parameter “G,” affects the magnitude of the curvature.
The parameters r and Λ indicate the type and magnitude of the different types of energy that cause the curvature. The parameter “r” represents the density (that is, the energy per unit volume) of the two types of “ordinary” energy: matter and radiation. It is “ordinary”
in the sense that we are familiar with it in daily life and it is of a form that makes gravity an attractive force. Λ represents an exotic type of energy density which can transform gravity from an attractive to a repulsive force.
Friedmann’s fi rst equation tells us how the scale factor changes as time elapses. Mathematically, this is the fi rst derivative of the scale factor “a,” known as “a-dot,” or å. One can see that the increase (or decrease) in the scale factor is strongly a function of the universe’s energy content. Now the “ordinary” energy density r will become smaller as the universe 25. Hence, the temperature of the universe would be seen to rise as one looked back in time.
128 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
expands, since one has the same amount of energy spread out over a greater volume. So its causal impact on the expansion will progressively diminish at ever later times (this works in reverse for contraction). By contrast, Λ, which represents the dark energy density, is constant. The dark energy does not become more dilute during expansion or concentrated during contraction. Hence, early in the life of an expanding universe, Λ is unimportant compared to r. But its impact “snowballs” as time goes on. As long as the impact of r in the early universe is not enough to overturn an expansion and begin a contraction, the effect of Λ will eventually lead to a runaway expansion of the universe. There will appear a moment in the history of the universe when the dark energy will begin to dominate the ordinary energy, and the universe’s expansion will begin to accelerate. Recent observations, in fact, seem to show precisely this effect in our own universe, with a transition age at 9
billion years (Overbye 2006).
Friedmann’s second equation gives the rate of change of the expansion rate:
a
4π G
= −
(ρ
Λ
+ 3 p) +
a
3
3
It determines whether the expansion itself is slowing or accelerating. This acceleration is referred to as “a-double-dot,” or ä. A new term “p” appears in the equation. This is the pressure (similar to the pressure of a gas inside a balloon). Pressure itself can produce gravitational force. Pressure is normally negligible for ordinary matter, although it can play a role in radiation dominated universes. Pressure, however, can have a tremendous impact given a universe dominated by dark energy. As Friedmann’s second equation shows, the rate at which the expansion of the universe accelerates is proportional to: (−r − 3 p). But the pressure in the vacuum is just equal to the negative of the energy density; this is called the equation of state. Hence, overall, the acceleration is positive (which will produce expansion) and proportional to twice the energy density.
Ordinary matter will exert positive pressure (which will keep a balloon infl ated, for example). This type of pressure will produce an attractive gravitational force, which supple-ments the attractive gravity that accrues from mass. Dark energy has the bizarre property that it generates negative pressure. But dark energy, while it has a positive energy density (which contributes to attractive gravity) will, on net, produce a repulsive gravitational effect. Looking at Friedmann’s second equation, one sees that an attractive gravitational contribution tends to slow down expansion (or accelerate contraction), while repulsive gravity will do the opposite.
The monumental signifi cance of the Friedmann–Lemaître model lay in its historization of the universe. As one commentator has remarked, up to this time the idea of the expansion of the universe “was absolutely beyond comprehension. Throughout all of human history the universe was regarded as fi xed and immutable and the idea that it might actually be changing was inconceivable” (Naber 1988, pp. 126–7). But if the Friedmann–
Lemaître model is correct, the universe can no longer be adequately treated as a static entity existing, in effect, timelessly. Rather the universe has a history, and time will not be a matter of indifference for our investigation of the cosmos.
In 1929, the American astronomer Edwin Hubble showed that the redshift in the optical spectra of light from distant galaxies was a common feature of all measured galaxies and was proportional to their distance from us (Hubble 1929, pp. 168–73). This redshift, fi rst
KA LA M C O S M O LO G I C A L A RG U M E N T 129
observed by Vesto Slipher at the Lowell Observatory,26 was taken to be a Doppler effect indicative of the recessional motion of the light source in the line of sight. Incredibly, what Hubble had discovered was the isotropic expansion of the universe predicted by Friedmann and Lemaître on the basis of Einstein’s GR. It was a veritable turning point in the history of science. “Of all the great predictions that science has ever made over the centuries,”
exclaims John Wheeler, “was there ever one greater than this, to predict, and predict correctly, and predict against all expectation a phenomenon so fantastic as the expansion of the universe?” (Wheeler 1980, p. 354).
2.31. The standard Hot Big Bang model
According to the Friedmann–Lemaître model, as time proceeds, the distances separating the ideal particles of the cosmological fl uid constituted by the matter and energy of the universe become greater. It is important to appreciate that as a GR-based theory, the model does not describe the expansion of the material content of the universe into a preexisting, empty, Newtonian space, but rather the expansion of space itself. The ideal particles of the cosmological fl uid are conceived to be at rest with respect to space but to recede progressively from one another as space itself expands or stretches, just as buttons glued to the surface of a balloon will recede from one another as the balloon infl ates. As the universe expands, its density progressively declines.
This has the astonishing implication that as one reverses the expansion and extrapolates back in time, the universe becomes progressively denser until one arrives at a state of infi -
nite density at some point in the fi nite past. This state represents a singularity at which space-time curvature, along with temperature, pressure, and density, becomes infi nite. To be more correct, the volume of the universe approaches zero in the limit as the scale factor of the universe approaches zero. The Friedmann–Lemaître model does not, in fact, describe what happens at the singularity, since Einstein’s GR breaks down at this limit.
The initial cosmological singularity is, therefore, not in space-time but constitutes an edge or boundary to space-time itself. Robert Wald describes how singular space-times are to be properly characterized:
By far the most satisfactory idea proposed thus far is basically to use the ‘holes’ left behind by the removal of singularities as the criterion for their presence. These ‘holes’ should be detectable by the fact that there will be geodesics which have fi nite affi ne length; that is, more precisely there should exist geodesics which are inextendible in at least one direction but have only a fi nite range of affi ne parameter. Such geodesics are said to be incomplete. (For timelike and spacelike geodesics, fi nite affi ne ‘length’ is equivalent to fi nite proper time or length so the use of affi ne parameter simply generalizes the notion of ‘fi nite length’ to null geodesics.) Thus, we could defi ne a spacetime to be singular if it possesses at least one incomplete geodesic.
Nevertheless, there is a serious physical pathology in any spacetime which is timelike or null geodesically incomplete. In such a spacetime, it is possible for at least one freely falling particle or photon to end its existence within a fi nite ‘time’ (that is, affi ne parameter) or to have begun its existence a fi nite time ago. Thus, even if one does not have a completely satisfactory general notion of singularities, one would be justifi ed in calling such spacetimes 26. Slipher’s early papers are now available online at http://www.roe.ac.uk/~jap/slipher/.
130 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
physically singular. It is this property that is proven by the singularity theorems to hold in a wide class of spacetimes. (Wald 1984, pp. 215–6)27
The existence of a boundary to space-time implies, not merely that the “stuff ” of the universe begins to exist but that space and time do as well (for in the Friedmann–Lemaître model all past-directed geodesics terminate at the singularity). P. C. W. Davies comments:
If we extrapolate this prediction to its extreme, we reach a point when all distances in the universe have shrunk to zero. An initial cosmological singularity therefore forms a past temporal extremity to the universe. We cannot continue physical reasoning, or even the concept of spacetime, through such an extremity. For this reason most cosmologists think of the initial singularity as the beginning of the universe. On this view the big bang represents the creation event; the creation not only of all the matter and energy in the universe, but also of spacetime itself. (Davies 1978, pp. 78–9)
The term “Big Bang,” originally a derisive expression coined by Fred Hoyle to characterize the beginning of the universe predicted by the Friedmann–Lemaître model, is thus potentially misleading, since the expansion cannot be visualized from the outside (there being no “outside,” just as there is no “before” with respect to the Big Bang).28
The standard Hot Big Bang model, as the Friedmann–Lemaître model came to be called, thus describes a universe which is not eternal in the past, but which came into being a fi nite time ago. Moreover – and this deserves underscoring – the origin it posits is an absolute origin ex nihilo. For not only all matter and energy but also space and time themselves come into being at the initial cosmological singularity. As Barrow and Tipler emphasize, “At this singularity, space and time came into existence; literally nothing existed before the singularity, so, if the Universe originated at such a singularity, we would truly have a creation ex nihilo” (Barrow and Tipler 1986, p. 442). On such a model the universe originates ex nihilo in the sense that it is false that something existed prior to the singularity .
27. A geodesic is the path that a freely falling particle traces out through space and time. A timelike geodesic is traveled by a massive particle. A null geodesic is traveled by a massless particle such as the photons that make up visible light.
28. As Gott et al. write:
The universe began from a state of infi nite density about one Hubble time ago. Space and time were created in that event and so was all the matter in the universe. It is not meaningful to ask what happened before the big bang; it is somewhat like asking what is north of the North Pole. Similarly, it is not sensible to ask where the big bang took place. The point-universe was not an object isolated in space; it was the entire universe, and so the only answer can be that the big bang happened everywhere. (1976, p. 65) The Hubble time is the time since the singularity if the rate of expansion has been constant. The singularity is a point only in the sense that the distance between any two points in the singularity is zero. Anyone who thinks that there must be a place in the universe where the Big Bang occurred still has not grasped that it is space itself which is expanding; it is the two-dimensional surface of an infl ating balloon which is analogous to three-dimensional space. The spherical surface has no center and so no location where the expansion begins. The analogy of the North Pole with the beginning of time should not be pressed, since the North Pole is not an edge to the surface of the globe; the beginning of time is more like the apex of a cone. But the idea is that just as one cannot go further north than the North Pole, so one cannot go earlier than the initial singularity.
