Reconciling the Asymmetry with Contemporary Physics

Abstract

As has previously been argued, the growing block theory of time (GBT), since it is essentially asymmetrical (necessarily sometimes the structure it describes is not reflection invariant), while it accepts Temporal Becoming (new things are created in the present), is better positioned than the traditional models of the temporal structure of the world (eternalism and presentism) to accommodate our intuition that the future is open and the past fixed. However, GBT (like any other A-theory of time) is often criticized for conflicting with some important results of contemporary physics (e.g., by requiring an absolute notion of objective simultaneity). In this chapter, I argue that GBT, far from being disqualified by contemporary physics, might be underpinned by some recent approaches to quantum gravity, especially CST.

4.1 Introduction

At first sight, GBT appears at odds with some important results of contemporary physics. In particular, since GBT is usually based on the idea that the layers of existence successively coming into being are slices of Newtonian absolute time, this theory could hardly be reconciled with the developments of the Special and General theory of relativity. For example, Hilary Putnam (1967) and Wim Rietdijk (1966) have both argued that, since the view that the future is unreal requires an objective notion of absolute simultaneity, it is incompatible with the Special theory of relativity (SR), according to which the simultaneity of space-like separated events is relative. However, not only the assumptions of these arguments can be disputed (cf. Bourne, 2006; Correia & Rosenkranz, 2018; Miller, 2013; Sklar, 1974; Stein, 1968, 1991; Tooley, 1997; Zimmerman, 2008), but recent approaches to quantum gravity have been formulated in terms that echo C. D. Broad’s theory. For example, Raphael Sorkin put forward a model of causal set dynamics according to which “[…] reality is more naturally seen as a ‘growing being’ than as a ‘static thing’” (2007: 153). In other words, even if GBT turns out to be inexpressible in a relativistic framework (which is doubtful), one can question the credentials of the theories of relativity (especially given their incompatibility with quantum mechanics), and argue that our most fundamental physics is rather to be found in the nascent theories of quantum gravity, while some of them seem compatible with GBT.

Specifically, the causal set approach to quantum gravity (CST), which dismisses the spacetime continuum as mere approximation¹ in favor of locally finite causal sets, seems consistent with C. D. Broad’s notion of temporal becoming introduced in the previous chapter. Temporal becoming could indeed be restored by a dynamics, called the ‘classical sequential growth dynamics’ (CSG), by which the growth in the causal sets takes place. Thus, although the search for a quantum theory of gravity is currently dominated by String Theory and Loop Quantum Gravity, CST might help solve some of the key problems encountered in trying to make GBT work in a relativistic setting. Of course, CST – or any other scientific research program – cannot settle the debate about the nature of time, but it can at least inform and frame the debate, so that metaphysicians will be offered new ideas and be prevented from saying too much nonsense.

This chapter is structured as follows. In the second section, I briefly introduce some basic elements of (neo-)Newtonian physics and show why it is a friendly environment for GBT. In the third section, I present the ‘relativity revolution’ that emerged out of Albert Einstein’s observations, by introducing the main postulates and consequences of the Special theory of relativity (SR); I lay particular emphasis on Minkowski’s attempt to make sense of SR as a spacetime theory. In the fourth section, I expose the threat that SR may represent for GBT, especially through its rejection of an objective notion of absolute simultaneity, and different strategies (both incompatibilist and compatibilist) that growing blockers may adopt to escape this threat. In the fifth section, I go beyond SR’s limited aspirations by introducing the General theory of relativity (GR) and consider some of the main difficulties in squaring it with quantum mechanics (especially with regard to causal relativity). In the sixth section, I expose one approach to quantum gravity which, since it encompasses both relativistic and quantum phenomena, aims to address these difficulties. Moreover, I show that this approach, the causal set approach, promises to provide a notion of temporal becoming, and could thereby make GBT an attractive model in the contemporary scientific context. I conclude against a widespread opinion that fundamental physics does not undermine any attempt to defend an open-future view. Finally, in the seventh section, I move from science to science fiction and show that GBT is, in principle, compatible with some scenarios such as time-travel.

4.2 (Neo-)Newtonian Basics

For more than two centuries, physicists have assumed, in accordance with Isaac Newton’s Principia (1687), that space was an infinite and immutable three-dimensional Euclidean continuum of points, all of which persist through time, which is also infinite.² The Newtonian view can be described, somewhat anachronistically, in terms of spacetime, provided that persisting spatial points are replaced with a succession of numerically distinct spacetime points. Newtonian spacetime is then to be conceived as a four-dimensional continuum consisting of a stack of three-dimensional volumes of space (or hyperplanes), each instantaneous and spread out continuously in a fourth, temporal, dimension. Intuitively, if one thinks of Newtonian space as a cube, then one should think of Newtonian spacetime as a succession of cubes. It is however often useful to pretend that space has just two dimensions, so that we can visualize Newtonian spacetime as a succession of flat-surfaces, each of them representing a three-dimensional hyperplane of absolutely simultaneous spacetime points (cf. Fig. 4.1). In such a diagram, there are spatial and temporal distance relations between every point (including spatial distance relations between every point in different hyperplanes). Furthermore, each persisting object is represented by a worldline (or a worldtube, when the object is not a point), which depicts its complete trajectory through a succession of different spacetime points. Specifically, Newton’s laws of motion³ allow for the following description, which refers to Fig. 4.1: objects that are at absolute rest (a) have vertical straight worldlines; objects in uniform motion (b) have non-vertical straight worldlines that cut through the flat-surfaces (the greater the degree of deviation from the vertical, the faster the velocity); and finally objects that undergo acceleration (c) have curved worldlines (the steeper the curve, the greater the acceleration – this is a little loose, since a motion could actually be such that acceleration could be greatest at point where tangent to curve is vertical).

Fig. 4.1

Newtonian spacetime

It is worth mentioning that if one abolishes the spatial distance relations between points in different hyperplanes, while leaving everything else unchanged, one turns a Newtonian spacetime into a neo-Newtonian (or Galilean) spacetime.⁴ This seems important, since the distinction between absolute rest and uniform motion, which is obtain through these cross-temporal relations, appears to be superfluous. After all, Newtonian physics itself predicts that no experiment whatsoever could determine the absolute rest of an object: “[l]aboratory experiments done in a room at absolute rest would have identical outcomes to those done in a room moving with constant velocity” (Maudlin, 2011: 36). Thus, whereas a neo-Newtonian spacetime preserves the distinction between straight and curved trajectories, it abandons cross-temporal spatial distances: only points that are simultaneous are located at a spatial distance from one another. It is therefore wrong to think of classical physics as entirely free from relativity. In a neo-Newtonian spacetime, for instance, statements such as ˹This object is moving has a speed of 10 km h⁻¹˺ or ˹This object is at rest˺ are meaningless; people in different frames of reference⁵ will disagree about what speeds objects have, and there is no objective fact of the matter as to which groups are right. Most interesting differences and similarities of Newtonian and neo-Newtonian spacetimes are summarized in the Fig. 4.2.

Fig. 4.2

Comparison between Newtonian and Neo-Newtonian spacetime

Newtonian and neo-Newtonian spacetimes show that, although the word ‘spacetime’ is spontaneously associated with the ‘block universe’ view of time, it is perfectly compatible with dynamic conceptions of time.⁶ In particular, since both Newtonian and neo-Newtonian spacetimes allow for an objective partition (or foliation) into slices of events that all happen at the same time, it seems that a dynamic theory such as GBT can fairly be expressed within classical physics. To put it another way, although classical physics does not postulate a now, it allows for the possibility of an objective spatially extended present, and even for temporal becoming. For example, John Earman (2008) argues that, since temporal becoming consists of the infinite accretion of layers of existence, which are naturally seen as slices of Newtonian absolute time, it can be implemented in the Newtonian setting, which thereby offers a growing model of the universe. Roughly, Earman’s idea is to define a model of GBT as a pair {$\mathfrak{N}$, ≾}, where $\mathfrak{N}$ is a set of spacetime models, each of which is isomorphic to a ‘future truncated’ version of a Newtonian spacetime, i.e. to a model that results from deleting from a Newtonian spacetime all points later than some particular time, and restricting the geometric and physical fields to that truncated spacetime. The relation ≾, which expresses temporal becoming, holds between two of the spacetime models in $\mathfrak{N}$ iff one can be isomorphically embedded as a submodel of the other.⁷ Thus, although expressing GBT in Newtonian setting requires some efforts, classical physics (with absolute simultaneity) appears to be a friendly environment: (at least) some versions of GBT seem respectful of its main imperatives.

However, at the beginning of the twentieth century, physics underwent a deep revolution, known as the ‘relativity revolution’, which entailed the rejection of the classical theory – though it is still considered as an accurate approximation at low velocities (relative to the speed of light). This revolution, which emerged out of Albert Einstein’s observation that the classical theory conflicts with Maxwell’s equations of electromagnetism and, experimentally, with the Michelson-Morley outcome,⁸ seems to exert significant pressure on some philosophical theories of time, and especially on GBT. In particular, both the Special and General theory of relativity rule out the Newtonian objective notion of absolute simultaneity, which seems required in the definition of GBT. According to both SR and GR, there is simply no objective fact of the matter as to whether two space-like separated events are simultaneous or not. The Putnam-Rietdijk argument is probably the most famous argument pointing in this direction. However, before detailing this argument and possible options to answer it, it is worth recalling what the relativity revolution is about. In the following pages, I will therefore introduce the main postulates and consequences of Einstein’s early work.

4.3 The Relativity Revolution

In 1873, James Clerk Maxwell published a work, entitled A Treatise on Electricity and Magnetism, in which he formulated the classical theory of electromagnetic radiation, bringing together electricity, magnetism, and light as different manifestations of the same phenomenon. Specifically, Maxwell’s equations of electromagnetism, which have been called the ‘second great unification in physics’ (after the first one realized by Newton), show how electrical and magnetic forces are intimately related, and especially how electrical and magnetic fields could combine to form self-propagating electromagnetic waves. The velocity of such waves, which can be predicted from the equations, turned out to be the speed of light: c = 300,000 km s⁻¹. This result led Maxwell to suppose that light itself was an electromagnetic wave (which turned out to be the case). Since waves had always been studied as perturbations in something (e.g., water, air, the surface of drums, etc.), physicists immediately assumed that light waves also consisted of perturbations in a medium: luminiferous aether. This was the advent of ‘aether-physics’, which was in charge of capturing the properties of this invisible, but all-pervasive medium. It was, for instance, commonly agreed that aether must be in a state of absolute rest. This consensus was to prove fruitful, since it opened the possibility of experimental tests for distinguishing absolute motion from absolute rest (cf. Dainton, 2010: 315-317).

Such an experimental test was first carried out in 1881 by Albert Michelson alone (and then in collaboration with Edward Morley in 1887), but its outcome defied the odds. Basically, the idea was to compare the speed of light in perpendicular directions, in attempt to detect variations through the absolute stationary aether. After all, given that waves are not affected by how fast their sources are moving (contrary to projectiles),⁹ if light rays travel in all directions at c = 300,000 kms⁻¹ (as Maxwell’s equations predict), this speed should only be obtained when the test is conducted at rest relative to the aether. If the speed of light is measured by someone who is moving through the aether, he should find that the light rays travelling in the same direction as him are moving slower than those travelling in the opposite direction, or so theory predicted. Thus, assuming that the Earth moves through the aether as it orbits the sun, Michelson measured the time it took for light to travel along two paths of equal distance; one of these paths was aligned in the direction of the Earth’s motion around the sun, the other at right angles to this direction. But, whereas Michelson expected the light ray travelling parallel to the flow of the aether to be slower than the one travelling perpendicular to it, no variations in the speed of light (through the presumed aether) were detected. Light turned out to always travel at the same speed, irrespective of the Earth’s motion.¹⁰ This could not be explained by classical physics (especially because of the Galilean transformations),¹¹ which therefore had to be (substantially) revised.

In that respect, Einstein’s radical proposal,¹² called ‘Special Relativity’ (SR), results from the willingness to follow through to the end the consequences of two postulates: (i) the Relativity Postulate, according to which the laws of nature do not distinguish between different observers undergoing inertial (i.e. force-free) motion, and (ii) the Light Postulate, according to which the numerical value of the speed of light is the same when measured in any direction, by any inertial observer.¹³¹⁴ This proposition involves a reformulation of mechanics in terms of the Lorentz transformations (instead of the Galilean transformations) that reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different ordering of events, but always such that the speed of light is the same in all inertial frames of reference. In short, one could say that the relativity revolution consists in replacing one absolute, Newton’s absolute space and time, with another, the velocity of light. This famously requires the notions of distance and time (both involved in the definition of velocity) to vary in systematic ways in the relevant frames. It is precisely these variations, sometimes misleadingly called ‘length contractions’ and ‘time dilations’, that the Lorentz transformations allow one to calculate.¹⁵ Surely the most natural comparison here is the absolute structure of Minkowski spacetime, which postulates a different pattern of spatial and temporal distances to (neo-)Newtonian spacetime. If one takes the light speed as the absolute, one could consider Harvey Brown’s Fable of Keinstein (2005: chap. 3), and see forces and masses as the absolutes that allow one to derive the Galilean transformations in a way analogous to Einstein’s derivation of the Lorentz transformations.

To get a feel for what this means in practice, imagine two people, Max and Mary, travelling outer space. Suppose that Max switches on a torch and sends a light signal in Mary’s direction; he therefore sees a light ray moving at c = 300,000 km s⁻¹ towards her. Furthermore, suppose that Max sees Mary moving away from him at the very high speed of 0.5c; if she measures the speed at which Max’s light signal passes her by using a clock, she will also find that it is moving at c. Indeed, given the Light Postulate, the speed of light is constant in all frames of reference, and independently on the motion of the source, which implies that moving away from its source does not make it appear greater. How can this story be possible? Roughly, Einstein’s answer is this: if Max could look at Mary’s clock, he would find that it is running slower than his. In other words, keeping the speed of light constant for Max and Mary requires their clocks to be desynchronized: a clock that is moving relative to an inertial frame of reference will be measured to tick slower than a clock that is at rest in this frame of reference. But, given the Relativity Postulate and the abandonment of absolute motion, Einstein goes a step further by arguing that each can regard themselves as at rest with as much right as the other. Counterintuitively, that means that if Mary could look at Max’s clock, she would also find that it is running slower than hers.¹⁶ Thus, although Max and Mary’s readings are different, Einstein argues that they are equally valid; there is no objective fact of the matter as to which of their clocks is accurate.¹⁷ This example is inspired by Dainton (2010: 317).

