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THE BLACK HOLE ORIGIN THEORY OF THE UNIVERSE: FRONTIERS OF SPECULATIVE, CURRENT PHYSICAL COSMOLOGY by Quentin Smith Western Michigan University Paper Read at Internal Conference on Physical Cosmology, Santa Barbara, 2000. 1. INTRODUCTION Victor Stenger, a professor of physics at the University of Hawaii, writes in chapter thirteen of his book, TIMELESS REALITY: SYMMETRY, SIMPLICITY AND MULTIPLE UNIVERSES [Prometheus, 2000]] that: "Philosopher Quentin Smith [in 1990] and physicist Lee Smolin [in992, 1997] have independently suggested a mechanism for the evolution of universes by natural selection. They propose a multi-universe scenario in which each universe is the residue of an exploding black hole that was previously formed in another universe". In this paper, I shall evaluate my [1990] theory, Smolin's [1992] and [1997] theory, and shall argue that both my ideas and Smolin's ideas suffer irreparable defects that requires a third version of the cosmological natural selection to be developed. I develop this third version in the last section of this paper. The most basic assumption of my theory is the physicist's Arthur Vilenkin's suggestion in his article in The Physical Review D, "Has Physical Cosmology Become Metaphysics?" Arthur Vilenkin invented the modern theory of quantum gravity in [1982] and in later papers (along with Hartle and Hawking [1983], leading to thousands (not merley hundreds) of articles published on this subject in physics journals. Roughly, the idea is that there are now and will be many competing cosmological theories, all of which are both observationally equivalent and underdetermined by the observational evidence. One reason for thinking this is that crucial observational tests will pertain to the Planck era, whose smallness and whose nature preclude it from being observed. The decision between cosmological theories will from now on have to be based on theoretical criteria such as explanatory power, conservativeness and parsimony. Thus my present theory leads to no new predictions but (like all other current cosmological theories, such as Hawking's, Linde's and Vilenkin's, merely predicts all the previously known observational data. I assume no complete and definitive theory of quantum gravity will be developed, merely proposals for parts of such a theory or proposals for theses that approximate such a theory. We have the Hartle-Hawking proposal, the Vilenkin propsal, and numerous proposals of string theory, superstring theory, Membrane theory, etc. I assume (I of course cannot prove this assumption, since I cannot forsee the future) that the era of hard science in physical cosmology is over (if there ever was such an era--perhaps the observations leading to the belief we live in a Friemandman universe in the late 1920s was an era of apparent, hard, cosmological science). I do not agree with the many physicists who think we should remain silent until a "hard science" quantum gravity theory is developed. From now on (I assume) cosmology is a metaphysical subject to be addressed by both physicists and philosophers. In conformity with this assumption, I shall develop a new cosmological theory that leads to no new predictions but merely has explanatory power and conservativeness in its favor (and lack of parsimony as its main weakness). The enormous gain in explanatory power, I believe, outweighs the loss of parsimony, especially when we add to this fact the close adherence to the criterion of conservativeness (changes one's basic theories as little as possible). The basic question now is not "which theory is most highly confirmed by the observational evidence?" but "which theory is the most plausible on theoretical grounds?" My challenge is to show that the great increase in explanatory power plausibly outweighs the great loss of parsimony. I would be happy for my basic assumption to be proven wrong by the advent of a new era of "hard science" in physical cosmology, but I see insufficient evidence that such an era is either upon us or is just over the horizon. (In addition to Vilenkin's article, Tim Maudlin's very plausible 1994 article in The Journal of Philosophy may be interpreted in a way that supports my assumption.) The point of this paper is to reach Part Three, which is a cosmology based on one of Hawking's quantum gravity proposals about black holes. But to reach Part Three, we need to examine the problems with my 1990 theory and with Smolin's 1992 and 1997 theory. 