KA LA M C O S M O LO G I C A L A RG U M E N T 131
2.32. Evidence for GR
The earliest evidence in favor of the Big Bang came from the consonance of theory and experiment. Einstein’s early papers proposed two tests that could be performed immediately. It had been known for some time that Newton’s gravitational theory could not adequately describe the orbit of the planet Mercury. The real orbit precessed around the sun (i.e. the ellipse itself rotates over time). In contrast to Newton’s theory of gravity, GR “predicted” that this precession should take place. Einstein’s theory also predicted that, since matter bends space, light rays should have their paths noticeably bent when they pass close to massive objects. A solar eclipse in 1919 provided the opportunity for a test of this prediction. An expedition led by Arthur Eddington confi rmed that light rays were indeed defl ected.
These tests were not suffi ciently accurate29 to ensure that small deviations from GR
were not possible. It was also suspected that, since the real universe is not completely homogeneous and isotropic at all scales, Friedmann and Lemaître’s prediction of a true singular beginning to the universe would ultimately fail. Perhaps a slight anisotropy could result in matter’s “sling-shotting” past itself at a minimum (but nonzero) radius condition, so that the present expansion was preceded by a cosmic contraction, thereby avoiding the absolute beginning of the universe. In 1970, however, Stephen Hawking and Roger Penrose proved that the homogeneity/isotropy assumption was irrelevant. The Hawking–Penrose singularity theorems showed that so long as the universe is governed by GR (with a few technical exceptions, which will become prominent in our discussion later into the chapter), our past must include a singularity (Hawking & Penrose 1970).
Wald comments:
[Hawking–Penrose 1970] gives us strong reason to believe that our universe is singular . . . the observational evidence strongly suggests that our universe – or, at least, the portion of our universe within our causal past – is well described by a Robertson-Walker model [standard hot Big Bang theory] at least back as far as the decoupling time of matter and radiation.
However, in these models, the expansion of the past directed null geodesics emanating from the event representing us at the present time becomes negative at a much more recent time than the decoupling time. Thus there is strong reason to believe that condition 4c of [Hawking–
Penrose 1970] is satisfi ed in our universe. Since we expect that conditions (1)-(3) also are satisfi ed, it appears that our universe must be singular. Thus, it appears that we must confront the breakdown of classical general relativity expected to occur near singularities if we are to understand the origin of our universe. (Wald 1984, p. 241)
The conditions Wald mentions are:
1. Satisfaction of the strong energy condition (typically obeyed by “normal” types of matter).
29. However, further experiments did defi nitively establish the correspondence between GR and nature. The 1993
Nobel Prize for Physics was awarded to two astronomers: Russell A. Hulse and Joseph H. Taylor, Jr. The award was given for their study of a distant solar system consisting of a binary pulsar – two neutron stars orbiting each other. GR predicted that the orbit would shrink over time due to the emission of gravitational waves. They proved that GR is accurate to a startling degree of one part in 1014. This makes GR perhaps the best-proved theory in all of physics.
132 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
2. Satisfaction of the generic energy condition (there is no exotic property of the spacetime that prevents gravitational focusing).
3. No closed time loops (the future does not bend back and become one’s own past).
4. There is a point p such that past-directed worldlines emanating from p have negative expansion; that is, they are focused back on each other (the worldlines of observers do not trace into an infi nite past given suffi ciently concentrated matter but are “focused,”
as by a lens, into a singular condition within a fi nite time).
2.33. Exceptions to the Hawking–Penrose theorems
Four possible exceptions to the Hawking–Penrose singularity theorems conveniently distinguish four classes of nonstandard models that provide possible alternatives to the standard Big Bang model (Figure 3.3). The Hawking–Penrose Theorem also has the obvious, but implicit, condition that GR is fundamental; that is, it is a complete as well as correct description of conditions within our universe (thereby defi ning a 5th condition).
The fi rst option (closed time loops) has been the subject of some exploration in cosmological circles. The next two – eternal infl ation and quantum gravity – represent areas of fertile cosmological investigation which merit our attention. The last two exception conditions are not expected to be part of “reasonable” physical models of the universe.
Hawking explains:
Between 1965 and 1970 Penrose and I used the techniques I have described to prove a number of singularity theorems. These theorems had three kinds of conditions. First there was an energy condition such as the weak, strong or generic energy conditions. Then there was some global condition on the causal structure such as that there shouldn’t be any closed time like curves. And fi nally was some condition that gravity was so strong in some region that nothing could escape. . . .
Five exception conditions to the Hawking–Penrose Theorem
Closed timelike loops
(Universe as time machine)
Strong energy condition violated
(Eternal inflation)
Falsity of general relativity
(Quantum gravity)
Generic energy condition violated
(“Exotic” space-time)
No closed trapped
surface in our past
Figure 3.3 Model classes based on exceptions to the Hawking–Penrose singularity theorems.
KA LA M C O S M O LO G I C A L A RG U M E N T 133
[The generic energy condition] says that fi rst 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 satisfi ed by a number of known exact solutions. But these are rather special. One would expect it to be satisfi ed by a solution that was “generic” in an appropriate sense [that is, a reasonable physical model]. If the generic energy condition holds, each geodesic will encounter a region of gravitational focusing. (Hawking & Penrose 1996, pp. 14–5)
We do expect our past to feature a closed, trapped surface, and there is no reason to postulate an exotic space-time construction that would just happen to have perfect defo-cusing characteristics so as to counter the effects of gravity. Hence, our discussion will revolve around the fi rst three options.
I. Closed timelike curves (CTCs)
A fi rst, exotic exception to the Hawking–Penrose theorems is the possible existence of CTCs. Permitted by Einstein’s GR, CTCs represent an observer tracing out a circular path through space and time.
J. Richard Gott and Li-Xin Li have proposed a model according to which the early universe (only) is a closed time loop that occasionally gives “birth” to a universe like ours (Figure 3.4).
They maintain that Alexander Vilenkin’s “tunneling from nothing” model (see section IVc) should properly be taken as tunneling from a previous state. As will be seen in this chapter, most cosmological models assert that the past terminates at a boundary a fi nite time ago.
One then wishes to explain what exists at that boundary. Vilenkin (and, independently, the team of Stephen Hawking and James Hartle) believes that the universe sprang into being “out of nothing.” Gott and Li believe, instead, that there is a CTC at this boundary.
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Figure 3.4 A Gott–Li universe time machine.
The region of closed timelike curves (CTC) exists at the bottom of the diagram. This region is separated from the future evolution of the universe by a Cauchy horizon (a boundary that separates space-time into CTC regions and non-CTC regions). The four branches to the top of the diagram can be thought of as infl ationary bubbles undergoing a de Sitter-like expansion (see sections II and IIIa for a discussion of infl ation and de Sitter spaces). (source: Gott & Li 1998)
134 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
The Gott–Li model could be an example of a universe timeline looping back upon
itself to allow the universe, in Gott and Li’s words, “to become its own mother.” They explain:
In this paper, we consider instead the notion that the Universe did not arise out of nothing, but rather created itself. One of the remarkable properties of the theory of general relativity is that in principle it allows solutions with CTCs. Why not apply this to the problem of the fi rst-cause?
Many infl ationary models allow creation of baby infl ationary universes inside black holes, either by tunneling across the Einstein-Rosen bridge, or by formation as one approaches the singularity. If one of these baby universes simply turns out to be the universe we started with, then a multiply connected model with early CTCs bounded by a Cauchy horizon is produced.
. . . Then the Universe neither tunneled from nothing, nor arose from a singularity; it created itself. (Gott & Li 1998, p. 39; emphasis added)
Some histories in our past were circular in nature (forming a multiply connected spacetime) and recreating the Big Bang. Further, this is not a cyclic universe; it is the same Big Bang.
The CTC scenario raises interesting philosophical questions about the nature of time (see below, p. 191). But here, our interest is in the model’s physical viability. The primary physical problem confronting CTC models in general is their violation of the so-called Chronology Protection Conjecture (CPC).
Gott and Li indicate that “the region of CTCs . . . should be in a pure vacuum state containing no real particles or Hawking radiation and no bubbles”(Gott & Li 1998, p. 39).
This is so because this stray radiation would destroy the CTC. The reason for this curious feature of a CTC model was discussed by Stephen Hawking (1992), where he formally suggested a “CPC.” His theory was that a time machine (CTC) would have characteristics that were so unstable that it would quickly destroy itself. Hence, nature conspires to prevent time machines. A popular level (and entertaining) description of this effect is given by GR
theorist Kip Thorne. He constructs a scenario that allows a local time machine to exist with one end on a spaceship departing Earth with his wife Carole and the other end on Earth with him in his living room (this is an exotic general relativistic structure called a wormhole).
Imagine that Carole is zooming back to Earth with one wormhole mouth in her spacecraft, and I am sitting at home on Earth with the other. When the spacecraft gets to within 10 light-years of Earth, it suddenly becomes possible for radiation (electromagnetic waves) to use the wormhole for time travel: any random bit of radiation that leaves our home in Pasadena traveling at the speed of light toward the spacecraft can arrive at the spacecraft after 10 years’
time (as seen on Earth), enter the wormhole mouth there, travel back in time by 10 years (as seen on Earth), and emerge from the mouth on Earth at precisely the same moment as it started its trip. The radiation piles right on top of its previous self, not just in space but in spacetime, doubling its strength. What’s more, during the trip each quantum of radiation (each photon) got boosted in energy due to the relative motion of the wormhole mouths (a “Doppler-shift” boost).
After the radiation’s next trip out to the spacecraft then back through the wormhole, it again returns at the same time as it left and again piles up on itself, again with a Doppler-boosted energy. Again and again this happens, making the beam of radiation infi nitely strong.
KA LA M C O S M O LO G I C A L A RG U M E N T 135
In this way, beginning with an arbitrarily tiny amount of radiation, a beam of infi nite energy is created, coursing through space between the two wormhole mouths. As the beam passes through the wormhole . . . it will produce infi nite spacetime curvature [i.e. a singularity] and probably destroy the wormhole, thereby preventing [a time machine from coming into being in the fi rst place]. (Thorne 1994, pp. 505–6)
Of interest to us is the general applicability of this effect to the Gott–Li model. Gott and Li are sensitive to this problem and have developed a solution30. They and others have found some specially constructed space-times that appear to elude Hawking’s CPC. To avoid the CPC, they have constructed a special initial state for the universe: a zero-temperature empty space called an “adapted Rindler vacuum.” It is specially built and balanced such that it does not develop the destructive effect suggested by Thorne earlier.
