How convincing are the above objections, and what conclusions should we draw from them? In terms of our four stages of identifying mathematical structures for physics (discussed in Sect. 3), to what stage do they belong? As mentioned in Sect. 6, non-Hausdorff manifolds enable one to define all the notions of differential geometry needed to pose the problem of the evolution-distribution of matter, which is enough to pass the first (kinematic) stage. Although the solution is not unique in some cases (as bifurcating geodesics can occur, leading to non-unique trajectories of free test particles), it does not matter at this stage because only the fact that the problem can be formulated matters. Its more specific properties should be considered at a later (i.e., the third) stage. With regard to the dynamical stage, nothing in the Einstein equations prohibits the occurrence of non-Hausdorff manifolds, so they can also pass this stage if the stress-energy tensor \(T_{\mu \nu }\) is appropriately chosen. Therefore, it seems that if non-Hausdorff manifolds were to be excluded, this should be done at the third or fourth stage. In fact, all of the arguments reviewed in Sect. 7 are associated with the third stage. Concerning the fourth stage, very little can be said. Currently, no successful usage of non-Hausdorff manifolds to represent some physical situation is known, and proposing such a model is beyond the scope of this paper.
The aim of this section is to discuss whether the presented objections are strong enough to lead to the judgement that non-Hausdorff manifolds do not pass the third stage. This is particularly difficult, as the criteria here are not clear: there are different ideas of what counts as physical reasonability conditions, and satisfying some of them does not imply that the others are satisfied as well, so they should be considered case by case. Therefore, the following analysis should be regarded as rather provisional.
Ad (i): First, observe that the two properties Earman mentions are not equivalent to the Hausdorff property, but only implied by it, so it is not the case that all non-Hausdorff spacetimes violate them. Second, and more importantly, as was suggested in Sect. 6, it is not clear whether these mathematical facts have a physical significance on their own. If they are important because they are connected with some more physical features of non-Hausdorff manifolds, such as the presence of bifurcating geodesics, then these physical features, and not the mathematical facts mentioned by Earman, form the proper basis of the argument against non-Hausdorff manifolds.
An alternative reading of Earman’s quote could be that the crucial thing is not the particular mathematical theorems mentioned by him, but a general fact (of a rather practical nature) that all the available literature assumes the Hausdorff condition and, therefore, we would be unable to recover the results that are important for scientific practice if we dropped it. However, this is not the case—although most of the literature on the subject indeed assumes the Hausdorff condition, not all textbooks do this. For example, as previously mentioned, [12] introduces all the basic notions of differential geometry without assuming the Hausdorff condition and starts using it substantially no earlier than in chapter 6 of his book.
Before we move on to the next objection, we should recall the exact relationship between non-Hausdorff manifolds and bifurcating geodesics. As Theorem 2 states, some non-Hausdorff manifolds do not admit bifurcating curves of the second kind (some of which may be bifurcating geodesics). Interestingly, most of the non-Hausdorff manifolds considered in the existing literature are of this type, for example, extensions of Misner spacetime and extensions of Taub-NUT spacetime (see [13, 14, 24, 25]). Therefore, some physicists do consider liberalizations of the Hausdorff condition. For example, Hawking [25, p. 174] allow for these non-Hausdorff manifolds that do not admit bifurcating geodesics. Similarly, Geroch [26, p. 465] allows for non-Hausdorff manifolds in which every geodesic has a unique extension and every curve has no more than one end point.
As a consequence, our initial question of whether non-Hausdorff manifolds are physically reasonable splits into two: (1) are non-Hausdorff manifolds with bifurcating geodesics physically reasonable (which is addressed in the objections (ii)-(v)) and (2) are non-Hausdorff manifolds without bifurcating geodesics, which, according to Theorem 4, violate strong causality, physically reasonable (which is addressed in the objection (vi))?