According to its standard interpretation, SR has many weird consequences; some of them directly impact the classical A- and B-theories of time, especially GBT. I will mention 3 of the most striking consequences. First, there is no objective fact of the matter about which events happen at the same time. Whether or not two events are simultaneous depends on the frame of reference from which they are considered; observers in different frames of reference will find different events simultaneous, and there is no sense in saying that one observer is right and another wrong. Second, space and time are not to be conceived as two separable and quite distinct entities, but much rather as entangled aspects of the same underlying four-dimensional continuum that fuses the two into a spacetime, the so-called ‘Minkowski spacetime’ (cf. ‘unitism’ contrasted with ‘separatism’ in Gilmore et al. (2016)). Although we are already familiar with the general idea of spacetime (cf. Sect. 4.2), Minkowski spacetime interestingly differs from the two types of structure we have encountered so far, Newtonian and neo-Newtonian, especially with respect to the quantities it takes to be invariant: neither spatial nor temporal distances are invariant, the speed of light and the spacetime interval are.¹⁸ Third, considering two twins, Max and Mary, if Mary departs on a space journey and subsequently returns, she will be younger than stay-at-home Max. Time has passed at a slower rate for her than for Max (which appears at odds with the Relativity Postulate). This is due to the geometry of Minkowski spacetime (which will be introduced below): if one calculates the length of Mary’s trajectory in a Minkowski diagram, one will find out that it is smaller than the length of Max’s trajectory, even though Euclidean representations of the case present Mary’s path as being longer than Max’s path (which explains why the case may seem paradoxical).¹⁹

At first sight, one might be surprised to find out that two seemingly innocuous postulates – the Relativity Postulate and the Light Postulate – have such enormous consequences. But, one has to remember that the difficult step, which Einstein first took, was to consider seriously the possibility that these postulates might be true at a time when this was far from obvious. Among these consequences, the relativity of simultaneity, which undermines the Newtonian assumption that there is an objective fact of the matter as to which events occur at the same time, is presumably the most problematic for GBT.²⁰ It is therefore worth considering it in greater detail. Basically, the idea behind the relativity of simultaneity is that whereas there is no problem in establishing a common time system for a group of people who are stationary with respect to one another, so that everyone in this group will agree on which events are simultaneous, things get harder for people who are moving with respect to one another: it is impossible for them to synchronize their clocks.²¹ Examples involving high-speed trains and lightnings are commonly used to illustrate the situation.

For instance, Einstein (1920: 25–27) invites us to consider a train, both ends of which are struck by bolts of lightning, producing two flashes. Whether or not these two flashes occur simultaneously depends on the motion of the observers relative to the location of the events. For example, if Max stands by the side of the track and, when the lightning strikes, is at the mid-point of the high-speed train, he will see the two flashes as occurring simultaneously. However, if Mary is sitting midway inside the train when the lighting strikes, she will see the two flashes as occurring successively. In other words, the two flashes have different time coordinates in frames of reference that are in motion relative to each other.²² From Max’s perspective, Mary’s deviant observation can easily be explained by the fact that she is moving towards the light travelling from the front of the train, and she thus decreases the distance the light has to travel to reach her. But, given the Relativity Postulate, Mary’s perspective, which takes the ground to be moving beneath the stationary train, is not less valid than Max’s one. She is therefore perfectly legitimate in saying that it is Max’s observation that is deviant, because he increases the distance between him and the light coming from the front of the train. We are therefore left with two flashes, the time ordering of which differs depending on who observes them, and there is no objective fact of the matter as to which observer is right.

It may be helpful to visualize the situation using Minkowski diagrams, which aim to graphically depict a portion of the new conception of spacetime that seems implied by SR. In such two-dimensional diagrams, where space has been curtailed to a single dimension, the vertical axis t (or t’) refers to temporal and the horizontal axis x (or x’) to spatial coordinate values. The introduction of a separate x’ axis is required by the Lorentz transformations, according to which observers moving at different velocities may measure different ordering of events. Intuitively, the lines parallel to x (or x’) correspond to the usual notion of simultaneous events for a stationary observer. Minkowski diagrams thus allow a qualitative understanding of the relativity of simultaneity (and other relativistic phenomena) without mathematical equations: each observer interprets all events on a line parallel to his x (or x’) axis as simultaneous. Specifically, considering the previous case, a Minkowski diagram (cf. Fig. 4.3) allows one to show that whereas the two flashes occur simultaneously in Max’s perspective (since they are both situated on x), they occur successively in Mary’s perspective (since they are not situated on a line parallel to x’). But, in both perspectives, the two flashes are spatially separated (even though their spatial distance is also frame-relative), which corresponds to the front and the back of the train.²³

Fig. 4.3

Two Minkowski diagrams

It is worth saying a bit more about the standard way of construing Minkowski’s attempt to make sense of SR as a theory of spacetime. Unlike Newtonian and neo-Newtonian structures, which both take some spatial and temporal distances as invariant (e.g., the spatial distance between points that occur at the same time), Minkowski spacetime is built around another invariant quantity: the speed of light. Indeed, since neither spatial nor temporal distances are invariant in SR, but the speed of light is, Minkowski’s proposal is to integrate it into the very structure of spacetime. This seems to be a natural response to the abolition of luminiferous aether: given that light is no perturbation within any medium whatsoever, it seems that the only candidate available for determining the paths light rays can take is spacetime itself. The notion of a light-cone is crucial to understand the luminal structure of Minkowski spacetime. A light-cone is all the possible light rays that can be sent from (and received by) a single event e (occupying a certain spacetime point), which together trace out a double cone. Specifically, if one imagines light confined to a two-dimensional plane, the light rays emanating from e spread out in a spherical surface; and if one depicts this expanding spherical surface with the vertical ‘time’ axis, the result is a cone, known as the future light-cone. The past light-cone behaves like the future light-cone in reverse: a spherical surface contracts in radius at the speed of light until it converges to the spacetime point occupied by e. When the future light-cone (representing light emitted by e) is combined with the past light-cone (representing light received by e), they together yield the characteristic hourglass shape shown in the Fig. 4.4.

Fig. 4.4

The light-cone structure

Given that light has the maximum possible speed (this can be derived from the Light Postulate and the Lorentz formula for velocity addition), the light-cone of e partitions the remainder of the universe into 3 separate (but topologically connected) regions (as indicated in Fig. 4.4): region I is known as the ‘absolute future’, region II as the ‘absolute past’, and region III as the ‘absolute elsewhere’ of e. In particular, the spacetime points lying on the surface of regions I and II are those that can be connected by light rays travelling in a vacuum, and are called ‘light-like separated’ from e – their spacetime interval is zero. The spacetime points lying inside regions I and II are those that can be connected by signals travelling slower than light, and are called ‘time-like separated’ from e – their spacetime interval is positive. Finally, the spacetime points lying in region III (i.e. outside the light-cone) are such that only a signal travelling faster than light could connect them; they are called ‘space-like separated’ from e – their spacetime interval is imaginary. Events that are very close together in time but spatially at a significant distance typically fall into this third region. It is worth mentioning that, for any space-like separated events, there is an inertial frame in which they are simultaneous, whereas time-like separated events are non-simultaneous in all inertial frames (though there are frames in which they occur at the same spatial coordinates). Thus, just as has been shown that classical physics is not entirely free from relativity, one finds that the relativity of simultaneity has limits within SR: relativization applies only to space-like separated events.

To illustrate, consider event e, which lies where the tips of the two cones meet (as indicated in Fig. 4.4). In its frame of reference, e is simultaneous with all the space-like separated events that compose a horizontal hyperplane (parallel to the x), slicing through the absolute elsewhere (region III). Yet one can imagine alternative hyperplanes centered on e, slicing through the absolute elsewhere at different angles. What I am assuming is a timeline direction at e that can be thought of as encoding the instantaneous state at e of an observer who has e on their worldline (e.g., the t axis in the diagram). Specifically, each of the alternative hyperplanes contains events that, from e’s frame of reference, lie either in the past or in the future, but not in the absolute past or future. Furthermore, all the events that are space-like separated from e, i.e. all the events that are in the absolute elsewhere of e, are simultaneous with e from some inertial frame. But there is no frame of reference in which events that are time-like separated from e, i.e. events that are in e’s absolute past or future, are simultaneous with e. Does that mean that, from the perspective of each spacetime point, the temporal ordering of half the entire universe is frame-relative? Not necessarily; “[…] the bulk of spacetime may well lie in our absolute past and future” (Dainton, 2010: 326). Indeed, one has to remember that indicating light-like connections with forty-five-degree lines results from a useful, but potentially misleading conventional choice.

Finally, it is worth mentioning that, since no physical influence can travel faster than light, light-cones allow for a causal interpretation: events that are time-like separated can be causally related, but no causal relation can hold between space-like events. To put it another way, whereas no event in region III can be the cause or the effect of e (since any influence would have had to travel faster than light), any event in region II can, in principle, influence (or be the cause of) e, and any event in region I can, in principle, be influenced (or be the effect of) e. Accordingly, regions I and II are sometimes renamed the ‘causal future’ and the ‘causal past’ of e. For example, two planets can causally interact only if they are linked by paths through spacetime that are always time-like (i.e. paths that always stay inside the relevant light-cones). If the spacetime paths between the two planets are space-like (even only partially), they cannot causally interact. Therefore, although the temporal ordering of events is affected by relativity, this cannot be extended to causal ordering: the relativity of simultaneity only applies to events that nothing travelling slower than light, such as causal influence, can connect. That is why Minkowski spacetime is often said to embody the causal structure of the universe.²⁴

In a nutshell, unlike Newtonian and neo-Newtonian spacetimes, Minkowski spacetime, which seems implied by SR, admits no uniquely correct foliation into three-dimensional simultaneity slices (or hyperplanes); the objective notion of absolutely simultaneous events is therefore meaningless in the relativistic context. In other words, whereas the (neo-)Newtonian approach assumes that absolute simultaneity of two events could be established by suitable measurements, the Einsteinian approach assumes that, since measurements of the time elapsed between two events rely on light, which has a finite speed, no measurement whatsoever could establish an objective notion of absolute simultaneity between them. In relativistic physics, the light-cone structure has thus replaced the objective foliation. It is important to keep in mind that the light-cone structure allows for a causal interpretation (events that can causally interact²⁵ with e are only located in regions I and II); this interpretation will remain true in GR (Sect. 4.5) and will play a crucial role in the causal set approach to quantum gravity (Sect. 4.6).

4.4 Relativity as a Threat to GBT

Given what has been said so far, it could be argued that SR exerts some pressure on GBT (and on other classical A- and B-theories of time), at least as defined in the previous chapter. Such an argument has originally been developed by Hilary Putnam (1967) and Wim Rietdijk (1966) who have famously concluded that the view that the future is unreal is incompatible with SR. Contrary to what one might think, this conclusion is not derived from the mere fact that SR is a spacetime theory, although past and future light-cones may give the impression that Minkowski spacetime is an eternal block of events. As has been shown, (neo-)Newtonian physics can also be formulated as a spacetime theory, while nothing a priori prevents from implementing distinctions between the future and the past in the (neo-)Newtonian setting (cf. Earman, 2008). More threatening for GBT is what SR has to say about the present itself. According to SR, the distinction between what is present and non-present has no ontological significance, but depends on an arbitrary choice of frame of reference, and the same seems to apply to the distinction between what is real and unreal.²⁶

Specifically, the Putnam-Rietdijk argument establishes that an observer here-now in Minkowski spacetime must, on pain of inconsistency with his base conventions, assign the label ‘real’ to events in the future light-cone of here-now (cf. also Savitt, 2000, Petkov, 2006, Dorato, 2008, Earman, 2008, Dainton, 2010, Wüthrich, 2010, Miller, 2013).²⁷ Roughly, the reason is that events in an observer’s future light-cone can be in another observer’s relative past or present, at a stage when the first observer judges the second observation to be real. This is an immediate consequence of the relativity of simultaneity: two observers in relative inertial motion (e.g., Max and Mary in the train example) will disagree about whether some set of events occur at the same time or not, and there is no sense in saying that one observer is right and another wrong. To put it another way: relative to inertially moving observers, spacetime will foliate differently into three-dimensional space-like hyperplanes, so that different sets of events will be simultaneous. Whereas this is no big deal for the B-theorist of time, since she does not ontologically discriminate between regions of spacetime,²⁸ things are more complicated for the growing block theorist (and other A-theorists). This can be illustrated by the following example.

Suppose that Max and Mary, two observers in motion relative to one another, make a fleeting contact. For example, they brush against each other as they pass. It clearly appears that Max and Mary are real with respect to each other. Then, suppose that event e₁ is intersected by Mary’s simultaneity hyperplane; it follows that e₁ is real with respect to her. The transitivity of the relation ‘x is real with respect to y’ entails that, since e₁ is real with respect to Mary, and Mary is real with respect to Max, e₁ is also real with respect to Max. However, given the relativity of simultaneity, it could be that, although e₁ is simultaneous to Mary, it is located in the absolute future of Max. Indeed, since Max and Mary are in motion relative to one another, SR predicts that they will have different perspectives on whether some events, such as e₁, lie in the present or in the future. The situation is depicted in Fig. 4.5: the unprimed frame represents Max’s perspective, and the primed frame (which is moving relative to the unprimed frame) represents Mary’s perspective. Now, assuming that GBT is true, Max finds himself in an impossible situation: on the one hand, he must take e₁ to be unreal, since it lies in his absolute future, but on the other hand, he must take e₁ to be real, since Mary takes it to be real and she is real with respect to him. This apparent contradiction leads Putnam to conclude that “[…] future things (or events) are already real” (1967: 242).²⁹

Fig. 4.5

The Putnam-Rietdijk argument illustrated

The Putnam-Rietdijk argument seems to threaten the 2 essential components of GBT introduced in the previous chapter: (i) the asymmetry between the future and the past, and (ii) temporal becoming (cf. Sect. 3.4). In particular, since no frame of reference is privileged, it seems that all of the hyperplanes in the representation of the world are metaphysically on a par and, therefore, that there is no way to make sense of the thesis according to which there must be a geometric asymmetry between the future and the past (in order to ground some widespread intuitions about the nature of time) – the asymmetry collapses. More specifically, since no frame is privileged, no single hyperplane in the representation can be picked out from the set as being the unique ‘present’ axis around which reflection symmetry can be operated. The asymmetric-theories family, to which GBT belongs, seems therefore wrongheaded. Moreover, it seems that there is no way to make sense of temporal becoming, according to which new events are created in the present. After all, to say that temporal becoming is real is to say that “[…] there is something privileged about one of the many hyper-planes represented by the four-dimensional manifold” (Miller, 2013: 353), namely the latest hyperplane added to the growing block, which seems incompatible with SR – temporal becoming collapses. Mauro Dorato expresses the same thing in a slightly different way: “[t]o the extent that the notion of temporal becoming presupposes the unreality of future events as its necessary condition, [SR] seems to rule out also temporal becoming” (2008: 59).