2. THE FIRST VERSION OF THE BLACK HOLE ORIGIN THEORY OF THE UNIVERSE: [SMITH, 1990] The basic idea behind the Smith [1990] theory is that some black holes have two properties, sucking in matter from one maximal spacetime S1 and spewing forth the matter in a different maximal spacetime S2. Smith's [1990] reads: ". . .one and the same singularity S has the property of sucking in matter (qua black hole singularity) and spewing forth matter (qua big bang singularity). [Smith, 1990: 42]. The big bang singularity in our spacetime is a black hole singularity in another spacetime, and this latter spacetime has a big bang singularity that is a blackhole singularity in yet a third spacetime, and so on ad infintim. This hypothesis, I argued, explains the existence, initial conditions and basic physical constants of our universe. The explanataion has the following structure. There is some singularity S that has the property of being the big bang singularity in our spacetime M1 and a second property of being a black hole singularity in a different spacetime M2. The singularity S in an endpoint of some timelike and null curves in M2 and the beginningpoint of the timelike and null curves in our spacetime M1. In the spacetime M2, the matter sucked into the black hole is crushed out of existence at the black hole singularity S. But in our spacetime M1, the singularity S explodes and spews forth mass-energy in an expanding spacetime. Not only the mass-energy in our spacetime begins to exist in this big bang explosion, but spacetime itself begins to itself. Smith's [1990] hypothesizes that the spacetime M2 contains not only the black hole that resulted in our spacetime M1, but many other black holes, say 10 billion. One billion of these ten billion black hole singularities in M2 also have the property of being a big bang singularity of another spacetime M', such that one billion spacetimes are a consequence of the black holes that belong to the spacetime M2. This provides for an inductive-statistical explanation of the initial conditions and basic laws unique to our spacetime M1. For the sake of familiarity let us adopt Hempel's theory of inductive-statistical explanations, I-S explanations, particularly as he later modified it in response to Salmon's argument [1971] that probabilities lower that 1/2 have explanatory relevance. Hempel's modified version [1976], which allows probabilities less than 1/2 into the explanans of I-S explanations, shall be called a neo I-S explanation. A neo I-S explanation explains something in the sense that it predicts it with some degree of probability. Contra Humphreys [1989] and Salmon's later theory [1984], Hempel, the early Salmon and I hold that the explanans need not cite a probabalistic cause of the explanandum. It is sufficient that the explanans lead us to predict the explanadum with some degree of probability. However, in the third, quantum gravity version of this theory, I shall provide a causal explanation of these factors. But for now, let us stay with the 1990 theory. In Smith [1990], the initial conditions and basic constants of our universe are explained in the following way. Let us explain, for example, why the electromagnetic force has the value 1/137, relative to the strong force, whose value is set at 1. Universe M2, we said, has ten billion black hole singularities, one billion of which are also big bang singularities of other universes. Let us suppose that of these one billion black hole singularities, 100 are big bang singularities of universes whose electromagnetic force has the value 1/137. In accordance with the Standard Model and the symmetry breaking account of the early universe, at 10-10sec after the big bang singularity the electroweak force split into the weak force and the electroweak force, with the electromagnetic force acquiring the value 1/137. This gives us a neo I-S explanation of the electromagnetic force constant of 1/137. 100 of the one billion black hole singularities in the universe M2 have the property of being big bang singularities of universes where the symmetry breaking results in a value of 1/137 for the electroweak force. Let
F = the property of being a black hole singularity in M2 G = the property of being both a black hole singularity in M2 and a big bang singularity of a universe that contains an electomagentic force with the value 1/137. P = probability
The explanation, cast in a deductive form by including the "therefore" line after the premises, is:
(1) P(G/F) = 0.00000001 [one out of ten million chances: 10-7] (2) (Ex) Fx
Therefore, it is probable to degree 0.00000001 (one ten-millionth) that (3) (Ex) Gx.
