Getting Philosophy from Symmetry - The Philosophy of Physics (2016)

The Philosophy of Physics (2016)

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Getting Philosophy from Symmetry

Many of the central debates in the history of philosophy of physics concern the nature of space, time, and motion (kinematics) and geometric symmetries (symmetries of space, time, and motion). Lessons from these debates flow readily into a host of other areas, not directly related to spacetime. Space and time are, quite independently of tricky issues to do with motion and symmetry, rather strange. As Frank Arntzenius points out, we can’t see them, nor smell them, hear them, taste them or touch them.1 For this reason there is much overlapping between debates about space and time in the philosophy of physics and in philosophy more generally - more so than, e.g. the quantum theory of fields, for obvious reasons. We have a kind of direct access (Arntzenius’ point aside) to space and time, that makes us feel like we know what we are talking about: space seems empty; time seems ‘flowy.’ But physics radically modifies our everyday views of space and time and so impacts on the more general philosophical debates.

This chapter builds on some of the core aspects of physics (kinematics, dynamics, states, observables, symmetries, etc.), developed in the previous chapters. We begin with the justly famous debate between Newton and Leibniz in which they agree about the world’s overall geometry, but disagree over the ontological status of space and time: Newton will say that the world has its geometry because there is a space and time with that geometry; Leibniz will say that the geometry comes about from the laws linking material objects and events so that space and time emerge from this. Next comes a similar argument that links ‘handedness’ with the reality (or not) of space and time. After this we switch to relativistic physics: firstly, in special relativity, we focus on the twins paradox and a related argument about the reality of past, present, and future events; and then finally comes general relativity and the so-called ‘hole argument.’

4.1 Leibniz Shifts and the Reality of Space and Time

I expect that for most people space is understood to be much like a really big room. Just as your bedroom contains your bed and wardrobe and so on, so space contains your bedroom, the Earth, galaxies, and all of the particles making up the universe’s objects. Time is more difficult, perhaps, but again it is most likely to also be understood along ‘container’ lines in which events occupy specific instants and intervals of time, much as objects occupy the parts of space. As with ordinary containers, it seems easy to imagine space being emptied of its contents, and time emptied of its contents. But is this correct? Does a space emptied of all objects make sense? Is it a genuinely possible situation? What might an alternative look like? This is the debate over the ontological nature of space and time: are they real like chairs and hares and other material objects, or are they some kind of construct from chairs, hares, and such; or perhaps they are purely mind-dependent aspects of reality?

Absolute and Relative Motion

Space and time for Newton were certainly real. They were viewed to be quite independent from material objects, which they could indeed be said to contain. The motion of objects in this space then happened within this container, which is understood to have the same structure as Euclidean three-dimensional space. There is an ‘absolute Cartesian reference frame’ (x, y, z) in this space in which measurements must be made to formulate properly the laws of Newtonian mechanics - this analogy between a Cartesian coordinate system for Euclidean space and an (inertial) frame goes very deep, as we shall see. There would of course be motion of the objects relative to one another too, but real motion is understood to be motion relative to the special frame picked out by the absolute container. As we will see, this kind of motion was required in order to account for certain otherwise inexplicable ‘inertial effects’ (in which one must invoke a non-inertial frame). After all, in otherwise empty space, if we pass one another at uniform velocity in our cosmic armchairs we wouldn’t be able to say which of us was truly moving and which at rest (or whether we were both in motion). But if you felt a force pushing you into your armchair, then we would know that you are truly in motion. For Newton, true motion simply meant absolute motion.

There is a problem, however: the special absolute frame is not unique. Rather, there is a special class of frames: the inertial ones. The laws (and the possible motions that satisfy them) do not depend on position, orientation, or velocity, which implies that given one ‘special’ frame, any other frame (axes: (x′, y′, z′)) that differs from it by some rotation, displacement (translation in space), or even by some uniform velocity (a Galilean boost) will do just as well as far as the laws are concerned. This freedom in the choice of frame stems from a set of symmetries of Newtonian mechanics. The transformations that map some frame to another, in such a way that one still has a frame that is suitable for formulating the laws (i.e. an inertial frame), are symmetries of the theory: if some motion is a solution of the equations in one frame, then it will be in any frame that is related to it by applying one of these transformations. Of course, this means that one cannot use the laws to determine whether one is at rest relative to absolute space, or gliding against it at some uniform velocity.

This is all readily understandable if one assumes that space really is just like a Euclidean container. After all, the points and regions of the latter look the same regardless of the orientation or position of the axes. One is simply transferring the symmetries of Euclidean space (i.e. the distance preserving transformations of space, or isometries) to a physical context. But why believe in this absolute structure? Why believe in the reality of something that leads to such unobservable quantities as absolute position, orientation, and velocity?

Globes and Buckets

Newton believed he had proven the existence of absolute space by reference to certain physical examples in which relational structure would not be sufficient to account for some so-called inertial effects (or forces). These involve rotations (and accelerated motion in general, such as the cosmic armchair example above), but not understood as frames at different but stationary orientations: actively rotating systems. Rotations, he thought, must involve absolute motion. One of these examples involved a pair of identical globes connected by some cord, such that the entire system is rotating about a central axis. The rotation would, of course, result in tension in the cord so that it is pulled tight by the centrifugal force (see fig. 4.1).

