Particle and Astro-physics Challenge Kant's Phenomenolism
Lawrence H. Starkey
In his Transcendental Dialectic, (1) Immanuel Kant poses a set of four Antinomies, issues for which he presents both theses and antitheses giving equally strong arguments both for and against each statement. Two of these Antinomies concern the nature of the spatiotemporal Universe-in particular, its alleged infinity in both space and time, and the supposed infinite divisibility of matter. I shall review, withconsiderable scientific documentation, the resolutions of the Antinomies afforded by progress in theoretical physics. (2)
Kant's purpose in the first Critique is to "put an end for ever to all objections to morality and religion" (B xxx). To accomplish this purpose he sets his sights upon three pure Ideas-those of God, of simple soul-substance, and of cosmic infinity-each of which involves the notion of utter completeness or totality. The pretensions of these Ideas to being knowledge, as objects of a scientific metaphysic, are quashed by Kant by displaying their illusory natures when held to be objects of possible empirical experience. Strawson cautions, however, that Kant is "securing the supersensible realm from our scepticism as well as from our knowledge." (3) Thus Kant will later be more positive and return to the first two Ideas (God and Soul) to employ them as regulative principles in his moral philosophy. Meanwhile, Kant replaces the allegedly specious metaphysic with a philosophical psychology reflecting the structure that any mind must necessarily have that can generate knowledge from the sensory manifold.
In all of this, as Weldon urges, "the only ideas in which [Kant] is interested are those of a first cause and a self-existent substance." (4) Clearly, then, the third Idea-that of cosmic infinity-is incidental; it is mainly an exercise, designed to provide a first introduction to the problem of totality and to demonstrate the self-contradictory nature of reason when its powers are, as he thought, misfocused upon things-in-themselves. The cosmological Ideas are merely principles of investigative procedure rather than disclosers of ontal objects. Having discharged this didactic function, Kant then returns to his chief concern, employing the insights gained through his exercises in cosmology to analyze the Ideas of the free soul and of God.
The First Antinomy is stated by Kant as follows (A426/B454 and A427/B455):
And the Second Antinomy is A434/B462 and A435/B463:
As early as his Dissertation of 1770, Kant was exploring these perplexities-questioning the infinity of space, time, and divisibility. And, as Weldon writes, "the existence of these contradictions . . .had seemed to Kant a scandal to philosophy." (5) Thus, feeling that he had already shown that cosmological totality can only be a regulative Idea, (6) Kant concluded that the other two Ideas must also be regulative -in the sense that, by operating as if God and soul-substance were authentic Ideas, we can apply to the various infinite series an iterative procedure that furthers the advance of theology and psychology.
Thus pure reason, rebuffed in its search for a transcendent metaphysic, forced Kant to resort, as Strawson expresses it, to a "'transcendental Idealism' [ending '-al'], according to which the whole world of Nature is merely appearance," (7) and to turn from transcendent to what Walsh calls 'immanent' metaphysics. (8) Thus Strawson correctly generalizes that "the idiom of the [Critique] is throughout a psychological" one. (9) This transcendental turn, however, was aberrant, as I shall show by thinking right past the limits that the Antinomies imposed upon Kant, arguing that today's physics enables us to surmount these constraints. Thus, drawing upon the general theory of relativity and returning to scientific Realism, I shall deal with the world's totalities in terms of geometrical insights inaccessible to Kant, viz., those of the Schwarzschild space-time, with its curved and hole-pocked space, and dilated time.
To Kant, the essential illusion in the concept of totality is that of "the unconditioned totality of an infinite series," as Strawson puts it, "all of whose members are conditioned." (11) Kantian scholarship has long held that the problem lies in the nature of our faculties for geometrical thinking. Not so! It lies instead in our choice of those intuitions making the best fit with the external world. Kant, whose intuitions yielded only paradox, urged that they cannot be so applied at all: Apart from our sensibility "space and time themselves, would vanish," wrote Kant. "As appearances, they cannot exist in themselves, but only in us" (A 42/B 59). Contrary to Kant, however, I shall show how our pure geometrical intuitions can be applied today to the deepest realities of the external world without resort to infinity.