Ad (ii)–(iii): The main problem that arises from the violation of the Hausdorff condition is indeterminism, both at the level of geodesics (i.e., curves followed by free test particles) and at the level of the entire manifold (i.e., non-Hausdorffly attached ‘branches’). However, it is far from obvious that the fact that a theory involves indeterminism should be considered as an argument against it. For this argument to be valid, the authors should provide some independent argument that indeterminism is not allowed in physical theories (which seems to be too strong a position, as physicists in general allow indeterministic theories) or that the specific form of indeterminism that occurs in the case under consideration is unacceptable. Without such further support, this argument comes down to a simple rejection of indeterminism.
One way of supplementing this argument is to say that indeterminism is problematic because it is a kind of arbitrariness and therefore should be avoided in science. The issue of whether some arbitrariness exists in nature (and if so, what types of arbitrariness are allowed) is interesting but somewhat speculative.Footnote 10 It seems that in our explanation of physical facts, we need to stop at some point anyway, which means that some arbitrary factor needs to be postulated. It can be either the value of some physical constant or the form of the equation; or, if we managed to derive both equations and constants, the arbitrary factor would be the assumption from which they are derived. This is arbitrariness at the level of general facts about our world (laws, constants, etc.). What about the level of particular facts (i.e., why this particular course of events have been realised from among the bigger set of courses of events allowed by what we called ‘general facts’)? Can we a priori exclude that something unexplainable is already at the level of particular facts as indeterminism postulates? Do we have reasons to insist that all the arbitrariness arises at the general level, that is, at the level of the structures that explain particular facts, such as laws and physical constants? These are difficult questions, but they have been intentionally posed in a way to suggest that there are no obvious answers to them. And such answers, if attainable at all, should not be found a priori, but on the basis of considering our best empirical theories; if some of them are best interpreted as indeterministic, we are allowed to ascribe (tentatively) indeterminism to the physical reality itself. The argument from indeterminism was presented by Earman and Hájíček in a way suggesting that the answer is known a priori or at least obvious on the basis of the empirical knowledge we already have, but this is clearly not the case.
How should bifurcating geodesics or new branches attached to a manifold be conceptualized? An interesting interpretative option is that a basic object of GR does not necessarily represent a single spacetime. In the case of non-Hausdorff manifolds, another interpretation, which can be called ‘modal’, is available, namely that they represent bundles of alternative possible spacetimes, only one of which is actual. This interpretation can be supported by the following theorem (for the proof, see [15]):
Theorem 5
Any non-Hausdorff manifold can be constructed by gluing together a family of Hausdorff manifolds.
The modal interpretation, based on the above theorem, goes as follows. Any non-Hausdorff manifold can be decomposed into a family of Hausdorff submanifoldsFootnote 11 (from which it can be obtained by gluing maps, see definition 6), and each of these Hausdorff manifolds represents a single possible spacetime.Footnote 12 A non-Hausdorff manifold represents all the possibilities, taken together, but exactly one of them actualises. Things look similar at the level of geodesics. A bifurcation of geodesics requires non-Hausdorffness to occur, which means that each branch of a geodesic can be associated with a single spacetime, and bifurcating geodesics as a whole can be interpreted as representing the possible trajectories of free test particles, exactly one of which actualises.Footnote 13 This is why the notion of the basic object of GR instead of just spacetime has been used throughout the text; according to the modal interpretation, a single spacetime is not the only object that can be represented by a manifold.
One may worry that such an interpretation leads us too far away from the original theory: if we include the subject matter of a theory in its identity conditions and assume that the subject matter of GR is spacetime, then the above change of the subject matter would amount to changing the theory into a new one. To avoid the mentioned assumption, I introduced in Sect. 2 the notion of the basic object of GR, allowing in principle that it can be different from spacetime. In Sect. 4, a functional restriction was placed on it, namely that it should play a certain role with respect to matter; and in Sect. 5 it was stated that non-Hausdorff manifolds can play this role. ‘Can play’ means ‘have enough mathematical structure’ to do so; whether they actually do can depend on the physical interpretation, but surely the modal interpretation does not prevent them from doing so. Another way to convince ourselves that this is still the same theory is to define its subject matter more ‘locally’, namely, instead of saying that GR is about spacetime (or about a basic object of GR as defined earlier), one can say that it describes spatiotemporal states of affairs (and every model of GR describes many such states). Whether a given state of affairs is spatiotemporal does not depend on whether it is actual, so merely possible spatiotemporal states of affairs are still subjects of GR; and under the modal interpretation, non-Hausdorff manifolds describe (actual and possible) spatiotemporal states of affairs no less than Hausdorff manifolds, interpreted as single spacetimes, do.