How is the relationist supposed to deal with this apparently perfectly possible situation? The globes themselves are postulated to be identical. Given that they are the only things in an otherwise empty world, they also share all of their relational properties. So if there is tension in the cord the relationist has no purely relational resources to account for it. There must be rotation given the tension: the tension will give a means of measuring the rotation, and thus distinguishes the state of motion from one of uniform motion. Newton argues that the only way to account for it is to suppose that they are rotating relative to the frame of reference provided by absolute space. In other words, motion cannot be merely relative in this case.

Fig. 4.1 Newton’s globes in an otherwise empty universe. The tension in the cord connecting them is an indicator of absolute rotation and a non-inertial frame.

Another thought experiment, more famous than the globes, is ‘Newton’s bucket’ from the Scholium to his Principia Mathematica. Here we are asked to consider a bucket filled with water suspended from some hanging point by rope. The idea is to consider what happens when we let the bucket spin. There are several stages through which the bucket/water system proceeds: first one imagines that the bucket has been twisted around many times so that there is potential energy in the rope that will spin the bucket when let go. What would we see? First the water would be flat and at rest, and the bucket would also be at rest. Then, as we let go of the bucket so that it can spin, the bucket will spin, but the water will remain flat and at rest. Then the bucket will be rotating and the water will be rotating, and will edge up the side of the bucket as it does so: again, giving an indication of absolute motion. Why absolute motion rather than merely relative motion? Because the bucket and the water are supposed to rotate at the same rate, and so will be at rest relative to one another. As Samuel Clarke (an ardent Newtonian put it, in his correspondence with Leibniz):

[Newton] shows from real effects that there may be real motion in the absence of relative motion, and relative motion in the absence of real motion.

But there is a strategy for denying Newton’s conclusion starting with Bishop Berkeley. Berkeley pointed out that if the spinning bucket, with water, were all that existed then it simply wouldn’t make sense to speak of it as rotating: relative to what? The same point can be applied to the globes. The positivist physicist Ernst Mach, in his The Science of Mechanics, leveled a similar attack in the nineteenth century, pointing out that in the case of the bucket experiment, rotation relative to the mass of the Earth and the other celestial objects might be responsible for the non-flatness of the water’s surface: the water is flat relative to these masses, and curved relative to these masses, rather than absolute space. In order to provide a decisive argument for absolute motion using the bucket (or the globes) we would have to empty the universe of all other matter. Though we might think we can do this in thought experiments, we can’t rely on our intuitions in such cases: our intuitions might differ, but how are we to decide between them in such cases? The globes might expand or contract for all we know. However, Mach’s response has its own difficulties: how, for example, do the masses of the various celestial objects in the universe act on our little bucket? They are very far away (some extremely far away), so does it take some time for the effect to occur? If not (if the effect is instantaneous, and so nonlocal), then surely this explanation is not as good as one that depends on local features of absolute space?

Ultimately, however, Mach was pointing out that since there is no way of distinguishing between a rotating bucket and the rotating heavens, the question of which is true is meaningless. That is, we have no way of assessing whether such inertial effects are the result of absolute motion or relative motion unless we could hold Newton’s bucket still while spinning the rest of the universe around it, to see if that generated the same curvature in the water’s surface by centrifugal forces. As Mach himself put it:

The Universe is not twice given, with an Earth at rest and an Earth in motion; but only once, with its relative motions, alone determinable.

Whether one sides with the Mach-Berkeley strategy or not, it cannot be doubted that Newton does have an empirical argument on his side that needs to be responded to by the relationist.

In the age of airplane travel we can add another effect (that does not require rotation) observed during take off: the force felt that pushes you back in your seat is an inertial effect (this time generated by a linear acceleration, rather than the rotations of the bucket and globes experiments). You are at rest relative to the plane, and yet there is some force - here your body is much like the water sloshing in the bucket. Accelerated motion can be noticed: one can tell an accelerated frame from one at rest or in uniform motion. Just try and drink a cup of tea in an accelerated frame (a car taking a sharp corner), compared to a uniformly moving frame (a peaceful train journey). Whether it comes about through motion against an absolute container (i.e. departure from inertial, straight-line motion in that container) is the issue, however. The problem for the relationalist is that only one reference frame feels the effects: the runway accelerating from your plane at the same rate feels nothing! Mach would, of course, point out that it is acceleration relative to the rest of the mass in the universe such that if we could hold you and the airplane still and whoosh the rest of this mass away you might still feel the same force. Hence, there is absolute motion, only it is relative to this ‘cosmic frame’ rather than absolute space. What is missing, however, is a theory embodying this idea.

