Einstein, Gödel, and the mathematics of time Francisco Antonio Doria and Manuel Doria Advanced Studies Research Group and Fuzzy Sets Laboratory PIT, Production Engineering Program COPPE, UFRJ P.O. Box 68507 21945–972 Rio RJ Brazil fadoria@gmail.com manueldoria@gmail.com Version 1.0 March 28, 2009 Abstract We consider exotic spacetimes in the sense of differential geometry, and then state genericity, undecidability and incompleteness results about general relativity, with the nature of time and the existence of some kind of “cosmic time” as central questions. 1 1 Introduction Discussions on the nature of time are as old as philosophical inquiry itself, and have always riddled scientists and philosophers alike with its many perplexities. 20th century physics, entertaining us with theoretically feasible and actual phenomena such as temporal dilation, time travel, timeless singularities — and, as we shall see in this article, the possibility that there is no global arrow of time — has only deepened the mysteries surrounding the concept of time. Progress has been made in the terrain of cognitive science, particularly in the description of cognitive mechanisms involved in the conceptualization of time [12]. Lakoff et al. suggest that it is “virtually impossible to conceptualize time without metaphor.” Even Kant, who in the 18th century championed the thesis that time was a pure a priori intuition that necessarily structured all our subjective experience admitted that we reasoned about it in terms of an iterated progression along a geometrical line (just like Galileo in the dawn of kinematics). Contra Kant, Lakoff claims that there is no such thing as a “pure intuition” of time: temporal concepts themselves have an internal structure that is largely assembled by our prior experiences of motion in space. General relativity itself conceptualizes time metaphorically as a space–akin dimension on the spacetime manifold. We take our cue from the fact that Einstein and Gödel were close friends, and yet the only ground which they eventually shared in scientific terms were Gödel’s papers on general relativity. No doubt those are landmark papers: they show that general relativity allows for the existence of an universe with intrinsic rotation; they suggest the possibility of a time machine — and they have a very counterintuitive kind of time, as we do not have a “global” time coordinate in the Gödel universes. It is meaningless to refer to a “beginning of time” in such universes. Is that an isolated phenomenon? Nonexistence of a global time coordinate is just a property of Gödel’s and Gödel–like models of the universe? Or can it be seen as the typical situation? This is the underlying question in the present paper, and in order to deal with it we concoct a potion that mixes up ingredients from differential geometry, from general relativity and from logic. We will argue at the end that: Nonexistence of a global time coordinate, from Big Bang to Big Crunch may well be the typical, generic situation in general relativity. We present a result whose interpretation may support that claim. The idea that there is no universal direction of time may sound cognitively abhorrent precisely because of the everyday metaphors involved in thinking about time (see [12], entry on “The Moving Time Metaphor”). A theory being counterintuitive may be a consequence of either taking as literal without sufficient ground metaphorical aspects of a certain concept or as not having apt conceptual metaphors for dealing with novel empirical phenomena. 2 Einstein, Gödel 3 The meaning of ‘generic’ in this paper We use the word ‘generic’ in several different senses in this paper: 1. Topologically generic sets. Given a topological space X, a subset Y ⊂ X is topologically generic if its complement is a first–category set (a meager set). 2. Measure–theoretically generic sets. Given a space X endowed with a measure µ, a subset Y ⊂ X is generic for measure µ if µ(X − Y ) = 0. 