Time Order, Direction, and the Presentist’s View on

Cord Friebe

Abstract The physical possibility of containing closed timelike curves (CTCs) challenges the of time in the way that temporal ordering is, at best, remarkably non-standard: events on CTCs precede themselves. Apparently, such universes do not possess a consistent time order but only a consistent time direction. Thus, temporal directionality seems to be more fundamental than ordering in earlier-later or --. I argue that this favors presentism as the adequate of spacetimes: only presentism consistently copes with the idea that temporal ordering depends on empirical constraints. The presentist Now is fundamentally undivided and productively directed towards . Time order arises by extending the Now, which can fail and in fact fails in universes containing CTCs.

Keywords: general relativity, philosophy of time, presentism

1 Introduction

In 1949, Kurt G¨odelfamously presented a solution of Einstein’s generally relativistic field equations that permits the occurrence of closed timelike curves (CTCs).1 According to G¨odelhimself, the possibility of CTCs confirms ‘the’ idealist philosophy of time, namely what he takes to be the view that time is unreal, i.e., that the objective world is timeless: [I]t seems that one obtains an unequivocal proof for the view of those philosophers who, like Parmenides, Kant, and the modern idealists, deny the of change and consider change as an illusion or an appearance due to our special mode of perception. ([6], 202)

Kriterion – Journal of Philosophy, 2016, 30(2): 91–106. http://www.kriterion-journal-of-philosophy.org c 2016 The author 92 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106

Consequently, the drawn above as to the non-ob- jectivity of change doubtless applies at least in these worlds [containing CTCs] [...] strengthening further the idealistic viewpoint. ([6], 205)

[N]o can [hence] be given why an objective lapse of time should be assumed at all. ([6], 206)

Nowadays, philosophers of physics disagree. They hold that even - containing CTCs have a temporal dimension (see, e.g., [4]; [5]; [9]). Thus, they are realists about time, at least as long as spacetime is ‘time orientable’. I agree that the occurrence of CTCs does not contradict the objectivity of time. It is unclear, however, in which sense time is ‘objective’ or ‘real’, and, correspondingly, in which sense CTC-spacetimes have a ‘temporal dimension’. The of time offers, roughly, two different views about what real time consists in: either it is “B-time” or “A-time”, i.e., either it is essentially the ordering of earlier-later or the directionality of a moving Now. I will argue that the possibility of CTCs is in conflict with B-time-order but requires A-time-direction. Then, the question arises of whether spacetime grounds A-time or rather the other way around: that presentness grounds spacetime, containing CTCs or not. I will suggest this latter option: universes containing CTCs motivate to develop a reasonable theory of presentism. The plan is as follows: In Section 2, I will briefly recap both the essentials of general relativity and of the analytic philosophy of time. Upon these physical and metaphysical basics, Section 3 provides the argument in favor of time directionality more fundamental than ordering. Finally, Section 4 argues that presentness is more fundamental than spacetime.

2 Physical and Philosophical Basics

The paper aims to demonstrate the relevance of the A- vs. B-time dis- tinction for the ontology of relativistic spacetimes. This section presents the basic elements of general relativity and central themes from the cur- rent philosophy of time, as they are relevant for my purpose. Cord Friebe: Presentist’s View on Spacetime 93

2.1 General Relativity To begin with, take a 4-dimensional (differentiable) manifold to rep- resent spacetime. Every physically possible spacetime has a lightcone structure, i.e, with every spacetime point p is associated a lightcone such that the distinction between timelike, lightlike, and spacelike vectors is ‘everywhere’, at every p, available. In particular, a timelike vector points inside the lightcone and is conventionally called “future-directed” iff it points into the upper lobe (and “past-directed” otherwise, i.e., iff it points downwards). Then, spacetimes are time orientable iff a con- tinuous designation of “future-directed” and “past-directed” for timelike vectors can be made over the entire manifold.2

Figure 1: Time orientability: any continuous transport of a timelike vector must preserve its orientation.