The problem of any infinite totality-of end-to-end yardsticks, say-can be generalized by adapting an analysis by Strawson, (12) which enables us to present Kant's strategy in two propositions: 1) If the series existed as a whole, it would be a limited whole; and 2) If it existed as a whole, it would be an infinite whole. The conjunction of these propositions can now be taken as the major premiss of a destructive dilemma, for which the Antinomies supply the minor premiss. If W symbolizes "existing as a whole," then ~W v ~W, which, by tautology is a denial of the wholeness. As Bennett suggests, Kant is saying in effect that "our only concept of x is our approach to x," (13) which reflects a transcendental Idealism.
Kant's phenomenalism is betrayed also in his discussion of totality directly in terms of an imagined regress of successively larger blocks of space: "Only by reference to the magnitude of the empirical [mind's eye] regress am I in a position to make for myself a concept of the magnitude of the world" (A 519/ B 547). Or, to cite Bennett again, "That the series of possible [world] explorations has such-and-such a length is not just a consequence of the world's having a certain size-[the series] is the world's having that size." (14) Put thus bluntly, one suspects a semantic evasion reflecting the fuzzy thinking of intellectual despair-for which, however, we can hardly blame either Kant or Bennett, nor even Broad (below), for we are witnessing here a foundering in problems beyond their means, means which became accessible only with Einstein, and the black-hole concepts of Penrose and Hawking. (15)
Broad sees, besides Kant's answer, two others equally confused and desperate: 2) that 'world' can be taken as equivocal (16) : a) If it means the sensuous, then Kant's Antitheses (asserting infinities) are true; b) If it means the intelligible, then his finitistic theses are true. (17) Broad's own recourse is 3) to charge either that the premisses are dubious or the arguments invalid. "If so," he concludes, "there is no genuine Antinomy and therefore no call for a solution." (18) I shall, on the contrary, acknowledge such a call and actually present solutions-thus exposing all three of the above approaches as muddled sophistry. My own view, then, is 4) to follow all modern physicists in judging anything erroneous where calculations yield infinities and to employ with elegance the insights of late-century physics, which offer us alternative graphs or frames of measurement in place of Kant's phenomenalism.
Turning specifically, now, to time, and to Kant's Antithesis (that the world had no beginning), Kant first defines 'infinity' as that which never comes to completion in an end term. But the present moment is in fact such a completion; hence the series could not have been infinite. Broad likens the flow of time to the steady pressing of toothpaste out of its tube and asks, "How could the strip ever have got to this . . . determinate point if the mouth of the tube had been infinitely remote?" (19) But this, he says, is nonsense; for, while it is meaningful to ask "How fast is the paste coming out of the tube?" it is meaningless to do so of years-as though their moments might flow by at all sorts of speeds.
In modern physics, as against Broad, the toothpaste metaphor is, in fact, apt; for time does flow at different speeds. This fact was predicted by Einstein who, as described in the "Gravitation" article in the Britannica, "was able to show that clocks would run slower when near massive bodies"; (20) and it has been confirmed by experiment to within one percent of the theoretical. (21) If Broad shrank from envisioning time-flow as like tooth-paste but with squeeze-outs at varying rates, we now know that such time dilatation, as it is called, is a real, absolute, and asymmetrical occurrence. And carrying the image of clocks near massive bodies to the extreme, we may cite Davies, who reports that "some stars are known where the grip of gravity is so ferocious that time there is slowed by several percent relative to us. In fact . . . if the gravity of such a star were a few times greater . . . time would grind to a halt altogether" as the star implodes into a black-hole. (22)
Kant was born too early to access the concept of a black-hole or that of 'proper time,' defined as that measured by the clock of a victim sucked in toward such a hole-which would run at a rate radically faster than the time ticked off by a clock far outside in the ordinary world. Such a victim-as described in Britannica's "Relativity" article-will, as measured in his own time frame, "penetrate the Schwarzschild radius [the hole's core] within a finite proper time. . . . To an outside observer [on the other hand], any objects approaching the Schwarzschild radius [will] appear to take an infinite time to penetrate toward the inside." (23) To the victim looking out, however, the entire Universe-because of the asymmetry-will appear to go through its paces at a fantastic rate, and in almost no time the end of the whole Universe will collapse down upon him. Thus, a nearly infinite time in the outer Universe will register as a mere moment on the clock displaying the proper time of the victim entering the sink.