So-called Barbour-Bertotti (after Julian Barbour and Bruno Bertotti) models constitute a genuine attempt at a relationalist mechanics where the action takes place in a relative configuration space (essentially a space of all possible shapes, with no redundancy). One of its key Machian features is a constraint outlawing the rotation of the universe - or rather postulating a symmetry between the universe undergoing rotation, while an object (subsystem) stays fixed, and that subsystem rotating instead, with the rest of the universe as a whole remaining fixed. This is needed to secure a relationalist dynamics, so that fixing some initial relative configuration (just specifying the relative distances of the objects) will be enough to fix the relative motions forever after (since the whole system is in inertial motion) - without this we would need to supply more data (an axis and rate of rotation) to get the subsequent evolution.2

Leibniz’s Principles

Leibniz wanted nothing of this absolute container, which he believed allowed for a multiplicity of indistinguishable yet physically distinct possible worlds. To understand why Leibniz thought this was a bad thing, we need to quickly say something about his general philosophical principles: the principle of the identity of indiscernibles [PII] and the principle of sufficient reason [PSR]. The former principle has become an integral part of many issues in the philosophy of physics relating to symmetries. It simply says that objects sharing all of their properties in common are really just one and the same object, perhaps given different names or labels, which merely serve to (over-) represent the object. We can write this using logical notation as follows:

(4.1)

In words, this says ‘for any properties F, and objects x and y; if x has property F if and only if y has the property F, then x and y are identical.’ Or, to put it another (converse) way: there is no distinction without a difference. One can also understand this principle to mean that there cannot exist two things that differ only in number, i.e. only in that there are two of them.

This sounds simple enough, but there are several subtleties involved in making sense of the principle. Firstly, there is an issue over what properties are to be included here: just qualitative ones? Spatial and temporal locations? There have even been proposed certain special non-qualitative properties known as ‘haecceities’ (primitive thisnesses). In ordinary language, a primitive thisness says that something is what it is because it just is, so there! ‘Haecciety’ is just a fancy word for this-ness. We can understand it as a certain kind of non-qualitative property (qualitative properties are ‘suchnesses’), which involves being identical with a certain individual (obviously only possessed by a unique individual). Denying that such things exist, as a PII-wielding Leibnizian naturally would, commits one to a ‘bundle’ theory of individuals, according to which an individual’s identity is just given by the various properties it has: there is no ultimate ‘pincushion’ sitting under all of the properties that gives each object its distinctive identity, so that even objects sharing every other property will differ in at least some way. Another issue is over what kinds of things are being compared: objects or possibilities? Or even entire worlds?

The PSR simply says that “for anything that is the case, there’s a reason why it should be so rather than otherwise.” If you have blue eyes, there has to be a reason for that: it can’t be a whim of nature. Same goes for every single element and aspect of our world: including the way the material universe is configured and, if absolute space should exist, this also includes location within it.

The Shift Argument(s)

Leibniz used these principles to great effect in an argument that philosophers like to label the ‘shift argument.’ There are really several types: static, kinematic, and dynamical (depending on whether we use translations and rotations, boosts, or accelerations). Recall that the symmetries of Newtonian space mean that the laws cannot be used to distinguish between frames related by Galilean transformations. There is no mechanical experiment one could perform to tell which situation was the ‘real’ one - accelerations are a different kettle of fish, of course, as we will return to.

The Static Shift

Now, let us suppose that this Newtonian space is real, and the theory correctly describes our universe. Let’s suppose that in this universe the tip of your nose is sitting at point x at some instant t. That means, given the symmetries of the theory, there is an infinitude of alternative possibilities (consistent with the theory) differing by some translation, or rotation, or boost, or some combination of them, leaving the tip of your nose at an entirely different (though no less real) point.

The problem (absurdity) that Leibniz draws out is that these possibilities would be qualitatively identical (indiscernible). The world in which the tip of your nose is at x and that in which it is at x + 2 feet westward (generated by rigidly shifting all of the matter in the universe by x + 2 feet westward: an element of the group Gal from the previous chapter) cannot be distinguished since all measuring devices have been shifted by the same amount: all relational material structure is preserved by the transformation (this is the meaning of isometry) - let’s call these shifted worlds ‘Galileomorphs.’ We could alternatively leave your nose at x and shift the time at which your nose sits there, from t to t + 2 seconds (generated by another element of Gal). Again, there is no way to distinguish between t and t + 2 (remember, they smell, sound, taste, feel, and look the same!), so the worlds are strictly indistinguishable. Yet for Newton (and the believer in absolute space and time) they are physically distinct.

For Leibniz this is a massive violation of his principles. One can’t have indistinguishable yet distinct entities. But, worse (as far as Leibniz was concerned), if it were true then it would mean that God created the universe when and where he did without having a good reason for doing so: not cool, says Leibniz. The argument can be ‘de-theologized’ quite straightforwardly. The fundamental point is that Newton’s theory generates a bunch of solutions that cannot be physically distinguished in any way - we don’t need these to fall under God’s gaze for this to be the case. This is epistemologically unsatisfactory, since it means we cannot ever tell which of these solutions is the ‘true’ solution. If we think of possibilities as in some sense real things (possible worlds), then we have a further problem, which is that it is difficult to see in what sense these are genuinely distinct possible worlds. Leibniz’s move, now called ‘relationism,’ was to cut out these indiscernible possibilities, collapsing them to one using his PII.3 As he put it himself: “two states indiscernible from each other are the same state” and the idea that one could shift the entire contents of the universe is a mere “fiction.”