It should be emphasized that any simply connected spacetime is time orientable, a spacetime that is not time orientable is hence topologi- cally strange. Consider, in particular, a timelike curve: a curve is called “timelike” iff its tangent vectors are ‘always’, at every point of the curve, timelike. Such a curve lies inside all of its lightcones. Given time ori- entability, one may say that such a curve is ‘always’ future(or, past)- directed, and only given this, monotonically increasing parameters may consistently run along timelike curves – the (even invariant) proper times along worldlines. Therefore, time orientability is generally taken to be a necessary condition for time to be real. 94 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106

Figure 2: τ consistently parametrize every timelike curve ω.

However, it is questionable whether time orientability is also sufficient for spacetime to have a temporal dimension. For it does not exclude the possibility that a timelike, even ‘always’ future-directed, curve can be closed (or, almost closed). Intuitively, in such a case, the located at a given starting point p precedes itself. G¨odel’schallenge precisely was that even Einstein’s field equations allow for such a possibility so that it is not only mathematically compatible with time orientability but also physically possible. He concluded that the objective world is timeless. Still, it is undoubtedly not the case that the events on a CTC are simultaneous. Rather, the distinction between events being simultaneous and being non-simultaneous – a distinction which seems to be crucially temporal – is still available. Given this, one has good reason to sustain that universes containing CTCs are temporal. To sharpen the problem, it is instructive to confront an event/observer on a CTC with a freely falling body along a geodesic (see Figure 3): the hypothetical observer on the CTC is not simultane- ous with any event on the CTC but, e.g., with the body/event located at p0. However, (s)he is located at p ‘more than once’, namely modulo (periodic) τω. Does that mean – and what exactly could that mean – that the observer is ‘again and again’ simultaneous with the body/event Cord Friebe: Presentist’s View on Spacetime 95 at p0? I will argue that the B-theorists of time, focusing on the (partial- )order of earlier-simultaneous-later, are in trouble here and, so, that the possibility of CTCs favors tensed (A-) theories of time.

Figure 3: Is the observer at p again and again, at periodic τCTC , simul- taneous with the event at p0?

2.2 Analytic of Time Before arguing in favor of tensism, let me rewrite carefully the A- vs. B-time distinction from the analytic philosophy of time. One might have the impression that, according to the B-theorists, the world is a rather static block universe, whereas, according to the A-theorists, the objective world is really dynamical – ‘growing’, or whatever – in of an ontologically distinguished present. So, seemingly, a B-theory is much closer to allegedly static CTC-universes than the A-theories. However, this impression is misleading. For, on the one side, self-declared B-theorists, such as Mellor ([8]), defend “real” time and “real” change and so apparently do not intend 96 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106 to be closest to static block universes. And, on the other side, there is a very prominent A-theory – namely, the so-called moving-spotlight view – which in fact assumes an existentially static block universe: the moving Now is rather an irreducible than an existentially distinguished event. Therefore, the distinction between A- and B-theories of time is fairly independent from the (alleged) dynamics of existence.3 It is rather about ordering and directionality. According to the B-theorists, real time consists in the relation of earlier-later. Events (et al.) are temporal in the B-theorist’s sense, i.e., they are in B-time iff they are earlier and/or later than some other events, perhaps also simultaneous with some other events, respectively. B-time is hence fundamentally time order. B-time direction, by contrast, is a derivative concept: nothing more than the convention or a physical fact of whether p is earlier than p0 or vice versa. According to the A-theorists, real time is grounded in the present. The meaning of “A-time” depends on that single point rather than on an infinite set of points to be ordered.4 Its motion is (somehow) directed, and A-time is hence fundamentally time direction. Time order emerges derivatively, in virtue of the directed motion of the present. Accordingly, the moving-spotlight view (usually) presupposes the existence of a so- called C-series of events – a non-temporal ordering of events. The present is in relative motion against this C-series and turns it into a (temporal) B-series. However, there is a tendency in the to downplay the provided sharp contrast between A- and B-theories of time. One prominent example is the metaphysics of time, found by Maudlin ([7], chap. 4) within physics. Maudlin argues for a block universe that – ‘nevertheless’ – contains the “passage of time”. NB, he does not defend a version of the moving-spotlight view but somehow undermines the traditional classification of theories of time. “Temporal passage” is taken to be primitive, unanalyzable. In particular, one cannot analyze whether such a passage consists in directionality or in ordering. Let me sketch why the analytic distinction still matters. Apparently, the notion of passage is used by Maudlin only in its general sense. It needs to be specified, A- and B-theorists would reply. In one specifi- cation of the , it means a (continuous) succession of numerically different points (or, events). In this line, “passage” is expressible by an ordering relation such that time differs from timeless manifolds (such as space) in the way that moments of time are ordered by the earlier- later relation as opposed to other, timeless, relations. According to this Cord Friebe: Presentist’s View on Spacetime 97 view, time is essentially time order which view one can call “B-theory”. In the other specification, by contrast, “passage” means a (continuous) motion of one single point, the present. According to this view, time differs from timeless entities (such as space) not (primarily) in virtue of a special ordering relation but due to a characteristic directedness of the moving Now. Understood in this way, time is essentially time direction which view one can call “A-theory”. These are exclusive specifications of “passage”: no synthesis is available, let alone by taking it primitively. Finally, there is a further, important variant of the tensed (A-) theo- ries of time which is most relevant for my purpose: presentism. Accord- ing to this view, the present is in fact existentially distinguished: only present entities exist. Time fundamentally consists in the present so that, more strikingly as before, time cannot consist in temporal ordering but only in its directionality. The presentist A-theorist would argue that passage is a productive ‘succession’, a continuously coming into being of the present. In contrast to directions in space, “time direction” has, then, an existentially productive sense.