In this fashion, every constant of the forces and particles in our universe M1 has an explanation. Indeed, the existence and initial conditions of our universe M1 has an explanation of this form as well. The same holds for the universe M2, which is not hypothesized to be a brute, unexplained given. Rather the existence of M2, and the value of its constants, is explained in terms of a black hole singularity in another universe M3. The same holds for M3, and so on ad infinitum. This enables the existence, initial conditions and law-forming constants of our universe to have a logically complete explanation. By this I mean that it is not logically possible that there is any why-question left unanswered in the explanatory hierarchy that begins with M2's explanation of M1. It may be asked, why does the infinite sequence of universes, M1. M2, M3, . . . etc., exist, rather than some other sequence? This question rests on an assumption that is a contradiction in second order logic. Second order logic includes the axiom of extensionality, which implies that sets have their members essentially. That is, two sets are identitical if and only if they have all and only the same members. Now a sequence is an ordered set and thus is a set. A set logically supervenes upon its members, where "logically" has the force of second order logic. If the members exist, that logically implies the set of the members exist. Consequently, to ask for an explanation of the sequence of universes M1, M2, M3, . . etc., after each member has been explained, is to betray a fallacy in logic. If each member of the sequence M1, M2, M3 has an explanation in terms of another member, that suffices to explain the existence of the sequence. The sequence exists because its existence is logically implied by the existence of its members, and each of its members exist because they are explained by another member. Introducing mereological sums will not help the critic of the claim that my theory gives a logically complete explanation. For merelogical sums, like sets, contain their members essentially, and thus logically supervene on their members. And the members have been explained. But why is there anything at all, rather than nothing? That question has already been answered. Anything that exists (such as a universe) is explained in terms of another universe in the sequence; for anything x, x exists rather than nothing at all since x is the probabalistic result of a black hole singularity in another universe. If there were nothing at all, x would not be the probabilistic result of a black hole singularity. But since it is a result, that explains why there is anything at all, rather than nothing. What about possibilities? Why are not other possibilities actualized? This also has been explained. For each possibility x, if x is unactualized, this is because no black hole singularity actualized x. For each actualized possibility x, x's actualization is explained in terms of a black hole in another universe. This answers the metaphysical question about why a given infinite sequence of states exist rather than some other sequence or nothing at all. To think there is still some logically possible explanatory question left unanswered is to misunderstand the logic of explanation. There is no possible non-poetic and logically clear criticism of this theory to the effect that some "ultimate Why-question" is left unanswered. One could object that Leibniz's principle of sufficient reason is not satisfied, since the explanations are probabalistic. But to assume that something is left unexplained by a probabilistic explanation is a fallacy in the logic of explanation, as has been well-documented in 20th century philosophy of science. In the above metaphysical discussion, I have shown how a logically complete explanation can be given. But in Smith [1990] I claimed only to give a logically complete explanation of the existence of our universe (and of every other universe) and of the law-constituting constants of the forces and particles. I did not explain the basic laws of nature that remained constant from universe to universe, that is, the basic and derived laws constituting the Standard Model (which is the combination of the electroweak theory and quantum chromodynamics) would include everything from quantum mechanics to relativity theory to the idea that there are four forces). The retention of the Standard Model, nonetheless, enabled the theory to rank highly on the criterion of conservativeness ("changes one basic ideas as little as possible"). I think, howver, that Smith's [1990] version of the theory is not only explanatorily incomplete (and thus does not have explanatory power that cleary outweighs its loss of parsimony) but fails on other grounds, grounds that will require a quantum gravity version of the theory. The basic problem is the theory's claim that the singularities exist as points on the boundaries of the spacetime manifolds. First, the c-boundary and g-boundry construction of the singular points, which allows them to be really existent points on the boundary of the spacetime manifold, and not merely idealizations that are limits to a mathematical series that belongs to the mathematical model of the theory. The b-boundary and g-boundary construction of singularities as points on the boundary of a spacetime manifold have both been shown to be unsatisfactory by Johnson [1977] and Gerroch, Liang and Wald [1982]. If there are not black hole singular points and no big bang singular points, then the entire edifice of the 1990 theory collapses. 3. The Second Version of the Natural Selection Theory of the Origin of the Universe [[Smolin,1992; 1997] The next appearance of the Black Hole Origination Theory appeared in Lee Smolin's work of [1992] and [1997], who independently arrived at the same basic ideas. But there are some differences, and these differences seem to me to be problematic. Smolin [1997] share in common with Smith's 1990 theory the following three basic theses, among many others, but differs in two respects that reveal the problems in his theory.
(1) First Similar Thesis: The first state bounding the big bang in our universe is the last state bounding the collapse state of a black hole in another universe.
(2) Second Similar Thesis. The black hole origin of our universe explains the basic physical constants, such as the strengths of the four forces.
(3) Third Similar Thesis. The second universe, containing the black hole to which our universe is connected, also has a first state bounding a big bang that is the last state bounding the collapse state of a black hole in a third universe. And this third universe has a first state bounding a big bang that is the last state bounding the collapse state of a black hole in a fourth universe, and so on.
(1') First Dissimilar Thesis. Smith [1990] postulates an infinite sequence of universes of this sort, and thus achieve a logically complete explanation of the existence of each universe in the sequence, whereas Smolin posits a merely finite sequence of universes of this sort, with a first universe that has a first big bang state This means Smolin's offers us a theory where we end up with a brute fact; there is an unexplained and arbitrary set of initial conditions and laws that belong to the first universe. Since Smolin's theory results in this brute fact, his theory does not increase sufficiently in explanatory power to compensate for its decrease in parsimony. On grounds of parsimony, we should let the brute facts be the initial conditions and laws of our universe. Thus, Smoluin postulate of finitude deprives his theory of sufficient justification to warrant the belief that is many-universe theory is epistemically preferable to the traditional one universe theory.