Ad (iv): Earman’s quoted argument against extendible manifolds relies on reluctance to accept arbitrariness, which is the issue addressed while discussing objections (ii)–(iii). However, two further points are to be made concerning this particular objection. First, some of the formal results suggest that in some cases, extendible manifolds should be preferable over inextendible ones, as is summarized by Manchak [5, pp. 415–416], who says:
But, however compelling the metaphysics, it is sometimes problematic to insist on inextendibility. For example, [16, p. 20] has shown that not every well-behaved space-time admits a well-behaved inextendible extension. Should we cling to inextendibility at the expense of other desirable space-time properties? The answer is far from clear. Additionally, a space-time does not always have just one inextendible extension [21, p. 9]. Thus, the principle of sufficient reason can actually be used to argue against the property of inextendibility. After all, why should one extension be preferred over another?
Therefore, even if we accept the type of reasoning based on the principle of sufficient reason, it is not clear how it should work in the case of extendible manifolds. In some cases, it is more important not to lose some other properties that our model of spacetime is expected to have than to save inextendibility; and in some other cases, more than one extension is available, and the choice between them is arbitrary, unless one of them is distinguished by special properties the others lack.
The second point I want to make is specific to the context of the modal interpretation of non-Hausdorff manifolds. Adding new branches (at the level of mathematical construction) to the already constructed manifold does not amount to enlarging the model of the actual spacetime, as only one of these branches, at most, is actualised under the modal interpretation. Therefore, what is extended is the structure of possible alternatives for a spacetime to evolve, not the spacetime itself. To what extent does this alleviate Earman’s worry? This worry can be formulated in two ways: one in terms of the ‘Creative Force’ and the other in terms of the existence of sufficient reason, but both formulations concern only the actualised part of the manifold; namely, they state in different ways the question, ‘why has this particular spacetime been actualised and not a larger one?’ Therefore, they ask about a single spacetime (the one that is actualised), not about the whole bunch of spacetimes. And for the actualised part, there is a good answer why it should not be extended to the whole non-Hausdorff structure: because we identified a single spacetime with a Hausdorff submanifold.
Ad (v): Another objection posed by Earman, concerning energy, turns out to be harmless if we carefully interpret branching structures as representing possible evolutions, where one branch at most can be actualised. In the actual world, the energy tensor is wholly contained in one branch, and the discontinuity concerns only branches that are not realised. Therefore, if the continuity condition is violated at all, it is not violated by any actual physical object, only by non-actual ones. What about a counterfactual situation in which a different branch had been actualised? Then energy would follow that other branch which is actualised in that counterfactual situation. There would never be a problem of discontinuity in the actual physical reality. As we do not need to postulate that energy follows all the branches, the problem with the conservation of energy does not arise.Footnote 14
Ad (vi): As already mentioned in the exposition of this objection, it is not the violation of strong causality that is problematic, but the violation of a weaker condition, called the chronology condition; strong causality is perceived as a desired feature because of its close relation with the chronology condition. According to the quote by Hawking, we should disregard manifolds that are not strongly causal because they are, is some sense, very close to manifolds that contain closed timelike loops.