3 What is Fundamental: Time Order or Time Direction?

On a CTC, every event is timelike separated from itself, it “chronolog- ically precedes itself” ([4], 259). Here, “chronological” can firstly be taken in its purely technical sense, i.e., there is a non-zero (negative, or timelike) spatiotemporal distance between p and p itself when measured along the closed curve. The interpretative question is whether events on a CTC temporally precede themselves. The first sense is at best purely physical (perhaps, merely geometrical), whereas the is metaphys- ical. One might be inclined to the view that the former implies the latter, but one also might be skeptic: for, all events on a CTC are equivalent – namely ordered by a reflexive, symmetric, and transitive relation –, so they are, apparently, temporally on a par. On the other hand, events on a CTC are not simultaneous: as said, figure3 shows that the event at p is simultaneous rather with the one at p0 than with some other event on its CTC.5 Thus, the equivalence does not mean that the events on a CTC are simultaneous and so not that they are temporally on a par in that (usual) sense. Again on another hand, no event on a CTC occurs before or after another such event, and a fortiori no event occurs before or after itself. Being non-simultaneous does not imply occurring before- or afterwards. According to Dorato, different events on a CTC are related by a (very) peculiar relation of “temporal 98 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106 betweenness” ([4], 262). So, perhaps, the event at p is indeed temporally separated from itself, not because it is earlier or later than itself but because other events are located temporally in between: between p and p itself. Following Dorato, the lesson from CTCs is that time order is some- how different from what the traditional B-theorists of time have in mind. Analytic metaphysicians may believe that it is an a priori that temporal order is earlier-later, namely an irreflexive, asymmetric, and transitive relation. Scientific metaphysicians, by contrast, believe, in a Quinean fashion, that meaning always has an empirical aspect: what a time order is, (partly) depends on scientific theories. In the case at hand: “time order” means something such that even CTCs can represent a pe- culiar temporal order. That’s precisely how anti-armchair-philosophy works! However, other scientifically minded philosophers (rightly) disagree: CTCs exist only for spacetimes that possess a globally consis- tent time direction but not a globally consistent time order. ([5], 94) Although a CTC is everywhere timelike – and, hence, every event on a CTC “chronologically” precedes itself –, one cannot consistently say that an event temporally precedes itself. The notion of temporal, they suggest, has – at least partly – a philosophical meaning that goes be- yond the purely technical concepts of general relativity. The possibility of CTCs allegedly is in contradiction with any reasonable meaning of temporal order. Consequently, such spacetimes do not have a “globally consistent” time order, and that means that they do not have a time order at all. At this point, one might object that according to Earman et al. such spacetimes indeed have locally consistent time order(s), within small neighborhoods around every point. However, this is so only for prag- matic motives, analogous to the spatial geometry of a sphere which pragmatically allows the speech of right and left in small neighborhoods around the points. In this case, one would say, I guess, that only for practical it is convenient to talk about right and left, locally. Globally, however, and so objectively, there is no consistent right-left or- der but spherical coordinates (angels) are the metaphysically adequate ones. Likewise, in universes containing CTCs it might be reasonable for practical purposes to talk of earlier and later, locally. Globally, and so objectively there is no earlier-later order on a CTC but a symmetric and reflexive order only. Cord Friebe: Presentist’s View on Spacetime 99