Now the resolution of Kant's Antinomy lies in imagining the exact reverse of this black-hole process, with the Big Bang functioning as a white hole, (24) which emits matter instead of gulping it up. Reality would then emerge from such a hole instead of collapsing into one. (25) In an outsider's time frame it would display an "infinite blue shift" (26) of its radiation, reflecting the incredible rush of time in the earliest inflation. Many astronomical objects, possibly outliers of the Big Bang, are actually observed pouring matter into the Universe, like white-holes, at fantastic rates. (27) The clear factuality of these extremes demonstrates places where time flows much faster. Indeed, the proper time-that of the participant himself emerging from such a throat-like source, particularly from the Big Bang-could reflect, in his time frame, an almost infinite past, which in our frame, however, might measure mere milliseconds. (28) Which horn of Kant's Antinomy we are on-whether that of a finite or an infinite past-clearly depends very physically upon where we are observing things from, upon its gravitational field or acceleration, rather than, with Kant's phenomenological Idealism, on what sorts of mental equipment we have.
As with space, so with time, Kant could not envision its curvature-nor can one who considers things only in the space-time frame of the victim sucked through wormholes or emerging from 'elsewhere' in the Bangs, whether Big or small. To him time advances straightforwardly. But outside observers (ourselves) see time as curved, viewing its arrow more and more end-on as it follows the curving away of space into the realm of the quasars or into the flattening gravity of a black-hole. Thus, in our frame its trajectory graphs like an arc in a plane and time's vector eventually runs off at right angles to time as we know it and plunges into and through the Universe's final 'Big Crunch.' By curving off it generates, along with a fourth dimension of space, (29) a new space-time expanse substituting for the absurdities of real backward time-flow. Particle experiments on decaying K0 mesons (30) prove the dire need for a new temporal order to counterbalance what would otherwise violate the laws of charge/handedness symmetry. To Bergmann, it is even conceivable that time flows in an entire circuit (31) -which I rationalize by appeal to my double-universe model, which accommodates the ψ-spin material particles by going twice around a loop before restoring its original symmetry (below Fig.3).
