On the Inextendibility of Spacetime∗ John Byron Manchak Abstract It has been argued that spacetime must be inextendible – that it must be “as large as it can be” in some sense. Here, we register some skepticism with respect to this position. 1 Introduction A spacetime is counted as inextendible if, intuitively, it is “as large as it can be”. It has been argued that inextendibility is a “reasonable physical condition to be imposed on models of the universe” (Geroch 1970, 262) and that a spacetime must be inextendible if it is “to be a serious candidate for describing actuality” (Earman 1995, 32). Here, in a variety of ways, we register some skepticism with respect to such positions. 2 Preliminaries We begin with a few preliminaries concerning the relevant background formalism of general relativity.1 An n-dimensional, relativistic spacetime (for n ≥ 2) is a pair of mathematical objects (M,gab) where M is a connected n-dimensional Hausdorff manifold (without boundary) that is smooth and gab is a smooth, non-degenerate, pseudo-Riemannian metric of Lorentz signature (−, +, ..., +) defined on M. We say two spacetimes (M,gab) and (M ′,g′ab) are isometric if there is a diffeomorphism ϕ : M → M′ such that ϕ∗gab = g ′ ab. Two spacetimes (M,gab) and (M ′,g′ab) are locally isometric if, for each point p ∈ M, there is an open neighborhood O of p and an open subset O′ of M′ such that O and O′ are isometric, and, correspondingly, with the roles of (M,gab) and (M ′,g′ab) interchanged. A spacetime (M,gab) is extendible if there exists a spacetime (M,gab) and a proper isometric embedding ϕ : M → M′. Here, the spacetime (M′,g′ab) is an extension of (M,gab). A spacetime is inextendible if it has no extension. A P-spacetime is a spacetime with property P. A P- spacetime (M′,g′ab) is a P-extension of a P-spacetime (M,gab) if (M ′,g′ab) ∗Thanks to Thomas Barrett, Ben Feintzeig, David Malament, Jim Weatherall, and Chris Wüthrich for help with earlier versions of this document. Special thanks go to David Malament for providing a sketch of the Lemma 2 proof. 1The reader is encouraged to consult Hawking and Ellis (1973), Wald (1984), and Malament (2012) for details. An outstanding (and less technical) survey of the global structure of spacetime is given by Geroch and Horowitz (1979). 1 is an extension of (M,gab). A P-spacetime is P-extendible if it has a P- extension and is P-inextendible otherwise. We say (M,gab,F) is an n- dimensional framed spacetime if (M,gab) is an n-dimensional spacetime and F is an orthonormal n-ad of vectors {ξ1, ...,ξn} at some point p ∈ M. We say an n-dimensional framed spacetime (M′,g′ab,F ′) is a framed extension of the n-dimensional framed spacetime (M,gab,F) if there is a proper isometric embedding ϕ : M → M′ which takes F into F ′. For each point p ∈ M, the metric assigns a cone structure to the tan- gent space Mp. Any tangent vector ξ a in Mp will be timelike if gabξ aξb < 0, null if gabξ aξb = 0, or spacelike if gabξ aξb > 0. Null vectors create the cone structure; timelike vectors are inside the cone while spacelike vectors are outside. A time orientable spacetime is one that has a continuous timelike vector field on M. In what follows, it is assumed that spacetimes are time orientable. For some connected interval I ⊆ R, a smooth curve γ : I → M is timelike if the tangent vector ξa at each point in γ[I] is timelike. Similarly, a curve is null (respectively, spacelike) if its tangent vector at each point is null (respectively, spacelike). A curve is causal if its tangent vector at each point is either null or timelike. A causal curve is future directed if its tangent vector at each point falls in or on the future lobe of the light cone. We say a timelike curve γ : [s,s′] → M is closed if γ(s) = γ(s′). A spacetime (M,gab) satisfies chronology if it does not contain a closed timelike curve. For any two points p,q ∈ M, we write p << q if there exists a future-directed timelike curve from p to q. This relation allows us to define the timelike past of a point p: I−(p) = {q : q << p}. We say a spacetime (M,gab) satisfies past distinguishability if there do not exist distinct points p,q ∈ M such that I−(p) = I−(q). We say a set S ⊂ M is achronal