After the publication of Gott and Li’s paper, William Hiscock developed a defense of the CPC that still appears to stand (Hiscock 2000). First, Hiscock argues that the Gott–Li choice of initial conditions is highly fi ne-tuned. In fact, Gott–Li’s vacuum is of “measure zero” in the set of all possible Rindler vacuums. This means that the scenario is just about as unlikely as is possible without ruling it out summarily. D. H. Coule agrees in his summary of quantum gravity models, referring to the Gott–Li model as “rather contrived”
(Coule 2005).31 Second, Hiscock argues that the Gott–Li vacuum is not stable, given more realistic physical force fi elds. He writes:
. . . the (Rindler) vacuum stress-energy of a nonconformally coupled scalar fi eld, or a conformally coupled massless fi eld with a . . . self-interaction will diverge on the chronology horizon for all values of the Misner identifi cation scale [this is the parameter that Gott–Li have fi ne-tuned]. In addition, the vacuum polarization of [the scalar fi eld considered in the Gott–Li model] diverges in all cases [leading to the Thorne effect cited earlier], even for the conformally invariant case examined by Li and Gott. Hence, the regular behavior found by Cassidy and Li and Gott holds only for a conformally invariant, non-interacting fi eld, and only for the stress-energy tensor. While some fi elds in nature (e.g., the electromagnetic fi eld, before interactions are added) are conformally invariant, others – notably gravity itself–are not; and interactions are the rule, not the exception. (Hiscock 2000, p. 4)
30. Gott indicates:
I think no one has been able to rule out CTC’s. There have been no signifi cant changes in quantum gravity since our paper. To understand whether one can create a time machine one may have to understand quantum gravity and we do not yet. Several loopholes in chronology protection have been found. Li Xin-Li and I and Cassidy, Hawking’s student have found examples of quantum vacuum states that do not blow up on the cauchy horizon. Li Xin-Li’s paper on the correct renormalization procedure [Phys. Rev. D.
084016 (1999)] showed that the vacuum fi eld did not blow up for electromagnetic fi elds and other fi elds as well as for scalar fi elds. This means the action or entropy does not blow up either–solving a trouble Hawking and Cassidy thought existed. For matter obeying the weak energy condition instabilities are cured if the time loop occurs at the beginning of the universe, as we are proposing. (pers. comm., March 1, 2008)
31. Coule has some additional objections of his own. For example, he criticizes the nature of the vacuum, indicating that thermal fl uctuations need to be of a precise form in order to avoid the radiation backreaction described earlier by Thorne. This expectation is inconsistent with the Planck scale physics employed by Gott and Li.
136 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Coule adds: “. . . in Misner space this state [Gott–Li model] was only possible with identifi cation scale b = 2p, or b = 2p r 0 for the multiple de Sitter case. Such an exact value is itself inconsistent with notions of quantum uncertainty” (Coule 2005, p. 31). So the Heisenberg uncertainty principle of quantum mechanics (QM) would guarantee that the relevant parameter could not be “just-so.” But if it is not “just-so,” then the universe collapses into a singular condition in the presence of a time machine. Coule also suggests that this parameter, called the “Misner identifi cation scale” is not a constant. Rather it is likely to change dynamically as a function of matter couplings or energy potentials. As soon as it does, the CTC will destabilize.
Interestingly, Gott and Li used similar objections when arguing for their model at the expense of the “creation from nothing” approach of Vilenkin and Hartle and Hawking
[section IVc]. Gott and Li criticize the “creation from nothing” approach on the grounds of the uncertainty principle and the fact that their competitors are not using realistic force fi elds; that is to say, the Vilenkin approach is not close enough to what we expect for the real universe. Yet their own model appears to break down when similar objections are leveled against it.32,33
CTC physics is interesting, and while some theorists still pursue it, it occupies only a small minority of ongoing cosmological investigation. While it is true that no one has been able defi nitively to rule out CTCs, the evidentiary burden lies upon those defending the viability of such space-times and models predicated upon their reality.
II. Eternal infl ation
Motivation
A more serious exception to the Hawking–Penrose singularity theorems is afforded by infl ationary theory. Although the Friedmann–Lemaître model had a great deal of evidential support, there were, nonetheless, observational anomalies which suggested that there was more to the story. There were also theoretical reasons to think that the description was not quite complete. These diffi culties, especially the horizon, fl atness, and cosmic relic problems, prompted theorists to propose a modifi cation of the standard Big Bang picture called
“infl ation.”
With respect to the horizon problem, cosmologists lacked an explanation as to why the universe should be so homogeneous and isotropic.34 Alan Guth explains:
The initial universe is assumed to be homogeneous, yet it consists of at least ~1083 separate regions which are causally disconnected ( i.e. , these regions have not yet had time to commu-32. In fairness to Gott and Li, it should be noted that Hiscock’s criticisms are based on a semiclassical approach (an approximation of quantum gravity), and it is possible that a full theory of quantum gravity could vindicate their idea.
33. Vilenkin has also criticized the Gott–Li model (see Vilenkin 2006, p. 219). He indicates that the Gott–Li space-time contains incomplete histories, so, “This means that the spacetime itself is past-incomplete, and therefore does not provide a satisfactory model of a universe without a beginning.”
34. The universe does appear different at various distances as we look at it. But this is due to the fact that we observe distant galaxies as they were in the past, given the time it takes for their light to reach us.
KA LA M C O S M O LO G I C A L A RG U M E N T 137
nicate with each other via light signals) . . . Thus, one must assume that the forces which created these initial conditions were capable of violating causality. (Guth 1981, p. 347) Cosmology had an appropriate “organizing” principle at hand – thermodynamic
equilibrium – yet mathematics showed that, in the limit, as one looked backward at the Big Bang, the different parts of the universe would lose causal communication. Without causal communication, all the parts of the observable universe could not have cooperated in energy transfer so as to make all parts look the same (in the present). Physicist Brian Greene describes the horizon problem:
Physicists defi ne a region’s cosmic horizon (or horizon for short) as the most distant surrounding regions of space that are close enough to the given region for the two to have exchanged light signals in the time since the [Big] bang. . . . The horizon problem, then is the puzzle, inherent in the observations, that regions whose horizons have always been separate – regions that could never have interacted, communicated, or exerted any kind of infl uence on each other – somehow have nearly identical temperatures. (Greene 2004, p. 289; emphasis in the original)
. . . imagine running the cosmic fi lm in reverse while focusing on two regions of space currently on opposite sides of the observable universe – regions that are so distant that they are beyond each other’s spheres of infl uence. If in order to halve their separation we have to roll the cosmic fi lm more than halfway back toward the beginning, then even though the regions of space were closer together, communication between them was still impossible: they were half as far apart, but the time since the bang was less than half of what it is today, so light could travel only less than half as far. Similarly, if from that point in the fi lm we have to run more than halfway back to the beginning in order to halve the separation between the regions once again, communication becomes more diffi cult still. With this kind of cosmic evolution, even though regions were closer together in the past, it becomes more puzzling – not less –that they somehow managed to equalize their temperatures. Relative to how far light can travel, the regions become increasingly cut off as we examine them ever farther back in time. This is exactly what happens in the standard big bang theory. (Greene 2004, p. 288; emphasis in the original)
A second problem was that the universe appears to be “fl at” (i.e. space is Euclidian: the angles of a triangle add up to 180 degrees; parallel lines do not intersect), while GR predicts that that is a wildly improbable outcome.
A typical closed universe will reach its maximum size on the order [of the Planck scale of 10−44
sec], while a typical open universe will dwindle to a ρ [density] much less than ρcr [critical density; the density for a long-lived universe]. A universe can only survive ~1010 years [approximately the age of our universe] only by extreme fi ne tuning. . . . For [the likely initial conditions for our universe] the value of H0 [the initial expansion rate of the universe] must be fi ne tuned to an accuracy of one part in 1055. In the standard model this incredibly precise initial relationship must be assumed without explanation. (Guth 1981, p. 348)
The third problem was that the supposition of an initial disorganized state of the universe led to the prediction of the presence of bizarre cosmic relics. Magnetic monopoles should appear in our universe at a density amenable to detection with our present means.
In Guth’s original paper on infl ation, he indicates that standard particle physics predicts a
138 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
monopole concentration 14 orders of magnitude greater than the upper bound observed in our universe. To date, we have seen none of these exotic structures.35
Guth’s solution to these three problems was to postulate a period of exponential expansion very early in the history of the universe. Again Greene:
In infl ationary cosmology, there was a brief instant during which gravity was repulsive and this drove space to expand faster and faster. During this part of the cosmic fi lm, you would have to wind the fi lm less than halfway back in order to halve the distance between the two regions. . . . the increasingly rapid separation of any two regions of space during infl ationary expansion implies that halving their separation requires winding the cosmic fi lm less – much less – than halfway back toward the beginning. As we go farther back in time, therefore, it becomes easier for any two regions of space to infl uence each other, because, proportionally speaking, there is more time for them to communicate. Calculations show that if the infl ationary expansion phase drove space to expand by at least a factor of 1030, an amount that is readily achieved in specifi c realizations of infl ationary expansion, all the regions in space that we currently see . . . were able to communicate . . . and hence effi ciently come to a common temperature in the earliest moments of the universe. (Greene 2004, pp. 289–90; emphasis in the original)
This infl ationary period began and ended in a fraction of a second, yet a typical infl ationary event could lead to 70 “e-folds.” An e-fold is a logarithmic measure of how large the universe grows during an infl ationary event. Here, N is the number of e-folds, and a( t) represents the scale factor of the universe at the beginning and end of infl ation.36
N t
( ) ≡ ln[ a t
(
]
end )/ a t
( beginning )
E-folds are a shorthand way of expressing the huge increase in size of the universe during an infl ationary event (recall Greene’s factor of 1030).