(2') Second Dissimilat Thesis. Smolin offers a certain postulate, which I shall quote: " . . quantum effects prevent the formation of singularities, at which time starts or stops. If this is true, then time does not end in the center of black holes, but continues into some region of space-time, connected to our universe only in its [i.e., our universe's] first moment." [[92-93] The explosion of a black hole that results in a big bang occurs when the density of matter reaches the Planck scale, which is appromiately 1079 denser than an atomic nucleus.
In order to see why this postulate renders untenble Smolin's version of the cosmological natural selection theory, we need to explain another postulate of Smolin. His second postulate is similar to a basic postulate of the first version of the theory in [Smith, 1990]. Smolin states it as follows: ". . . the basic forms of the laws don't change during the bounce, so that the standard model of particle physics describes the world both before and after the bounce [at the end of the black hole]. However, I will assume that the parameters of the standard model do change during the bounce. . . These changes are small and random." [1997: 94]. If we ignore the part of the quotation that involves the temporal words "before" and "after" and interpret these words in terms of some nontemporal relation, then we can see the similarity with Smith [1990]. But the problem with Smolin's theory is precisely that we cannot ignore the temporal meanings he gives to these words. Smolin argues that the bounce occurs when the density of matter in a black hole collpase reaches the Planck density. But it is a consequence of of some of the underlying quantum mechanical and general relativistic ideas of the Standard Model that Smolin assumes that at the Plack density, (a) time does not exist in the linear fashion postulated by Smolin, and (b) the dense matter does not explode in a big bang but instead forms miniblack holes. When matter reaches the Planck density, something happens to regions of matter at the Planck size, 10-33 centimeter. What happens at this length scale is that the Planck energy is so great that space-time regions of the Planck size curve right around themselves and become mini black holes. This is what physicists mean when they say that "time breaks down as the Planck era at 10-43second after the big bang". Thus, Smolin's theory contradicts the Standard Model he assumes as the basis for his theory. I agree with Steven Weinstein and Arthur Fine that "it will be clear to every reader that Smolin is not merely speculating and that his arguments are grounded in science . . ." [Weinstein and Fine, 1998: 267]. However, the open-minded attitude that Weinstein and Fine display in their essay on Smolin suggests that they would encourage critical engagement with Smolin's ideas, even if Winstein and Fine may not agree with all of the criticisms of Smolin's theory. I believe the two criticisms I made suggest we need a new and improved version of the Black Hole Origin Theory of the Universe. 4. A THIRD AND SUCCESSFUL VERSION OF BLACK HOLE ORIGIN COSMOLOGY Smith [1990] and Smolin [1997] both failed to develop succesful versions of the cosmological natural selection theory. I shall here present in barest outline form a version that I believe can be successfully developed in a detailed way. Criteria for theory-formation include simplicity, explanatory power, predicive novelty, conservatiness, nonarbitrarness and symmetry, among others. I shall here make use specifically of conservatiness and symmetrry. A brief and intuitive understanding of conservatiness may be obtaining by saying that a theory T1 is more conservative than a theory T2 if T1 requires less changes in our background or prior scientific beliefs than T2. For example, if theory T2 requires a change only at the periphery of our web of belief, to borrow Quine's metaphor, and T1 requires only a change at the center of our web of belief than, then T2 is more conservative than T1 and, all else being equal, we ought to believe T2 rather than T1. The symmetry criterion is illustrated by the fact that if we know a situation S1 that instantiates the property F also instantiates the property G, and we know there is another situation S2 that instantiates F and that is all we know about S2 except for the fact that it must instiate either G or H, but not both, then, all else being equal, we ought to belief that situation S2 insitantiates the property G rather than the property H. In our Black Hole Origin Theory of the universe, we have at the core of this theory the hypothesis that the big bang in our universe is connected to a black hole in another universe. One question we need to ask, but which Smolin does not ask, is what kind of black hole is connected to our big bang? The principles of conservativness and symmetry imply that we should postulate that it is a Kerr black hole (which is a rotating rather than non-rotating black hole). The reason for this is that all observed black holes in our universe, such as Cygnus X-1, are Kerr Black holes. The other theoretical criteria, explanatory power, preditive novelty, simplicity, etc., do not counterbalance this hypothesis since nothing is gained (in respest of any these criteria of theory formation) by hypothesizing that the black hole is a Schwarschild black hole or a Reissner-Nordstrom black hole. If all else is equal regarding our knowledge of black holes, we ought to believe the black hole