if there do not exist p,q ∈ S such that p ∈ I−(q). An extension of a curve γ : I → M is a curve γ′ : I′ → M such that I is a proper subset of I′ and γ(s) = γ′(s) for all s ∈ I. A curve is maximal if it has no extension. A curve γ : I → M in a spacetime (M,gab) is a geodesic if ξa∇aξb = 0 where ξa is the tangent vector and ∇a is the unique derivative operator compatible with gab. A point p ∈ M is a future endpoint of a future directed causal curve γ : I → M if, for every neighborhood O of p, there exists a point t0 ∈ I such that γ(t) ∈ O for all t > t0. A past endpoint is defined similarly. A causal curve is inextendible if it has no future or past endpoint. A causal geodesic γ : I → M in a spacetime (M,g) is past incomplete if it is maximal and there is an r ∈ R such that r < s for all s ∈ I. For any set S ⊆ M, we define the past domain of dependence of S, written D−(S), to be the set of points p ∈ M such that every causal curve with past endpoint p and no future endpoint intersects S. The future domain of dependence of S, written D+(S), is defined analogously. The entire domain of dependence of S, written D(S), is just the set D−(S) ∪ D+(S). The edge of an achronal set S ⊂ M is the collection of points p ∈ S such that every open neighborhood O of p contains a point q ∈ I+(p), a point r ∈ I−(p), and a timelike curve from r to q which does not intersect S. A set S ⊂ M is a slice if it is closed, achronal, and without edge. A spacetime (M,gab) which contains a slice S such that D(S) = M is said to be globally hyperbolic. 2 Given a spacetime (M,gab), let Tab be defined by 1 8π (Rab − 12Rgab) where Rab is the Ricci tensor and R the scalar curvature associated with gab. We say that (M,gab) satisfies the weak energy condition if, for each timelike vector ξa, we have Tabξ aξb ≥ 0. We say that (M,gab) is a vacuum solution if Tab = 0. Let S be a set. A relation ≤ on S is a partial order if, for all a,b,c ∈ S: (i) a ≤ a, (ii) if a ≤ b and b ≤ c, then a ≤ c, and (iii) if a ≤ b and b ≤ a, then a = b. If ≤ is a partial ordering on a set S, we say a subset T ⊆ S is totally ordered if, for all a,b ∈ T, either a ≤ b or b ≤ a. Let ≤ be a partial ordering on S and let T ⊆ S be totally ordered. An upper bound for T is an element u ∈ S such that for all a ∈ T, a ≤ u. A maximal element of S is an element m ∈ S such that for all c ∈ S, if m ≤ c, then c = m. Zorn’s lemma (which is equivalent to the axiom of choice) is the following: Let ≤ be a partial order on S. If each totally ordered subset T ⊆ S has an upper bound, there is a maximal element of S. 3 Definition Recall the standard definition of spacetime inextendibility. Definition. A spacetime (M,gab) is inextendible if there does not ex- ist a spacetime (M′,g′ab) such that there is a proper isometric embedding ϕ : M → M′. The definition requires that an inextendible spacetime be “as large as it can be” in the sense that one compares it to a background class of all “possible” spacetimes. Standardly, one takes this class to be the set of all (smooth, Hausdorff) Lorentzian manifolds as defined in the previous section. But what should this class be? The answer is unclear. Consider Misner spacetime (Hawking and Ellis 1973). Let Misner* be the globally hyperbolic “bottom half” of Misner spacetime. By the standard definition of inextendibility, Misner* is extendible and cannot be extended and remain globally hyperbolic (see below). But suppose that a version of the cosmic censorship conjecture is correct and all physi- cally reasonable spacetimes are globally hyperbolic (Penrose 1979). Then shouldn’t Misner* be considered “as large as it can be”? It follows that whether or not Misner* spacetime should count as inextendible depends crucially on the outcome of this version of the cosmic censorship conjec- ture – a conjecture which is far from settled (Earman 1995, Penrose 1999) and perhaps may never be settled (Manchak, 2011). Because of examples like these, one is tempted to revise the definition of inextendibility.2 But Geroch (1970, 278) has argued that a revision is less urgent if one can show, for a variety of physically reasonable proper- ties P, that the following is true. (*) Every P-inextendible P-spacetime is inextendible. 