Notes on Holes. Bryan W. Roberts. November 4, 2009

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1 Notes on Holes Bryan W. Roberts November 4, 2009 Abstract These notes are a brief overview of the debate about holes in relativistic spacetimes, in addressing the question: what counts as a physically admissible spacetime in general relativity? 1 Worry About Nothing Determinism in spacetime theories can be easily wrecked by the presence of holes. Example 1. Take the flat spacetime (R 4,η ab ) with a vertical line deleted a hole. Any trajectory that runs into the hole comes to a stop, disappearing from space and time. And, since our best spacetime theories are time-reversible, the reverse trajectory is possible as well: there are trajectories in which an object pops out of the hole and begins its journey through spacetime, as in Figure 1 (Earman 1995, p.66). Thus, holes evidently entail the rampant violation of determinism. Figure 1: A naked singularity disgorges. It would be nice to have a simple characterization of what counts as a holey spacetime. There are a number of motivations for this. One is militant: one might like to excise such dishonerable spacetimes our list of physically possible models. 1

2 If we could describe a simple, self-respecting law of physics that rules out these spacetimes, then perhaps this would provide a better demarcation of what general relativity is perhaps it might even make the world safe for determinism! But there is also a more modest motivation: we would also (quite simply) like a concise characterization of what holey spacetimes are like. Whatever our motivation, characterizing holes is easier said than done. 2 Attempts to Define Holes 2.1 Extendibility: Global and Local Definition 1. Let (M,g ab ) be a spacetime. We say that (M,g ab ) is extendible if there is an isometry µ : M M which takes (M,g ab ) to a proper subset of another spacetime (M,g ab ). Attempt 1. A spacetime has holes if it is extendible. An extendible spacetime is not a good model of the world. Think of it this way: spacetime points should represent all the possible locations that an object can be in space and time. An extendible spacetime fails to describe all possible locations, because we can attach more points to it. So, Attempt 1 seems like a fair first shot at understanding holes. Unfortunately, it is not enough to restore determinism. Example 2. Deutsch-Politzer Spacetime is constructing by identifying two closed intervals of 2D Minkowski spacetime and deleting the endpoints, as shown in Figure 2. This spacetime is inextendible: the deleted endpoints cannot be cannot be Figure 2: Deutsch-Politzer Spacetime inextendible but not locally inextendible. smoothly put back in (Chamblin, Gibbons, and Steif 1994). So, Attempt 1 fails to excise it from the list of physically possible spacetimes. But the existence of such a spacetime gives rise to a certain kind of indeterminism. Suppose we began on a surface to the past of the handle. The data on this surface fails to determine if we are in plain old Minkowski spacetime, or whacky Deutsch-Politzer spacetime. 2

3 But here s a hopeful observation. Although Deutsch-Politzer spacetime is not extendible, it is does have the following strange property. Take a little neighborhood around one of the deleted points. The closure of this region is not compact because it does not contain all its limit points (in particular, the one we deleted). But the region can be extended to regular Minkowski spacetime, in which its closure is compact. This is a strange local property of Deutsch-Politzer spacetime, which Minkowski spacetime seems (on the face of it) to lack. Such considerations led Hawking and Ellis to the following attempt. Definition 2. Let (M,g ab ) be a spacetime. We say that (M,g ab ) is locally extendible if there exists any neighborhood U M with an extension (U,g ab ) of (U,g ab U ), such thatu has compact closure andu does not. Attempt 2. (Hawking and Ellis 1973) A spacetime has holes if it is locally extendible. In a locally extendible spacetime, the closure of some region is not as big as it could be. In other words, we ve failed to describe all the possible limit points of spacetime. Clearly, this attempt is much in the spirit of the previous one. Unfortunately, this definition turns out to be far too strong. As Beem (1980) showed, even Minkowski spacetime is locally extendible. So you can t excise locally extendible spacetimes without excising good ol Minkowski spacetime as well! Example 3. Here is Beem s construction. Consider a curveγ that begins atx 1 = 2, asymptotically approaches the x 1 = 0 axis in Minkowski spacetime, as shown in Figure 3 below (Beem 1980). Let U be a region surrounding that curve that gets thinner as x 2, so that U always stays above the x 2 axis. The mapping Figure 3: Minkowski spacetime is locally extendible. 3