All of the indiscernible worlds match up with respect to the material relations they embody: they are relationally identical. If one has an ontology of relations, then the worlds are viewed as one and the same: the apparent differences are not really physical. This is relationism: space and time are viewed as kinds of ‘secondary qualities’ that depend on distance and direction relations between material bodies and/or events. One also shrinks down the kinds of ‘fundamental stuff’ that exists in this way: there is only matter standing in various relationships, not matter and space and time (understood as equally fundamental) as in Newton’s universe. Of course, if the multiplicity of indiscernible possibilities are not seen to be real, then absolute space cannot be real either. QED!

This pair of positions forms the basis of the modern debate over the ontological status of space and time, and has spread into other debates in which some symmetry is at the root of possibility generation: wherever there is a physical symmetry, there is the potential to generate indiscernible possibilities, which translate, in terms of physics, into empirically inaccessible structure. Whenever one has indiscernible possibilities one has an interpretive fork: treat the possibilities as distinct or as identical (so that, e.g. the differences are merely arbitrary features of the mathematical representation used). The former is generally termed a substantivalist approach and the latter a relationist approach - these terms stretch beyond positions concerning space and time.

There are problems with Leibniz’s strategy of response. For example, Sklar ([46], p. 180) points out a way that the substantivalist might respond in a way that offers a sufficient reason: the material content is where it was yesterday and no force interfered with this during that time, so here it remains sitting perfectly consistent with Newton’s laws. To be otherwise would violate the principle of sufficient reason. The problem with this is that it misunderstands Leibniz’s point. Even given that it was in the same location yesterday and no forces shifted it, the question remains “why wasn’t it at some other point yesterday and, given the absence of forces, today?” That is, why should it have spent its entire existence (even for all eternity) at some undistinguished point, given that all others would be just as suitable? Pointing to prior causes does not yield an answer to the issue of the contingency of the specific points at which the matter sits. Might there be another possible world, also in which we can say the same thing about its location (it’s always been there, with no forces to disturb it), yet this location is distinct from the other in absolute space?

Or there might just be some irreducible randomness involved in the fact that since all points are the same any will do as well as another. Perhaps God threw a cosmic dart to find the central point, or did the trick where one closes one’s eyes and simply drops a fingertip on a map to decide where to go, only here the decision was where to place the universe’s contents. We are used to symmetries being broken ‘spontaneously’ in modern physics (though it is not without controversy); but the point is that we are less likely to be swayed by Leibniz’s talk of ‘God’s sufficient reasons’ for action.

The Kinematic Shift

Absolute locations might be passed off as detectable in a way (though not literally observable): objects are where they are in absolute space, period. We can’t get a high-powered microscope and look at the spatial points, but if they are there then the body will be stationed over some particular points at any instant. Absolute velocities are more difficult since they require motion in a direction. Though we can say where we are in absolute space at any instant (namely right here!), we can’t say where we are going and at what speed. The kinematic shift, involving Galilean boosts, causes similar problems to the static case, but there is a difference then: although there will indeed be infinitely many possible indiscernible solutions (being isometries), each with a different uniform velocity through absolute space, there is, in terms of the PSR at least, a reason to pick one as ‘privileged’ in some sense: the state of absolute rest has certain features that make it special and worth realizing if you are a creator that likes to act rationally. Because it does not involve any motion it has no direction that needs to be selected and so the isotropy of space needn’t be a cause for concern. So while there is not a distinguished origin in terms of location, there is a kind of distinguished origin for velocity. But this leads to another shift problem.

The shift problems all spring from the independence of Newton’s laws of motion (his equations) from time, position, velocity, and orientation, all of which spring themselves from the homogeneity and isotopy of space and time. Instead the equations depend on relative properties: configurations and velocities. But they do depend on accelerations. What we need is a way to distinguish accelerated motion from unaccelerated motion. One way to do this is with spacetime diagrams, in which lines (worldlines) represent trajectories of particles (see fig. 4.2): covering some spatial distance in some interval of time. The faster an object moves the smaller the angle between the worldline and the spatial axis: infinite speed would correspond to the worldline’s being parallel with the spatial axis and rest corresponds to the worldline’s being parallel to the time axis. Straight lines represent constant velocity. Therefore bent lines will indicate departure from constant velocity: acceleration!

Fig. 4.2 Trajectory of two particles in absolute space (restricted to one spatial dimension and time): the straight line represents a particle with uniform speed while the curved line represents an accelerating particle.

The Galilean symmetry of Newton’s laws translates into this spacetime diagram picture into an equivalence of tilted, yet still parallel worldlines generated by a Galilean boost. Hence, in fig. 4.3 (now with an additional dimension shown to reveal the different instants of time) the straight trajectories are indistinguishable yet distinct if absolute spacetime exists.