However, although the Putnam-Rietdijk argument looks powerful, there are various options to avoid its conclusion. These options can be grouped into two families: the compatibilist and the incompatibilist options. Whereas the compatibilist options state that any metaphysical view of the world must be compatible with the fact that SR is approximately true, the incompatibilist options deny this. In the rest of the section, I will review some of these options.³⁰ The aim is not be to identify one specific option as our best chance to defeat the Putnam-Rietdijk argument, but rather to show that the great variety of options strongly suggests that this argument is not definitive; there are plenty of ways that growing blockers can take to escape the pressure it exerts. In particular, I will discuss the incompatibilist options to be found in Zimmerman (2008), Bourne (2006), Miller (2013), and Tooley (1997), and the compatibilist options to be found in Sklar (1974), Stein (1968, 1991), and Correia and Rosenkranz (2018). At the end of the section, I will acknowledge that my preference goes to the incompatibilist family of options, because it seems better suited to accommodate both some of our intuitions about time, and some considerations from quantum mechanics (e.g., experimental results connected with John Bell’s theorem). This preference will lead me to introduce, in the sixth section of the chapter, the causal set theory (CST), which, although still in its infancy, offers hope for reconciliation between GBT and our best physics.

First, let us focus on the incompatibilist options, which contest the metaphysical relevance of SR. These options come in 2 forms: (i) some philosophers argue that, although SR speaks to the geometry of spacetime, it has no ontological import, whereas (ii) some others go a step further by arguing that SR (at least as formulated above) is not approximately true of the world.³¹ Dean Zimmerman (2008) choses an incompatibilist option of the first form. He argues that the four-dimensional manifold of spacetime points, which is posited by SR, is merely a theoretical entity and, therefore, should not lead to any ontological conclusion. For instance, the straightness of a worldline, he says, provides no information as to what exists; it merely indicates where an object would go (and where it could come from), if it were undergoing no accelerations (or decelerations). The fundamental information that the four-dimensional manifold gives is therefore not about ontology, but accessibility: it merely tells an object “[…] ‘where to go next’ if it is located at a series of points on the line, and no other forces are at work” (2008: 219). Interpreted in this way (rather than as a contribution to ontology), SR does no longer appear as a threat to GBT. An immediate objection could be that the geometry of spacetime seems fundamental to explain, e.g., the twin paradox (cf. Sect. 4.3); it is therefore not clear how Zimmerman can account for it.

In a similar vein, Craig Bourne (2006) and Kristie Miller (2013) argue that, although SR says that there is no privileged hyperplane, growing blockers are free to reject the idea that this entails that “[…] all of the hyperplanes in our representation of the four-dimensional manifold are metaphysically on a par and hence that each corresponds to an existing hyperplane” (Miller, 2013: 353). Indeed, it is one thing to say that our best physics does not privilege any hyperplane, it is another to say that no hyperplane is metaphysically privileged. After all, the claim that there is a metaphysically privileged hyperplane (which seems both required by the geometric and the dynamic component of GBT) does not necessitate that we have (or could have) any physical or empirical access to which hyperplane is privileged. As Miller sums it up: “[…] what SR tells us is that it is in principle impossible to determine which plane is the metaphysically privileged one. But it does not tell us that no plane is in fact metaphysically privileged” (2013: 353). Growing blockers can therefore argue that, although SR entails that no hyperplane is physically privileged, there actually is a metaphysically privileged hyperplane that no physical or empirical method allows one to detect.

This first option sounds acceptable, but since our defense of GBT rests on our concern for accommodating a basic intuition we have regarding the nature of time (the future is open, while the past is fixed), it would surely be problematic to end up with a theory that says that, although there is a privileged hyperplane, it is empirically impossible to determine which one it is (cf. Prosser, 2000, 2007). As Miller herself notices: “[…] if there is no way to detect which plane is privileged and its being metaphysically privileged makes no empirical difference in the world, then it is hard to see how the fact that a plane is metaphysically privileged could ground our temporal phenomenology” (2013: 353). Indeed, how could it be that something that is empirically undetectable accounts for the manifest image of the world? It therefore seems that either the metaphysically privileged hyperplane is empirically salient, or GBT fails to account for our pre-theoretic thoughts about the nature of time. Thus, since one takes (at least part of) our temporal phenomenology seriously (i.e. as describing how the world truly is), and that this constitutes the core motivation for accepting GBT, it seems that one should reject Bourne-Miller’s way of addressing the challenge raised by the Putnam-Rietdijk argument.

Instead, one could adopt an incompatibilist option of the second form and argue that, since there clearly seems to be a single universe-wide border between the past and the future, which coincides with the present, any theory that (frame-)relativizes it must be false (or incomplete). This seems corroborated by various considerations from quantum mechanics, especially by experimental results connected with John Bell’s theorem, which suggest that some tenets of SR must be given up.³² Without going into details, it indeed seems that if there are truly instantaneous connections between two correlated particles, A and B, at different places in space, then the objective notion of absolute simultaneity has a real application. For example, if B acquires determinate values (e.g., a determinate spin) by an independent measurement performed simultaneously on A (irrespective of the spatial separation of the particles), then, since this seems to require A and B to have a superluminal causal connection (sometimes called ‘spooky-action-at-a-distance’), it can be concluded that SR, which explicitly rules out such connections, fails to provide a complete account of the spatiotemporal connections that actually exist.³³ In such circumstances, a natural belief is that SR will one day meet the same fate as Newtonian mechanics, i.e. being replaced by a theory with superior predictive and explanatory power. Such a theory will crucially diverge from SR in entailing that some events in our world do stand in relation of absolute simultaneity. Of course, this latter claim can be criticized, especially on the ground that relativistic empirical effects (e.g., ‘length contractions’ and ‘time dilations’) are well confirmed, and therefore that the successor theory will also have to account for them. The question is then whether this can be done with an objective notion of absolute simultaneity.

Some philosophers answer ‘yes’ to the latter question. For example, Michael Tooley sets himself the task of developing an alternative to SR, since “[…] the Special theory of Relativity does not provide a complete account of the spatiotemporal relations that obtain between events” (1997: 338). In his book Time, Tense and Causation, Tooley presents a theory that, although compatible with all the empirical data that are usually taken to confirm SR, entails that some events stand in relation of absolute simultaneity with respect to each other. This theory depicts an absolute substantival spacetime (i.e. a spacetime that is not reducible to spatiotemporal relations between events) that is causally self-propagating over time. As Tooley puts it: “[…] the fundamental argument for the central thesis that the world is a dynamic one in which the past and the present are real, but the future is not, rests upon a claim concerning the nature of causation – namely, that causation is a theoretical relation between events, and one whose basic postulates can only be satisfied in a dynamic world of the type in question” (1997: 376). Specifically, Tooley’s theory is constructed from two well-established Einsteinian definitions:

• (i) The speed of light, which is constant relative to absolute space,

• (ii) The relation of simultaneity, according to which two events, e₁ and e₂, occur simultaneously relative to some frame of reference F iff light emanating from each object would arrive at an object O – which is both equidistant and at rest within F – at the same time.

But, whereas Einstein assumes that all inertial frames are on an equal footing and, therefore, that simultaneity is relative, Tooley denies this. According to him, if space is absolute, then some inertial frames should be at rest relative to it, so that one can define an objective notion of absolute simultaneity: “[t]wo events, [e₁] and [e₂], are absolutely simultaneous means the same as [e₁] and [e₂] are simultaneous relative to some frame of reference that is at rest with respect to absolute space” (1997: 343–344).

Of course, the possibility of absolute simultaneity comes with consequences. The most important concerns the speed of light: whereas Einstein assumes that the numerical value of the speed of light is the same when measured in all directions in all frames of references (cf. the Light Postulate), Tooley notices that this can no longer be the case. In his theory, a light signal will have a speed of (c - v) in a frame moving at velocity v in the same direction as the signal, whereas the signal will have a speed of (c + v) in a frame moving at velocity v in the opposite direction to it (cf. 1997: 345–346). However, as Tooley insists, the rejection of the Light Postulate is not contradicted by empirical data. Actually, what empirical data support is a weaker postulate, the Round-Trip Light Postulate, according to which observers in all inertial frames will agree on the round-trip speed of a light signal travelling from any location L₁ to any other location L₂ and back again. By contrast, the Light Postulate rests, as Einstein (1905) himself concedes, on a mere convention, namely that the one-way speed of light (from L₁ to L₂) is constant in all inertial frames. Whereas many experiments have been undertaken to prove that one-way speed of light is constant, none has been successful yet; it is merely regarded as reasonable assumption.³⁴ Tooley feels therefore free to conclude that light might well travel at different speeds in inertial frames that are moving relative to absolute space.

This conclusion forces Tooley to explain why no such variations in the speed of light have ever been detected – recall the Michelson-Morley outcome. His answer relies on a Lorentz-style compensatory theory, according to which the variations in light speed are systematically concealed by the way natural processes are affected in moving frames. In short, since the original Lorentz transformations presuppose that the one-way speed of light is the same in all inertial frames, Tooley replaces them with new transformations, commonly referred to as ‘∈ − Lorentz transformations’ (cf. Reichenbach, 1957: 127). Following John Winnie (1970), these new transformations are derived from a suitably modified version of SR (which has been shown to be inconsistent only if the original version of SR is itself inconsistent) that does not entail the one-way Light Principle. The ∈ − Lorentz transformations allow Tooley to calculate the necessary compensation corresponding to different assumptions (captured by the variable ‘∈’) concerning the relevant one-way speed of light (cf. Tooley, 1997: 349, and also Dainton, 2010: 337–342).

The details would bring us too far, but it is worth knowing that Tooley’s neo-Lorentzian theory, enriched with further principles,³⁵ can account for all the same empirical effects as SR (including ‘length contractions’ and ‘time dilations’). Now, although there is no definitive physical proof for there being an objective relation of absolute simultaneity,³⁶ Tooley’s theory might seem preferable to SR, since it is better suited to accommodate (i) a single universe-wide border between the past and the future (which seems reflected by our basic intuitions about time), and (ii) some considerations from quantum mechanics, especially experimental results connected with John Bell’s theorem, which provide strong reasons to believe that “[…] there is sometimes no temporal gap between spatially separated events that are nomologically connected” (1997: 354).³⁷ However, despite these attractive features, Tooley’s theory is often criticized for being both too revisionist and too costly, especially regarding its neo-Lorentzian nature. Christian Wüthrich, for instance, writes that “[Tooley’s theory] violates Ockham’s razor so crassly that the move cannot be justified by putting some post-verificationist philosophy of science on one’s flag” (2010: 264). Ideally, the incompatibilist strategy would therefore be better served by a theory that retains the Lorentz invariance; this is precisely one of the qualities of the causal set theory, which will be explored in Sect. 4.6.

Second, let us focus on the compatibilist options, which take the metaphysical relevance of SR for granted. These options also come in 2 forms: (i) some philosophers challenge the premises of the Putnam-Rietdijk argument; (ii) some others reformulate GBT to make it expressible in relativistic settings. Lawrence Sklar (1974) choses a compatibilist option of the first form. He resists the claim that the non-physical relation ‘x is real with respect to y’ is transitive. In that sense, from the fact that e₁ is real with respect to Mary (via the primed frame) and that Mary is real with respect to Max, it does not follow that e₁ is real with respect to Max (cf. Fig. 4.5). It indeed seems that, as far as the relation ‘x is real with respect to y’ is intransitive, a growing blocker can acknowledge that different events exist relative to different frames, without being committed to the existence of events that are, from her frame of reference, located in the future. Sklar’s move, which implies that what exists at a distance depends on a state of motion, seems thus to immediately block the Putnam-Rietdijk argument.

Of course, one might be reluctant to treat the relation ‘x is real with respect to y’ as intransitive. One reason is that the rejection of transitivity comes with counterintuitive consequences. For example, “[…] two observers zooming past each other would share the same present without sharing what is real at a distance, and by simply changing reference frame (getting off a bus or jumping on an airplane), we would change what counts as real for us at a distance” (Dorato, 2008: 60). But one has to remember what SR taught us: all of our talk must be frame-relativized. Keeping this in mind, there seems to be no a priori reason to think that our talk about existence should be an exception – though Gödel famously claims that “[t]he concept of existence […] cannot be relativized without destroying its meaning completely” (1949b: 558).³⁸ This reply may seem disappointing, but it must be recognized that one cannot have one’s cake and eat it too: if one chooses the compatibilist path and, therefore, if one takes the metaphysical relevance of counterintuitive SR for granted, then there will be counterintuitive consequences, and the intransitivity of the relation ‘x is real with respect to y’ might be one of them.