Hence, prior to infl ation all parts of the present, observable universe could be in causal communication with one another. Infl ationary expansion would also smooth the curvature of the present-day universe to be fl at or nearly fl at, similar to the way the curvature of a basketball would appear to vanish if it suddenly grew to the size of the Earth. Further, since our present observable universe would be only a microscopic part of the original generic manifold, the density of exotic cosmic relics would be expected to be so small that we should not see them.
Infl ation was a remarkable fi x to a set of serious anomalies; but it also had one more feature in store. The Hawking–Penrose singularity theorems had as one of their requirements that gravity is always attractive – just as it is for ordinary matter. But the 35. To understand what these exotic structures represent, consider the analogy of a pond freezing in wintertime.
If the pond starts freezing in one place and the ice simply grows until it encompasses the whole pond, you will have a smooth surface. But if different parts of the pond start freezing separately, ultimately, these growing “ice-bergs” must meet at a boundary. (Imagine taking big rocks and cracking holes in the ice; then letting it refreeze.
The boundaries will be rough.) The early universe was similar. These boundaries are called “defects” and can be zero-, one-, or two-dimensional. Zero-dimensional defects are called magnetic monopoles. One-dimensional defects are called cosmic strings. Two-dimensional boundaries are called domain walls.
36. Defi nition of e-fold available at http://astro.uchicago.edu/~cunha/infl ation/node4.html.
KA LA M C O S M O LO G I C A L A RG U M E N T 139
most likely physical candidate that could account for an infl ationary event was a type of energy similar to the original cosmological constant that Einstein had proposed (Einstein 1917). This bizarre type of energy would act like repulsive gravity. This led to a philosophically desired outcome. If this “repulsive gravity” was present in the early universe and could dominate attractive gravity, then the possibility arises that the Hawking–Penrose singularity theorems did not apply to the real universe. Perhaps the universe is past eternal after all.
Infl ationary theorizing eventually led to a yet grander theory, according to which the gravitationally repulsive material may, in fact, be the norm in the universe rather than the exception. Cosmologist Andrei Linde explains:
This process, which I have called eternal infl ation, keeps going as a chain reaction, producing a fractal-like pattern of universes. In this scenario the universe as a whole is immortal. Each particular part of the universe may stem from a singularity somewhere in the past, and it may end up in a singularity somewhere in the future. There is, however, no end for the evolution of the entire universe.
The situation with the very beginning is less certain. There is a chance that all parts of the universe were created simultaneously in an initial big bang singularity. The necessity of this assumption, however, is no longer obvious.
Furthermore, the total number of infl ationary bubbles on our ‘cosmic tree’ grows exponentially in time. Therefore, most bubbles (including our own part of the universe) grow indefi nitely far away from the trunk of this tree. Although this scenario makes the existence of the initial big bang almost irrelevant, for all practical purposes, one can consider the moment of formation of each infl ationary bubble as a new ‘big bang.’ From this perspective, infl ation is not a part of the big bang theory, as we thought 15 years ago. On the contrary, the big bang is a part of the infl ationary model. (Linde 1998, p. 103; emphasis added) Linde’s chaotic infl ation was one of two competing views for the theory. The competitor, called “new” infl ation, featured the idea that there is a “false vacuum,” which represents (meta)stable vacuum with a high Einstein-like cosmological constant (compared with the
“true vacuum” in which we live). In “new infl ation,” this eternally expanding false vacuum regionally decays into the slower expanding true vacuum. It expands faster than it decays, so the process never stops.
Chaotic infl ation is rather a different idea. Here, the universe starts from a “generic manifold,” which is a state of maximal entropy chaos. An energy fi eld of different regional values pervades this manifold. Where the fi eld is large, infl ation occurs. The fi eld can undergo quantum fl uctuation to high values as well; thereby giving onset to infl ation. Locally, a region will have a tendency to seek out the minimum allowed value of the energy fi eld. This leads to a process called “reheating,” which creates the ordinary matter and energy that we see around us. Meanwhile, in locations of the universe where the fi eld energy density is high, the quantum fl uctuations will tend to offset the minimizing tendency and make perpetual (globally, but not regionally) the infl ationary process (Figure 3.5).
Eternal infl ationary models
The 1980s and 1990s witnessed a proliferation of infl ationary models that theoretically allowed for a projection into an eternal past (Linde 2005). The infl ationary phase mentione d earlier was not viewed in these models as an isolated event. Theorists began to describe the
140 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Figure 3.5 In chaotic infl ation, an initial generic manifold (a global space with random characteristics; that is, inhomogeneous and anisotropic in energy density and curvature) undergoes regional infl ation. An observer sitting on an infl ating spot will eventually see infl ation come to a stop, but conditions are always suitable – somewhere – for infl ation to proceed.
exotic energy that produces infl ation as a fi eld that pervades otherwise empty space. A key assumption was that the density37 of the energy throughout space never changes, so that it resembles Einstein’s cosmological constant. It does not depend on space or time, that is , it is constant. In that case, as space expands, more energy must continually be produced in order to maintain a constant energy density (where this energy comes from is still a matter of controversy). Space “clones” itself.38 Occasionally, parts of this rapidly expanding space decay (convert) into the type of “empty” space that we live in. This space has a much lower energy density, so there is now a great deal of excess energy that pervades our new
“bubble.” This excess energy is thought to convert into the normal matter that we see around us.
In the latest version of infl ationary theory (Figure 3.6), these decays take the form of quantum tunneling events. Every state that possesses a positive cosmological constant is
“metastable.” That means that, similar to a radioactive isotope, the state lasts for a while and then changes to a different (usually lower) allowed value of the cosmological constant.
This lower-energy state is initially confi ned to a tiny portion of space, but given that the cosmological constant makes space expand, it becomes a rapidly growing bubble that is nested within the original space.
But what happens to the original space, part of which decayed to form our universe?
It is still there, continuing to expand at enormous speed. Since it (usually) has a cosmological constant larger than the new bubble, its growth outpaces that of the new bubble.
Since the false vacuum expands faster than it decays, infl ation is eternal into the future. New bubbles of low-energy vacuum will continue to decay out of the expanding space.
37. In Linde’s chaotic infl ation, the energy fi eld does feature quantum fl uctuations that are critical to the onset of new infl ationary patches.
38. Since only the “material” cause is missing, this process is an example of genuine creatio ex nihilo seen by physical theorists in the present day. Such recognition of effi cient causation in the absence of material causation may serve to mute objections to theistic creatio ex nihilo as featured in the kalam cosmological argument.
KA LA M C O S M O LO G I C A L A RG U M E N T 141
Publisher's Note:
Permission to reproduce this image
online was not granted by the
copyright holder. Readers are kindly
requested to refer to the printed v ersion
of this chapter.
Figure 3.6 The String Landscape infl ationary model. The Big Bang is just a regional event within a larger multiverse. There are different kinds of “empty space,” which feature different values of the cosmological constant (different bubbles). The larger the constant, the faster the universe expands. Our universe decayed from one of these “false vacuum” regions. (inset adapted from the National Aeronautics and Space Administration, available at: http://map.gsfc.nasa.gov/news/index.
html)
Theorists wondered whether this process could be infi nitely extended into the past.
Interestingly, Guth himself, along with collaborators Alexander Vilenkin and Arvind Borde, has likely closed the door on that possibility. In 2003, Borde, Guth, and Vilenkin published an updated singularity theorem far grander in scope than the Hawking–Penrose theorems.
They explain,
Our argument shows that null and time-like geodesics are, in general, past-incomplete in infl ationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics. (Borde, Guth, & Vilenkin 2003, p. 3)39
A remarkable thing about this theorem is its sweeping generality. We made no assumptions about the material content of the universe. We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modifi cation, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe 39. H av refers to the average value of the Hubble constant throughout history.
142 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
never gets below some nonzero value, no matter how small. This assumption should certainly be satisfi ed in the infl ating false vacuum. The conclusion is that past-eternal infl ation without a beginning is impossible. (Vilenkin 2006, p. 175)
Vilenkin affi rms that any universe (including universes modeled by higher dimensional cosmology, pre–Big Bang cosmology, and so forth,) which, on average, expands has to connect, in a fi nite time, to a past boundary (pers. comm., March 4, 2004).
Intuitively, the reason that the universe must have a beginning in the fi nite past is that, in an expanding space, an observer tracing out a worldline (to the future) slows down. This is the redshift. Vilenkin explains:
Let us now introduce another observer who is moving relative to the spectators [each of whom is motionless except for the expansion of space]. We shall call him the space traveler. He is moving by inertia, with the engines of his spaceship turned off, and has been doing so for all eternity. As he passes the spectators, they register his velocity.
Since the spectators are fl ying apart [i.e. the universe is expanding], the space traveler’s velocity relative to each successive spectator will be smaller than his velocity relative to the preceding one. Suppose, for example, that the space traveler has just zoomed by the Earth at the speed of 100,000 kilometers per hour and is now headed toward a distant galaxy, about a billion light years away. That galaxy is moving away from us at a speed of 20,000 kilometers per second, so when the space traveler catches up with it, the observers there will see him moving at 80,000 kilometers per second.
If the velocity of the space traveler relative to the spectators gets smaller and smaller into the future, then it follows that his velocity should get larger and larger as we follow his history into the past. In the limit, his velocity should get arbitrarily close to the speed of light. (Vilenkin 2006)40
So, looking into the past, the observer must be seen to speed up. But one cannot exceed the speed of light. The implication of this is that the past worldline of this observer has a fi nite length. This is the symptom of singularity; the “pathology” that Robert Wald referred to earlier. The observer will have “begun its existence a fi nite time ago.”
The Borde–Vilenkin–Guth (BVG) singularity theorem is now widely accepted within the physics community. As of this writing, it has gone largely unchallenged.41 Instead a new round of model building has resulted based on exceptions to this theorem. Four alternatives present themselves (Figure 3.7).