2See Manchak (2016a) for an extended discussion. 3 The significance of (*) is this: If a property P satisfies (*), then any P-spacetime is inextendible if and only if it is P-inextendible. Effectively, it makes no difference in such cases whether one defines inextendibility relative to the standard class of all spacetimes or a revised class of all P-spacetimes. Accordingly, one would like to investigate (*) with respect to a variety of properties P. Already from the Misner* example above, we have the following (a proof is provided in the Appendix). Proposition 1. If P is global hyperbolicity, (*) is false. Are there physically reasonable properties P which render (*) true? Geroch (1970, 289) has suggested a number of good candidates including: being a vacuum solution, satisfying chronology, and satisfying an energy condition. The first two cases are still open. Here, we settle the case where P is the weak energy condition (a proof is provided in the Appendix). Proposition 2. If P is the weak energy condition, (*) is false. We see that the prospect of avoiding the need to revise to the definition of inextendibility does not look good. In the meantime, we may conclude that it is not yet clear that the standard definition captures the intuitive idea that an inextendible spacetime is “as large as it can be”. 4 Metaphysics A number of experts in general relativity (Penrose 1969, Geroch 1970, Clarke 1976), seem to be committed to the following. “Metaphysical considerations suggest that to be a serious candidate for describing actuality, a spacetime should be [in- extendible]. For example, for the Creative Force to actualize a proper subpart of a larger spacetime would seem to be a viola- tion of Leibniz’s principles of sufficient reason and plenitude. If one adopts the image of spacetime as being generated or built up as time passes then the dynamical version of the principle of sufficient reason would ask why the Creative Force would stop building if it is possible to continue” (Earman 1995, 32). These metaphysical views are underpinned by an important result due to Geroch (1970). Proposition 3. Every extendible spacetime has an inextendible ex- tension. The result (which makes use of Zorn’s lemma) seems to show that the Creative Force can always build spacetime until it is no longer possi- ble to build. But of course, this interpretation presupposes that we have been working with the proper definition of inextendibility. And as we have noted, it is not yet clear that we are. Accordingly, one would like to know, for a variety of physically reasonable properties P, whether the 4 following version of the Geroch (1970) result is true. (**) Every P-extendible P-spacetime has a P-inextendible P-extension. With a bit of work (and Zorn’s lemma), one can show the following (a proof is provided in the Appendix). Proposition 4. If P is chronology, (**) is true. We see that if we revise the definition of inextendibility to be relative to the class of all chronological spacetimes (rather than the standard class of all spacetimes), we have an analogue of the Geroch (1970) result. This is certainly good news for those committed to the metaphysical views expressed above. But are there physically reasonable properties P which render (**) false? There are. Of course, spacetime properties may be considered physically reason- able in various senses. Let us conservatively restrict attention a property usually taken to be satisfied by models of our own universe: the property of having every inextendible timelike geodesic be past incomplete. Let us call this the big bang property given that it is satisfied by all of the standard “big bang” cosmological models. We are now in a position to state the following (Manchak 2016b). Proposition 5. If P is the big bang property, (**) is false. We see that if we revise the definition of inextendibility to be relative to the class of all spacetimes with the big bang property (rather than the standard class of all spacetimes), we do not have an analogue of the Geroch (1970) result. It is not yet clear that the Creative Force always has the option of building spacetime to be “as large as it can be”. 