For those who regard the intransitivity of the relation ‘x is real with respect to y’ as unacceptable but, at the same time, are inclined towards the compatibilist strategy, the situation is not altogether desperate. A compatibilist option of the second form has been developed by Howard Stein (1968, 1991), who aims to reconcile the unreality of future events with the idea that SR forces us to abandon the objective notion of absolute simultaneity. Basically, his idea is to show that events can be partitioned into past, present, and future, in a way that respects the geometric structure of Minkowski spacetime. To make that case, Stein establishes that, for any given point x, the only points that are real (or definite)³⁹ with respect to x are those that lie in x’s past light-cone. Conversely, x’s future light-cone and x’s absolute elsewhere contain points that are unreal (or indefinite) with respect to x.⁴⁰ In this context, ‘being real (or definite)’ is a binary relation between point-like events: Rxy = _df ‘y is real (or definite) with respect to x’. This relation is non-universal (for all events x of spacetime, there are events y such that ¬Rxy), reflexive, and transitive. Then, Stein identifies the spatially extended present with the set of events on the past light-cone of the here-now. The latter proposition is motivated by an appreciation of epistemic accessibility: causal signals reaching us now emanate from the events on the past light-cone, and thus appear to us as being present. An immediate consequence of this option is the loss of a single universe-wide border between what is real (or definite) and what is not: two observers will not fully agree on what is real, unless their past light-cones fully coincide. There are therefore different perspectives for each observer, and what is real (or definite) for each of them is different.

Of course, Stein’s theory provides no evidence that temporal becoming actually occurs in Minkowski spacetime, but it shows at least that, contrary to what the Putnam-Rietdijk argument suggests, the possibility of temporal becoming is not ruled out by SR. In particular, if one accepts to conceptually distinguish the notions of ‘temporal becoming’ and ‘spatially extended present’, so that the former does not necessarily requires the latter, SR no longer appears as a hostile environment for the creation of new things (e.g., spacetime points, events) in the present. Moreover, as suggested above, Stein’s theory allows one to preserve the transitivity of the relation ‘x is real with respect to y’. Indeed, given that a point can only be real with respect to points that are in its causal past, although it follows from Rab and Rbc that Rac (transitivity), there is no risk of ‘spreading reality’ from the past to the future, since c will also be in the absolute past with respect to a. By contrast, the relation ‘x is real with respect to y’ is not symmetric: if b is real with respect to a (because b lies in a’s causal past), it does not follow that a is real with respect to b; otherwise, we would end up in the situation where all points are equally real with respect to each other. This is no big deal, however, since every version of GBT (including Broad and Tooley’s versions) must anyway take the relation ‘x is real with respect to y’ to be non-symmetric: it is a milestone of GBT that “[…] 1900 is real as of 2000, but as of 1900, the year 2000 is not real” (Dainton, 2010: 333).

At first sight, Stein’s theory appears to be coherent, but it comes with important costs. First, although Dorato (1995, 2008), Dainton (2010), and Correia and Rosenkranz (2018) provide some suggestive descriptions, it might be difficult to visualize what this theory amounts to; intuitive pictorial flesh can hardly be put on it, or so it could be argued.⁴¹ Second, as Christian Wüthrich and Craig Callender point out, Stein’s theory has deeply counterintuitive consequences. For instance, it entails that “[…] if we take the past lightcone as the present, then the big bang counts […] as ‘now’” (2016: 5). However, the latter objection can be avoided simply by denying that the present is identical to the set of events in the past light-cone of the here-now. Indeed, following Dorato (2008) and Correia and Rosenkranz (2018), Stein’s theory is more charitably interpreted as a pointy relativistic version of GBT. Third, following Savitt (2000: 568), Correia and Rosenkranz (2018: 151–152) argue that this pointy relativistic version of GBT leaves no room for the existence of spatiotemporally-extended things that one can perceive. They provide the following example. I (understood as an embodied consciousness occupying a single spacetime point) can here-now perceive my limbs to be located at distinct spacetime points. Obviously, since the light signals involved in this perception take time to reach me, what I actually perceive is that, somewhere in the immediate causal past, my limbs occupy distinct spacetime points. This might not appear problematic since, unlike here-now, the causal past is thick enough to comprise several spacetime points, but the situation is quite different when persistence comes into the picture. Since I have every reason to believe that my limbs have persisted and, therefore, that they exist here-now, the question that arises is: ‘Where are my limbs located?’. No convincing answer is available to Stein. Whereas I here-now perceive my limbs to have been located at distinct points in the causal past of here-now, Stein’s theory provides no plurality of points in the causal future of those points that my limbs could occupy here-now.

Finally, a third compatibilist option has been developed by Fabrice Correia and Sven Rosenkranz (2018). In their book Nothing to come, they reformulate GBT without using the problematic notion of presentness, so the theory can accommodate relativistic spacetime structure. To make that case, Correia and Rosenkranz define GBT as describing a world in constant creation in which nothing ever gets annihilated: while new things continuously come into existence, nothing ever goes out of existence. This results from the acceptance of 2 principles that we have already encountered in Sect. 3.3:

• (P1) Always everything will always in the future be something (E!x →GE!x)

• (P2) Every time is new at itself’ (Tx → At x, H¬E!x).⁴²

This reformulation brings many benefits. First, (P1) and (P2) are apparently sufficient to capture the idea of a single edge of becoming beyond which nothing exists (cf. 2018: 44–45). Second, since neither (P1) nor (P2) invokes a notion of ‘being present’, this version of GBT avoids the main pitfalls encountered by traditional attempts to spell out this notion (cf. Sect. 3.3). Third, when physics come into the picture, this version of GBT seems well suited to meet the relativistic requirements. In particular, since no objective notion of absolute simultaneity is required to express (P1) and (P2), it dissipates the tension highlighted by the Putnam-Rietdijk argument. However, this cannot be the end of the story: the process of ‘constant creation’, involved in this version of GBT, rests on an absolute temporal order relation, which seems inconsistent with relativistic conceptions of spacetime. That is why Correia and Rosenkranz propose to characterize GBT in a spacetime-sensitive language definable in relativistic terms. Roughly, this means that (P1) and (P2) should be replaced with the following two relativistic principles:

• (P1_R) Everywhere everything everywhere in the causal future still exists (E!x → ▾E!x, where ‘▾φ’ stands for ‘Everywhere in the causal future, φ’).

• (P2_R) For any spacetime point s, at s, everywhere in the causal past of s, s did not yet exist’ (Sx → @ x▴¬E!x, where ‘S’ abbreviates the predicate ‘is a spacetime-point’, and ‘@ m’ abbreviates ‘At spacetime-point m’).

Specifically, (P1_R) and (P2_R) imply that, for any spacetime point s, there are spacetime points in the causal past of s, but no spacetime points in its causal future (cf. 2018: 149). As a last step, Correia and Rosenkranz add to (P1_R) and (P2_R) the principle (BO) ‘for any spacetime point s, at s, everywhere in the elsewhere region of s, s exists’ (Sx → @ x◂E!x, where ‘◂’ stands for ‘Everywhere in the elsewhere region’), so they can derive that, for any spacetime point s, the elsewhere region of s may be populated.⁴³ This brings a further benefit: unlike Stein’s theory, this theory can account for the persistence of non-causally separated things that we can perceive. In particular, it allows for the existence, here-now, of several spacetime points that my limbs, which I perceive to have been located at distinct points in the causal past, can occupy. The similarities and differences between Stein’s relativistic version of GBT and that of Correia and Rosenkranz are pictured in Fig. 4.6.

Fig. 4.6

Stein’s pointy relativistic model, and Correia and Rosenkranz’s bow-tie relativistic model. (This figure is freely inspired from Correia & Rosenkranz’s figures 9.3. and 9.4 (cf. 2018: 150))

However, here again, one ends up with a theory, so-called ‘Bow-tie relativistic GBT’, which seems immune to the usual objections drawn from relativistic physics, but which requires us to abandon some of our basic intuitions about time. This is what prompts Ulrich Meyer, for instance, to say that “[…] I am not sure that [Correia and Rosenkranz’s] proposal is what other people had in mind when they endorsed the growing block theory” (2019). For example, since it allows for two points, which are causally separated from here-now, to be in a relation of causal precedence, Bow-tie relativistic GBT entails that the relation ‘x is real with respect to y’ is intransitive. This clearly betrays some of our common intuitions, especially the intuition that “[i]f it is the case that all and only the things that stand in a certain relation R to me-now are real, and you-now are also real, then it is also the case that all and only the things that stand in the relation R to you-now are real” (Putnam, 1967: 241). Although Correia and Rosenkranz contest the appeal to common intuitions in those circumstances,⁴⁴ one might be reluctant to pay such a price for the reconciliation of GBT with our best physics. For, to sacrifice our intuitions is to sacrifice what made GBT attractive in the first place.

At this step, the idea is not to defend one option (either compatibilist or incompatibilist) as the best way to meet the challenge raised by the Putnam-Rietdijk argument. Rather, the idea is to show that, although the objection to GBT based on relativity is troubling – mainly because the progress of science has taught us to be extremely wary about the deliverances of our intuitions – there are various reasons to think that SR does not extinguish all hope of adequately describing reality as growing. Both incompatibilist and compatibilist options are available to defuse the tension between SR and GBT; they are summarized in Fig. 4.7. Nonetheless, I am inclined to think that incompatibilist options are more promising than compatibilist ones, because, as far as our intuitions are concerned, accepting SR (at least as formulated above) requires a tremendous amount of sacrifices. Does the previous statement undermine any attempt to reconcile the manifest image with our best physics? The answer is ‘not necessarily’, since it is far from clear that SR belongs to our best physics. As has been said, many considerations from quantum mechanics (e.g., experimental results connected with John Bell’s theorem) lead us to think that SR should be replaced by a theory with superior predictive and explanatory power. A reply might be that SR is anyway not the final answer that Einstein provides to the question of the nature of spacetime. But, as will be shown in the next section, the General theory of relativity (GR), which has been proved incomplete anyway,⁴⁵ also encounters important difficulties when squaring it with quantum mechanics. That is (partly) why we will turn our attention to the nascent theories of quantum gravity in Sect. 4.6, and especially to the causal set theory (CST). As will become clear, this final move should be regarded as a representative of the incompatibilist options.⁴⁶

Fig. 4.7

Incompatibilist and compatibilist options summarized

4.5 Beyond Special Relativity

In the brief history of modern physics that has been told so far, two successive phases were identified. The first phase was (neo-)Newtonian mechanics, which, since it takes B-relations like ‘earlier than, later than or simultaneous with’ to be absolute (or frame-invariant), allows for an objective and absolute temporal ordering of events. The second phase was Special Relativity (SR), which, since it admits no objective notion of absolute simultaneity, comes with a loss of comparability: for two space-like separated events, e₁ and e₂, there no longer is a frame-independent fact of the matter as to whether e₁ is, for instance, earlier than e₂. Accordingly, the temporal ordering of events is only partial. In this section, one will introduce a third phase, General Relativity (GR), which makes things even more radical: since the topology of GR allows for the possibility of closed time-like curves, it entails that the temporal ordering of events is not even partial. As Christian Wüthrich puts it: “[i]n fact, there is no global time deserving this title in general relativity (GR), a fact that finds a particularly vivid expression in the so-called ‘problem of time’ arising in the Hamiltonian formulation of GR” (2010: 260). There therefore is no reason to believe that GR may help the compatibilists, who rightly point out that counterintuitive SR is not the final answer that Einstein provides to the question of the nature of spacetime, to restore the manifest image. Worse, since GR also requires ‘relativistic causality’ (i.e. that causal signals can travel at most as fast as light), it seems to be in no better position than SR when squaring it with quantum mechanics. As a reminder, Bell’s theorem predicts that quantum mechanics is non-local, in the sense that a measurement on a system by an observer at one location has an instantaneous effect on a distant correlated system (one with which the original system has interacted).

But let’s start with some basics. Ten years of hard work after the development of SR, Einstein completed a new theory, General Relativity (GR), which is not to be understood as a disavowal of his early work, but rather as an attempt to overcome its limited aspirations. Whereas SR provides a rigorous account of the difference between accelerated and inertial motions (reflected by Minkowski spacetime’s ‘affine’ structure) and of the dynamics of moving objects, it remains silent on one crucial phenomenon: gravitation. The reason is roughly that Newton’s theory of gravitation, which was then widely accepted, could hardly be slotted into SR’s framework, because it is incompatible with (i) some relativistic effects (such as ‘length contractions’ and ‘time dilations’), and (ii) the Light Postulate. This is easily understandable. The strength of Newtonian gravitational force between two objects depends on their distance apart, while SR states that this distance will be different in the inertial frames of the relevant objects (assuming that they are moving relative to one another). Therefore, two observers on the two objects will obtain different results when measuring the strength of the gravitational attraction between them. Moreover, Newtonian gravitation acts instantaneously between all material objects, irrespective of their spatial separation, while this is incompatible with SR’s limited causal propagation – recall the light-cone structure of Minkowski spacetime (cf. Sect. 4.3). Behind the apparent incompatibility between Newton’s theory of gravitation and relativity lies a fundamental disagreement on the nature of gravitation: unlike Newton, Einstein does not regard it as a force (that allows for instantaneous ‘action-at-a-distance’). Instead, Einstein will propose a new conception of gravitation, according to which gravitation is a mass-induced spacetime curvature, that is at the core of GR.

Basically, GR rests on the extension of the Relativity Postulate from inertial motion to accelerated motion (or so Einstein thought): the laws of nature do not distinguish between frames that are freely falling in gravitational fields and inertial frames. The reason is that “[…] the inertia of a test mass is increased if it is surrounded by a shell of inertial masses and that, if these same masses are accelerated, they tend to drag the test mass with it” (Norton, 1993: 799). For example, if Mary is in a windowless spaceship (enjoying weightless conditions), there is nothing she can do to determine her state of motion: perhaps she is accelerating towards a nearby black hole, perhaps she is drifting in space. In either case, all the experiments she could perform within her spaceship will yield exactly the same results. This is known as the Equivalence Principle, which was originally expressed as follows:

[…] we assume that the systems K [inertial system in a homogeneous gravitational field] and K’ [uniformly accelerated system in gravitation free space] are physically exactly equivalent, that is, […] we assume that we may just as well regard the system K a s being in a space free from gravitational fields, if we then regard K as uniformly accelerated (Einstein, 1911, §1).⁴⁷

A significant consequence of this principle (which is also true in Newton’s theory) is that it makes distinguishing between gravitational and acceleration effects impossible. For example, if Mary realizes that she is no longer weightless (and that objects around her are no longer floating in mid-air, but are lying on the floor), no experiment she could carry out in her windowless spaceship would help her to determine whether this be a consequence of gravity (e.g., her spaceship is motionless near a planet) or acceleration (e.g., her spaceship is accelerated by a rocket). This led Einstein to infer that gravity and acceleration might essentially be the same phenomenon: a modification of the very structure of spacetime.