Under which assumptions should this closeness be regarded as problematic? Hawking appeals to the idea of ‘nature judging accurately’, which seems to be a kind of fine-tuning, and to quantum fluctuations. We have to deal with fine-tuning if, in a space of possible models, the model that is supposed to be the actual one is very close to other models that are radically different, where ‘closeness’ can be measured, for example, in terms of values of some physical parameter with respect to which the models differ. This situation is regarded as problematic because it makes some important characteristics of the actual model improbable; to get the actual model, the values of the appropriate parameters should be ‘fine-tuned’. This type of reasoning is often invoked in the context of particle physics.Footnote 15 In our context, it is not clear which parameter would be fine-tuned, but it is clear that almost-closed timelike curves are, in some sense, ‘close’ to closed timelike curves.
A typical objection to fine-tuning arguments is that they require an assumption about a probability measure on the space of possible models, and there is no way to infer such a measure from what is observed. Furthermore, if the choice of parameters is a single, unrepeatable ‘event’, then the whole idea of describing it probabilistically is suspicious. Perhaps the origin of a probability measure, in our context, should be from quantum theory as the invoking of quantum fluctuations by Hawking suggests. If we allowed for manifolds violating strong causality—the reasoning can go—then we would also allow for all the manifolds that are obtainable from them by quantum fluctuations, and manifolds that violate chronology are probably among them. But this ‘probably’ is supported by intuition rather than calculation; one cannot exclude that quantum laws prevent somehow quantum fluctuations from entering the chronology-violating region (in the space of models) even if one starts from manifolds violating strong causality.
This ananlysis is clearly inconclusive, but at least two messages can be drawn from it. First, the violation of strong causality alone does not lead to the interpretative problems connected with causal loops. Second, starting from the chronology condition, we need some quite substantial additional assumptions to justify strong causality as a physical reasonability condition.
Ad (vii): Penrose’s arguments do not concern non-Hausdorff manifolds as such, but only one possible way of using them in physics. The first argument concerns the fact that no detailed model connecting non-Hausdorff manifolds with the known forms of indeterminism has been built, but the possibility of such a model is not excluded by Penrose. It is also not obvious that the only motivation for using non-Hausdorff manifolds can come from the Everettian interpretation of Quantum Mechanics, as Penrose seems to implicitly assume; other indeterministic interpretations of Quantum Mechanics can provide an equally good motivation, and of course, one does not need to restrict to Quantum Mechanics (although it is currently the most serious candidate for a theory capturing genuinely indeterministic phenomena).
Regarding the second argument, solving the issue of the direction of time does not need to be a role of non-Hausdorff manifolds. The question under investigation is whether they may form a basis for physical models of the actual world (or at least of some physically possible world), not what further problems can be solved by using them. The idea that some relationship between indeterminism and the direction of time exists is attractive because indeterminism is a temporally asymmetric phenomenon: there is only one way for the past to be (namely, the actual one), but there is more than one way for the future to be (as it is not unambiguously determined). However, it is not yet clear whether the two asymmetries are indeed the same or even deeply related, and more popular discussions of the direction of time do not refer to indeterminism at all (see, e.g., [37, part II]). However, attempts have been made to try to relate the asymmetry of past and future with the asymmetry involved in indeterminism (see, e.g., [38, 39]), so if they are accurate and if the correct representation of indeterminism involves non-Hausdorff manifolds, Penrose’s claim may turn out to be false.
Penrose’s third objection can be decomposed into three steps: (1) the only motivation for using non-Hausdorff manifolds in physics is the Everettian interpretation of quantum mechanics; (2) this interpretation commits us to paradoxical theses about minds and their relationship to the world; from (1) and (2), it follows that (3) non-Hausdorff manifolds have no place in physics. We have already seen that the first premise is highly contestable. And the second premise is clearly false: the Everettian interpretation should not be read as assuming anything about the nature of mind and consciousness—it is not the same as the ‘many-minds’ interpretation. Instead, it should be called the many-worlds interpretation or the emergent multiverse interpretation, as in Wallace’s [40] book. Therefore, the argument is more about potential problems for the many-minds interpretation of Quantum Mechanics and does not threaten non-Hausdorff manifolds as such, as using models based on non-Hausdorff manifolds does not commit us in any way to any hypotheses about the relationship between the physical world and our minds.