3. Set–theoretically generic sets. Let L be Gödel’s constructive universe of sets, and let LB be a forcing extension of L, or a Boolean extension of it. Then a set x ∈ LB − L is a generic set. Set–theoretically generic sets may be collected in measure–theoretically or in topologically generic sets, given adequate axioms (see below the discussion of Martin’s Axiom). We will sometimes speak of generic sets without qualification; context will make clear the intended meaning of the word. We will sometimes use “typical” as a loose, informal way to describe sets that can be made generic in one of the senses above. Preliminary concepts and results We summarize here the axiomatics for general relativity that has been introduced in [7] and more recently described in detail in [6]. That axiomatics is “natural” in the sense that we simply rebuild the usual mathematical background for gravitation theory within Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). (For details see [6].) Roughly, we take general relativity to be a theory so that: • Its arena is an arbitrary 4–dimensional noncompact real differentiable manifold, which we identify to spacetime. Therefore we must consider in our characterization of general relativity, the collection of all 4–dimensional real noncompact manifolds with a differentiable structure, a notoriously complicated object. • To each such 4–dimensional real differentiable manifold we add a smooth pseudo–Riemannian metric of signature +2, and then the Einstein gravitational equations, with or without the interaction of matter fields. • We also add as much extra structures as required for the description of the fields that appear in the energy–momentum tensor. General relativity is a theory of gravitation that interpretes this basic force as originated in the pseudo–Riemannian structure of spacetime. That is to say: in general relativity we start from a spacetime manifold (a 4–dimensional, real, adequately smooth manifold) which is endowed with an pseudo–Riemannian metric tensor. Gravitational effects originate in that tensor. 4 Doria, Doria Given any 4–dimensional, noncompact, real, differentiable manifold M , we can endow it with an infinite set of different, nonequivalent pseudo–Riemannian metric tensors with a Lorentzian signature (that is, − + ++). That set is uncountable and has the power of the continuum. (By nonequivalent metric tensors we mean the following: form the set of all such metric tensors and factor it by the group of diffeomorphisms of M ; we get a set that has the cardinality of the continuum. Each element of the quotient set is a different gravitational field for M .) Therefore, neither the underlying structure of M as a topological manifold, nor its differentiable structure determines a particular pseudo–Riemannian metric tensor, that is, a specific gravitational field. From the strictly geometrical viewpoint, when we choose a particular metric tensor g of Lorentzian signature, we determine a g–dependent reduction of the general linear tensor bundle over M to one of its pseudo–orthogonal bundles. The relation g 7→ g–dependent reduction of the linear bundle to a pseudo–orthogonal bundle is 1–1. This is equivalent to endowing spacetime with a smooth 1–foliation. Spacetimes with cosmic time Definition 1.1 A spacetime M has a global time coordinate whenever: 1. M is diffeomorphic to N ×R, where N is a differentiable, real, 3–manifold. 2. M is endowed with a pseudo–Riemannian metric tensor of signature (−1, +1, +1, +1) so that there is a coordinate system where it has the form g00 dx0 + gij dxij , with coordinate 0 being that of R and i, j roaming over N . Condition 1 excludes exotic [10, 13] spacetimes, and Condition 2 essentially means that there is a trivial foliation of M “parallel” to R which behaves as the global time coordinate. So, we can reasonably talk about, say, the universe having begun 14 billion years ago, if our universe has a global time coordinate, or global time for short. We say that a spacetime has the “cosmic time property” if it exhibits a global time coordinate. From here on we suppose that Zermelo–Fraenkel set theory is consistent. Moreover, if required, we suppose that it has a model with standard arithmetic. The ZFC set of all spacetimes This is a side remark, but how do we make precise the ZFC set of all spacetimes? • A (topological or differentiable) manifold is described by coordinate domains and transition functions. If the manifold is noncompact, there are denumerable many such domains. Einstein and Gödel 5 • So, we can code each manifold (in many different ways) by a real number. • We can therefore define a 1–1 function from the reals to the manifolds (see [7] on that function). • Use the Axiom of Replacement to define the set of all manifolds out of that function. 