4 f(x 1,x 2 ) = (x 1,x 2 mod1) from Minkowski spacetime to a cylinder is a locally isometric covering map. This mapping embeds the region U on the cylinder, in such a way that its image f(u) spirals down ever closer to the disc x 1 = 0. The closure of the embeddedf(u) is compact, because it s a subset of0 x 1 2, and closed subsets of compact sets are compact. But closure of the original U is not compact (since it s open toward x 2 = + ). Thus, Minkowski spacetime is locally extendible. 2.2 Hole-Freedom Three years before Beem discovered his counterexample and put local extendibility to shame, Geroch suggested another approach to characterizing holes. Definition 3. Let (M,g ab ) be a spacetime. We say that S M is an achronal set if no two points of S are timelike separated: I + (S) S =. Definition 4. Let S be an achronal surface of a spacetime (M,g ab ). We say that the future domain of dependence of S, written D + (S), is the set of points p M such that every past-inextendible causal curve through p intersects S. Definition 5. Let S be open, achronal surface of a spacetime (M,g ab ). We say that (M,g ab ) is hole free if every isometry φ : S M into another spacetime (M,g ab ) is such thatd+ (φ(s)) = φ(d + (S)). Attempt 3. (Geroch 1977) A spacetime has holes if it is not hole-free. The domain of dependence of a surface describes the locations of certain possible interactions: those which are causal consequences of the initial data on the surface. So, one way to understand a hole-free spacetime is as one in which each space of possible causal interactions is as large as it can be. In this sense, Attempt 3 is in the same spirit as our previous attempts to define holes. Hole-free spacetimes have also been called determinism maximal (Earman 1995, p.98). In particular: if a spacetime is hole free, then every initial data surface fixes a domain of dependence. Example 4. Consider again Deutsch-Politzer spacetime, and let S be an achronal segment that enters the handle, as shown in Figure 4. The domain of dependence of S is indicated by the gray regions. However, this region can be enlarged by isometrically embedding S into Minkowski spacetime via the identity map. Thus, Deutsch-Politzer spacetime is not hole free. So if local extendibility isn t enough to excommunicate Deutsch-Politzer spacetime, one might still hope that the failure of hole-freeness is. Unfortunately, hole-freedom has also has recently been proven too strong: even Minkowski spacetime is not hole-free. Krasnikov showed this in a recent correspondence with Manchak, as an alternative to his example in (Krasnikov 2009) 1. 1 This published result made use of a non-standard definition of D + (S), which (arguably) made the result uninteresting. 4

5 Figure 4: Deutsch-Politzer spacetime is not hole free: the domain of dependence of S can be enlarged by embedding it in Minkowski spacetime. Example 5. Consider any open achronal surface of Minkowski spacetime, and let D + (S) be its domain of dependence. Let φ be an isometric embedding of this surface onto a cylinder. Let us choose φ such that the right-boundary of the embedded surface intersects the left-boundary of the embedded domain of dependence, as shown in Figure 5. Since S is open, a lightlike curve extended backward from a Figure 5: The boundary of the gray region is not in the domain of dependence of the surface S. But it is in the domain of dependence of the embedded surface. boundary point will never intersects. So the points on the boundary of the gray region are not ind + (S). That s not the case for the surface embedded on the cylinder. The black dotted line illustrates how such a lightlike curve will indeed intersect the embedded surface φ(s). Since every past-inextendible causal curve beginning on the right-boundary of the gray area will so intersect φ(s), it follows that this boundary is in the domain of dependence of the embedded surface. So, hole-freedom fails, because we have grown the domain of dependence ofs through an isometric embedding. Nevertheless, it seems one can resist Krasnikov s argument by noticing a strange feature of the embedded surface φ(s): it s not achronal. (Just observe 5