40. Alan Guth, in a 2003 lecture at the University of California Santa Barbara’s Kavli Institute, says: “If we follow the observer backwards in an expanding universe, she speeds up. But the calculation shows that if Haverage > 0 in the past, then she will reach the speed of light in a fi nite proper time.” (See http://online.kitp.ucsb.edu/online/
strings_c03/guth/pdf/KITPGuth_2up.pdf.)
41. Andrei Linde has offered a critique, suggesting that BVG imply that all the individual parts of the universe have a beginning, but perhaps the WHOLE does not. This seems misconstrued, however, since BVG are not claiming that each past inextendible geodesic is related to a regional singularity. Rather, they claim that Linde’s universe description contains an internal contradiction. As we look backward along the geodesic, it must extend to the infi nite past if the universe is to be past eternal. But it does not (for the observer comoving with the expansion). Rather, past inextendible geodesics are the “symptom,” not the “disease.” As Robert Wald (1984, p. 216) says, “Unfortunately, the singularity theorems give virtually no information about the nature of the singularities of which they prove existence.” So we do not know the nature of the singularity that the BVG Theorem indicates; we know only that Linde’s description of an infi nite past is in error.
KA LA M C O S M O LO G I C A L A RG U M E N T 143
Four exception conditions to the
Borde-Vilenkin-Guth singularity theorem
Infinite contraction ( H avg < 0)
Example: de Sitter cosmology
Asymptotically static ( H avg = 0)
Example: Emergent model class
Infinite cyclicity( H avg = 0)
Example: Baum–Frampton “phantom bounce”
Time reversal at singularity
Example: Aguirre–Gratton
Figure 3.7 Post-2003 cosmological model building based on fi nding exceptions to the Borde–
Vilenkin–Guth Theorem.
IIIa. Infi nite contraction
Assume that a spatially infi nite universe contracted down to a singularity and then
“bounced” into our present expansion. In such a case, the universe cannot be said to be, on average, in a state of cosmic expansion throughout its history since the expansion phase, even if infi nite, is canceled out by the contraction phase. While permissible under the BVG
Theorem, this option is not, however, a popular option among contemporary cosmologists.
George Ellis identifi es two problems that bedevil such an approach:
The problems are related: fi rst, initial conditions have to be set in an extremely special way at the start of the collapse phase in order that it is a Robertson-Walker universe collapsing; and these conditions have to be set in an acausal way (in the infi nite past). It is possible, but a great deal of inexplicable fi ne tuning is taking place: how does the matter in widely separated causally disconnected places at the start of the universe know how to correlate its motions (and densities) so that they will come together correctly in a spatially homogeneous way in the future??
Secondly, if one gets that right, the collapse phase is unstable, with perturbations increasing rapidly, so only a very fi ne-tuned collapse phase remains close to Robertson-Walker even if it started off so, and will be able to turn around as a whole (in general many black holes will form locally and collapse to a singularity).
So, yes, it is possible, but who focused the collapse so well that it turns around nicely? (pers.
comm., January 25, 2006)
So there is a signifi cant problem of acausal fi ne-tuning. One asserts not just brute contingency but also a rather curious form of it. In the face of apparent fi ne-tuning, physicists usually prefer to offer some type of explanation. Consider, for example, multiverse models as an explanation of the apparent fi ne-tuning of the fundamental physical constants, or Guth’s infl ationary resolution of the horizon problem (past thermodynamic equilibrium).
Second, there is the problem that the collapse becomes chaotic as it approaches the singularity. This will produce a preexpansion start condition that is known to be dramatically different from our actual “Big Bang.” This phenomenon is referred to as “BKL
144 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
chaos” after its discoverers (see Belinsky, Khalatnikov, & Lifshitz 1970).42 This problem will appear for all attempts at a past-eternal timeline that seek to introduce a pre–Big Bang phase that “bounces” into the present expansion. In fact, the true implication of BKL may well be that it is physically impossible to “bounce” through a singularity.
In stating that the initial conditions “have to be set in an acausal way (in the infi nite past),” Ellis puts his fi nger on a nettlesome philosophical issue in cosmological models featuring an infi nite past, namely, they often seem to treat the infi nite past as though it featured an infi nitely distant beginning point. Several of these models are discussed in this essay. But, as we have already seen in our discussion of philosophical kalam arguments, such a supposition is illicit, since such an infi nitely distant point is merely an ideal limit characteristic of the potential infi nite, not a moment that actually once was present.43 If we are allowed to speak of the condition of the universe at past infi nity, then Zenonian paradoxes (see p. 119) are unavoidable.
If the past condition of the universe is acausal, then of course there was no “setting” of the condition; it just “is.” Ellis is referring merely to the construction of the mathematical model. But suppose we do imagine that the boundary conditions were literally set at past infi nity. Something like this was a feature of Charles Misner’s old “Mixmaster” universe: In reality we don’t expect universes to expand at exactly the same rate in every direction, and when they become asymmetrical like this they behave in a very complicated way. Although they expand in volume, one direction tends to contract while the other two expand, tending to create an expanding ‘pancake’. But soon the contracting direction switches to expansion and one of the other two expanding directions switches into contraction. Over a long period of time, the effect is a sequence of oscillations . . . The striking thing about the sequence of oscillations of the volume of the universe as it shrinks to zero, when one runs its history back into the Big Bang at time-zero, or on into the Big Crunch at crunch-time, is that an infi nite number of oscillations occur. . . . The difference between the Mixmaster Universe and Zeno’s paradox is that an infi nite number of physically distinct, real events happen in any fi nite interval of time that includes time-zero or crunch time. Measured by a clock that ‘ticks’ on this oscillatory time, the Mixmaster Universe would be judged to be infi nitely old, because an infi nite number of things have happened to the past in this time, and it will ‘live’ forever because an infi nite number of things are still to happen in the future. (Barrow 2005, pp. 242–3)
The Mixmaster universe is interesting in that it appears to offer a past infi nite timeline that nonetheless features a clear past boundary to that timeline; that is, an infi nitely distant 42. Also, see Damour and Henneaux (2000): “. . . our fi ndings suggest that the spatial inhomogeneity continuously increases toward a singularity, as all quasi-uniform patches of space get broken up into smaller and smaller ones by the chaotic oscillatory evolution. In other words, the spacetime structure tends to develop a kind of
‘turbulence’.”
43. In response to the question, “Are c-boundaries [see Figure 3.8 and the following discussion for explanation of these terms] such as past and future timelike infi nity and scri+ physically real edges to spacetime (real, as a black hole is an ontologically real entity) or are they merely mathematical conveniences? But if infi nity is ‘actual’
and reachable, then a c-boundary must be an actual edge to spacetime, physically real in its ontology,” Ellis (pers.
comm.) responds curtly:
1. no
2. maths – after all a spacetime diagram is just a representation of physical reality 3. in my view infi nity is neither actual nor reachable.
KA LA M C O S M O LO G I C A L A RG U M E N T 145
beginning point. There is a question of judging the most physically appropriate measure of time. By proper time, Mixmaster arose a fi nite time ago from a singularity and will end its existence a fi nite time to the future. Time measured by oscillatory “ticks” would report a timeline that is infi nite to the past and future. Barrow and Tipler elucidate: It is always possible to fi nd a conformal transformation which will convert an infi nite universe to a fi nite one and vise-versa. One can always fi nd a time coordinate in which a universe that exists for a fi nite proper time . . . exists for an infi nite time in the new time coordinate, and a time coordinate in which a universe that exists for an infi nite proper time . . . exists for only a fi nite time. The most appropriate physical time may or may not be the proper time coordinate. (Barrow & Tipler 1986, p. 636)
Physicists routinely consider an infi nitely distant past “beginning” point, in effect bringing infi nity into their physical models through a process called a conformal transformation.
Consider Barrow and Tipler, here explaining simple Friedmann–Robertson–Walker (FRW) cosmological models using a device called a Penrose diagram.
The boundaries of a Penrose diagram represent what are termed c-boundaries of the cosmological models. The c-boundaries are composed of the singularities and the points at infi nity; the c-boundary of a cosmology is the edge of space-time, the ‘place’ at which space and time begin. By convention, singularities are represented by double lines in Penrose diagrams. [For example] the initial and fi nal singularities are the only c-boundaries in a closed Friedmann universe. An open Friedmann universe, on the other hand, has four distinct c-boundary structures: an initial singularity out of which the entire space-time arose, a single point i0
representing spatial infi nity, a 45° line ϑ+ (called ‘scri plus’) representing ‘null infi nity’ which are the points at infi nity that light rays (null curves) reach after infi nite time, and a single point i+ which all timelike curves approach for all fi nite times, and reach after infi nite time (with the exception of those timelike curves that accelerate forever and thus approach arbitrarily close to the speed of light. These curves hit scri plus rather that i+ at temporal infi nity).
A Penrose diagram allows us to defi ne rigorously ‘an achieved infi nity’, a concept whose logical consistency philosophers have been doubtful about for thousands of years. Using the c-boundary, it is possible to discuss the topology of the ‘achieved infi nity’ and the ‘beginning of time’ in cosmological models. (Barrow & Tipler 1986, pp. 635–6)
Models such as Mixmaster, the problem of supertasks44 in general, and the meaning of conformal transformations raise the question of whether an infi nite past implies the absence of a past boundary.
Figure 3.8 shows a Penrose diagram for the universe type under consideration in this section; that is, one that contracts from infi nite size down to a singularity, and then bounces into an expanding universe (see right side of the diagram). Figure 3.9 shows another type of simplifi ed model (a de Sitter universe) that has this type of behavior. The de Sitter model includes “dark energy,” while the contracting model in Figure 3.8 includes only ordinary matter. A more realistic physical model would include both ordinary matter and “dark energy.” The behavior of the universe at large size would be dominated 44. A supertask is an infi nite series of subtasks that can be completed in a fi nite time.
i+ future timelike infinity
i+ future timelike infinity
ϑ + (wher
ϑ +
e null geodesics te
rminate; i.e. light rays)
em
“Bounce” singularity
Space
i0
Ti
spatial infinity
emTi
i0 spatial infinity
Initial singularity
Space
−ϑ
Infinite contraction,
bounce,
Flat or open FRW universe
infinite expansion
(no cosmological constant)
(no cosmological constant)
i− past timelike infinity
Figure 3.8 Penrose depiction of Friedmann–Robertson–Walker (FRW) cosmology.
de Sitter universe
A universe with no matter but with a positive cosmological constant Λ.