2 Exoticisms We are interested in 4–dimensional real differentiable manifolds as those are the arena where the game of general relativity is played. The situation is, however, extremely complicated due to the peculiarities of the geometry of 4–dimensional manifolds. A very brief introduction to smooth exotic 4–manifolds Let’s start from topological manifolds. • Consider a topological real n–dimensional manifold, that is, a separable metrizable space endowed with a maximal atlas that makes it locally like Rn . • If it admits a differentiable maximal atlas, then it can be endowed with a differentiable structure. • The number of differentiable structures may be > 1 modulo diffeomorphisms. • In that case, if there is some atlas that may be taken as a standard differentiable structure (say, like the usual structures for R4 or S 7 ), we say that the remaining differentiable structures are exotic [13]. The next summary comes from several sources [1, 10, 13]. Below there is a list of concepts and results that we require here: • Given a smooth manifold, its possible submanifolds determine the manifold. Given a closed differential 1–form α∗ , its (local) integral gives a parametric family of submanifolds of our manifold (the family is parametrized by the integration constant). That idea can be generalized to encompass higher–order forms. • The intersection form arises out of the possible submanifolds of a given manifold in a way that we are going to specify. Restrict the attention to 2–forms on four manifolds. These forms can be seen to determine 6 Doria, Doria submanifolds of the 4–manifold M , as explained above (see also [13], p. 115 ff). Then we define the intersection form as: Z α∗ ∧ β ∗ . QM (α∗ , β ∗ ) = M The intersection form arises out of elements (α∗ , β ∗ ) of the second DeRham cohomology group H 2 (M ; R) for manifold M . One usually says that the solutions for the Einstein equations “determine the geometry of spacetime.” That’s not correct. The fact that one can use DeRham cohomology to handle intersection forms [13], together with the fact that mesonic and electromagnetic test fields over spacetime can be used to characterize its DeRham cohomology provides another link between the geometric structure of spacetime and the physics one does over it [9]. α∗ and β ∗ as above are 2–forms over the manifold M , which can be interpreted as mesonic test fields, or even electromagnetic test fields over spacetime M . So, these fields are the ones whose classes determine the global structure of a spacetime. • So, we can say that given an intersection form, there is a (topological) manifold that corresponds to that form. And if we classify intersection forms, we get a classification for manifolds. • More precisely we have Freedman’s Classification Theorem : for any integral symmetric unimodular form Q there is a closed simply–connected topological 4–manifold that has Q as its intersection form. – If Q is even, there is exactly one such manifold. – If Q is odd, there are exactly two such manifolds, at least one of which does not admit any smooth structure. • Follows the very interesting result: the odd intersection form noted [+1] (see the references) represents projective space CP2 . It must also represent “fake CP2 ,” a nonsmoothable 4–manifold which is homotopy equivalent to CP2 , as both share the same form [+1]. • Donaldson’s Theorem. Another fundamental result in this domain is due to S. K. Donaldson, who proved it in 1982: The bilinear symmetric unimodular forms ⊕m[+1] and ⊕m[−1] are the only definite forms that can be realized as intersection forms of a smooth 4–manifold. • Notice that this and similar partial results for indefinite forms give the global topological structure of possible spacetimes, which can very precisely be said to arise out of the spacetime’s intersection form. The result that interests us here is: 7 Einstein and Gödel Proposition 2.1 There is an exotic R4 with a compact set C so that no smooth embedded S 3 encloses C. For the proof see [13], p. 250. It is one of the two main tools required to prove Taubes’ Theorem: Proposition 2.2 There are uncountably many non–diffeomorphic exotic ER4 s. We will actually require one of the consequences of Taubes’ Theorem: Proposition 2.3 If ER4 is an exotic R4 and h as below is an homeomorphism: h : R4 → ER4 then given an open ball D(ρ) ⊂ R4 of radius ρ, there is a value ρ0 so that for a compact set C ⊂ ER4 , for no ρ > ρ0 does a smooth image h(D(ρ)) encloses C. (It is actually a consequence of the result we gave above.) 