that is connected to our universe is a Kerr black hole. Now comes the most important obstacle to the cosmological natural selection theory, the obstacle that defeated versions one and two of this theory. This is to explain the nature of the connection between the black hole and the big bang of our universe. We recall that version one failed since Johnson, Geroch, Liang and Wald demonstrated the unsuccesfulness of the b-boundary and c-boundary constructions of "singularties" as really existent points on the boundary of a spacetime, leaving version one without an existent reality, a point, that could both be the end-point of a black hole and the beginning-point of a big bang explosion. Version two failed because its postulate that time continues through a black hole into a big bang at the Planck density is inconsistent with the accepted scientific notions that linear time breaks down at the Planck density. A Kerr black hole is rotating and has a singularity at r = 0. As my first attempt to develop a more satisfactory third version of the Black Hole Origin Theory, I will allow the theoretical criterion of explanatory power to override the criterion of symmetry and conservativeness, since if I postulate that the black hole connected to our big bang is not a normal Kerr black hole, i.e., the type we observe in our universe, but is instead a Kerr vacuum black hole (or, if you prefer, a Kerr white hole), then we will have a type of black hole that can lead to a big bang explosion. A vacuum black hole does not arise from a gravitation collapse If we apply the metric of a normal Kerr black hole to a Kerr vacuum black hole, we will have at r = 0 a disc singularity. The edge of this disk has an infinite curvature, and the rest of the disc is an extendible edge of spacetime. Call the spacetime in which this Kerr vaccuum black hoile exists S1. From the viewpoint of the spacetime S2 into which worldlines can be extended, the Kerr vacuum black hole singularity will be the extendible part of the disk. I hypothesize that this edge of this disk is the big bang singularity of the new spacetime S2. This Kerr vacuum black hole is not to be confused with the popularly discussed Einstein-Rosen bridge that connects two pre-existing universes [Smith, 1990: 41]; such a bridge invovles two Schwarschild vacuum black hole singularities in two asympotically flat regions of a Minkowski spacetime that connect and form a transient worm-hole between the two spacetime regions. This familiar idea will not help us, since it presupposes rather than explains the existence of the two spactimes that are connected by the wormhole. The Kerr vacuum black hol gives us a basis to work from, since neither [Smith, 1990] nor [Smolin, 1997] ever explained how a theory of a black hole exploding as a big bang can be mathematically described by a solution to Einstein's equation. For example, Smith [1990: 41] says the major problem with his theory is that "there is no known solution of the equations of GTR that shows a singularity of one of these sorts [i.e. a black hole singularity] to also be a singularity of the other sort [i.e. a big bang singularity]" Smith [1990] is right in only one respect; the Kerr vacuum black hole solution to the Einstein equation does not require that the disk is in fact extended into another spacetime. It implies merely that it is extendible, i.e. that it is physically or nomologically possible for it to be extended. But Smith [1990] is wrong in another respect; he thought there was no solution to the Einstein equation that allowed for the physical or nomological possibility of a singularity that had the property of being both a black hole singuality in one universe and a big bang singularity in a second universe. However, the disk is precisely such a singularity. Furthermore, Smith was wrong in [1990] in asserting that all vacuum black holes (white holes) are retarded pieces of a big bang singularity. This is required of Schwarzschild vacuum black holes [Misner, Thorne and Wheeler, 1973: 842-843], but not of Kerr vacuum black holes. But note that this issue is in fact irrelevant, which is a point Smith did not grasp in [1990]. For even if all vacumm black hole singularities are retarded pieces of a big bang singularity, the big bang singularity in our universe could have the property of being a vacuum black hole that is a retarded part of the big bang singularity in another universe. There is no necessity for the big bang singularities to be gravitational collapse black hole singularities rather than vacuum black hole singularities. By identifying the Kerr vacuum black holes as the class of black holes relevant to big bangs, we seem to have made the first step towards developing a third and improved theory of the Black Origin Cosmological Theory. The second step is to attempt to surmount the obstacle consisting of the failure of the b-boundary and g-boundary constructions of singularities as really existent entities on the boundaries of spacetime. I must eliminate the disk that has the two properties of having an edge with an infinite curvature and having an extendible spacetime. But once the disk singularity is eliminated, what do I have left? A contradiction. There is an insurmountable problem here. Given my explanatory goal, I need the Kerr vacuum black hole to be part of a spacetime S1 that has different values of its physical constants than the spacetime S2 that extends from the Kerr vacuum black hole. However, imagine a worldline