Hence, we have an awful lot of redundant structure in this representation. We have a unique way of splitting time into three-dimensional snapshots (Nows), corresponding to the unique, privileged inertial frame (with zero velocity), but the laws cannot determine which points of space the particles travel through on each slice. The kinematic shift simply employs the various families of parallel lines in place of the worlds shifted in terms of their absolute locations in the static case and Leibniz’s two-pronged attacked, using his two principles, can get a purchase.

But all we really need to do justice to Newton’s laws is a notion of when a line is straight and when it’s bent (in the technical jargon, we need an ‘affine structure’) to distinguish inertial from non-inertial motion. This requires absolute acceleration and simultaneity, but reference frames (families of lines above) that differ by Galilean boosts are relative. So-called Galilean spacetime encodes this reduced structure, preventing the shift arguments from gaining a foothold. In effect, PII is being imposed on the spacetime but in a very restricted way (to absolute rest and velocity).

This eliminates a serious epistemological defect in Newton’s version. In order to get a foothold there needs to be a notion of sameness of points from slice to slice (so that we can speak of different constant velocities), which Newton assumes, but this goes beyond the notion of straightness of lines across slices. Our ability to detect absolute accelerations does not depend on this additional structure. Removing this renders the world and its boosted counterpart utterly indistinguishable (not just undetectable, as it is for Newton’s spacetime). Galilean spacetime only cares about relative velocities, in keeping with the symmetry of Newton’s laws, so that infinitely many states in Newton’s framework correspond to a single state in this new and improved framework.4

Fig. 4.3 Indistinguishable motions (straight lines) of particles in Newtonian spacetime representing physically distinct velocities, with the untilted lines in the middle representing particles at rest. The curved dotted line shows an accelerated particle once again.

The ‘dynamical shift’ is a different matter entirely, since it involves perfectly detectable forces. These give Newton’s system its power. We turn to these issues next, but strictly speaking the notion of a shift argument in this sense threatens relationalism (along Leibnizian lines): a world and an accelerated counterpart are not indiscernible, so none of the usual tools for ridiculing absolute space (PII and PSR) are available. However, in purely relational terms they are indiscernible: accelerations are only relative motions. In the shifted world version of this, one world is at rest, say, and the other receives a global acceleration. Given the global nature of the transformation nothing relational would change, yet we might expect to feel a force in the accelerated case. It is possible to dig one’s heels in and point out that we can’t say what would happen in the global situation. Still, the law of inertia remains a problem for relationists.

Real Motion

We started with a discussion of motion, rather than the reality of space and time. It turns out that Leibniz agreed with Newton that certain states of motion (the non-inertial ones) are ‘true (i.e. non-relative) motions,’ but seemingly he doesn’t see this as significant for the ontology of spacetime debate:

I find nothing … that proves, or can prove, the reality of space in itself. However, I grant there is a difference between an absolute true motion of a body, and a mere relative change of its situation with respect to another body. For when the immediate cause of the change is in the body, that body is truly in motion; and then the situation of other bodies, with respect to it, will be changed consequently, though the cause of that change not be in them. (LV.53)

But how can this be squared with his overarching relationism about space and time (which must surely infect his view of motion, given that motion is couched in terms of space and time)? Newton had a response, of course. Indeed, his very reason for postulating absolute space and time was to provide a response to the reality of inertial effects (in which real forces are felt because of motion): absolute space is needed to make sense of inertial motion (undisturbed motion in a straight line); absolute time is needed to make sense of constant (i.e. unaccelerated) speed. If we can’t ground inertial motion then how do we ground non-inertial motion? Leibniz grounds it in the bodies themselves, independently of relations to space and time, but also of other bodies. Real motion is that caused by force, and this, for Leibniz, is grounded in what he calls “vis viva” (‘living force’ or that which is responsible for force and so for real motion). This suggests that if we want to know which object is in true motion (and don’t want to rely on the inertial effects themselves: looking for an explanation instead) then we must search for the causal origin of the motion - in the case of the airplane and the runway, we can see that the engines caused the motion. No mention of absolute space or time here. But there is no real story of how the idea works, and what it means to contain vis viva.

Similarly, Lawrence Sklar ([46], pp. 229-234) has argued that relationalism and inertial effects can in fact be squared (and the debate about motion separated from the debate about the existence of space and time) if we take absolute states of motion to be brute, ‘intrinsic’ features of the objects that have them (not in need of deeper explanation), rather than being extrinsically related to absolute space and time (or even other objects, à la Berkeley and Mach) - monadic properties of objects rather than relations to spacetime. Hence, the relationist can believe that space and time are constructs from material objects, but add absolute acceleration as one of the possible intrinsic properties to be had by these objects.5

Mach’s response is simply not to take the bait of these ‘real motion’ examples. They all rest on unverifiable thought experiments, as we saw above. But, as mentioned earlier, finding an actual theoretical scheme that can do everything Newton’s absolutist scheme can do is a difficult challenge, the Barbour-Bertotti models notwithstanding.6

4.2 A Handy Argument for the Substantivalist?

The shift arguments in the preceding section made use of the isometries of Euclidean space (as well as Galilean boosts) to generate indiscernible possibilities. Another isometry is that of reflection. This cannot be achieved by a rigid motion (it is a discrete symmetry), so it is distinguished from the other motions accordingly. However, we can still draw similar philosophical consequences from it.