As for space, Kant correlates it with time, arguing that it would take an infinite time to lay out in one's mind an infinite succession of spaces. But the argument is specious; for Kant himself had urged, in his Dissertation of 1770 (§1n), that at least one mind, that of God, could "perceive an infinite multiplicity in one intuition." (32) The experience of cosmic space is handled somewhat better by Parsons under the heading of "Infinity and Kant's Conception of the 'Possibility of Experience.'" (33) Since we indeed intuit mathematics, Kant felt that its concept of 'infinity' must stem from "the way objects present themselves . . . in perception." Parsons, then, appealing to Gestalt psychology and its flip-flops between figure and ground and seeing the process as iteration, concluded that "for Kant, mathematical induction,[which is] a quite abstract kind" of experience, must be involved. (34)
Little did Kant realize how abstract it could get-that we would someday see our world as but one of two or three totally disconnected cross-sections through a hyperdimensional Universe. "If we speak of diverse spaces," wrote the Euclidean Kant, "we mean thereby only parts of one and the same unique space, [parts that] can be thought only as in it" (A 25).Even if the world were finite, it would have to exist "in an empty space which is unlimited" (A 428/ B 456). But since there would be "no correlate with which the world stands in relation, this would be a relation of it to no object" (A 429/ B 457). (35) Thus it must be itself infinite. Today, however, we have Riemannian geometry, in which space itself is curved enough to close in on itself and be finite, yet have no boundaries. Kant's finite world would not have to be related to an infinite space. And what Kant shunned as "diverse spaces" I espoused in 1968 as 'the double universe' (47 pp.) (36) -since confirmed in separate papers by a half-dozen key relativists as comprising "two distinct asymptotically flat regions" or spaces connected by an Einstein-Rosen bridge. (37)
On the absoluteness of space, Parsons faults Kant for not employing the 'analytic' method of his Prolegomena, where he discussed handedness. Kant there urged that space, contrary to his later view, must be absolute and not merely a tool for defining relations between bodies, for such a tool could not even distinguish a right from a left hand. (38) To his credit, Kant here formulates a perplexity still vexing physicists, whose beta-decay experiments, which should theoretically yield both left- and right-handed neutrinos in equal numbers, show that our world has only left-handed ones. Since their speed is always c, their handedness must be always the same. (39) Perhaps the others are denizens of the twin universe that I posited in discussing the K° mesons that demanded a veering-off time line. If space thus comes in different kinds, it must be absolute.
A couple such kinds, and the special par-ticles they accommodate, are displayed by a transparent Möbius strip (Fig.1) formed with half a twist. If, every couple inches along it, one writes, say, an RN crosswise, one finds after once around that one is not done, but that a series of Russian RNs begins to show through the paper. Only after one more once around that one is not done, but that a series of Russian RNs begins to show through the paper. circuit will one merge in again with the RNs. One has written all along both sides without lifting one's pen Two such strips stretched and seamed together form a Klein bottle (Fig.2) (40) . the inside and outside surfaces of which flow similarly into each other-as any wandering bug can attest. (One has to imagine the fourth dimension of space not shown here.)
To depict what I feel to be the actual world, I finally add a Schwarzschild throat (41) (Fig.3)-which one tunnels through twice before restoring the original figure. With the first plunge one enters the twin universe, where one dimension of time gives way to a space dimension and vice versa and where all particles turn into their antiparticles and mirror images-which explains neatly time/charge/parity images-which explains neatly time/charge/parity conservation. (42) And, as Hawking explains, (43) a topology carrying a particle through two circuits exactly defines the meaning of its having a spin of one-half. In dimensions higher yet, a third such universe (44) is also predicted such that the world's total dimensionality is 3 + 3 + 3 of space and 1 (or 2) of time, to total 10 or 11. The triple 3 enables the time vector to appropriate one space dimension in sequence from each universe and provides one universe for each of the three families of particles: our own plus those at the strange/bottom and top/b' levels. (45) A schema is also conceivable for alternating particle and antiparticle universes, bringing the total to six at most.
Whether speaking of the final Big Crunch or smaller black-holes, we know (Fig.4), as Kant did not, about the Minkowski light cones-graph- ing space-time between two limits, those of the velocity of light(±c)-and how they tilt more and more toward the hole so that c becomes at its edge the only velocity possible. (46) Everything then goes through, and out the other side at decelerating speeds to open up the twin universe. This model addresses Kant's misgivings about the world's having a relation "to no object ." In my cosmology, however, if our world should perchance be a rotating one (as Gödel arargued (47) and Kerr has formulated (48) ) its object of reference could be the twin universe and Kant's problem would evaporate. His spatial Antinomy is therefore resolved in favor of his finitistic thesis.