of an electron E1, where E1 is interacting with (e.g., being repelled from) another electron E2 by means of an electromagnetic force with the value 1/136 compared to the value of the strong force in our universe. If I am to explain why one of the seemingly arbitrary physical constants of our universe exists, namely, that the value of the electromagnetic force is 1/137,then in the present scenario I must suppose that the electromagnetic force changes its value from 1/136 to 1/137 as it passess through the Kerr vacuum black hole into the new universe. But this suggestion contradicts the Standard Model and its implications for the early phases of our universe. One of these implications is that the electromagnetic force does not even exist until 10-10 seconds after the beginning of our universe. Before this time, there existed the electroweak force and (according to the Grand Unified Theory) there existed even earlier (between 10-43 seconds and 10-32 seconds after the beginning of our universe), there existed only the strong-electroweak force and the gravitational force, and the nongravitational, unified bosonic-fermion particles and gravitons. Electrons did not even exist. Consequently, eliminating the Kerr disk-singularity is not sufficient to give me a theory consistent with the Standard Model. I think we must now face the fact that we cannot rest content with the Standard Model. The Standard Model is not a quantum gravity theory but a semi-classical theory. "Semi-classical" is a technical term used by quantum gravity physicists to refer to theories that quantize the matter and energy in spacetime, but do not quantize spacetime itself. Spacetime is regarded as a classical general relativistic spacetime. But we cannot rest content here, since the Standard Model contains an internal contradiction. It contains quantum field theories that are nonlocal (i.e., allow instantaneous or spacelike causal connections) and also contains General Relativity, which is local (it dissallows as nomically impossibly instantaneous or spacelike causal connections). It is because we cannot rest content with a theory that implies locality and nonlocality in the same respect that physicists are working on proposals for a quantum gravity theory. And this is a major reason, in my opinion, why cosmological has become metaphysical; a quantum gravity theory is about the Planck dimensions and yet the Planck dimensions are unobservable. We have to decide among observationally underdetermined and equavalent theories which theory best satisfies the theoretical criteria of explanatory power, conservativeness, parsimony, symmetry and the like. Quantum gravity ideas are inherently sketchy and imcomplete. Which shall we choose? I will select the Euclidean quantum gravity approach of Hartle and Hawking, since this theory, by virtue of Hawking's application of it to black holes, will allow us to have the theory we have been seeking. But I shall use this theory to develop a different and competiting cosmological theory than that developed by Hartle and Hawking. According to their theory, our universe does not emerge from a black hole but is topologically connected, at its temporal beginning, to a Euclidean spacetime (a timeless, four dimensional space). (See Isham; Smith 1997). The Hartle-Hawking approach permits summing over the solutions of the Einstein equation of general relativity (or summing over approximations to these solutions). The Kerr black hole is described by such a solution and we can retain it in the third version of the Black Hole Origin Theory. This allows to retain a significant conservatiness and symmetry to the theory by allowing that the originating black hole is not a Kerr vacuum black hole, but a normal gravitational collapse Kerr black hole of the sort we observe in our universe. At the same time, I can retain the explanatory power I earlier associated with the hypothesis of a Kerr vacuum black hole origin of our universe, for the two sorts of Kerr black holes do not differ in this respect. The first theoretical step is to resolve about the connection between imaginary time and real time. A few years ago, there was some savergy (as Hawking called it) between one of the authors of the Hartle-Hawking theory of imaginary time and the Craig-Smith critique of the Hartle-Hawking theory, a critique that can be found in the book I co-authored with William Lane Craig, THEISM, ATHEISM AND BIG BANG COSMOLOGY [1993]. Stephen Hawking, politely declining to mention names, responds to Craig's and my criticism of the idea that there can exist imaginary time. Craig first put forth this criticism in an article in 1990 in the British Journal for the Philosophy of Science, and this article was reprinted in the Craig-Smith book THEISM, ATHEISM AND BIG BANG COSMOLOGY, along with an extensive development of Craig's criticism by myself. Hawking responds as follows:
"Jim Hartle and I suggested the universe may not have had a beginning and end [in imaginary time]. I was savagely attacked by a philosopher of science for talking about imaginary time. He said: How can a mathematical trick like imaginary time have anything to do with the real universe? I think this philosopher was confusing the technical mathematical terms real and imaginary numbers the way that real and imaginary are used in everyday life." [BLACK HOLES, BABY UNIVERSE, 1993: 46]. fn1