‘Nevertheless’, spacetimes containing CTCs allegedly have a tem- poral dimension also according to Earman, Smeenk and W¨uthrich. In contrast to G¨odel,they do not conclude that general relativity denies the of time, namely because they do not conclude that universes with- out time-order are timeless. The reason for that is neither geometrical or physical, nor Quinean: Earman et al. neither recall some geometrical or physical facts (such as lightcone structure) that also G¨odelwas aware of, nor argue that “time order” gets a different meaning in modern physics. Instead, they introduce a further philosophical notion to interpret the geometrical/physical situation, namely “time direction”. Universes with CTCs have a temporal dimension exactly because they possess a glob- ally consistent time direction, they argue. Hence, here “time” does not mean time order but time direction. This is a remarkable claim because, consequently, the meaning of time direction should not depend on the meaning of time order: “temporal” essentially means time directionality – time direction is more fundamental than time order. B-theoretically, however, time direction depends on (traditional) time order.6 Given the irreflexive, asymmetric, and transitive relation < of earlier-later, time direction is derived when it is consistently possible to assign whether p < p0 or p0 < p. This might be a mere convention or is grounded in physical facts such as the past hypothesis:7 in any case, B-time is directed iff an arrow consistently points from (fundamen- tal) earlier to later. I don’t see how this concept can be applied with “temporal betweenness” alone, as Dorato must have it. In that case, the challenge for the B-theorists would be: How can mere temporal be- tweenness (without before- and afterwardness) ground time direction? From ‘where’ to ‘where’ points the arrow? As it seems to me, the time directionality from Earman et al. is rather independent from Dorato’s non-traditional time order, so at least equally fundamental anyway. Taking time direction as a (or, even the) fundamental concept, as Earman et al. in fact do, apparently requires tensed time (A-time). A- theoretically, the meaning of temporal depends on the objectively ‘mov- ing’ present. Its motion is directed so that A-time essentially consists in that time directionality. According to the moving-spotlight variant, the transitory present is not existentially distinguished but by being an intrinsic and changing (‘moving’) property of presentness – metaphor- ically represented by a spotlight. This spotlight is in relative motion against an existing series of objects or events, traditionally understood as ordered by an irreflexive, asymmetric, and transitive relation (Mc- Taggart’s timeless C-series). In that case, the timeless series turns into 100 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106 a temporal B-series, in virtue of the transitory present. It is its directed motion that gives the relation < the sense of earlier-later. Time direc- tionality – which is given by the single point of presentness – grounds time order (of a set of points), and not the other way around. In the case at hand, however, there is not even a C-series of events: on a CTC the points are ordered by an equivalence relation. Thus, along a CTC the moving spotlight cannot (and should not) produce a temporal B- series. One has its directionality without time order, in accordance with the claim that there is a globally consistent time direction but no time order. However, the problem with this view is that it requires su- per(space)time dimension(s) in order to give room for the peculiar change of the Now. It threatens to be McTaggart-contradictory, i.e., each event threatens to be equally past, present, and future, unless one introduces an infinite, perhaps even vicious, regress of meta-dimensions. Despite of this argument, Skow ([9]) provides a moving-spotlight interpretation of relativistic spacetimes.8 Personally, I do not see how Skow super- sedes McTaggart’s worries and, consequently, will not provide a further argument against the moving-spotlight variant. Instead, I will focus on an important difference between moving-spotlight and presentism: both A-theoretic options take time directionality as the fundamental essence of real time, but the first take spacetime as more fundamental than time whereas the latter must take time as more fundamental than spacetime. The moving-spotlight A-theory of time presupposes the existence of the whole of spacetime upon which and against which one (or, many inde- pendent) Now-points can perform its (their) motion(s). According to presentism, no thing eternally exists, a fortiori no spacetime as a four- dimensional block, but only the present. So, the presentist’s ‘moving’ Now underlies spacetime rather than the other way around.