5 Epistemology What observational evidence is there (or could there be) in support of the position that spacetime is “as large as it can be”? Following Malament (1977), let us say that a spacetime (M,gab) is observationally indistin- guishable from another spacetime (M′,g′ab) if, for every point p ∈ M, there is a point p′ ∈ M′ such that I−(p) and I−(p′) are isometric. One can show the following (Manchak 2011). Proposition 6. Every chronological spacetime is observationally in- distinguishable from some other (non-isometric) spacetime which is ex- tendible. Under the standard definition of inextendibility, it seems that any ob- server in a chronological spacetime is not in a position to know that her spacetime is “as large as it can be”. But this interpretation presupposes that we have been working with the proper definition of inextendibility. And as we have noted, it is not yet clear that we are. Accordingly, one 5 would like to know, for a variety of physically reasonable properties P, whether the following version of the Manchak (2011) result is true. (***) Every chronological P-spacetime is observationally indistinguish- able from some other (non-isometric) P-spacetime which is P-extendible. It turns out that a large class of physically reasonable properties ren- der (***) true. Following Manchak (2011), let us say that a property P on a spacetime is local if, given any two locally isometric spacetimes (M,gab) and (M ′,g′ab), (M,gab) has P if and only if (M ′,g′ab) has P. Lo- cal properties include being a vacuum solution and satisfying the weak energy condition. We are now in a position to state the following (a proof is provided in the Appendix). Proposition 7. If P is a local property, (***) is true. We see that, whenever P is a local property, if we revise the definition of inextendibility to be relative to the class of all P-spacetimes (rather than the standard class of all spacetimes), we have an analogue of the Manchak (2011) result. It is not yet clear that we can ever have observa- tional evidence that spacetime is “as large as it can be”. 6 Conclusion In the preceding, we have registered some skepticism with respect to the position that spacetime must be inextendible – that it must be “as large as it can be” in some sense. We have done this in a variety of ways. First we have shown that it is not yet clear that the standard definition of inextendibility captures the intuitive idea that an inextendible spacetime is “as large as it can be”. Second we have shown, by exploring some plausible revisions to the definition of inextendibility, that it is not yet clear that a spacetime can always be extended to be “as large as it can be”. Finally we have shown, by exploring a class of plausible revisions to the definition of inextendibility, that it is not yet clear that we can ever know that spacetime is “as large as it can be”. 7 Appendix Proposition 1. If P is global hyperbolicity, (*) is false. Proof. Let (N,gab) be Misner spacetime. So, N = R × S and gab = 2∇(at∇b)ϕ − t∇aϕ∇bϕ where the points (t,ϕ) are identified with the points (t,ϕ + 2πn) for all integers n. Now, let M = {(t,ϕ) ∈ N : t < 0} and consider the spacetime (M,gab). Clearly, it is extendible. It is also globally hyperbolic since the slice S = {(t,ϕ) ∈ M : t = −1} is such that D(S) = M. We need only show that any extension to (M,gab) fails to be globally hyperbolic. 6 Let (M′,g′ab) be any extension of (M,gab) and let p be a point in ∂M ∩M′. In any neighborhood of p, there will be a point q ∈ ∂M ∩M′ such that q 6= p. One can verify that I−(p) = M = I−(q). Thus, (M′,g′ab) is not past distinguishing and therefore not globally hyperbolic (Hawking and Ellis 1973). � Proposition 2. If P is the weak energy condition, (*) is false. Proof. Consider Minkowski spacetime (R4,ηab) in standard (t,x,y,z) coordinates where ηab = −∇at∇bt + ∇ax∇bx + ∇ay∇by + ∇az∇bz. Let Ω : R4 → R be the function defined by Ω(t,x,y,z) =exp(t3). Consider the spacetime (R4,gab) where gab = Ω2ηab. Associated with gab we have (Wald 1984, 446): Rab = −2∇a∇bt3 −ηabηcd∇c∇dt3 + 2(∇at3)(∇bt3) −2ηabηcd(∇ct3)(∇dt3) R = 1 Ω2 [−6ηab∇a∇bt3 − 6ηab(∇at3)(∇bt3)]. We note that ∇at3 = 3t2∇at and ∇a∇bt3 = 6t(∇at)(∇bt). Of course, ηab(∇at)(∇bt) = −1. Simplifying, we