I think Hawking's response is off the mark. The point that Craig and I made is that (a) Hartle and Hawking argued in their [1983] article that a quantum gravity theory can be developed on analogy with existent quantum field theories. Jim Hartle and Stephen Hawking write in THE PHYSICAL REVIEW D: "The case of quantum fields is a striaghtforward generalization of quantum particle mechanics . . .", and quantum field theory includes the idea that there is an intergral "over all Euclidean field configurations. . . In the case of quantum gravity new features enter" [1983: 2961]. But the idea of integrating over Euclidean field configurations (now, Euclidean spacetimes) is retained. The point Craig and I had in mind is that in quantum field theory, the integral over Euclidean field configurataions, where imaginary numbers appear, there is no physical interpretation given to imaginary time. Indeed, Hawking himself, in his [1998] says this: ". . . as far as everyday quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean spacetime as merely a matahematical device (or trick) to calculate answers about real spacetime" [1998: 139]. But in Hartle's and Hawking's [1983], and especially in the later writings either singly authored by Hawking, or co-authored by Hawking and someone other than Jim Hartle, the Euclidean regime is given a physical interpretation. (For clarity's sake, a "Euclidean region" is a term used to refer to a solution to Einstein's equation where the metric has a positive definite signature (++++), which in effect means there are spatial dimensions but no temporal dimensions. Imaginary time is a fourth spatial dimension, in addition to height, width and depth. A four dimensional Euclidean spacetime is a four dimensional Riemannian spacetime and thus can be curved in various ways. Some philosophers have been (quite naturally) confused by quantum physicists' use of this phrase, thinking the physicists were referring to a flat Euclidean space, such as the one Newton postulated. Philosophers can substitute "Riemannian" whenever the term "Euclidian" is used in quantum gravity theories.) One problem that both Craig and I had with the Hartle-Hawking theory is that they provided no justification for interpreting realistically (rather than instrumentally) the Euclidean equations in quantum gravity theory. If ordinary quantum field theory uses Euclidean equations instrumentally, and if Hartle and Hawking are developing a quantum gravity theory as an analogue to ordinary quantum field theory, then how or why can they intrepret Euclidean equations realisltically, without any attempt to justify this noninstrumentalist use of the Euclidean equations? Furthermore, both Craig and I argued that realistic interpretations of these equations resulted in a logically incoherent ontology. Craig presented some ontological arguments, but my different ontological argument (with which Craig agreed) was that a realistic interpretation implied that imaginary time was related to the real time of our Lorentzian universe and that no such relationship was logically possible. However, I have since developed a theory that seems to me to privide a logically and physically coherent interpretation of the connection between imaginary time and real time.(See [Smith 1997]) and thus I retract my criticism in Chapter Eleven of Theism, Atheism and Big Bang Cosmology. I think the Hartle-Hawking- "no boundary" proposal can be conjoined with axioms from Riemanian geometery, point-set topology, theory of manifolds, and a three-valued logic of vagueness, and other resources to give a physically coherent and plausible interpretation of the Euclidean regieme and its connection to our Lorentzian spacetime. This is where Craig and I take different paths, however, since Craig still believes the Hartle-Hawking imaginary time to be metaphysically impossible and he does not agree with my theory of how it can be physically real (See Craig [1998] [1997: 290, n.1]). It is this theory, presented in my [1997] article, "An Ontological Interpretation of the Wave Function of the Universe" that may provide part of the missing key to a successful version of the Black Hole Origin Theory. I will show how they apply to the Kerr black hole. First, it should be noted, to the surprise of many, that Euclidean quantum gravity approach does not remove the singularity in the black hole but actually predicts the singularity, As Hawking points out, the Euclidean or positive definie metrics do not go inside the event horizon. This is why the action of the Euclidean metric remains well-defined. Hawking remarks: "One could regard this as a quantum version of cosmic censorship; the breakdown of the structure at a singularity should not affect any physical measurement" [Hawking and Penrose, 1996:76]. The Euclidean quantum gravity programm does not eliminate black hole singualities but shows how measurements to be only of well-defined metrics. As Hawking writes: ". . . the universe could be finite in imaginary time but without boundaries or singularities. When one goes back to the real time in which we live, however, there will still appear to be singularities. The poor astronaut who falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter no singularities. . . .". [1998: 144]. But since the imaginary time region, the Euclidean spacetime, is a timeless four dimensional space, it is of course impossible for an astronaut or anyone else to live in the imaginary time regions of our universe. As Hawking says, "the oscillating component of the wave function should be interpreted as corresponding