4 Fundamental Time Direction: the Presentist’s View

In my view, the only consistent philosophy of time that copes with the idea that time direction is more fundamental than time order is pre- sentism. So, if the lack of a (globally consistent) time order does not imply the timelessness of those spacetimes (containing CTCs), and if its temporality is essentially directionality – as it is claimed by Ear- man/Smeenk/W¨uthrich –, one should take it as an indicator for the view that the metaphysics of time is presentism. Then, spacetime is ontologically derivative, existentially grounded in a ‘moving’ Now which Cord Friebe: Presentist’s View on Spacetime 101 underlies spacetime (containing CTCs or not). In order to spell out a bit the essential idea of presentism, look, for contrast, again at the moving spotlight: its motion is relative, namely against an ‘already’ existing series of spacetime points or events located there. Accordingly, the presentist Now must be in some radical non- relative motion, so in an absolute motion neither against ‘already’ ex- isting bodies/events nor against ‘already’ existing space or spacetime. Although against nothing, it still must be ‘in motion’, otherwise the transitory aspect of time is lost. The presentist Now is not an isolated moment of B-time which would be indistinguishable from an isolated spatial (timeless) point. An isolated moment of B-time would be struc- tureless, while something happens with (or, rather, within) the presentist Now. The logical question is whether this happening is intelligible, and the metaphysical question is if it is compatible with relativity, in partic- ular, with universes containing CTCs. The presentist Now is not only no isolated moment of B-time but not a moment of time at all: it is simply time. There is (on the fundamen- tal) level neither the relational structure of earlier-later nor that of past- present-future: A-time is fundamentally undivided. This has a twofold advantage: logically, a McTaggart-contradiction cannot arise simply be- cause there is no past nor future; physically, presentism is prima facie compatible with CTC-universes since those universes are not divided into earlier and later but ‘nevertheless’ have a temporal dimension. Although being undivided, the presentist Now cannot be structureless: for, as said, it must be distinguishable from a timeless point. As I would suggest, its structure consists in the distinction between coming into being and annihilation. The crucial difference to a static point (and to a point in relative motion) is that the presentist Now does not persist. It rather comes into existence to vanish immediately. So, its peculiar internal motion consist in the productive activity of continuously coming into existence and going out of existence. These two aspects, however, are related to each other asymmetrically: it firstly comes into existence and secondly goes out of existence, not the other way around, and so on and so on. Therefore, time (i.e., the present) is essentially directed, namely towards existence, and therefore directionality is the fundamental essence of time. In order to create spacetime, the presentist Now has to be extended. This has firstly a spatial sense: the point (in its internal motion) has to be extended to create space, i.e., a simultaneity slice of, say, three dimensions. At this point, the challenge from special relativity enters: 102 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106 it forbids an absolute (meaning here: frame-independent) foliation of spacetime so that the continuously produced simultaneity slices must not be invariant. As argued elsewhere, presentism can but can only made compatible with special relativity if the (objective) extending of the Now-point is observer-dependent.9 There is secondly a temporal sense of extending the Now-point:10 presentist time, which is fundamentally not directed upwards or down- wards in spacetime but points productively into the peculiar, singular direction of existence, must be extended in a second sense to create the temporal relation of past-present-future or earlier-later. Extending the presentist Now in the temporal sense (objectively) produces new struc- ture, i.e., renders time dividing into past and future. Also this must be observer-dependent, already to make presentism compatible with spe- cial relativity. At this point enters the challenge from general relativity: derivative time, i.e. time-order, further depends on empirical conditions. In the case of G¨odelianuniverses, there are empirical conditions that strike against the division of time so that no time order arises. Under these circumstances, it is physically impossible to objectively extend the presentist Now in the temporal sense. Here, presentist time (presentness, the Now) remains undivided (“no order”), purely directed. Therefore, there is only a globally consistent time direction but no time order in spacetimes containing CTCs.