Another consequence of the Equivalence Principle directly impacts light: although light is known to be massless, the paths it takes are deflected by gravity. In particular, given that a light signal travelling in an accelerating spaceship will follow a curved path (e.g., it will hit the opposing wall at a lower position than if the spaceship were not moving), and that gravity has the same (local) effects as acceleration, a light signal travelling in a stationary spaceship within a gravitational field will follow a similar trajectory (e.g., it will also hit the opposing wall lower down). This observation has enormous consequences. As a reminder, Minkowski spacetime is built around the paths that light can take (cf. Sect. 4.3). In that sense, these paths do not only define the absolute future and past of every point, but they are also the paths of the shortest possible distance: “[…] the interval between light-like connected points is zero (and so there is, from the perspective of light, no spatial or temporal distance between them at all)” (Dainton, 2010: 346). Therefore, supposing that (i) Minkowski spacetime is approximately true of the world, and (ii) gravity has the same (local) effects as acceleration, it can be inferred that gravity affects the structure of spacetime itself. Actually, Einstein even goes a step further by arguing that gravity is nothing but the warping of spacetime: the effects that Newton explained in terms of an attractive force operating over material objects are to be understood as the result of matter bending spacetime in its vicinity – the more matter, the greater the distortion. And this matter-induced distortion is not transmitted through spacetime instantaneously, but at the speed of light; Einstein thus gets rid of action-at-a-distance.

Specifically, a large concentration of matter induces a strong curvature of spacetime in its immediate vicinity, and the curvature is transmitted through spacetime from region to region (which are not directly affected by the matter) in a gradually weaker way. As Dainton puts it: “[a]s you get further away from the mass, the curvature of each successive spherical region of spacetime diminished” (2010: 347). The overall shape of spacetime is therefore the product of the combined influence of all the material objects, and this shape evolves as the objects move. The following example might help to clarify the situation. As has been seen, the natural path of an object, which is not subject to any force, is a straight line (cf. Sects. 4.2 and 4.3). A spaceship moving along its natural path will therefore continue to do so (without slowing or speeding up) until some external force acts on it. Now, suppose that this spaceship quickly flies past a planet; its trajectory will be deflected. The planet will cause the spaceship to move in a curve rather than a straight line. Yet, GR states that the spaceship does not experience any external force (assuming that it is not accelerated by its engines). How is that possible? GR’s answer is this: the planet’s mass alters the natural path of the spaceship. Counterintuitively, the planet does not pull the spaceship toward it (pace Newton), but affects the geodesics⁴⁸ of spacetime (which deviate from Euclidean straight lines), and hence modifies its shape. As a consequence, gravity produces not only space-bending effects, but also time-dilation effects: “[…] clocks tick more slowly in the vicinity of large material objects; the stronger the spacetime curvature, the slower the clocks tick” (Dainton, 2010: 350).

This could make GR’s spacetime look very different from both Newton and Minkowski’s spacetimes, which are entirely unaffected by the presence and distribution of matter within them. But it is worth noting that, in GR, the metric can always be approximated to the Minkowski form, at least in small regions of spacetime. However, on the large scale, gravity can no longer be neglected and, therefore, the manifold should not be expected to be flat. The light-cones must be bent toward the location of mass, with greater curvature near larger masses. As will become clear, many different topologies are consistent with what equations of GR (or Einstein field equations, thereafter) tell us about matter-distributions and spacetime curvatures, and some of them are really bizarre. For instance, GR does not rule out the possibility of ‘closed time-like curves’ (i.e. paths through spacetime that loop back upon themselves), although such non-orientable scenarios are often regarded as “not physically real” or “pathological” (but not “unphysical”) (cf. Zimmerman, 2011: 188).

But first things first: what is a spacetime within GR? A spacetime is a four-dimensional manifold with a Lorentzian geometry. The Lorentzian geometry gives spacetime its light-cone structure. Assuming that spacetime is globally hyperbolic (which corresponds to a certain condition on the causal structure of a variably curved spacetime), the causal interpretation of light-cones is still valid in GR: the future light-cone of a certain event includes the boundary of its causal future (and similarly for the past). But, given gravitational lensing, which occurs when a huge amount of matter (e.g., a cluster of galaxies) creates a gravitational field that distorts light, it might be that the light-cone folds on itself. The various ways in which spacetime can bend and twist are captured by complex equations: Einstein’s field equations. They were first formulated in the form of a tensor equation, which relates the curvature of spacetime (described by the Einstein tensor) with local energy, momentum, and stress, within that spacetime (described by the stress-energy tensor). In other words, Einstein’s field equations establish a systematic relationship between the geometry of spacetime and the distribution of mass-energy through spacetime. They thus determine the metric tensor of spacetime (i.e. all the geometric and causal structure) for a given arrangement of stress-energy-momentum in the spacetime. The solutions of Einstein’s field equations are the components of the metric tensor. It is worth noting that, since the Einstein tensor has ten independent components, the Einstein field equations can be written as a set of ten non-linear partial differential equations; this will be useful in Sect. 4.6 to understand how causal set theorists retrieve the metric, the topology, and the differential structure of the manifold from the causal structure and volume information.

One of the basic principles of GR is its general covariance: its laws remain unchanged under an arbitrary transformation of the spacetime coordinates. This means that, in GR, “[…] coordinates on spacetime have no physical significance, no more significance than a choice of coordinate grid on a map of Mexico City, say” (Dowker, 2020: 145). This is, in particular, the aversion to violation of general covariance, which leads most philosophers of physics to claim that GR forces a ‘block universe’ view on us. It must indeed be acknowledged that GBT, for instance, contradicts general covariance by providing a foliation of spacetime into space-like hypersurfaces: “[i]t picks out a special time coordinate labelling the leaves of the foliation and gives it physical significance” (Dowker, 2020: 145). In that respect, GR seems just as at odds with our intuitions as its predecessor. After all, whether it is in GR or SR, simultaneity (when defined by Einstein’s light-signaling method) is relative, although this relativity is expressed differently.⁴⁹ Therefore, a hypersurface of becoming, as depicted by traditional versions of GBT (to account for some of our intuitions), is equally problematic in both theories.⁵⁰ Worse, since GR allows for the possibility of ‘closed time-like curves’ (and hence entails that the temporal ordering of events is not even partial), the situation even appears more unfavorable for GBT than it was in SR. For example, Kurt Gödel (1949a) argues that the mere fact that closed time-like curves are possible suffices to establish the ‘block universe’ view of time: if closed time-like curves can be produced in our universe by rearranging matter,⁵¹ then the past and future times to which these curves would provide access must be real. This does not mean that there is no version of GBT that can accommodate relativistic spacetime (although most versions of GBT take the causal precedence relation to be irreflexive, and hence exclude closed causal curves),⁵² but this means that these versions of GBT must make so many concessions to relativity that they inevitably lose the ‘intuitiveness’ that made GBT attractive in the first place.

Thus, it seems that the compatibilist strategy – which rests on the idea that the theory of relativity is (approximately) true of the world and, therefore, that any metaphysical view of the world must be compatible with it – condemns us to sacrifice our most basic intuitions on the nature of time. Taking that for granted, it seems that the only real hope to preserve our intuitions from the pressure exerted by physics is to opt for an incompatibilist strategy: although GR is undoubtedly closer to the truth of the world than SR (it accommodates the gravitational effects, after all), it fails to provide an accurate description of it. But is there any naturalistic reason to privilege such an incompatibilist strategy? The answer seems to be ‘yes’. Given that the causal interpretation of light-cones is still valid in GR, causal processes or signals can travel at most as fast as light, while this requirement, sometimes referred to as ‘relativistic causality’, can apparently be violated in the quantum context. Indeed, the interpretative problems of any quantum theory about non-locality and the measurement problem⁵³ are widely regarded as threatening relativistic causality. As, for instance, Jeremy Butterfield puts it: “[…] non-locality looks like ‘spooky action-at-a-distance’; and if measurement involves a ‘collapse of the wave-packet’ perhaps the collapse is superluminal’” (2007: 302, cf. also Maudlin, 2011: 187).

Specifically, since relativistic spacetime typically has no preferred foliation or ‘slicing’, it conflicts with at least 2 considerations from quantum mechanics. First, the correlation (or entanglement) of distant systems, which is widely accepted as a quantifiable and exploitable physical resource, produces a violation of Bell’s inequality in quantum theory. In that sense, whereas Erwin Schrödinger (in agreement with Albert Einstein) claims that “[m]easurements on separated systems cannot directly influence each other – that would be magic” (1935: 161), John Bell proved that the magic is real: although GR prohibits anything from traveling faster than light, it must be recognized that non-locality is inherent in the quantum theory; a measurement on a system at one location has an instantaneous effect on a distant correlated system. Second, a collapse of the wave-packet, which refers to an irreducible change in the state of an isolated quantum system (contravening the deterministic and continuous evolution prescribed by the Schrödinger equation),⁵⁴ makes use of absolute simultaneity in specifying the collapses of the quantum state. For example, as Maudlin (2019) makes clear, the dynamics of GRW (the most prominent collapse theory) employs the notion of absolute simultaneity in specifying the collapse dynamics.⁵⁵ Admittedly, the collapse of the wave-packet represents just one of the several families of interpretations of quantum theory, but it is not clear that the other families of interpretations are in a better position. For instance, Bohmian mechanics, which adds to quantum theory’s deterministic evolution of the wave-function the postulate that certain preferred quantities have at all times a definite value, presupposes an absolute time structure: it “[…] makes essential use of the objective time order of distant events” (Maudlin, 2019: 205). These considerations strongly suggest that GR, which explicitly rules out the possibility of absolute simultaneity, is not the final word on the nature of spacetime. Interestingly, the issue of non-locality and that of measurement are interrelated: non-locality appears when one solves the measurement problem. For example, “[i]f one resolves the measurement problem by allowing a real physical process of wave collapse, it is the collapse dynamics which manifests the non-locality […]. If one resolves the measurement problem by postulating additional variables beside the wave function, it is the dynamics of these variables which manifests the non-locality […]” (Maudlin, 2011: xi).

Before we go any further, it is important to recall the current state of knowledge in physics. SR is, at best, only approximately true. For more than one century now, GR has looked like our best theory of the structure of spacetime. Progressively, the difficulties in squaring GR with quantum mechanics have become salient to more and more researchers working on the fundamental structure of reality. An idea might therefore be that our most fundamental physics is rather to be found in the nascent theories of quantum gravity. This idea serves the incompatibilist strategy: perhaps, as the Putnam-Rietdijk argument points out, GBT (or more accurately the unreality of the future) is at odds with Einstein’s theories, but the fact that GR is probably not the final word on spacetime structure suggests that a more relevant question to ask is whether GBT is compatible with the development of a quantum theory of gravity. After all, it might be that both SR and GR are hostile to (at least intuitive versions of) GBT, but that a specific quantum theory of gravity is hospitable to it. To put it another way, even if GBT turns out to be incompatible with both SR and GR (which again is doubtful), it in no way follows that GBT is incompatible with our most fundamental physics. In this respect, the next section will be devoted to studying of the causal sets theory, which promises to offer (through a still-classical dynamics) a naturalistic basis to GBT (without betraying the principle of general covariance).

4.6 Quantum Gravity and the Revival of Temporal Becoming

According to many theoretical physicists, one of the yet outstanding tasks in fundamental physics is the development of a quantum theory of gravity. The so far unsuccessful attempt to develop such a theory is an attempt to unify the General theory of relativity (GR) with the principles of quantum mechanics (QM). In brief, Quantum Gravity aims to describe everything in the universe in terms of Quantum Theory.⁵⁶ Although the quest for unification is often presented as the main motivation behind the search for Quantum Gravity, it is worth noting that it is not the only motivation, and perhaps not even a good motivation (since it rests on a purely inductive basis).⁵⁷ As many physicists insist, the most compelling reason why one wants to pursue a quantum theory of gravity is that there are phenomena in which both gravitational and quantum effects should play an irreducible role. For example, it seems that the very early universe and the dynamics of black holes, which both combine high energy densities and strong gravitational fields, cannot be understood without a theory that coherently models the interaction of quantum matter with strong gravitational fields.

Although it should be acknowledged that the search for a quantum theory of gravity is currently dominated by two research programs, String Theory and Loop Quantum Gravity, a third program, called ‘Causal Set Theory’ (CST), arouses increasing interest, especially among those who aim to rescue the manifest image from relativity (cf. Dowker et al., 2004, Sorkin, 2007).⁵⁸ These various research programs reflect divergences of opinion regarding the nature of time; but it seems that these divergences have diminished in the last few years, and that many conclusions have become reasonably clear to most. According to Carlo Rovelli, for example, “[w]hat has been clarified is that the residual temporal scaffolding of general relativity […] falls away if we take quanta into account” (2018: 94). In that sense, there seems to be a minimum scale for all phenomena, including time. The value of minimum time, called ‘Planck time’, is estimated at 10⁻⁴⁴ seconds. This implies that if we could measure the duration of an interval with the most precise clock imaginable, we should find that the time measured takes only certain discrete, special values. As Rovelli says, time should therefore not be thought as “[…] something that flows uniformly but as something that in a certain sense jumps, kangaroo-like, from one value to another” (2018: 96). This is comprehensible: given that granularity is ubiquitous in nature (e.g., light is made of photons, the electrons in atoms can only take on certain discrete values of energy, etc.), it seems natural to suppose that space and time are granular too – though there are also many continuous quantities (at least in quantum mechanics), e.g., position, momentum and the like.⁵⁹

The idea that space and time are fundamentally discrete is not new. It goes back at least to Zeno’s paradoxes (fifth century BC), which aim to show that the limitless divisibility of space and time leads to a contradiction and, therefore, that the apparently evolving reality should rather be conceived as a static, changeless unity. In reply, Leucippus and Democritus argued that the successive division of space and time is not limitless, but terminates in atoms (understood as particles incapable of being further divided) and, therefore, that the continuous is reducible to the discrete (cf. Bell, 2013: §1). In the seventeenth century, Leibniz famously held that space and time, as continua, are ideal, and that anything real, in particular matter, is discrete, compounded of simple unit substances called ‘monads’. In the 19th century, Bernhard Riemann listed some benefits of taking the deep structure of space to be discrete rather than continuous. In particular, he argued that whereas counting the elements composing a region of discrete space provides a natural measure of that region’s volume, a continuous space lacks this possibility and therefore requires that the origin of the metric relationship be explained in some other way. Finally, the subsequent development of physics has provided compelling reasons for questioning the continuum, including the singularities and infinities of GR, QM and black hole thermodynamics. Einstein was, by the way, one of the first to voice doubts of this sort. As he put it: “[i]f the molecular view of matter is the correct (appropriate) one, i.e., if a part of the universe is to be represented by a finite number of moving points, then the continuum of the present theory contains too great a manifold of possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with the quantum theory” (1916: 379). Nowadays, Joseph Henson (2009), for instance, argues that a lack of short-distance cut-offs prevents us from obtaining the finiteness of the semi-classical black hole entropy (cf. Wüthrich, 2013 for discussion).