6 that a timelike curve beginning near the lower-left edge of φ(s) can intersect φ(s) again in its future.) So, we can make an obvious and well-motivated repair to the definition of hole-freedom, by requiring that both the initial data surface and the embedded surface be achronal. Unfortunately, there is a counterexample to this definition, too. Example 6. Consider Krasnikov s same embedding but with an open set deleted from the cylindrical spacetime, which goes right up to the boundary of φ(s) but does not include it (see Figure 6 below). Now the embedded surface is achronal. Thus, Minkowski spacetime is not hole-free under the revised definition, either. Figure 6: A slightly modified version of the previous example thwarts the revised definition of hole-freedom as well. The future of hole-freedom is starting to appear pretty tenuous. However, if one is willing to engage in shameless ad-hocery, there are several possible ways to proceed that would avoid the counterexamples. Demand that not just D + (S), but an open neighborhood of the closure of D + (S) be considered in our isometric embedding. Then the second counterexample would be avoided, since the deleted points near the closure of φ(s) prevent us from constructing an isometry. Demand that the closure of S and φ(s) be not just achronal but acausal 2 as well. Since the closure of the embedded surface fails to be acausal, both counterexamples would be avoided. Restrict our attention to the interior IntD + (S), instead of the domain of dependence. Since the interior of the gray region is the same in both Minkowski spacetime and the cylindrical embedding, the counterexample would be avoided. The problem is of course to provide motivation for such restrictions. Whereas we were able to provide some (perhaps) well-motivated reasons for hole-freedom, that justification does not obviously apply here. More interestingly, some of these worries about hole-freedom were anticipated by Manchak and Geroch (Manchak 2009). Manchak constructed a spacetime 2 A set S is acausal if no two points are connected by a causal curve: J + (S) S =. 6

7 that is globally hyperbolic and conformally Minkowski, but fails to be hole-free. This led him and Geroch to suggest the following definition of hole-freedom. Definition 6. (Manchak) A spacetime(m,g ab ) is hole-free* if, whenever there exists an extension of a globally hyperbolic subsetk with Cauchy surfaces, such that in the extension (i)s remains achronal and (ii)k is a proper subset ofintd(s), it follows that M is such an extension. It s similarly unclear what independent motivation can be given for this modification, except to avoid counterexamples like Manchak s construction. However, we can at least rest assured that hole-free*dom will not suffer the same ignominious fate of local inextendibility and hole-freedom: Manchak was able to prove that every inextendible, globally hyperbolic spacetime is hole-free* (Manchak 2009, Proposition 3). As a consequence, Minkowski spacetime is guaranteed to be hole-free*. On the other hand, it is not hard to see that both Deutsch-Politzer spacetime (Example 2), and Minkowski spacetime with a hole punched out (Example 1) fail to be hole-free*. So the prospectus for hole-free*dom appear to be promising. 3 Conclusion There is a intuitive character to all these holey spacetimes, which is that they aren t as large as they could be in one sense or another. In the first attempt (extendibility), our description of possible points wasn t as large as it could be. In the second (local extendibility), our description of possible limit points wasn t as large as it could be. In the third (hole-freedom), our description of possible interactions wasn t as large as it could be. Unfortunately, none of these attempts suffice: the first was far too weak, and the second and third too strong. The bullet-pointed ad-hoc responses to Krasnikov s counter-example above avoid these problems, but lack motivation. In particular, its hard to see how they capture the intuitive character of holey spacetimes, in that they aren t as large as they could be. Manchak s hole-free*dom may be more promising, although its intuitive character isn t clear either. And until these matters are settled, our ability to fruitfully ostracize spacetimes on the basis of hole-freedom is severely (sadly?) limited. References Beem, John K Minkowski space-time is locally extendible. Communications in Mathematical Physics 72 (3): Chamblin, Andrew, G. W. Gibbons, and Alan R. Steif Kinks and time machines. Physical Review D 50 (4): R2353 R2355. Earman, John Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetime. New York: Oxford University Press. 7

8 Geroch, Robert Predictions in General Relativity. In Foundations of Spacetime Theories, edited by Clark Glymour John Earman and John Stachel, Volume VIII of Minnesota Studies in the Philosophy of Science, University of Minnesota Press. Hawking, S. W., and G. F. R. Ellis The large scale structure of space-time. New York: Cambridge University Press. Krasnikov, S Even the Minkowski space is holed. Physical Review D (Particles, Fields, Gravitation, and Cosmology) 79 (12): Manchak, John Byron Is spacetime hole-free? General Relativity and Gravitation 41 (7):