5 Conclusion

The paper intends to provide a scientifically minded motivation for a presentist reading of spacetime theories, a stimulant for developing a theory of relativistic presentism. The physical possibility of spacetimes containing CTCs leads to the following reasoning: if one is after a metaphysically-temporal interpretation of all (time-orientable) space- times, the meaning of temporal essentially consists in directionality. Only presentism consistently copes with the idea that time direction is more fundamental than time order. Spacetime, then, is ontologically derivative, dependent on a fundamental Now which essentially points to- wards existence. Its essence is an existentially productive directionality. The theories of relativity offer empirical constraints under which the fundamental presentist Now can be objectively extended, both in a spa- tial as well as in a temporal sense. In its temporal sense, (observer- dependent) extending the Now leads to the temporal division into past- present-future or earlier-later. Under certain empirical circumstances, Cord Friebe: Presentist’s View on Spacetime 103 realized in the G¨odeliancases, the temporal extending of the Now ob- jectively fails so that no objective time order arises. Only time direction- ality is real in such worlds. Since those worlds are physically possible, time directionality also is fundamental in the actual world. (This mimics G¨odel’sown reasoning, according to which the actual world is timeless since CTC-universes are physically possible.)

Notes

1 To be more precise, in G¨odel’suniverse a CTC goes through every spacetime point, but nothing hangs on this: the challenge for the philosophy of time consists in the mere occurrence of CTCs through some spacetime points (which is, by the way, the more realistic possibility). 2 The precise definition of time-orientability (including lightlike vectors) doesn’t matter here. 3 Indeed, the existentially dynamic is B-theoretic in the sense that “presentness” is reduced to ‘nothing exists that is later than’ (see [3], 69f.). 4 Perhaps, there is more than one single present, but, also then, time surely does not consist in the ordering of these points. 5 For more on “simultaneity” and (corresponding) measurements in curved space- times see [2] (chap. 4 and 6). 6 Note that the usual way in which the ‘arrow’ of time has been debated in the philosophy of physics is based on temporal order. Usually, one presupposes the view of a tenseless block universe and asks whether such a universe physically allows a specific time orientation. With universes containing CTCs in mind, one should change the perspective and ask the following question: given a temporal direction, what are the physical constraints to get also a temporal order? 7 According to the past hypothesis, the early universe was in a low entropy state, which apparently saves the time asymmetry of the second law of thermodynamics; see, e.g., [1]. 8 Note that, in the CTC-case, superspacetime is likewise G¨odelianso that the directed motion of the spotlight in fact does not produce a temporal B-series. 9 The observer-dependence of space must, hence, not be in contradiction with objectivity. How this might work, has been shown on other occasions, e.g., in Anonymous. 10 The following sounds similar to Bergson’s mind-dependency of the B-theoretic earlier-later relation (see Anonymous2, this volume).

Acknowledgements

I would like to thank the audience of the Philosophy of Time Symposium, held at SOPhiA 2014 (Salzburg), and two anonymous referees of this 104 KRITERION – Journal of Philosophy, 2016, 30(2): 91–106 journal for helpful comments on earlier versions of the paper. Special thanks go to the SPoT Discussion Group at Bonn. Financial support stems from the Deutsche Forschungsgemeinschaft, project FR 1461/6-1.

Cord Friebe University of Bonn, Germany Institute for Philosophy Am Hof 1 D-53113 Bonn

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