We tend to take the phenomenon of handedness (sometimes called ‘enantiomorphy’) for granted, since most of us are born with a pair of hands and feet. Our very DNA too comes in both left and right-handed flavours. Some fundamental processes in physics appear to favor a particular orientation: “God is a weak left-hander” as Wolfgang Pauli once said. But how is it the case that there are left and right hands? It seems to have something to do with space, but what precisely, if anything, does it tell us about the nature of space? Also, is the switching of the handedness of something a fundamental symmetry? How do we explain the difference between left and right? The situation poses formidable problems for those that wish to be relationists about space, for if handedness really is a spatial feature, then presumably, in principle, by their lights left and right hands are identical. After all, they can share all of their internal relational properties (same distance from base of thumb to knuckle of forefinger and so on) and yet they are evidently not quite the same, as can quickly be seen by trying to put a left-handed glove on right hand, to use an example due to Immanuel Kant.

Incongruent Counterparts

Kant repeatedly returned to the problem of left and right - ‘incongruent counterparts’ as he called left and right-handed entities: “[a]n object which is completely like and similar to another, although it cannot be included exactly within the same limits” - over several papers spanning more than a decade. So formidable was the problem, that Kant used it (initially) to argue for (Newtonian) absolute space (and against Leibnizian relationism) - he later viewed the same phenomenon as pointing toward a more relational view, as we will see. To see why the problem of incongruent counterparts is so tricky, let’s consider a thought experiment.

Firstly, suppose that God (or the Flying Spaghetti Monster, if you are ‘Pastafarianly’ inclined) wanted to create a catalogue of all possible (instantaneous) relative configurations (relative distances) for the various objects (and their parts) in the world. This would say such things as d(apple, orange) = 2 cm, d(Earth, Mars) = 401,000,000 km, and so on. Now imagine that the full catalogue is made: essentially it should be able to function as a construction manual for (the spatial facts of) the world. Is that really enough to pin down all of the spatial facts of the world, such that any other world that was built following the specifications of this catalogue would be identical? Kant says No, because one has failed to specify how the various objects (the apples and oranges) are oriented relative to a global (worldwide) system of handedness.7 One could imagine two of the Flying Spaghetti Monster’s colleagues, each given a copy of the catalogue, constructing non-identical worlds from the same relative configurations that are mirror images of one another, in the sense that the one cannot be superimposed on the other. They would be what Kant called ‘incongruent counterparts’ (see fig. 4.4).

Fig. 4.4 Two possibilities constructed by following the ‘relative-distance instruction manual.’ Clearly distinct, since incongruent by any rigid motion, yet identical by the relationalist’s lights.

[Image source: 'Immanuel Kant. Aquatint silhouette.' The Wellcome Library and The European Library, CC BY-NC]

This example clearly has a flavour of the Leibniz shift argument about it. We have a pair of worlds, that share all of their relational properties (under the reflection operation) and yet, there seems to be something genuinely different about them in this case (and unlike the case of the static shifts generated by translations). Yet Leibniz writes directly of this scenario that to ask why God created the cosmos as it is, rather than its mirror image, is to ask “a quite inadmissible question.” He views the operations of (global) reflection and (global) translation as much the same: neither generates a discernible difference. We have been a little unfair on Leibniz since he would consider the entire universe reflected so that we don’t have the luxury of comparing it with another, with both embedded in some larger space. Yet the relationalist is nonetheless left with the challenge of explaining in virtue of what a right hand is different from a left hand. So we have two issues: one concerning handedness in the context of whole possible worlds and one concerning handed objects within a world.

The Lone Hand

Kant presents the following thought experiment that appears to demonstrate that contrary to Leibniz, even the possible world version is an admissible question:

Let it be imagined that the first created thing were a human hand, then it must necessarily be either a right hand or a left hand. In order to produce the one a different action of the creative cause is necessary from that, by means of which its counterpart could be produced. ([28], p. 42)

Hence, Leibniz had assumed that nothing could hang on simply reflecting everything in the same way: it would generate the same physical possibility and therefore is redundant. This redundancy would land God into a predicament in which there was no rational reason for actualizing one or the other. But Kant points out that handedness has a peculiar feature: there does seem to be an observable difference. If, for example, we assume that another entity is introduced into such a ‘lone hand’ world then it must break any Leibnizian-assumed symmetry. A hand would be shown to have been left or right all along, and hence there is a real difference between the left and right worlds. The alternative, for the relationalist, would have been that before the introduction of other handed objects, the handedness of the hand would have been indeterminate so that it would fit equally well on either side of a human body, which Kant views as absurd (it would surely only be congruent with one of the body’s hands). Facts about the orientation of things must go beyond purely relative facts. Kant claims that there is some “inner difference” between objects of different handedness. Kant put this difference down to their absolute spatial properties. Hence, Kant argued that handedness reveals the existence of absolute space:

[T]he determinations of space are not consequences of the situations of the parts of matter relative to each other; rather are the latter consequences of the former. It is also clear that in the constitution of bodies, differences, and real differences at that, can be found; and these differences are connected purely with absolute and original space, for it is only through it that the relation of physical things is possible. ([28], p. 43)

We are left with explaining in virtue of what left and right hands differ. Is it some internal, intrinsic feature they possess, or something external to them? Kant himself suggests that it involves reference to the space as a whole. If we have absolute space at our disposal then we can use the points of absolute space to point to some real difference between incongruent counterparts, even though it is a non-qualitative grounding.8 But a simple passing of the buck to absolute space must do more than have a set of points underlying left and right hands; after all, the configuration of points will face the same problem: why are they left and right-handed? Hence Kant’s reference to global properties of a space: it is a relation between the hand and (some structure of) absolute space that is supposed to do the work. This structure should allow for two ways to embed objects in the space - this is precisely why Hoefer believes that primitive identities are needed for the points in considering the two embeddings as two embeddings (see note 8).

Can Relationalists Handle Hands?

Is the situation for the relationalist as dire as Kant makes out? Or does the relationalist have some way of accounting for handedness in the world, and the role it plays? Recall that Kant’s argument says that incongruent counterparts can’t be explained relationally because there is simply no relational difference to be found in them. His lone hand scenario was supposed to clinch this. Yet the relationalist is capable of pointing out that handedness is an extrinsic property of objects: a lone hand has no external relations yet. In introducing, say, an opposite hand, then we have a pair of hands, but neither has its handedness intrinsically. Indeed, incongruent counterparts are intrinsically identical, but the reason for their incongruence comes from relations holding between them.

So a lone hand simply isn’t left or right when alone in the universe: it has no handedness. Left picks out the class of things that have a family resemblance (‘fits’) with whatever was (conventionally) the first left. Nick Huggett ([26], §16.2) calls this the ‘fitting account.’ The idea is that ‘congruence’ is really an equivalence relation on the universe that partitions all of the objects it contains into equivalence classes {Leftys} and {Rightys} with ‘left-handed’ and ‘right-handed’ simply designating the members of these respective classes. One checks for the handedness of some particular object by checking the fit between it and the classes. Clearly we need more than a lone hand to do this, so Kant’s premise is evaded. The problem was in supposing that handedness was an intrinsic property. Incongruence is then explained in terms of the spatial relation holding between them as material objects. (In the case of the Möbius strip world, in such a case the fitting account would be perfectly consistent with the idea that left and right handedness is a local property of the world, so that the partition into lefts and rights cannot be extended throughout the space. The substantivalist account might not fare as well in explaining left and right handedness in such a non-orientable world.)

Broken Mirrors

This highfalutin metaphysics can be linked rather directly with ‘real physics’ by simply asking whether the kinds of mirror reflections considered above are symmetries of physical processes and laws in the same way that, e.g. translations are. What this would mean is that there is no preference given by the laws of physics for some orientation. The laws would operate obliviously to switchings of left and right-handed versions of processes if this were so. It seems like a reasonable assumption, given the isotropy and homogeneity of space: why would physics care if we switched left to right? This was certainly the default position of most scientists in the first half of the twentieth century (until 1957: see below). For example, in his popular book on symmetry, the great mathematician-physicist-philosopher Hermann Weyl wrote:

The net result is that in all physics nothing has shown up indicating an intrinsic difference of left and right. Just as all points and all directions in space are equivalent, so are left and right. Position, direction, left, and right are relative concepts. ([54], p. 20)

Weyl, by his own admission in later paragraphs, was following in the footsteps of Leibniz in saying this. Martin Gardner writing more forcefully (with reference to the Ozma problem of note 7) states:

We are forced, therefore, to concede that our original problem is insoluble. There is neither a formal nor operational definition of left; no means by which it could be communicated to our sister planet. Another way of formulating this surprising conclusion is as follows: Every known inorganic asymmetric structure or phenomenon exists in two mirror image forms identical in all respects except left-right orientations. Mother Nature is ambidextrous. Apart from living organisms, she has no right or left-handed habits; whatever she does asymmetrically, she does in mirror image forms. ([17], p. 210)

He rather presciently adds that “[t]here is no a priori reason why science might not tomorrow discover some type of structure or natural law which throughout the cosmos would invariably possess a left-handed twist” (ibid.).

Counterintuitively, this symmetry (parity symmetry) is indeed violated, for certain lawlike processes (namely, those involving the so-called weak interaction). What we find is that electrons (or beta particles) in a beta decay process will preferentially be shot out of the South side (relative to a strong magnetic field) than the North side. The original experiment was carried out with Cobalt-60 atoms. When such atoms are cooled close to absolute zero, the usually random scattering of electrons from the nucleus is focused into North and South channels. This setup clearly allows one to look at the rates of electrons going in both directions. A world with mirror symmetry would see no difference: why should the world prefer one direction in space than another? Surely the atom is much like Jean Buridan’s donkey between a pair of identical bales of hay? But the electrons did prefer a direction in space allowing for a physical definition of a South Pole (that in which electron rates are highest).