ϑ+
O+
South Pole
T
O– O+
overlap
North Pole
O–
Contracts from infinite size to a
ϑ–
minimum radius, then expands
exponentially. A more realistic model
Penrose diagram: South and North “Pole”
would include radiation and matter, which
represent observers tracing out
would cause a singularity at time zero,
timelike paths. O– indicates limits of causal
while the distant past and future would
past of North Pole observer. O+ indicates
behave like the de Sitter solution.
causal future. No observer sees the full
space-time; hence the mystery of the
acausal fine-tuning.
Figure 3.9 A more realistic rendering of a universe that infi nitely collapses down to a Big Bang, and then expands. The de Sitter cosmology takes into account the dominant behavior of a cosmological constant for a universe of large size.
KA LA M C O S M O LO G I C A L A RG U M E N T 147
by the dark energy, and so a de Sitter model yields good insight into the behavior of an infi nite contraction model for the asymptotic past. By contrast, the behavior of the universe at the Big Bang would be best described by the Friedmann–Lemaître models in Figure 3.8.
It appears there is a dilemma. On the one hand, one could have the reality of a past infi nite timeline without a beginning. But then one must assert brute contingency. “Things are as they are because they were as they were.”45 Further, one must do this with respect to apparent fi ne-tuning. This seems implausible. One can at least say that it is unpopular, given that cosmologists are avidly seeking an explanation for apparent fi ne tuning in the form of a multiverse or a superdeterministic Theory of Everything. If we are going to give up explanation, then what was wrong with leaving cosmology as it was prior to 1980, namely, the standard Hot Big Bang model (with associated breakdown of physics at the singularity)?46
The other horn involves an infi nitely distant beginning point and allows the possibility of an explanation. But it opens the door for a supernatural explanation of the beginning.
The kalam cosmological argument’s second premise would be upheld, for that premise does not require that the beginning lie in the fi nite past.
IIIb. Asymptotically static space-time
An asymptotically static space is one in which the average expansion rate of the universe over its history is equal to zero, since the expansion rate of the universe “at” infi nity is zero. Hence, the universe, perhaps in the asymptotic past, is in a static state (neither expanding nor contracting). This allows the model to escape the BVG singularity theorem.
At fi rst blush, it would seem that the universe could hardly be said to have zero average expansion throughout its history if, as we know from observation, it has indeed been expanding! Would not the average expansion rate have to be greater than zero? No, not when we include “infi nity” in the average. Consider an analogy in which the local government decides that, henceforth, everyone will pay property taxes according to the average value of property (per acre) in the county instead of on one’s individual assessment. This might be good or bad for you, depending on whether you live in the high end district. But suppose that your county suddenly expanded to include the Sahara Desert. The Sahara is worthless and big, hence the average value of property, by the square mile, dives precipitously. Further, the larger the Sahara is, the closer to zero one’s property taxes will be. In the limit as the Sahara grows to infi nite size, one’s property taxes will go to zero. In a similar way, a zero expansion condition at infi nity would have the same impact on the average expansion rate. And the BVG Theorem only applies to a positive average expansion universe. George Ellis and his colleagues have been active in this type of model building.
Models of this sort belong to what is called the “Emergent” model class. They rehabilitate Einstein’s static model by postulating that the universe initially existed in such a phase and then transitioned via an infl ationary phase into the universe we see around us today. Ellis and Maartens explain:
45. Barrow and Tipler (1986, p. 408), attributed to cosmologist Thomas Gold.
46. See, for example, Earman and Mosterin (1999) for a related argument.
148 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
We show here that when K = +1 [recall the curvature parameter from Friedmann’s equation]
there are closed infl ationary models that do not bounce, but infl ate from a static beginning, and then reheat in the usual way. [Recall Guth’s infl ation] The infl ationary universe emerges from a small static state that has within it the seeds for the development of the macroscopic universe, and we call this the “Emergent Universe” scenario. (This can be seen as a modern version and extension of the Eddington universe.) The universe has a fi nite initial size, with a fi nite amount of infl ation occurring over an infi nite time in the past, and with infl ation then coming to an end via reheating in the standard way. (Ellis & Maartens 2004; emphasis added)47
As such, it is a manifestly nonsingular closed infl ationary cosmology that begins from a meta-stable Einstein static state and decays into a de Sitter phase and subsequently into standard hot Big Bang evolution. (Ellis, Murugan, & Tsagas 2004; emphasis added) A second, equally intriguing, possibility is that the initial Einstein static universe is created from “nothing” by some quantum tunneling process. Indeed, fi niteness of the tunneling action requires that the universe created through instantonic tunneling be closed. It is not implausible, then, that through spontaneous quantum fl uctuations, a closed universe could be created in a long lived but transient Einstein static state which then makes a transition to a fi nite lifetime de-Sitter and subsequent marginally closed FRW phase along the lines described above. (Ellis, Murugan, & Tsagas 2004; emphasis added)48
Now the question that interests us is whether the past of this model is perceived as eternal. A certain amount of ambiguity attends the answer. In some accounts (such as the above), it seems pretty clear that the Emergent models do have a beginning, namely, the Einstein static state (ESS). It is also stated explicitly that the model can be constructed with ESS occurring a fi nite time to the past (Ellis & Maartens 2004, sec. V). However, in the relevant papers ESS is usually described as asymptotically approached for past infi nite time:
“Here . . . we consider a universe fi lled with a dynamical scalar fi eld, which is past asymptotic to an Einstein static model with a radius determined by the fi eld’s kinetic energy”
(Ellis & Maartens 2004, p. 1). Some philosophers who have written on the topic have a problem with the contrived nature of the past infi nity in models of this type. For example, Rüdiger Vaas characterizes the Emergent models as “soft-bang/pseudobeginning” in nature.
47. Now we just showed in the previous section that Ellis has a philosophical problem with models that suggest an infi nitely distant beginning point and even, in fact, the notion of a realized infi nity in nature. Yet here we have a family of models developed by Ellis et al. that seem to suggest precisely that. This is explained via the following:
1.
Infi nity is so deeply ingrained in GR that pure pragmatism demands that one include the concept within one’s work.
2. It is not cognitively dissonant to consider that one might be wrong and research accordingly. In fact, good scientifi c procedure includes an attempt to falsify one’s own theories.
3. Ellis’s collaborators may not have the same philosophical commitments.
4.
An
infi nity that appears due to a technical interpretation of GR can disappear given a generalization of the theory (say, by considering quantum gravity).
A full look at Ellis’s recent work indicates a bias toward models with compact spaces (i.e. spatially fi nite either through closed curvature or topology), a skepticism with regard to infi nite multiverses, and openness toward the idea of a “pseudobeginning” in the fi nite past. In short, the pseudobeginning idea is that there is timeless reality where time “switches on,” producing our present state of affairs.
48. This is related to “creation from nothing” models; see section IVc.
KA LA M C O S M O LO G I C A L A RG U M E N T 149
He views the asymptotic approach toward ESS as something of a mathematical artifact (Vaas 2004, p. 18).
It is worth focusing on the issue of the instability of ESS. The Einstein static universe itself was originally viewed as past eternal. But there are obvious problems with this interpretation. The reason Einstein himself originally dropped the model was its feature of unstable equilibrium. Although, in pure nonquantum GR, one can consider a static state with worldlines that trace to negative infi nite time, in reality we know that gravity is a quantum force. As Vilenkin notes, “Small fl uctuations in the size of the universe are inevitable according to the quantum theory, and thus Einstein’s universe cannot remain in balance for an infi nite time” (Vilenkin 2006, p. 209).49 On the other hand, the current observable universe is demonstrably not in a static state. A quantum (or perhaps a thermal) fl uctuation is necessary to force a transition to an expanding universe. A fl uctuation is, in fact, necessary for the two phase model to work. But this very mechanism implies that the initial state is not past eternal.
The best that can be done is the latest version of the Emergent model, which uses a
“low-energy” solution of loop quantum gravity (LQG) to make the Einstein state stable against perturbations of a limited size (Figure 3.10). In response to the question, “Is the initial state metastable and therefore fi nite in its lifetime?,” Ellis answers that the Einstein state can persist at most for a “long” but apparently fi nite time.50
LQG theorist Martin Bojowald explains that any perturbation, even if not of suffi cient initial size to cause the system to escape the metastable potential, will eventually cause the system to escape it:
Static solutions do not evolve, and so are clearly ill-suited as a model for the Universe. But by introducing a perturbation to a static solution, one can slightly change it and thereby start a more interesting history. Unfortunately, the classical solution [ESS] is unstable: any disturbance grows rapidly, leaving little of the initial state behind. The insight of Mulryne and colleagues is that quantum effects could supply all the necessary ingredients where classical solutions do not. Within the framework of loop quantum gravity, repulsion also implies static solutions at small size, but these – in contrast to the classical case – are stable. According to the authors’
model, perturbing such a state leads to small cycles of interchanging expansion and contraction.
During this process, matter will evolve slowly, and the cycles will gradually change their behavior. By itself, this perpetual recurrence and incremental change seems to lack the spark necessary for so momentous an event as the birth of the Universe. And indeed, Mulryne and colleagues identify one fi nal theoretical ingredient that lights this spark: mediated through repulsive effects, potential energy is gradually pushed into the matter during its slow evolution.
At the point when potential energy starts to dominate kinetic energy, the mundane cycling is broken by a sudden, dramatic infl ationary explosion – the emergent Universe. (Bojowald 2005, pp. 920–1; emphasis added)
49. We note, as well, that a perturbation to a near-ESS state should be just as effective at disrupting the universe as a perturbation to a genuine ESS. Hence, a model which is only past asymptotic ESS does not escape the problem.
In fact, given past infi nite time, and the variety of exotic quantum universe transitions postulated throughout the cosmological literature, it seems inconceivable that any universe could possibly maintain a conserved structure over time periods “long” compared with the interval since the Big Bang.