We find in the previous paragraph some of the basic intuitions that motivated the development of CST: (i) spacetime continuum is not the ultimate reality, (ii) spacetime continuum emerges from a discrete structure (a collection of discrete spacetime points, called the elements of the causal set), (iii) spacetime points (or events) are related by a partial order, and (iv) this partial order has the meaning of the causality relations between spacetime events. Specifically, CST postulates that the fundamental description of spacetime is not a continuum, but some discrete structure of elementary events ordered by a relation of causality, to which the continuum is only an approximation. To put it another way, CST’s main hypothesis is that the spacetime continuum disappears on sufficiently small scales and is superseded by an ordered discrete structure, the causal set (or causet for short), the relation of which with the continuum is conceived as one of coarse-grained, macroscopic representation. CST is, in this sense, nothing more than an attempt to show that “[…] at appropriately large scales, this discrete quantum structure approximates⁶⁰ the smooth metric manifolds that represent spacetime in general relativity” (Wüthrich, 2013: 227). Furthermore, the causet is generally thought as ‘growing’ as new elements are added one by one to the future of already existing elements. This process of ‘growth’, which echoes C. D. Broad’s temporal becoming, is said to unfold in a ‘covariant’ manner, such that it seems compatible with relativity. The claim is then that CST (when augmented with a dynamics) allows one to ‘rescue’ temporal becoming from relativity, and hence to provide a naturalistic basis for GBT. In a nutshell, CST stands out from other approaches to quantum gravity (i) by its conceptual and logical simplicity, (ii) by the fact that it incorporates “[…] the assumption of an underlying spacetime discreteness organically and from the very beginning” and (iii) by the fact that it gives rise to a framework “[…] in which time is an active process of ‘becoming’ that can be identified with the continual birth of new elements of the causal set” (Sorkin, 2006: 1007).

But, what exactly is a causet? A causet is simply an ordered pair < C, ≤> constituted of a set C of elementary events and a binary relation ‘≤’ on C satisfying the following conditions:

• (i) Reflexivity: For all x ∈ C, we have x ≤ x.

• (ii) Anti-symmetry: For all x, y ∈ C, we have x ≤ y and y ≤ x implies x = y.

• (iii) Transitivity: For all x, y, z ∈ C, we have x ≤ y and y ≤ z implies x ≤ z.

• (iv) Locally finiteness: For all x, z ∈ C, we have | {y ∈ C | x ≤ y ≤ z} | < ℵ ₀ (i.e. the cardinality of C has to be less than ℵ ₀).

These simple conditions constitute the basic kinematic assumptions of CST. In particular, the condition of Anti-symmetry prevents the causet from being the equivalent of closed time-like curves,⁶¹ and the condition of Locally finiteness implies that the causet is a discrete structure. The relationship induced by the basic order relation ‘≤’ allows for a variety of interpretations. For example, the relationship ‘x ≤ y’ can variously be described as ‘x precedes y’, ‘x is an ancestor (or a parent) of y’, ‘y is a descendant (or a child) of x’, ‘y lies to the future of x’, or ‘x lies to the past of y’. Physically, this ordering can be thought as a microscopic counterpart of the macroscopic ‘earlier-later than’ relation. The reason why the word ‘causal’ comes into the picture is that, as has been explained in Sect. 4.2, to say that an event e₁ is earlier than an event e₂ is to say that e₁ could exert a causal influence on e₂. These kinematic assumptions suggest that CST is naturally interpreted in structuralist terms, which corroborates the thesis defended in Sect. 3.4, according to which structures are ontologically primary, while individuals (such as spacetime points, events, and objects) have a mere derivative status. Indeed, the elements of a causet naturally appear as “completely featureless events”,⁶² while the relation ‘≤’ is “the only concrete physical relation”. As Wüthrich puts it: “[…] it is thus evident that causal sets offer what is arguably the most straightforwardly structuralist example of a physical entity postulated by any physical theory” (2013: 233).

What is remarkable is that this structure seems sufficient to reproduce the geometry of four-dimensional spacetime (cf. Reichenbach, 1969; Robb, 1936; Zeeman, 1964).⁶³ The details would bring us too far, but the main idea is easy to grasp: given that light-cones can be defined in causal terms and that, in the continuum, the light-cones determine the metric up to a conformal rescaling (cf. Sect. 4.2), it appears that (given minimal regularity conditions)⁶⁴ the causal order of a Lorentzian manifold captures fully the conformal metric, as well as the topology and the differential structure. To understand this, it is worth remembering that, according to GR, the geometry of spacetime (but not its size) is determined by its causal structure, and hence can be defined by ten numbers to be specified at each spacetime point (cf. Sect. 4.5). These ten numbers correspond to the Einstein tensor, which can be represented schematically as a matrix that possesses four rows and four columns (16 numbers) but that is symmetric in the indices (i.e. along the diagonal), so that it comprises 10 different numbers: four along the diagonal, six on the top right (and six on the bottom left, but they are the same). To capture the geometry of spacetime is therefore to capture these ten numbers.⁶⁵ Nine of these numbers are given by the light-cones themselves (which trace all the possible light signals that can be sent from, or received, at a certain spacetime point). Given that these light-cones allow for a causal interpretation – for any event e, no event that lie outside the absolute past of e can be the cause of anything that would influence e, and no event outside the absolute future of e can be causally influenced by e (cf. Sect. 4.2) – it seems that all the information light-cones contain is encoded in the partial ordering of the causet elements. Thus, since a causet specifies the causal ordering among events, it defines light-cones and, thereby, provides nine of the ten numbers necessary to describe the geometry of spacetime.

The missing number, which corresponds to spacetime volume, cannot be recovered from the causal order of a Lorentzian manifold. However, as Riemann suggested (see above), in the context of a discrete order, the volume can be obtained in another way: “[…] by equating the number of causet elements to the volume of the corresponding region of the spacetime continuum that approximates C” (Sorkin, 2002: 7). To get a feel for the numbers involved in such kind of counting, Sorkin (2006) estimates that a region of spacetime of spatial volume 1³ cm and a temporal extent of 1 sec is composed of around 10¹³⁹ elements. Although it is an incredible number, which might explain why the granularity of spacetime has not yet been observed in laboratories, it is still a finite number, which exemplifies the idea that the structure of reality is fundamentally discrete. Thus, the order carries 9/10 information and the number (which corresponds to the volume) 1/10. Together they add up to 10/10 (this is what Sorkin’s famous slogan ‘order + number = geometry’ is meant to express). This underpins the claim that a causal set can indeed be approximated by a Lorentzian geometry: “[…] the causal set’s order relation proved the approximating continuums causal order and the local physical scale is set by the causal set’s discreteness” (Dowker, 2020: 147).⁶⁶

As has been said, these basic considerations provide the kinematical starting point for a theory of discrete quantum gravity based on causal sets. But, this cannot be the end of the story, since the vast majority of causets sanctioned merely by the kinematic axiom “[…] do not stand in a relation of faithful approximation to spacetimes in low-dimensional manifolds” (Wüthrich & Callender, 2016: 4). This is sometimes called ‘the inverse problem’. What is additionally required is a dynamics, which allows one to select, among the great variety of kinematically possible causets, those that are approximated by a relativistic spacetime. The basic idea is that the dynamics should be specified by some further axioms, which are intended to perform the selection: only the causets that satisfy them can successfully be reasonable models of the theory.⁶⁷ This is absolutely necessary if one wants to show how classical spacetimes emerge and, thereby, why GR is as successful as it is. From a phenomenological point of view, this step will turn out to be the most interesting aspect of CST since, taken seriously, the dynamics might help rescue temporal becoming and, therefore, make GBT work in a relativistic spacetime setting. Intuitively, this dynamics depicts a process of ‘growth’ or ‘cosmological accretion’, at each step of which an element of the causet comes into being. This new element is regarded as the “[…] ‘offspring’ of a definite set of the existing elements – the elements that form its past” (Sorkin & Rideout 2000: 3). Thus, the process of growth has always been explicitly linked to the passage of time by the proponents of CST. A few lines further on, Sorkin and Rideout say that “[t]he phenomenological passage of time is taken as a manifestation of this continuing growth of the causet. Thus, we do not think of this process as happening ‘in time’, but rather as ‘constituting’ time […]” (2000: 3).⁶⁸ It is worth noting that, although the process of growth entails a succession of births in a definite order, it does not presuppose any notion of absolute simultaneity (cf. Sorkin, 2007: 156).

Specifically, a popular dynamics by which the growth in the causets might take place is called the ‘classical sequential growth dynamics’ (CSG) (cf. Rideout & Sorkin, 2000; Varadarajan & Rideout, 2006). CSG is to be understood as a “[…] stochastic process starting from the empty set and adding elements one by one, with the transitions governed by probabilities satisfying a Markov condition⁶⁹” (Earman, 2008: 155). In addition to the Markov condition, CSG also respects the following three conditions: (i) internal temporality (which prevents events from ‘birthing’ in the past of events that have already become), (ii) discrete general covariance, and (iii) Bell causality. In that sense, CSG assigns probabilities to each transition from every finite causal set to its possible ‘children’ in accordance with certain physical principles. For example, considering Fig. 4.8, the birth of a can be construed as a transition from the empty causet to the (unique) causet of one element; it occurs with probability 1. The birth of b, however, can occur in two different ways: either b will be a child of a (as depicted in the diagram), or it will not; each of these two events occur with non-zero probability. At subsequent stages the number of possible causets rises quickly: as Sorkin calculated, after the fourth birth, one can have “[…] any of 16 non-isomorphic causal sets, while after the tenth there are already over two million distinct possibilities (2567284 to be precise)” (2007: 154). The logical space of all these possibilities is delimited by two extreme cases: (i) each new element acquires all the previous ones as ancestors, and the result is a chain, a causet that corresponds to one-dimensional Minkowski space, (ii) none of the elements has ancestors (they are space-like separated from each other), and the result is an antichain, a causet that corresponds to no spacetime whatsoever. In between (i) and (ii), one finds the most interesting cases, where, for example, there are CSG analogs of cyclical cosmologies in such a way that reality grows with successive cycles of collapse and reexpansion.

Fig. 4.8

A causet as a Hasse diagram. (A similar figure can be found in Sorkin (2006: §3))

Although CSG is, as its name indicates, classical (which means that no allowance is made for quantum interference between possible distinct transitions from any causal set to its ‘children’),⁷⁰ a quantum version of the theory is expected within the near future. This would, however, not affect the underlying kinematics or ‘ontology’ of the theory; as Sorkin makes it clear, “[…] the criterion of ‘discrete general covariance’ could carry over essentially unchanged from the classical to the quantal case” (2007: 155). In particular, the quantum version of CSG is expected to arise from a decoherence functional (which is itself an “[…] indeterministic, yet mostly irreversible process” (Weinert, 2004: 275)), defined on sets of histories (causal sets).⁷¹ Decoherence functional would thus generalize “[…] the notion of probability measure to allow for interference of distinct possibilities” (Varadarajan & Rideout, 2006: 2). This is good news, since the rejection of determinism has been depicted as a necessary (though non-sufficient) condition for the future to be called ‘open’ (cf. Sect. 2.5). Of course, conceiving causation in a non-deterministic context is not a straightforward matter⁷²; but, as many have insisted,⁷³ essential quantum mechanical experiments (e.g., the double-slit experiment) suggest that this is required to establish the validity of quantum theory. Furthermore, this suggests that, although the quantum interpretation that at first sight deals better with relativistic causality is actually deterministic, ‘going deeper’ allows indeterminism to be compatible with relativistic causality, and thus with the open future.

The most exciting aspect of CSG is that it seems to provide an objective correlate of our pre-theoretic intuition that new things are created in the present (cf. Temporal Becoming).⁷⁴ And, more generally, the unceasing occurrence of birth-events that build up the causet seems to explain why we experience time as we do (e.g., the arrow of time points from past to future, the future is open while the past is fixed, etc.). Furthermore, since CSG is said to unfold in a generally covariant manner, it seems perfectly compatible with relativity. Therefore, not only could CSG restore temporal becoming within physics, but it could do so “[…] without paying the price of a return to the absolute simultaneity of pre-relativistic days” (Sorkin, 2006: §4). In this last respect, CST is preferable to Tooley’s theory, which has been shown incompatible with Lorentzian covariance (cf. Sect. 4.4). If CST turns out to be correct, it might therefore have significant consequences for the philosophy of time. In particular, as Sorkin (2007) and Earman (2008) point out, since CST (augmented with CSG) encodes a ‘birthing’ process akin to C. D. Broad’s notion of temporal becoming, this model might underwrite a growing block theory of time (GBT). This appears very plausible for the three reasons that I detail now.