have: Rab = (18t 4 − 12t)(∇at)(∇bt) + (18t4 + 6t)ηab R = 1 Ω2 [36t + 54t 4 ]. Einstein’s equation Rab − 12Rgab = 8πTab requires that: Tab = 1 8π [(18t 4 − 12t)(∇at)(∇bt) − (9t4 + 12t)ηab] In (R4,gab), consider any timelike vector ξa = k0(∂/∂t)a+k1(∂/∂x)a+ k2(∂/∂y) a +k3(∂/∂z) a where k0,k1,k2,k3 ∈ R and k20 > k21 +k22 +k23. We have: Tabξ a ξ b = 1 8π [(18t 4 − 12t)k20 + (9t 4 + 12t)(k 2 0 −k 2 1 −k 2 2 −k 2 3)] = 1 8π [t 4 (27k 2 0 − 9(k 2 1 + k 2 2 + k 2 3)) − 12t(k 2 1 + k 2 2 + k 2 3)] Because k20 > k 2 1 +k 2 2 +k 2 3, we know t 4(27k20−9(k21 +k22 +k23)) ≥ 0. And for t ≤ 0, we know −12t(k21 + k22 + k23) ≥ 0. It follows that Tabξaξb ≥ 0 for t ≤ 0. Let M = {(t,x,y,z) ∈ R4 : t < 0}. We have shown that the spacetime (M,gab|M ) is such that it satisfies the weak energy condition and is ex- tendible. It remains for us to show that any extension to (M,gab|M ) fails to satisfy the weak energy condition. 7 Let (M′,g′ab) be any extension to (M,gab|M ). Let p be a point in ∂M ∩M′. Let (O,ϕ) be a chart with p ∈ O such that we can extend the coordinates (t,x,y,z) on M to M ∪O ⊂ M′. So, for some p1,p2,p3 ∈ R we have p = (0,p1,p2,p3). Find some δ > 0 such that (δ,p1,p2,p3) ∈ O. For t ∈ (−δ,δ), let p(t) = (t,p1,p2,p3) ∈ M′. Consider the smooth function f : (−δ,δ) → R given by f(t) = g′abζ aζb|p(t) where ζa = √ 2(∂/∂t)a + (∂/∂x)a. Of course, for all t < 0, we have: g ′ abζ a ζ b = gabζ a ζ b = Ω 2 ηabζ a ζ b = −Ω2 = −exp(2t3) Smoothness requires that f(0) = −1. This allows us to find an � ∈ (0,δ) such that f(t) < 0 for all t ∈ (−�,�). So ζa is timelike at p(t) for all t ∈ (−�,�). Consider the smooth function g : (−�,�) → R given by g(t) = T ′abζ aζb|p(t) where T ′ab is defined on M ′ in the natural way (using the metric g′ab and Einstein’s equation). Of course, for all t < 0, we have: T ′ ab = Tab = 1 8π [(18t 4 − 12t)(∇at)(∇bt) − (9t4 + 12t)ηab] Because ∇at∇btζaζb = 2 and ηabζaζb = −1, we have for all t < 0: T ′ abζ a ζ b = Tabζ a ζ b = 1 8π [(36t 4 − 24t) + (9t4 + 12t)] = 1 8π [45t 4 − 12t] Smoothness requires that g(0) = 0 and d dt g(0) = − 3 2π . This allows us to find a γ ∈ (0,�) such that g(t) < 0 for t ∈ (0,γ). Thus, the weak energy condition is violated at p(t) for all t ∈ (0,γ). � Definition. Let F denote the set of framed spacetimes. Let ≤ denote the relation on F such that (M,gab,F) ≤ (M′,g′ab,F ′) iff (M′,g′ab,F ′) is a framed extension of (M,gab,F). Lemma 1. The relation ≤ is a partial ordering on F. Proof. See Geroch (1969, 188-189). � Lemma 2. Let C denote the set of framed spacetimes which satisfiy chronology. C is partially ordered by ≤. Every subset T ⊂ C which is totally ordered by ≤ has an upper bound in C. Proof. Since C ⊂ F, it follows from Lemma 1 that C is partially ordered by ≤. Let T = {(Mi,gi,Fi)} be a subset of C which is totally ordered by ≤. Following Hawking and Ellis (1973, 249), let M be the union of all the Mi where, for (Mi,gi,Fi) ≤ (Mj,gj,Fj), each pi ∈ Mi is identified with ϕij(pi) where ϕij : Mi → Mj is the unique isometric embedding which takes Fi into Fj. The manifold M will have an induced metric g equal to ϕi∗gi on each ϕi[Mi] where ϕi : Mi → M is the natural isometric 8 embedding. Finally, take F to be the result of carrying along a chosen Fi using ϕi : Mi → M. Consider the framed spacetime (M,g,F). We claim it is an upper bound for T. Clearly, for all i, we have (Mi,gi,Fi) ≤ (M,g,F). We need only show that (M,g,F) ∈ C. Suppose (M,g,F) /∈ C and let γ ⊂ M be (the image of) a closed time- like curve. As a topological space (with induced topology from M), γ is compact. For all i, let γi be γ ∩ Mi. So, A = {γi} is an open cover of γ. By compactness, there must be a finite subset A′ ⊂ A which is also a cover of γ. One can use the relation ≤ on T to order the finite number of elements in A′ into a nested sequence of subsets γj ⊆ ... ⊆ γk. It follows that γk = γ. So, (Mk,gk,Fk) /∈ C: a contradiction. � Proposition 4. If P is chronology, (**) is true. Proof. Let P be chronology and let (M,gab) be a P-spacetime which is P-extendible. Let F be an orthonormal n-ad at some point p ∈ M. So, (M,gab,F) ∈ C where C is the set of framed spacetimes which satisfiy chronology. 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