Modern physics also resolves Kant's Second Antinomy, that of divisibility. Here the peppering of the Schwarzschild space-time with holes (49) marks a limit to the divisibility of matter. Such holes, as seen in our coordinate frame, cannot be further divided even though they are not infinitesimally small. To Eddington, their insides "must be cut out of space-time altogether." (50) In the frame of the victim, however, such a hole is a point. But it opens up to another whole universe with its own particles, black-holes for him to resume cutting up, which lead then into the third universe and eventually back to the first (or its equal).
In this paper I have shown how the entire Universe can be conceived as curved and reëntrant to itself in a fourth (or higher) spatial dimension. The most successful way of suppressing infinities is to bend them into circles, spheres, or hyperspheres. I have here shown how modern physics has made all three-time, space, and divisibility-finite in the same sense that the circumference of a circle must be 2pR; and I have thus resolved the Kantian Antinomies in a perfectly physical manner. In so doing I have undercut Kant's reasons for distrusting pure reason and have rescued cosmology and with it the pure Ideas of God and soul-substance from their reduction to mere regulative principles and restored them to their roots in transcendent metaphysics.
(1) Immanuel Kant's Critique of Pure Reason, trans. Norman Kemp Smith (London, 1929). Kant references in parentheses (A, First Ed.; B, Second Ed.).
(2) For much expanded treatment and pursuit of related issues, see my article, "Kant's Cosmological Antinomies and Modern Physics," in Prairie Home Philosophy, ed. Arnold Johanson, et al. (Moorhead, Minn.: Moorhead State University, 1987), pp. 146-157.
(3) P.F. Strawson, The Bounds of Sense: An Essay on Kant's Critique of Pure Reason (London, 1966), p. 22.
(4) T.D. Weldon, Introduction to Kant's Critique of Pure Reason (Oxford, 1945), p. 113.
(5) Ibid., p. 118.
(6) Strawson, p. 33.
(7) Ibid.,p. 21.
(8) W.H. Walsh, Metaphysics ("A Harbinger Book"; New York, 1963), pp. 84f.
(9) Strawson, p. 19.
(10) A cosmic geometry based upon the metric tensor in Einstein's equations. Robert M. Wald, General Relativity (Chicago, 1984), Part I, chap. vi.
(11) Strawson, p. 158.
(12) Ibid., pp. 187ff. Thus, as Broad notes, "Kant does not attempt to prove [but] to refute in turn the antithesis and the thesis." C.D. Broad, "Kant's Mathematical Antinomies: The Presidential Address," Procs. of the Aristotelian Soc., Vol. LV (1954-1955), pp. 1-2.
(13) Jonathan Bennett, in Ralph C.S. Walker (ed.), Kant on Pure Reason ("Oxford Readings in Philosophy"; (Oxford, 1982), p. 191.
(14) Ibid., p. 188.
(15) Stephen Hawking and Roger Penrose, The Nature of Space and Time ("The Isaac Newton Institute Series of Lectures"; Princeton, 1996). Their original papers are: R. Penrose, "Gravitational Collapse and Space-Time Singularities." Phys. Rev. Lett., Vol.14 (1965), pp. 57-59; and S.W. Hawking, "Black Holes in General Relativity," Communications in Mathematical Phys. Vol. 25 (1972), pp. 152-166.
(16) Broad, p. 10.
(17) Norman Kemp Smith, A Commentary to Kant's Critique of Pure Reason (2nd ed., rev. and enlarged; New York, 1962), pp. 479-491; esp. p. 481.
(18) Broad, p. 22.
(19) Ibid., p. 6.
(20) Kenneth L. Nordvedt, Jr., "Gravitation," Encyc. Brit., 15th ed., Vol. VIII, p. 290D.
(21) R.V. Pound and G.A. Rebka, "Apparent Weight of Photons," Phys. Rev. Lett., Vol. IV (Apr. 1, 1960) pp. 337-341.
(22) Peter Davies, God and the New Physics, (New York, 1983), p. 122.
(23) Peter Gabriel Bergmann, "Relativity," Encyc. Brit., Vol. XV, p. 587C.