First, CST provides a natural way to construct a ‘spatially extended’ present, which seems required by both the geometric and the dynamic component of GBT: “[…] the events co-present with the ‘here-now’ are those events on a space-like slice – technically, a ‘maximal antichain’ – that is, a maximal set of events such that any two events are incomparable in terms of the relation ≤. A sequence of presents would then be a partition of a causet into such maximal antichains” (Wüthrich & Callender, 2016: 5).⁷⁵ Second, CST pictures a discrete spatiotemporal substructure which is “[…] four-dimensional from the very beginning, but which at any stage of its growth is still incomplete” (Sorkin, 2007: 157). In that sense, at any stage of the process of ‘growth’ we stop, we are left with the maximal elements of C, which form a ‘future boundary’ of the growing causet. Third, and more generally, what makes it difficult to express GBT within relativistic settings is that the ontological growth it entails is generally conceived as an accretion of thin layers of Newtonian absolute time, which therefore betrays general covariance. But why should accretion be conceived this way? What the causal set approach to quantum gravity (augmented with CSG) tells us is that the accretion does not proceed with respect to one particular cosmic time function, but rather by the birthing of events through a discrete stochastic process – this highlights that the coming into existence in a particular order is a different issue from a universe wide layer of increase. And, when the outcome of this stochastic process is approximated by a sequence of classical general relativistic spacetimes, the result may (or may not) look like a growing block model of time, as described in the previous chapter.

Of course, this approach could be criticized for being too speculative. After all, CST remains in an incomplete stage of development.⁷⁶ For example, as has been acknowledged, one of the major aims of current work on CST is to develop an appropriate quantum version of CSG. But, against a widespread opinion, there is at least one important thing that CST teaches us: intuitive phenomena, such as temporal becoming, the passage of time, or the openness of the future, are logically consistent with the four-dimensional Lorentzian manifold of relativity.⁷⁷ As, for instance, Fay Dowker puts it: “[…] CSG models are counterexamples to the claim that Relativity implies a block universe view of time” (2020: 148). It is therefore wrong to claim that contemporary physics forces a ‘block universe’ view on us, since it can accommodate an objective (i.e. mind-independent) form of temporal becoming. Specifically, when a CSG model produces a causet C, well-approximated by a continuum spacetime like our universe, a relativistic form of temporal becoming arises, whereas no comparable phenomenon could be found in the ‘block universe’ view. This suggests that philosophers of physics might have been wrong to neglect our intuitions, which have always spoken in favor of a becoming conception of time. Perhaps some of the elements for understanding the inner structure of time were pre-theoretically given to us, but they have been ignored, because of a misleading conception of science, according to which scientific concepts are completely divorced from our actual experience (cf. Sect. 1.1).

Another criticism concerns the ‘spatially extended’ present that a maximal antichain within the structure of a causal set may seem to offer. Wüthrich and Callender argue that such a way of constructing the present from the resources of CST is problematic, mainly because, for any given event here-now, “[…] there are in general many maximal antichains of which it is an element” (2016: 5). As a consequence, the present cannot be uniquely defined (various sets of co-present events can be created); and privileging one antichain over the others seems just as problematic as privileging one particular foliation in Minkowski spacetime: “[…] a partition of a causal set into such maximal antichains would not be invariant under automorphisms of its structure” (id). Supposing that the latter criticism is well-founded, does it undermine the way GBT has been defined in the previous chapter, i.e. as the only asymmetric A-theory of time that accepts Temporal Becoming (Sect. 3.5)? The answer is ‘not necessarily’, but surely some adjustments would be needed. For example, the reflection symmetry, which allows one to distinguish between the symmetric and the asymmetric structures, would have to be operated, not around ‘the present’ axis (since no unique maximal antichain could play this role), but around a spatiotemporal point. The operation in question is a central (not axial) symmetry; from a fixed point Ω, it transforms any point M into an image point M’, such that Ω is the midpoint of the segment [MM’]. Further, temporal becoming would have to be conceived as a local (rather than global) phenomenon, sometimes called ‘asynchronous becoming’ (composed of multiplicity of ‘nows’). Sorkin, for instance, explicitly adopts this conception when he says: “[…] our ‘now’ is (approximately) local and if we ask whether a distant event space-like to us has or has not happened yet, this question lacks intuitive sense. […] the supposition of [a ‘super observer’, who would take in all of existence at a glance] would lead to a distinguished ‘slicing’ of the causet, contradicting the principle that such a slicing lacks objective meaning (‘covariance’)” (2007: 158).

This obviously leads to a further criticism: the dynamic picture of the world offered by CST (through a CSG model) is exotic and, therefore, does not answer to the typical GBT’s demands. Indeed, although a CSG model allows for a (perhaps localized) form of becoming, it rules out the possibility of a single physical world that grows and change, while this seems to be at the heart of C. D. Broad’s original project. Oliver Pooley (2013), for instance, claims that Sorkin’s view is best understood as a version of non-standard A-theory of time (in Kit Fine’s sense), but this kind of view as been shown to be not fully intelligible (cf. Sect. 3.2). However, things might be less dramatic than they seem, especially because Earman (2008), and Wüthrich and Callender (2016) have developed two concurrent objective and global forms of becoming that seem both respectful of CST-cum-dynamics and the structure of relativity, so that the above objection does not seem definitive. But, before introducing these two options, it is worth examining a further challenge that they might face. For that purpose, consider the singleton set, and suppose that it births a time-like separated element, Max’s birthday, at label time l = 1. Then, suppose that this two-element causet births a third element, Mary’s birthday, which is space-like separated from the other two elements, at label time l = 2. This is path α. By contrast, path β births Mary’s birthday space-like separated from the singleton set, and then births Max’s birthday, which is time-like separated only from the singleton set – the situation is depicted in Fig. 4.9. General covariance (which is used as a condition to derive the dynamics) requires that the probability of any particular causet arising is independent of the path to get to that causet and, therefore, that “[…] the product of the transition probabilities along the links of α is the same as that for β (and any other such path)” (Wüthrich & Callender, 2016: 13).

Fig. 4.9

Stochastic sequential growth. (A similar figure can be found in Earman (2008: 157), and Wüthrich & Callender (2016: 14))

This situation is sufficient to make apparent a difficulty: according to relativity, there is no fact of the matter as to which event – either Max or Mary’s birthday – came first. As Wüthrich and Callender put it: “[t]o say which one happened ‘first’ is to invoke non-relativistic concepts” (2016: 13). To put it another way, the world grows from C₁ (the singleton set) to C₂ to C₃, but there is not a determinate fact as to whether C₂ consists of the singleton plus Max’s birthday or the singleton plus Mary’s birthday. General covariance entails that, if a causal set can be reached via two different paths (and hence can occur with two inequivalent labellings), the transition probabilities along each path must be the same. In that sense, it is often said that general covariance ensures that the labels used in the growth process are ‘pure gauge’. This simple example shows that conceiving a physically consistent growth (within the causets) happening in time is not a straightforward matter. Fortunately, there seem to be promising solutions on the market. For instance, John Earman (2008) assumes (against a many-worlds interpretation of the stochastic birthing process) that only one of the possible paths, α or β, is actualized. Accordingly, in Fig. 4.9, the becoming of the actual world is modeled by path α (say), but not by path β, since the causet on the right is not actualized. As Earman puts it: “[s]ince quantum aspects are ignored in classical sequential growth dynamics, it seems fair […] to assume that each stage only one of the possible alternatives is actualized” (2008: 158). This ‘philosophical’ addition to the causets allows one to regain an objective and global form of becoming (cf. Fig. 4.10, on the left).

Fig. 4.10

Earman’s distinction between actualized and non-actualized alternatives, and Wüthrich and Callender’s bizarre form of indeterminacy

Of course, one might be reluctant to add such a hidden variable moving up the causet, especially because this move seem to betray the ‘standard’ interpretation of the labels as ‘pure gauge’.⁷⁸ If so, another option seems available: “[t]here simply is no determinate fact as to whether C₂ contains a or b; but there is a determinate fact that it contains one of them” (Wüthrich & Callender, 2016: 15). In other words, whereas it is determinate that C₂ consists either of the singleton plus Max’s birthday or of the singleton plus Mary’s birthday, it is indeterminate whether C₂ has Max’s birthday in it, and it is indeterminate whether C₂ has Mary’s birthday in it. This kind of indeterminacy should remind us how Elizabeth Barnes and Ross Cameron (2009, 2011) characterize the openness of the future (and borderline cases): there are multiple determinate (precise) states between which the world is unsettled (cf. Sect. 2.7). The logic is the same here: there are two possibilities, C₂-cum-Max’s birthday, and C₂-cum-Mary’s birthday, and the world is unsettled as to which one obtains. In that sense, indeterminacy should not be regarded as a mere epistemic phenomenon, but as a metaphysical phenomenon, which may concern the extension of a (finite) causet (i.e. what elements are to be found in it), but not its cardinality (i.e. how many elements are to be found in it). If this idea is coherent, “[…] CST does permit a new kind of – admittedly, radical and bizarre – temporal becoming” (Wüthrich & Callender, 2016: 15): things get determinate at every finite stage of becoming. For instance, in Fig. 4.9 (on the right), all the ‘ancestors’ of C₃ must have determinately been obtained, with the notable exception of its two immediate ancestors. At this stage, C₃ is neither determinate nor indeterminate, since it has not yet come into existence (the two structures below are therefore not at the same stage of development). Then, at the next stages (not shown in Fig. 4.9), C₃ comes into existence, and one of its two immediate ancestors – it is indeterminate which one – gets determinate. And so on.

In a nutshell, GBT implies an increase in ontology: new thin layers come successively into being. Admittedly, if these layers correspond to slices of Newtonian absolute time, GBT is at odds with contemporary physics. But why should one think of the layers in such an old-fashioned way? CST allows for a birthing of events through a discrete stochastic process, which has to be quantal to generate a truly manifold-like causet. Although we do not possess such a quantal dynamics yet, we can imagine how it will formally look like: Varadarajan and Rideout (2006), for instance, claim that it will be expressed in terms of a decoherence functional. The outcome of this process, when it is approximated by a sequence of general relativistic spacetimes, may resemble a hypersurface Becoming model. Classical relativistic spacetime would, in that sense, be an emergent feature of temporal becoming (though the quantal nature of this process remains mysterious). Although temporal becoming within CST is often regarded as a local phenomenon (cf. Sorkin, 2007), and therefore poses a threat to the intuitive picture of a single physical world that grows and changes, Earman (2008) and Wüthrich and Callender (2016) show that CST allows for global forms of temporal becoming. In particular, they found some generally covariant senses in which one could say that there is becoming, but not to be conceived as a ‘asynchronous’ or frame-relative phenomenon, but as an absolute phenomenon. Although this phenomenon could simply be expressed by the fact that cardinality of the causets grows, Earman (2008), Wüthrich and Callender (2016) went looking for more interesting facts about causets: either by distinguishing between actualized and non-actualized alternatives (Earman), or by putting forward a form of indeterminacy about what elements are to be found in the causets (Wüthrich & Callender). Taking these propositions seriously, it seems that CST might offer a naturalistic basis to GBT (at least as defined in the previous chapter). And, since GBT is well-designed to accommodate some of our basic intuitions about time (e.g., the future is open, while the past is fixed), CST might play a crucial role in the reconciliation of the manifest image and contemporary science.

4.7 Reconciling GBT with Science Fiction: The Case of Time-Travel

In this section, the proposal is to move from science to science fiction by considering the possibility of time-travel. Although this possibility became a real possibility with the advent of SR and GR, the expression ‘science fiction’ seems justified, since no such travel has yet been undertaken by any of our contemporaries. The possibility of time-travel has always captured the popular imagination; one can find hundreds of books and movies which explore it. The best example is perhaps H. G. Wells’ The Time Machine, in which the protagonist leaves his own time (1895) to travel in to the distant future and then return to his own present. Of course, some philosophers refuse to take these scenarios seriously, mainly because travels into the future involve certain oddities (e.g., a mathematician can bring back from the future a mathematical proof that he decides to publish in the present), and travels into the past involve paradoxes (e.g., the grand-father paradox). But, as has been argued in the second chapter (cf. Sect. 2.4), these difficulties may find some solution through an appropriate treatment of the question of ‘What we can and cannot do in the future or the past?’ (cf. Lewis, 1976). The possibility of time-travel has already been considered in the present book, especially in Sect. 2.8, where it was argued that dynamic branching-tree models cannot allow for it. Arguably, presentism is in no better position, since it recognizes no non-present location to which to travel (but see Dowe, 2000, and Baron & Miller, 2019: §8). This argument, which highlights the incompatibility of presentism with the possibility of time-travel, is commonly called the ‘no destination’ argument. It might therefore seem that if one can show that GBT allows for the possibility of time-travel, it would have a certain advantage over at least both of these two competing theories. It is worth noting that, in what follows, the question of the compatibility of GBT and time-travel will be asked on a non-relativistic conception of time / spacetime, and with no reference to the ‘bare particular’ view developed in the third chapter (since it is unclear whether it allows for time-travel). Furthermore, this section will offer the opportunity to address the thorny question of persistence: ‘How can things exist at different moments of time within GBT?’. It will be shown that GBT is a priori compatible with both endurantism, i.e. the view that things are wholly present whenever they exist, and perdurantism (or four-dimensionalism), i.e. the view that things have temporal parts (or stages).

At first sight, GBT appears to be better positioned than presentism when it comes to accommodating the possibility of time-travel. Specifically, GBT partially resists to the ‘no destination’ argument: although GBT a priori excludes time-travel to the future (since it implies that future temporal locations are unreal), it offers various past locations to which to travel. In particular, given GBT, if t₅ is the objective present, then there seems no reason to suppose that one could not travel to all temporal locations that are in the past relative to t₅. Yet, Kristie Miller contests the latter claim; she argues that GBT (just as presentism) is incompatible with backward time-travel, because “[…] it requires that present states be caused by non-existent indeterminate future states” (2005: 229). Let me explain. As usually conceived, time-travel is ruled by (at least) the following three principles: (i) P is a genuine time-traveler only if all of P’s temporal parts are united by some causal relation, (ii) it is not possible to change the past (cf. Sect. 2.4), and (iii) it is not possible to travel from a non-existent location to an existing one (cf. Miller, 2005: 227). The first principle, which demands a causal continuity among the stages of a time-traveler, is intended to rule out cases of counterfeit time-travel: if Fred is randomly created by a demon at a time t and it happens to be a duplicate of a stage of Sam destroyed at a later time t₅, this should not count as a case of time-travel (cf. Lewis, 1976: 148). The second principle implies that “[…] if one can travel to some past location t, it will be true at all times subsequent to t, that one existed at t. And it will be true at t, that one exists at t” (Miller, 2005: 227). In that sense, if Max does not exist at t when t is the present, then Max did not exist at t when t is in the past (and, therefore, Max has not traveled to t). The third principle is the converse of the ‘no destination’ argument. It is partly justified by the fact that a future non-existent event allegedly cannot be the cause of a current event (e.g., a time-traveler existing now), particularly given that it is supposed to be indeterminate whether the future event in question will occur (cf. Miller, 2005: 228).