to a lorentzian geometry and the exponentially growing component should be interpreted as corresponding to a euclidean geometry [with imaginary time]. We live in a lorentzian geomtry and therefore we are interested really only in the oscillating part of the wave function" [Hawking, 1984: 272]. When Hawking discusses black holes in his 1996 book with Penrose, he applies Euclidean quantum gravity to the event horizon of black holes and believes that there are black hole singularities, where r = 0, where there is a "breakdown of structure at a singularity" [Hawking and Penrose, 1996: 76] But we need not posit the singularity as an existent point attached to the manifold, in opposition to the refutations of the b-boundary and g-boundary constructions of singularities as existent points. Talk of singularities is now talk of an inextendible manifold, with its inextendibility being due to the "missing point" that would otherwise have been postulated as the point that exists on the edge of the manifold. We recognize singularities by the existence of incomplete geodesics that cannot be extended to infinite values of the affine parameter. [Haking and Penrose, 1996: 15] This is precisely what we need for our Black Hole Origination Cosmology. Since the Euclidean metric is defined on the event horizon of black holes, there are wave functions which are probability amplitudes for black holes. The break down of structure within the event horizon implies that the elementary particles and the four forces break down, i.e. become undefined (which, ontologically, means they cease to exist). We do not need an existent singular point that is forbidden by the failure of the b-boundary and g-boundary constructions. Rather, the breakdown of structure implies there is no metric and thus no gravitational force at the breakdown phase. Since the metric is the curvature of spacetime, there is no spacetime and thus no elementary particles and no electromagnetic force, no weak force, no strong force and no gravitational force. There remains merely a differential manifold with topological properties and a tangent vector field. The breakdown applies to the metric ("the metric becomes ill-defined"), not to the tangent vector field and underlying topology of the differential manifold. This unmetricated manifold (a manifold without spatial and temporal intervals or well-defined distances in space or time) is topologically connected to the big bang explosion, at which the manifold has a metric. The existence of this new spacetime brings with it elementary particles and forces that have randomly selected values. But how are the basic laws explained? If laws are regularities, they are explained causally in David Lewis's counterfactual definition causality. Let c be all the lawlike regularties in one universe and e be the lawlike regularities in an universe that results from a black hole in the first universe. c and e both exists and if c had not existed, e would not have existed. But if e would not have existed, c would still have existed (and caused nothing). This gives the direction of causation. So we have a causal explanation of the existence of each universe, its physical contstants, its initial conditions and its basic laws. It is not logically possible that there is anything left to explain. Note that we need a quantum gravity theory of black holes for this explanation to work. A classical or semi-classical theory implies that space and time does not break down inside a black hole, not even at a singular point since there exists no singular point, merely a missing point on the manifold. We avoid problems about Planck dimensions since space and time exist only at sizes greater than the Planck dimensions. The breakdown region is what would correspond to the Planck time, the Planck spatial radiues, the Planck density and temperature, etc, if there were Planck dimensional realities in this region. But there are none, since Planck dimensions require a metric of spacetime and there is no metric in the breakdown region. This solves the problem that bedeviled Smolin's quantum gravity proposals about how the black hole is connected to the big bang (namely, at a metricated spacetime region of the mass-energy density of the Plank scale. Smolin placed all his cards on his hope that "hard science cosmology" would return, with a mathematically complete, definitive, observationally well-confirmed and physically well-interpreted theory of quantum gravity--and that this theory would imply time is linear at Planck dimensions. But my assumption is that once cosmology hits the Planck dimensions, it has gone forever beyond "hard science" and becomes permanently metaphysical, with only competeting, incomplete, observationally equivalent, sketchy proposals about the nature of quantum gravity available to add onto and modify the Standard Model and the Grand Unified theory. Given this assumption, I think that the enormous explanatory power of the Black Hole Origin theory, an explanatory power that is the highest that is logically possible and that has not even been attempted by other theoriests, combined with its very degree of conservativeness, outweighs the enormous loss of parisomony that this many-universe theory brings. But since most cosmologies today are many-universe cosmologies (Linde's, Hawking's, the parsimony of a one-universe theory may be exceedingly difficult to attain. I suggest we have many universes and that everything is completely explained. |