(24) Rather than being sinks, white, holes, if they exist, would be sources, "from which anything can come spewing out," Stuart L. Shapiro and Saul A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects ("A Wiley-Interscience Publication"; New York, 1983), p. 355. See also John Gribbin, Unveiling the Edge of Time: Black Holes, White Holes, Wormholes (New York, 1992), chaps. 5 and 6; and Jean-Pierre Luminet, Black Holes, trans. Alison Bullough and Andrew King (Cambridge, 1992), pp. 160-179 (lucid figures).
(25) Wald, pp. 155, 300n, and 316 and, more recently (1992), both Gribbin and Luminet (operae cit.) have described space-time as having a white-hole in the past and a black-hole in the future.
(26) S. Chandrasekkar and J.B. Hartle, "On Crossing the Cauchy Horizon of a Reissner-Nordstrøm Black Hole," Procs. Royal Soc. of London: A. Mathematical and Physical Sciences, Vol. 384, No. 1787 (8 Dec., 1982), pp. 301-315. Cf. also Wald, pp. 404-405.
(27) Among such sources are quasi-stellar and BL Lac objects and Seyfert galaxies that appear to be gushing out torrents of matter in such amounts and at such speeds as to confound the mind. A. Cavaliere, "Compact Radiation Sources in Active Galactic Nuclei," in C. Edwards (ed.), Gravitational Radiation Collapsed Objects and Exact Solutions: Proceedings, Perth 1979, Vol. CXXIV of Lecture Notes in Physics, ed. J. Ehlers, et al.(New York, 1980), p. 88; and Daniel W. Weedman, "Seyfert Galaxies," Ann. Rev. of Astron. and Astrophys., ed. Geoffrey Burbidge, David Layzer, and John G. Phillips, Vol. XV (1977), pp. 69-95. It has even been proposed that jets may exist with velocities on the order of that of light: A.M. Wolfe, ed., Pittsburgh Conf. On BL Lac Objects, (Pittsburgh, 1978), p. 328.
(28) The vicissitudes of Alan Guth's concept of a primordial "inflationary expansion" are discussed by Stephen W. Hawking in A Brief History of Time: From the Big Bang to Black Holes (New York, 1988), pp.127-132. As a general relativity effect, opposite to black-hole collapse, its timing could be comparable. Luminet clocks the collapse-victim's world, for a mass of 10 suns, at one thousandth of a second and, for the heart of a galaxy, roughly an hour, op. cit., p. 136. Milne's cosmology may here be recalled, which featured two different time scales, one reflecting a finite past time and the other a universe of infinite age. E.A. Milne, Kinematic Relativity: A Sequel to Relativity, Gravitation and World Structure ("The International Series of Monographs on Physics," ed. Ralph Fowler, et al.; Oxford, 1948).
(29) See above, in my Note 2, Johanson (ed.), p. 152a, which urges that the cosmological red-shift may, in fact, betray a curving off of time.
(30) All such mesons were found to be left-handed: See original paper by Tsung-Dao Lee and Chen Ning Yang, Phys. Rev. (Oct. 1, 1956), pp. 254-258; and discussion in Martin Gardner, The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings (3rd rev. ed.; New York, 1990), pp. 213, 217, and 218. In default of mirror-image K0s in our own Universe, a physically possible alternative in a universe gushing out from the far sides of our black-holes in deceleration from the velocity of light-alluded to in Mitchell Begelman and Martin Rees, Gravity's Fatal Attraction: Black Holes in the Universe ("Scientific American Library," No. 58; New York, 1995), p. 228. Its time dimension would be normal to ours. See discussion on handedness below (and Note 39 on neutrinos).
(31) Bergmann, p. 588F. Also, Wald , p. 196, where (in Fig. 8.7) he shows tipped-over light-cones in a circle.
(32) Kant's Inaugural Dissertation and Early Writings on Space, trans. John Handyside (Chicago, 1929), Section 1, §1, note pp. II, 388-389.