3 Conjectures, speculations, more counterintuitive results Recall that ER4 is an exotic 4–plane. We first state: Proposition 3.1 No ER4 with the property spelled out in Proposition 2.1 has a global time–coordinate. Proof : If it had such a coordinate, then it would be diffeomorphic to R3 × R, which is impossible, since no R3 has an exotic differential structure. However it is homeomorphic to R3 × R. This means: there is a global, albeit sometimes nondifferentiable global time–coordinate. But we have that the global time coordinate, if it exists, must be differentiable. Corollary 3.2 For the family ER4 (ρ), absence of a global time structure is generic in the topological and measure–theoretic senses. Proof : Immediate: from the map ρ ∈ (ρ0 , ∞) 7→ ER4 (ρ) one can induce the corresponding concepts of genericity’ etc. in the space of all those manifolds. Since there is just one standard R4 , the set of all such exotic 4–planes will be generic in the (induced) senses. Now, for set–theoretic genericity (we require the axiomarization of general relativity here): 8 Doria, Doria Proposition 3.3 For B an adequate complete Boolean algebra, for L |= ZFC, being Gödel’s constructive universe, for ρ ∈ L a real number so that LB |= ρ > ρˆ0 , then ρ can be chosen a set–theoretically generic real number so that LB |= ER4 (ρ). That ER4 (ρ) is a set–theoretically generic exotic spacetime. There are other examples of similar beasties. The next result is given rather loosely: Proposition 3.4 Set theoretic genericity doens’t imply absence of global time coordinate. Sketch of proof : For adequate forcing extensions VB there are set–theoretically generic noncompact differentiable 3–manifolds [7], and given one such, noted M , M × R is a generic differentiable 4–manifold in the same forcing extension. Set theory with Martin’s Axiom For a review of Martin’s Axiom see [11]. Roughly speaking, Martin’s Axiom acts as a “regularizing tool,” that is, the sets that should be of zero measure, or of first category, or both, can be proved to be so given Martin’s Axiom. Proposition 3.5 If model MM A is such that it makes true the theory ZFC + ¬CH + MA then MM A makes true the formal version of the sentence “every constructible subset of the reals is a first–category set and a zero–Lebesgue– measure set.” CH is the Continuum Hypothesis, and MA is Martin’s Axiom. We will use that result in what follows. Category and measure We now go back to the question: which is the typical situation in Nature? Global time or its absence? What can we make out of the fact that there will be spacetimes so that we have no decision procedure to ascertain whether they have local or global time? How frequent is that situation? Results about the nongenericity of global time We again deal here with topological and measure–theoretic genericity. Some results that suggest that global time isn’t generic in the sense of topology or measure follow from Theorem 9.4.24 and Corollary 9.4.25 in Gompf and Stipsicz ([10], p. 378 s). Define a topologically cylindrical spacetime to be homeomorphic to S 3 × R. Then: Proposition 3.6 For a reasonable topology and measure, there is a generic set of spacetimes homeomorphic to a cylinder C × R which do not have a global time coordinate. Einstein and Gödel 9 Proof : It is again immediate: there are 2ℵ0 many non–diffeomorphically– equivalent, diverse, structures which are smooth for those spacetimes. Code each one by a binary irrational in some possible way and induce measure and category from the pullback map. The set of exotic topologically cylindrical spacetimes is of measure 1 and of the second category. A second, more general result, goes as follows. Consider the set of all connected topological real 4–manifolds and pick up those that admit a smooth structure; factor them out by homeomorphisms. We then have a set of nonequivalent (modulo homeomorphisms) topological real 4–manifolds which can be given a smooth atlas. Code them (via the function that maps spacetimes over some set of cardinality 2ℵ0 onto, say, the binary irrationals. Call that binary irrational λ; choose a particular smooth structure for it and call the resulting differentiable manifold Xλ . From the above quoted result (see the reference) we have that Xλ − {∗}, where {∗} is a point, has uncountably many nonequivalent differentiable structures. Then form the set of all pairs hXλ , Eµ (Xλ − {∗})i, where Eµ (. . .) represents the exotic structure denoted by µ; that set is coded by the λ, µ. In the induced topology and measure the set of exotic spacetimes is both set– theoretically and measure–theoretically generic. We can picture that construction as follows: over each “point” Xλ there is a “fiber” Eµ (Xλ − {∗}) to which we add (we code) all extra differentiable structures for Xλ , if any. If Y denotes that space: Proposition 3.7 The set Y of spacetimes without a global time coordinate is set–theoretically and measure–theoretically generic in the above–described topology and measure. Follows: Proposition 3.8 Spacetimes without global time are set–theoretically and measure–theoretically generic in the above described topology and measure. Proof : Follows from the fact that spacetimes with global time must have a standard structure. Martin’s Axiom again Follows from Propositions 3.5 and 3.8 that: Proposition 3.9 Model MM A makes true the formal version of the sentence “Given the above topologies and measures, the set of exotic set–theoretically generic spacetimes has measure 1 and is of second category.” 