50. He says, “note the later version of our model (astro-ph/0502589) based in the semi-classical approximation to loop quantum gravity where the static model is stable for a long while” (Private communication, January 24, 2006).
150 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Emergent universe via a loop quantum gravity approach
(figure not to scale)
The present day; a “relaxed” expansion that will accelerate due to
the future influence of a nonzero cosmological constant.
Eventually, the system
escapes the metastable
potential well and begins a
Guth-type inflation.
A small perturbation to the
system causes the size of
the universe to oscillate.
The amplitude of the
oscillation increases. This
Loop quantum gravity predicts a
state persists for an indefinite
correction to Einstein’s General
but finite time.
Relativity, which allows a metastable
e
balance between repulsive and attractive
mTi
gravity.
Size of universe
Figure 3.10 Evolution of an Emergent universe from a metastable loop quantum gravity state.
Metastability implies a fi nite life for the associated state. Either something must have come before it or it was “created.” This issue of metastability is a general problem across a wide array of model classes. Vaas elaborates:
Meta-stable states have a local, but not a global minimum in their potential landscape and, hence, they can decay; ground states might also change due to quantum uncertainty, i.e., due to local tunneling events. Some still speculative theories of quantum gravity permit the assumption of such a global, macroscopically time-less ground state ( e.g., quantum or string vacuum, spin networks, twistors). Due to accidental fl uctuations, which exceed a certain threshold value, universes can emerge out of that state. Due to some also speculative physical mechanism (like cosmic infl ation) they acquire – and, thus, are characterized by – directed non-equilibrium dynamics, specifi c initial conditions, and, hence, an arrow of time. (Vaas 2004, p. 10; emphasis added)
It therefore seems that metastable (and, for that matter, unstable) states must have a merely fi nite lifetime. Metastable states leave unexplained how they came to exist. Universes with a metastable initial state must therefore have a beginning, consistent with the second premise of the kalam cosmological argument.
IIIc. Cyclic universe
According to these models, the universe goes through a cycle in which it grows from zero (or near-zero) size to a maximum and then contracts back to its starting condition.
The universe itself is periodic, in the sense that it undergoes many such cycles, perhaps an
KA LA M C O S M O LO G I C A L A RG U M E N T 151
infi nite number. The average expansion of the universe would be zero in a “pure” cyclic model since cycle by cycle, one always experiences precisely equal amounts of expansion and contraction. Hence, a cyclic model evades the BVG Theorem. The past is featureless.
Unlike the previous two model classes, it is not the case that the universe asymptotically approaches some particular state in the infi nite past.
As Vilenkin indicates, however, cyclic models face a thermodynamic problem: “A truly cyclic universe has a problem with entropy increase: it should have reached thermodynamic equilibrium by now” (pers. comm., January 19, 2007). Our observation of the present day universe indicates that we are not at a condition of thermodynamic equilibrium – a good thing for us, as life requires nonequilibrium conditions to exist! As one looks into the past, the size of each cycle is also thought to decrease (due to radiation effect on entropy).
Eventually the cycles are so small that one ends up with a different physics – which would preclude the cycling and imply a beginning to the universe.
So how does one overcome this problem? Paul Frampton and Lauris Baum have recently proposed an ingenious mechanism that breaks genuinely new ground in cosmological studies. It is surprising that they base their model on a scenario that is generally thought to imply quite the opposite of cycling. They assume that a type of dark energy pervades the universe where its equation of state (the ratio between pressure and energy density) is less than −1. This would be different from the cosmological constant mentioned earlier (equation of state equal to −1). This type of expansion is thought to lead to an event called the Big Rip. Dark energy (also called phantom energy in this context) causes the acceleration in the expansion of the universe to become so great that our visible horizon shrinks over time. Eventually, this causal horizon shrinks so much that cosmological objects of smaller and smaller size become causally unbound. Galaxies, solar systems, planets, and, eventually, even atoms get ripped apart as the expansion rate of the universe tends toward infi nity. This would stop at a spatial singularity in the fi nite future. Baum and Frampton propose a “mosaic” model51 to overcome the problem of entropy buildup in a single universe:
We consider a model where, as we approach the rip, expansion stops due to a brane contribution just short of the big rip and there is a turnaround time t = t T when the scale factor is defl ated to a very tiny fraction ( f) of itself and only one causal patch is retained, while the other 1/ f 3 patches contract independently to separate universes. Turnaround takes place an extremely short time (<10−27 s) before the big rip would have occurred, at a time when the Universe is fractionated into many independent causal patches. (Baum & Frampton 2007, p. 1) What happens in the Baum–Frampton approach is that very close to the Big Rip event, the universe splits into noninteracting (causally disconnected) patches. The universe has expanded so much at this point that nearly all of these patches are empty of (normal) matter and radiation. They contain only phantom52 energy. It turns out that the entropy content of the universe (which is what interferes with cycling) is contained within the thinly spread matter and radiation. Those patches that contain only phantom energy are supposed never to undergo the Big Rip. Instead they separately undergo a defl ation event; contracting 51. Mosaic model: undesirable features of a model universe may be regional in scope; consideration of a multiverse may remove those features.
52. Phantom energy: dark energy with a supernegative equation of state, that is, p/r < −1.
152 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Baum–Frampton “phantom bounce” cyclic cosmology
(figure not to scale)
Multiverse of new Big Bangs; process begins anew
Turnaround (just prior to Big Rip)
A few rare baby universes
Expansion has made the
Today
will fail to cycle due to the
matter and radiation density
presence of residual matter
practically zero. Most baby
and radiation.
universes are empty except
for phantom energy.
Big Bang
Contracting parent universe
Time
Size of universe
Figure 3.11 Baum–Frampton phantom bounce model.
an amount exactly equal to the expansion that the universe experienced since the Big Bang (thereby avoiding the implications of the BVG Theorem). Prior to reaching a singularity, the contracting patch rebounds due to the effects of phantom energy. It then repeats the same cycle endlessly. Every patch that undergoes this fractionates into new “universes” that themselves begin to propagate (think of dandelions spreading over your lawn). Hence, the Baum–Frampton model is said to feature an infi nite multiverse in addition to a beginningless cyclic behavior (Figure 3.11). But is the model viable?
Several challenges remain to be addressed if the model is to be a viable option. First, in order to avoid the BVG singularity theorem, the average contraction must equal exactly the average expansion (for every geodesic). But how is this to be done without introducing explicit fi ne-tuning? There is no reason that defl ation of scale factor will exactly match post–Big Bang expansion. Frampton admits:
I have no idea why it [the BVG constraint] is satisfi ed because it does relate expansion with dark matter to contraction without dark matter. . . . I suspect it is not fi ne tuning but that may ultimately depend on one’s perspective: if BVG were not satisfi ed, infi nite cyclicity is impossible. (pers. comm., February 5, 2007)
The problem has not yet been addressed.53
53. In June 2007, Paul Frampton did put a paper on the Web preprint (Frampton 2007a) that partially addressed the issue. But it appears to show only that a generic cyclic model avoids the BVG Theorem. That point was not in dispute. The original question still seems to persist.
KA LA M C O S M O LO G I C A L A RG U M E N T 153
Second, globally, entropy should have already grown to an infi nite value. How is it that the various regions remain causally disconnected? The key factor in this model is the method for jettisoning the universe’s entropy. As Baum and Frampton emphasize, if matter is retained during a contraction phase:
. . . presence of dust or matter would require that our universe go in reverse through several phase transitions (recombination, QCD and electroweak to name a few) which would violate the second law of thermodynamics. We thus require that our universe comes back empty!
(Baum & Frampton 2007, p. 4)
Now, globally, over infi nite past time, the model posits that an infi nite amount of matter and radiation (hence, infi nite entropy) has been produced. How is it, then, that the entropy density avoids achieving an infi nite quantity? Frampton responds:
It is true that if we retain all the separate patches entropy continues to increase. Our key idea is to retain only one causal patch for our universe whose entropy thereby drops precipitously in what we for obvious reasons call defl ation. (pers. comm., February 5, 2007) The sole mention of where this entropy goes occurs in the following passage: “The old problem confronting Tolman is avoided by removing entropy to an unobservable exterior region; one may say in hindsight that the problem lay in considering only one universe”
(Frampton 2007a, p. 4).
Simply shoving the entropy into other “universes” raises the question whether, given infi nite time, a static space and a countable infi nity of realms within the multiverse, the realms must not eventually collide. Frampton responds:
. . . the ‘causal patches’ remain disjoint spawning separate universes which do not interact or collide again as they start out and remain causally disconnected. This more defi nite answer
. . . results from our better understanding of the turnaround from subsequent technical calculations. (pers. comm., October 10, 2007)
In our initial communication (February 2007), Frampton indicated that the problem of collision was a valid one and would be investigated. These calculations have not yet been published, hence the issue remains troubling.
Cosmologist Xin Zhang, in a recent paper (Zhang 2007a), argues that the causal disconnection mechanism at turnaround does not work precisely because the disconnected patches do come back into causal contact. Frampton has pointed out a possible error in Zhang’s critique (Frampton 2007b). This appears, however, to have been rectifi ed in a new communication (Zhang 2007b). Zhang’s (new) critique is the following: Frampton uses the following form of the modifi ed Friedmann equation:54
8π G ⎛
⎞
H 2 =
ρ
ρ
1 −
3
⎝⎜
ρc ⎠⎟
54. Phantom bounce models operate on the assumption that the standard Friedmann equation is no longer valid for universe conditions where the phantom energy density is near a critical value.
154 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
Here, “H” is the Hubble parameter, which is defi ned as the time derivative of the scale factor divided by the scale factor. Recall that the scale factor is the factor by which one multiplies the size of the universe in order to represent expansion or contraction.