(33) Charles D. Parsons in Philosophical Rev., Vol. LXXXIII (1964), pp. 182-197.
(34) Ibid., pp. 196 and 194.
(35) A contemporary relativist likewise sees finite matter as having "no reference body" thus violating Mach's principle: H.A. Atwater, Introduction to General Relativity ("International Series in Natural Philosophy," ed. D. Ter Haar, Vol. LXII; New York, 1974), p. 187
(36) My paper, "A Double-Universe Cosmology with Nil Total Mass-Energy," 1968; program brochure for Amer. Sci. Affil. convention listing my paper on the theory, and photo of that lecture showing couple dozen illustrative wall posters, 1970 (MS and lecture unpublished, c/o Steve Rank, 3725 Bay Shore Drive, Sturgeon Bay, WI 54235, U.S.A.). For my published summaries see Note 2 above, Johanson (1987), pp. 153a, 153b, and 154a; and Vortrage: V. Internationaler Leibniz-Kongress (Hannover, Germany, 1988), S. 943-945.
(37) M.D. Kruskal, "Maximal Extension of Schwarzschild Metric," Phys. Rev., Vol. 119 (Sept. 1960), pp. 1743-1745; and Peter Szekeres, "Naked Singularities," in C. Edwards (ed.) as detailed in Note 27 above, pp. 477f; Wald, op. cit. (1984), pp. 154-155 and 316; Gribbin (1992) ibid.; Luminet (1920), ibid.; and P.D. D'eath, Black Holes: Gravitational Interactions ("Oxford Mathematical Monographs," ed. J.M. Ball, et al.; Oxford, 1996), p. 18 and Fig. 2.5.
(38) Immanuel Kant, Prolegomena to Any Future Metaphysics, trans. P.G. Lucas (Manchester, 1953), Sec. 13. For a paragraph summarizing the argument, see J.J.C. Smart, "Space," The Encyclopedia of Philosophy, ed. Paul Edwards, Vol. VII, p. 506.
(39) This is because there can be no perspective other than that of an observer lagging behind: Gardner, p. 227.
(40) Brilliantly diagramed by David Hilbert and Stephen Cohn-Vossen, Geometry and the Imagination, trans. P. Nemenyi (New York, 1952), p. 310, Fig. 298b.
(41) From Wald, p. 155, Fig. 6.10.
(42) Gardner, chap. 25.
(43) Hawking, History, p. 67.
(44) A third universe would seem to be accommodated mathematically by Hilbert when he shows that the flat pattern for generating a Klein bottle lends itself also to a second twist normal to the first that gives rise to a more complex one-sided figure, that of the projective plane: Hilbert and Cohn-Vossen, pp. 237, 299, 309, and 313-321.
(45) Gordon Kane, The Particle Garden: Our Universe as Understood by Particle Physicists ("Helix Books"; Reading, Mass., 1995), pp. 54-60, especially Fig. 4.1.
(46) Begelman and Rees, pp. 11 and 12. The radical narrowing of light-cones is nicely depicted in a figure on p. 12.
(47) Kurt Gödel, "A Remark about the Relationship Between Relativity Theory and Idealistic Philosophy," in Albert Einstein: Philosopher-Scientist, ed. Paul Arthur Schilpp ("Harper Torchbooks: The Science Library"; New York, 1959), Vol. II, pp. 555-562, espec. Note 10.
(48) O'Neill Barrett, The Geometry of Kerr Black Holes (Wellesley, 1995).
(49) Eddington spoke of space-time as "studded, with holes," Sir Arthur Eddington, Space, Time and Gravitation: An Outline of the General Relativity Theory ("Harper Torchbooks: The Science Library"; New York, 1959), p. 92. Cf. G.C. McVittie, General Relativity and Cosmology (2nd ed.; Urbana, Ill., 1965), pp. 80-85.
(50) Eddington, Space, Time, p. 91.