10 Doria, Doria So, if our spacetimes are to be found in a — mathematical — universe where the Continuum Hypothesis doesn’t hold and where Martin’s Axiom is true, then (loosely speaking) the typical spacetime is a chimaera–like object; it is exotic and set–theoretically generic, and obviously without a global time coordinate. 4 Can we decide whether an arbitrary spacetime has a global time coordinate? The answer to that query is, no: Proposition 4.1 There is a family gn of metric tensors for a spacetime M so that: 1. There is no algorithm to decide, in the general case, whether gn , for each n, has the cosmic time property. 2. The decision problem for that question may be as difficult as one wishes in the arithmetic hierarchy. Proposition 4.2 Given any axiomatization for general relativity within ZFC, there is a metric tensor g over R4 with the usual differential structure so that: 1. ZFC 6⊢ g has global time. If h is Gödel’s metric tensor, then g = h holds of all models for ZFC with standard arithmetic. 2. ZFC 6⊢ g doesn’t have global time. If η is Minkowski’s tensor, then g = η will hold of some models with nonstandard arithmetic and of no model with standard arithmetic, for ZFC. 3. To sum it up: for any model with standard arithmetic N for ZFC, N |= g doesn’t have global time. Proposition 4.3 Given any axiomatization for general relativity within ZFC, there is a metric tensor g over R4 with the usual differential structure so that: 1. ZFC 6⊢ g has global time. 2. ZFC 6⊢ g doesn’t have global time. 3. L |= g doesn’t have global time. Here L is Gödel’s constructive universe. We can obtain an undecidability result as in the previous results. About the preceding result: there will be models with standard arithmetic for both sentences in the undecidable pair we have considered. Einstein, Gödel 5 11 Conclusion We may summarize our conclusions as follows: Spacetime may well be a cylinder S 3 × R with the standard topology and differentiable structure, and with global time. However that very specific geometry doesn’t follow from the Einstein gravitational equations, and is in fact very far from what a typical spacetime should look like: an exotic, set–theoretically generic 4–manifold, endowed with a very complicated time structure. We have here two sorts of results: • Category and measure. We have exhibited results about topological and measure–theoretic genericity of the non–existence of a global time coordinate. • Undecidability and incompleteness. There is no general algorithm to decide whether an arbitrary spacetime exhibits the cosmic time property (whether it has a global time coordinate). And there are formal sentences that translate as “spacetime X has the cosmic time property,” which can neither be proved nor disproved in, say, ZFC. The question is: can we take our arguments here as arguments that give a “natural” zero probability for the existence of global time? How are we to interprete the preceding results? Does our result on the genericity of spacetimes without a global time coordinate reflect the actual situation in the real world? In the world of possible spacetimes? Is our probability evaluation a “physical world” probability? Even if it includes a wide range of conceivable measure attributions? 6 Acknowledgments This paper collects some results from an ongoing research program with N. C. A. da Costa, whom we heartily thank for criticisms and comments. We must also thank C. M. Doria for his remarks on our results. The ongoing research program that led to this text has been sponsored by the Advanced Studies Group, Production Engineering Program, COPPE–UFRJ, Rio, Brazil. FAD wishes to thank the Institute for Advanced Studies at the University of São Paulo for partial support of this research project; we wish to acknowledge support from the Brazilian Academy of Philosophy and its chairman Professor J. R. Moderno. Both authors thank Professors R. Bartholo, C. A. Cosenza and S. Fuks for their invitation to join the Fuzzy Sets Lab at COPPE–UFRJ and the Philosophy of Science Program at the same institution. FAD acknowledges partial support from CNPq, Philosophy Section. 12 Doria, Doria References [1] T. Asselmeyer–Maluga and C. H. Brans, Exotic Smoothness and Physics, World Scientific (2007). [2] W. A. Carnielli and F. A. Doria, “Is computer science logic–dependent?” to appear in Festschrift in Honor of Prof. Shahid Rahman (2007). [3] N. C. 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