Now, suppose that t₅ is the objective present, and at t₅ Max travels back in time to t₁. Since it is not possible to change the past, “[i]f [Max] exists at t₁ when t₁ is the objective past, then [Max] must exist at t₁, when t₁ is the objective present. [But, according to GBT] when t₁ is the objective present, t₅ does not exist. So [given that it is not possible to travel from a non-existent location to an existing one], it is not possible for any time traveller to have travelled from t₅ to t₁” (Miller, 2005: 229). To put it another way, it seems that, given GBT, there are only two ways things could turn out: either (i) Max does not exist at t₁ when t₁ is the objective present, and hence he does not exist at t₁ when t₁ is the objective past (since it is not possible to change the past), or (ii) Max does exist at t₁ when t₁ is the objective present, but he is not a time-traveler, since his t₁ temporal part cannot be causally connected to any temporal part that exists in the future (such a temporal part does exist, after all). Neither of these options allows for time-travel. It might therefore be concluded that GBT is incompatible with this possibility.

This objection may seem powerful, but only if one assumes the Lewisian conception of time-travel, according to which time-travelers are perduring entities that have temporal parts at their point of departure (as well as at any other time at which they exist). One can therefore distinguish two potential options to resist the objection: (i) to reject perdurantism in favor of endurantism (time-travelers do not have temporal parts; they are wholly present whenever they exist), and (ii) to accept a revised conception of perdurantism which allows one to say that, in some sense, Max exists at t₁ when t₁ is present, and has a future temporal part that will exist at t₂. The first option is supported by orthodoxy: perdurantism does not go well with presentism, and for similar reasons, it might seem that perdurantism does not go well with GBT either. After all, why should one claim that an object has temporal parts at other times than the present (perdurantism) if these parts do not exist? Of course, a presentist could say that these temporal parts existed and exist no longer, but in what sense would they be parts of the object? It might seem, as Trenton Merricks puts it, that “[a]n object cannot have another object as a part if that other object does not exist” (1995: 524).⁷⁹ To put it another way, according to perdurantism (at least in its classical conception), an object is an aggregate of all its temporal parts while, given non-eternalist ontologies, there might be times at which such an aggregate does not exists. For example, since the Eiffel Tower will probably still be standing tomorrow, growing blockers have every reason to believe that some of its temporal parts do not exist (but will exist). It might therefore seem preferable for growing blockers (and other non-eternalists) to reject temporal parts and, thus, to endorse endurantism.⁸⁰

However, although this first option allows one to reject the second principle of time-travel scenarios, it is not clear that it is a way out for growing blockers, since this principle can be replaced by another one, upon which an ‘endurantist’ version of Miller’s argument can be built: “[…] P is a genuine time traveller only if for every times t and t* at which P exists, there is some causal relation that holds between P at t and P at t*” (Miller, 2005: 225). A better option might therefore be to argue that the classical conception of perdurantism (or four-dimensionalism) is misleading. Of course, if to perdure an object must exist at different times by having parts at those times, then perdurance is at odds with non-eternalist ontologies (at least regarding objects that will be partially located in the future). But, it seems coherent to say that a persisting object consists of its present and past parts, and of those parts that it will have in the future (pace Merricks). As, for instance, Lawrence Lombard (1999) points out, one must carefully distinguish between two senses of ‘exist’ if one is a perdurantist: (i) the ‘straightforward sense’ in which instantaneous temporal parts exist at a time, and (ii) the ‘derivative sense’ in which an object, i.e. a whole composed of all of its temporal parts, exists at some time. According to (i), if temporal parts exist at a certain time, they exist at this time entirely (they are three-dimensional entities) and they have all of their (spatial) parts at this time. According to (ii), objects (e.g., material objects, people, etc.) exist at some time in virtue of having a temporal part that does; but one is enough, it does not need to have all of their parts at this time. It is obviously this second, derivative, sense of ‘exist’ that is the interesting one for perdurantists; the first one being accepted by everyone: “[…] if there are any three-dimensional instantaneous entities, it is uncontroversial that they exist entirely at the time they do” (Benovsky, 2007: 84).

Criticizing the classical conception of perdurantism, Lombard argues that “[…] what is obvious is only that an object that exists at a time, t, cannot have, at t, another object as a part, if that other part does not exist at t. But what the perdurantist wishes to say is not inconsistent with that. What, in [the straightforward] sense, exists now – e.g., the present temporal part of a computer – is something that does not (ever) have as parts anything that does not exist now. But what exists now in [the derivative] sense – the computer – is something that does (at some time or other) have parts that do not exist now; but what exists now in that [derivative] sense does not now have those parts” (1999: 256). In short, an object, such as a computer, construed as composed of temporal parts (perdurantism), exists now in the derivative sense of having a part that exists now (in its entirety). Of course, it might be objected that, since what exists now, in the derivative sense, does not have its non-present temporal parts now, it does not, if eternalism is false, have them at all (cf. Merricks’s principle). This objection rests on the idea that objects must have their temporal parts in the same way that they have their spatial parts: temporal parts, like spatial parts, must exist in their entirety. But this does not apply to temporal parts in general. As Berit Brogaard observes: “[…] events are commonly understood as having temporally extended parts even though these never exist as a whole but only though their successive stages” (2000: 346). Similarly, although an object exists at t in virtue of having some temporal parts (perdurantism), it is not required that all of its temporal parts exist at t. The existence of an object at t (in the derivative sense) merely requires that one of its temporal parts exists at t (in the straightforward sense). Fabrice Correia and Sven Rosenkranz support a similar idea when they write: “[m]ereological fusions, if they exist, exist whenever, and wherever, one of their parts exist. […] residents of spacetime may, for all that, be mereological fusions of spatiotemporal parts, as long as one of their spatiotemporal part is located here-now” (2018: 151).

In the light of the distinction between the straightforward and derivative senses of ‘exist’, Merricks’s principle appears clearly problematic. On one reading, in which ‘exist now’ means ‘exists in its entirety at the present time’ (the straightforward sense), the principle is inconsistent with the existence of entities that have temporal parts. On a second reading, in which ‘exist now’ means ‘now has a temporal part that exists in its entirety at the present time’ (the derivative sense), the principle is simply false (cf. Lombard, 1999: 257). Once again, what exists now, in the straightforward sense, is a temporal part that does not have as parts anything that does not exist now; what exists now, in the derivative sense, is an object (material object, people, etc.) that does have parts (temporal parts) that do not all exist now, but this object does now have those parts. This revised conception of perdurantism allows one to conclude that the argument for the incompatibility of GBT and time-travel fails. To return to the previous example, Max can travel from t₅ to t₁ since, although his t₅ temporal part does not exist when t₁ is the objective present (in the straightforward sense), he still has it as a part, i.e. he still exists partially in virtue of having his t₅ temporal part in the future. Max can therefore be a genuine time-traveler, even if GBT is true, since there is a sense in which Max is composed of all of his parts (including the future ones), one of them (his t₅ temporal part) being the reason why he is now visiting t₁ when t₁ is the objective present. Does this undermine (i) the general causal picture and (ii) the openness of the future, as Miller (2005) claims? The answer seems to be ‘no’. What matters is that (i) there will be a cause at t₅ such that it will explain why, at t₁, there was a time-traveler coming from the future (which seems guaranteed by the fact that Max partially exists in virtue of having a t₅ temporal part in the future), and (ii) this cause is not predetermined (nothing there is or was, in conjunction with how it is or was, makes Max’s time-travel inevitable): it was not inevitable for Max to visit the past.

By the way, taking McTaggart’s conception of change seriously (the only way in which events can genuinely change is by first being future, then present and finally past), it seems that this revised conception of perdurantism is capable of avoiding what is generally introduced as the main objection against perdurantism (at least in its four-dimensionalist version), namely that it entails a changeless world. The objection runs as follows. Consider an apple that is green at t₁ and brown at t₂. What this amounts to, according to four-dimensionalists, is that one of the apple’s temporal parts is green, and another is brown. However, when considering this account of change, it might be objected that what we are looking for is an account of how a single object (the apple) can change, while four-dimensionalists are telling us a story about different objects (different temporal parts) having different properties. In the four-dimensionalist picture, “[w]hat we have is not change of an individual, but replacement of one changeless object (one temporal part) by another changeless one” (Benovsky, 2007: 81). Specifically, instead of saying that the apple has changed from being green to being brown between t₁ and t₂, four-dimensionalists say that the t₁ temporal part of the apple has changelessly the property of being green and the t₂ temporal part of the apple has changelessly the property of being brown. Since the apple itself cannot lose or gain any such properties, it seems that four-dimensionalism leaves no room for genuine change. As Peter Simons puts it: “[…] Lewis’s favoured four-dimensional alternative is not an explanation of change but an elimination of it, since nothing survives the change which has the contrary properties” (2000: 65).

Now, it seems that the revised conception of perdurantism might allow us to overcome the elimination of change by bringing back the passage of time into the four-dimensional picture. In particular, combined with GBT (rather than eternalism), perdurantism becomes the view according to which (i) objects have temporal parts, and (ii) at any given time, only past and present temporal parts of objects exist (in the straightforward sense). Of course, just as four-dimensionalism, GBT-perdurantism entails that “[…] an object does not gain or lose properties; rather, different properties are possessed by different [temporal parts]” (Brogaard, 2000: 348). But, contrary to classical-perdurantism, GBT-perdurantism entails that new temporal parts are coming into existence – and this in a way that seems to capture our most basic intuitions according to which change has taken place. Specifically, a change of x has taken place if and only if “[…] (i) there is an entity z which is a present [temporal part] of x; and (ii) there was an entity y which was a previous [temporal part] of x; and (iii) z has a different set of [intrinsic] properties than y had” (Brogaard id). The creation of new temporal parts plus new intrinsic properties (e.g., being red, being square, etc.) seems thus sufficient for a change to take place. Moreover, GBT-perdurantism definitely seems better suited to account for the asymmetry between the ‘open future’ and the ‘fixed past’ than the classical alternative. Classical-perdurantism implies that it is settled that people have the temporal parts that they have; these parts (including future ones) exist tenselessly, after all. By contrast, on GBT-perdurantism, whereas it is settled that people have the temporal parts that they in fact had and now have, it might be unsettled what parts people will have in the future (at least assuming that physical determinism is false), because these parts do not exist yet.

4.8 Conclusion

Physics informs and frames the metaphysical debate on the nature of time. In (neo-)Newtonian mechanics, time (as well as space) is absolute; it is regarded as an empty container of discrete, atomistic nows, which cannot be affected by any material agency. This reflects the intuitive idea that the world evolves in time in an objective manner. The (neo-)Newtonian picture allows for a four-dimensionalist interpretation, according to which spacetime can be foliated into three-dimensional hyperplanes. (Neo-)Newtonian mechanics therefore admits an absolute notion of objective simultaneity: it allows one to establish a common time system for a group of people (even if these people are moving with respect to each other), so that everyone in this group will agree on which events are simultaneous. This makes the (neo-)Newtonian mechanics a friendly environment for expressing GBT, provided that the layers of existence successively coming into being are seen as slices of Newtonian absolute time.

However, the (neo-)Newtonian mechanics has been shown to be deficient both theoretically (cf. Maxwell’s equations of electromagnetism) and experimentally (cf. the Michelson-Morley outcome). It has therefore been replaced with relativistic physics, which recognizes that the notion of absolute simultaneity is unfounded. This makes GBT notoriously at odds with relativity since, as the Putnam-Rietdijk argument points out, the unreality of the future requires an objective notion of absolute simultaneity. Fortunately, many options – both compatibilist and incompatibilist – are available to growing blockers to escape the pressure exerted by the Putnam-Rietdijk argument. Compatibilist options consist of either challenging some of the premises of the argument (Sklar, 1974), or re-conceptualizing GBT to make it expressible in relativistic settings (Correia & Rosenkranz, 2018; Stein, 1968). Incompatibilist options consist of either rejecting the metaphysical relevance of SR (Bourne, 2006; Zimmerman, 2008), or denying that SR is approximately true of the world (Tooley, 1997). Although both families of options come with consequences, those of the compatibilist options seem more problematic. Specifically, the compatibilist options seem in tension with both our intuitions (e.g., the intuition that there is single universe-wide border between the past and the future), and some considerations from quantum mechanics (e.g., experimental results connected with John Bell’s theorem). This is what led us to privilege an incompatibilist option in the remainder of the chapter.

The incompatibilist option in question took the form of a quantum theory of gravity, the causal set theory, which, since it encodes a ‘birthing’ process akin to the notion of temporal becoming, might give a second wind to GBT. This ‘birthing’ process (understood as a discrete stochastic process) is expressed by a dynamics, the so-called ‘classical sequential growth dynamics’ (CGS), which is in charge of selecting, among the great variety of kinematically possible causal sets, those that are approximated by a relativistic spacetime. Although a version of CGS that incorporates quantum aspects still has to be developed, this approach promises to offer a naturalistic basis to GBT, which was so far regarded as purely speculative. Although, becoming within CST is often regarded as a local phenomenon (cf. Sorkin, 2007), Earman (2008), and Wüthrich and Callender (2016) have shown that CST permits global forms of temporal becoming, which allows one to preserve the intuitive picture of a single physical world that grows and changes. Taking their propositions seriously, CST may play a predominant role in the reconciliation of the manifest image and contemporary science (at least supposing that it is to be successful as a theory of quantum gravity). Finally, it has been shown that GBT is (at least in principle) compatible with time-travel scenarios, provided that one accepts, for instance, a revised conception of perdurantism.