Zhang clarifi es:
. . . ρc is the critical energy density set by quantum gravity, which is the maximal density of the universe. Such a modifi ed Friedmann equation with a phantom energy component leads to a cyclic universe scenario in which the universe oscillates through a series of expansions and contractions. In the usual universe [that is one governed by the ordinary Friedmann equation], the phantom dark energy leads to a “big rip” singularity; however, in this peculiar cyclic universe, the big-rip singularity can be avoided because when ρ reaches ρc the universe will turn around due to [the modifi ed Friedmann equation]. (Zhang 2007b)
So the rate of change of the size of the universe is governed by the density of this special type of “phantom” energy. Zhang continues:
When the universe approaches the turnaround point (ρ → ρc), we have H → 0. Therefore, obviously, at the turnaround, we have H−1 → ∞. This implies that the Hubble radius becomes infi nity at the turnaround point, because at that time the universe becomes static instantaneously (that is, it ceases to expand at the turnaround). Obviously, at the turnaround, the universe would not be fragmented into many disconnected causal patches. (Zhang 2007b) The inverse of the Hubble parameter ( H−1, called the Hubble radius) governs the scale at which microphysics can act, that is, the scale of causal connection. While this is near zero as one nears a “Big Rip” event, it seems clear that as one approaches Baum and Frampton’s turnaround, the Hubble radius grows again to an infi nite value. This means that all the separate patches of the universe are not disconnected (i.e. light signals can now propagate among them and allow them to interact). Thus, a subsequent collapse phase should include all of the matter and radiation then present, which essentially prevents a working cyclic scenario (since all the ordinary matter and radiation from the previous expansion would now be included). Frampton offers the following comment with respect to Zhang’s criticism:
“defl ation must occur at a time before turnaround when the Hubble radius is small, before it reaches its minimum value (x = 1/2). Defl ation remains a plausible conjecture which still requires further technical calculation to confi rm” (pers. comm., Feb 7, 2008).
This seems diffi cult to square, as it would seem defl ation = contraction, and turnaround is the time at which expansion ends and contraction begins. Baum and Frampton, in fact, issued a preprint in 2006 entitled “Defl ation at Turnaround for Oscillatory Cosmology,” in which they state:
A key ingredient in our cyclic model is that at turnaround t = tT (mod τ) our universe defl ates dramatically with scale factor a(tT) shrinking to â(tT) = fa(tT) where f < 10−28. This jettisoning of almost all, a fraction (1-f), of the accumulated entropy is permitted by the exceptional causal structure of the universe. (Baum & Frampton 2006, p. 4)55
Frampton clarifi es:
The beginning of contraction is the turnaround. Defl ation is where the causal patches separate and our entropy drops to zero.
55. Here t indicates periodicy; it indicates which cycle the universe is presently in.
KA LA M C O S M O LO G I C A L A RG U M E N T 155
By the way the time difference between defl ation and later turnaround is a trillion trillionth of a second or less!!!
It [Hubble radius divergence at turnaround] does have the signifi cance that each spawned universe is separately one causal patch at turnaround. (pers. comm., February 7, 2008) So defl ation is not contraction. Instead it refers to causal disconnection. The Hubble radius does diverge at turnaround. But for some reason the causal horizons frozen in at defl ation remain intact. This seems problematic. After all, causal horizons are not real physical barriers. They are observer dependent. For example, we on Earth have a causal horizon that stretches out some 46 billion light years. So does a space traveler in orbit around Alpha Centauri. But the space traveler sees a different portion of the universe than we do here on Earth. There is not actually a physical barrier 46 billion light years away from each of us. This is distinct from the event horizon of a black hole, which is an objective physical barrier. How exactly do Baum and Frampton understand this causal disconnection? A study of Frampton’s early work could perhaps clarify the situation:
. . . the time when a system becomes gravitationally unbound corresponds approximately to the time when the growing dark energy density matches the mean density of the bound system.
For a “typical” object like the Earth (or a hydrogen atom where the mean density happens to be about the density of water ρH2O = 1 g/cm3 since 10−24 g/(10−8 cm)3 = 1 g/cm3) water’s density ρH2O is an unlikely but practical unit for cosmic density in the oscillatory universe.
. . . the unimaginable dark energy density at turnaround of ρΛ(tT) > 1027ρH2O. By the time the dark energy density reaches such a value, according to the Big Rip analysis,56 the smallest known bound systems of particles have become unbound. Additionally the constituents will be causally disconnected, meaning that if the expansion had, instead, continued to the Big Rip the particles could no longer causally communicate. (Baum & Frampton 2006, pp. 3–4; emphasis added)
This is the key. If the density of the phantom energy is rΛ( tT) > 1027rH2O, then in a Big Rip scenario,57 the universe would be causally unbound. But we are not in a Big Rip scenario.
Instead, according to Baum and Frampton, the laws of physics have changed (we now have a modifi ed Friedmann equation) where something (perhaps extradimensional brane dynamics) acts to stop the expansion and leads to a contraction. In this case, it is hard to see why Frampton and Takahashi’s analysis of causally unbound systems still applies. It is the runaway expansion in the Big Rip scenario that leads to and maintains shrinking causal horizons. If that stops, and even reverses, then it would appear reasonable to assume that the causal horizon grows along with it.
Frampton and Takahashi’s explanation seems akin to a personal causal horizon as opposed to some physical barrier that results in permanent causal disconnection. What is lacking is a discussion of what causes “turnaround” and maintains causal disconnection.
Without those details, it seems reasonable to entertain Zhang’s misgiving concerning the viability of the universe-fractionating mechanism.
We should note Frampton’s emphasis on the limited time between defl ation, turnaround, and contraction. That would make the relevant issue the behavior of phantom 56. Frampton and Takahashi (2003, 2004).
57. In a “Big Rip,” the expansion rate of the universe becomes infi nite and leads to a future singularity.
156 WILLIAM LANE CRAIG AND JAMES D. SINCLAIR
energy right at turnaround (given that there is no time for ordinary matter and radiation to reestablish contact, given the amount of contraction that occurs in the fi rst fraction of a second after turnaround). The question seems to be: given the homogeneity of the phantom energy at turnaround, and its ability to interact with its surroundings (given the unbounded Hubble horizon right at turnaround), why would the universe split into separate domains rather than precipitate a single, global contraction?
Third, the presence of any matter or radiation (during contraction) will prevent cycling.
This could be a problem, given that spontaneous structure can form as thermal fl uctuations (even if the contraction stage begins without any matter or radiation). Bousso and Freivogel explain:
In a long-lived vacuum with positive cosmological constant, structure can form in two ways.
Structure can form in the conventional way (through a period of infl ation followed by reheating), or it can form spontaneously as a rare thermal fl uctuation. Because deSitter space is thermal, if the vacuum is suffi ciently long-lived spontaneous structure formation will occur.
(Bousso & Freivogel 2007, p. 4)
The Baum–Frampton space is not a de Sitter space, but it is also thermal. Hence, one would expect that matter would still fl uctuate into existence spontaneously. If so, then (at a reasonable probability) the Baum–Frampton cyclicity would not work.
Cosmologist Thomas Banks contends that a contracting space fi lled with quantum fi elds will have an “ergodic” property as the space shrinks. Its fi elds become highly excited as one approaches the end of contraction and these fi elds will produce chaotic fl uctuations. Spontaneously created matter with a different equation of state will dominate the energy density.
That, and the inhomogeneity of the fl uctuations, will prevent cycling. Banks and Fischler even suggest that the fi elds will spontaneously produce a dense “fl uid” of black holes leading to a condition they call a “Black Crunch” (Banks & Fischler 2002) for arbitrary states approaching full contraction.58 Hence, it appears that the Baum–Frampton cyclicity will not work.59
58. Banks complains:
I have a problem with ALL cyclic cosmologies. . . . The collapsing phase of these models always have a time dependent Hamiltonian for the quantum fi eld fl uctuations around the classical background. Furthermore the classical backgrounds are becoming singular. This means that the fi eld theories will be excited to higher and higher energy states (defi ne energy in some adiabatic fashion during the era when the cosmology is still fairly slowly varying, and use this to classify the states, even though it is not conserved).
High energy states in fi eld theory have the ergodic property–they thermalize rapidly, in the sense that the system explores all of its states. Willy Fischler and I proposed that in this situation you would again tend to maximize the entropy. We called this a Black Crunch and suggested the equation of state of matter would again tend toward p = ρ. It seems silly to imagine that, even if this is followed by a re-expansion, that one would start that expansion with a low entropy initial state, or that one had any control over the initial state at all. (pers. comm., October 12, 2007)
59. We note that Xin Zhang has his own competing cyclic model, in which he admits that thermal fl uctuations pose a serious problem for phantom bounce cosmologies:
It is noteworthy that the cyclic universe discussed in this paper is an ideal case, and there are still several severe obstacles existing in the cyclic cosmology, such as the density fl uctuation growth in the contraction phase, black hole formation, and entropy increase, which can obstruct the realization of a truly cyclic cosmology. (Zhang, Zhang, & Liu 2007; emphasis added)
KA LA M C O S M O LO G I C A L A RG U M E N T 157
While phantom bounce cosmologies (such as Baum–Frampton and Xin Zhang’s own
model) do represent a frontier worth exploring, there seem to be unanswered questions as to the viability of such an approach. The fi eld is too young to pass full judgment. But some questions that can be answered (such as the ergodic/chaotic approach to a singular bounce) seem to indicate that problems native to cyclic cosmologies remain.
IIId. A fourth alternative?: time deconstruction
As Borde et al. point out in their seminal paper, one of their primary assumptions was the following:
The intuitive reason why de Sitter infl ation cannot be past-eternal is that, in the full de Sitter space, exponential expansion is preceded by exponential contraction. Such a contracting phase is not part of standard infl ationary models, and does not appear to be consistent with the physics of infl ation. If thermalized regions were able to form all the way to past infi nity in the contracting spacetime, the whole universe would have been thermalized before infl ationary expansion could begin. In our analysis we will exclude the possibility of such a contracting phase by considering spacetimes for which the past region obeys an averaged expansion condition, by which we mean that the average expansion rate in the past is greater than zero: Hav > 0. (Borde, Guth, & Vilenkin 2003, p. 1)