To return to the ‘Ozma problem,’ we can now see how it might be possible to communicate what we mean by left and right, thus enabling any technologically advanced civilization to reproduce any pictures we might send in the right orientation. In six ‘easy’ steps: (1) get some Cobalt-60 atoms, (2) cool them near to absolute zero, (3) align the nuclei spins with a strong magnetic field, (4) count the emitted electrons, (5) call “south” the end with the most electrons emitted, (6) label the ends of the applied field accordingly, transfer these labels to the ends of a magnetic needle, position the needle over a wire in which current flows away from you: left is then where the north pole of the needle points.

John Earman argues that the existence of such a lawlike left-right (or parity) asymmetry (i.e. as opposed to the mere contingent existence of lefts and rights) makes life far more difficult for the relationalist interpretation of handedness. Indeed, he views the failure of mirror symmetry for the laws of physics as “an embarrassment for the relationist account”! As he explains:

Putting some 20th century words into Kant’s mouth, let it be imagined that the first created process is π− − + p → Λ0 +K0, Λ0 → π− − + p. The absolutist has no problem in writing laws in which [one process] is more probable than [its mirror process], but the relationist … certainly does, since for him [they] are supposed to be merely different modes of presentation of the same relational model. Evidently, to accommodate the new physics, relational models must be more variegated that initially thought. ([9], p. 148)

That is, the usual ‘Leibniz equivalence’ manoeuvre (i.e. viewing situations with no intrinsic differences as physically identical) simply fails here since there are non-trivial differences. But there is nothing preventing the application of the ‘fitting account’ here. The “first process” of which Earman speaks is no different to the lone hand of course, and we can say that without comparative processes it has no orientation. The relationalist can, then, describe parity violating phenomena (a spatial asymmetry in the ejection of electrons), and so encompass the laws of such processes, yet they do not explain the asymmetry, treating it as a brute fact about reality. We might have lingering ‘principle of sufficient reasons’-based doubts about whether this is good enough. As Carl Hoefer quite rightly points out, the hidden assumption that makes the relationalist’s response seem underwhelming is that the processes are taken to happen against a background space that allows for multiple possibilities so that the nagging question “how do those subsequent decaying pions know which direction is supposed to be the more-probable one?” is faced (ibid., p. 252). It seems like a mystery (pre-established harmony) how all of those electrons know to go south given they don’t have any intrinsic quality within them that makes it so. But ultimate explanation is not on the table.9

4.3 Special Relativity: From Twins to the Block Universe

There was a young lady named Bright,

Whose speed was far faster than light;

She started one day

In a relative way,

And returned on the previous night.

[A. H. Reginald Buller in Punch Magazine, 1923].

The Classic Twins Paradox

The twins paradox of special relativity is one of the classic thought experiments in philosophy of physics. It appears to show that according to special relativity, for a pair of twins, one of which undergoes a round-trip into space at high speed, they will both appear to have aged less relative to the other when they meet again. Hence the initial paradox: one cannot be both older and younger simultaneously! The problem is, of course, that according to special relativity (in which only relative motions matter) either twin can be considered to be the one that remains at rest (in the ‘rest frame’) while the other dashes off. Though it might seem unnatural to suppose that the spacebound twin is at rest while the other twin (along with the Earth!) whooshes away, from the point of view of the physics, there is no absolute rest frame to ground the truth of one description over the other: all inertial reference frames (roughly, those in which Newton’s first law holds) are equivalent from the point of view of describing physical processes.

But this apparent paradox is easily dissolved: relativistic time dilation will occur, because of the high speed of the journey, one needs to figure out what feature is responsible for the decreased ageing of the space-traveling twin rather than her Earthbound counterpart: what is the nature of the asymmetry? The solution lies in the fact that only one twin will complete a journey in which there is a ‘turnaround’ to make the return journey. Perhaps they slingshot around a star or hit the reverse thrusters. This simple fact means that the spacebound twin must occupy multiple frames of reference (i.e. they will not be in an inertial frame, characterized by constant velocity, for the entire journey), while the Earthbound twin stays in a single inertial frame (since the Earth is in free fall). In which case the spacebound twin indeed ages less. The symmetry that would otherwise allow us freely to use either description (spacebound twin at rest or in motion) is therefore broken, since that only holds for inertial frames. But we still need to say exactly why the traveller ages less, and what changing frames has to do with it.

This solution makes the problem look rather trivial, and you’re perhaps wondering why I referred to it as “a classic.” However, there are still interesting features to probe, including some that have only recently emerged in which the turnaround manoeuvre is removed by a clever topological trick. This ‘topological twins’ scenario allows us to dispose of a common answer to the question of what causes the difference in ageing: accelerations (or the physical nature of the turnaround process itself) during the switch from an outbound to an inbound trajectory - the latter is closer to the truth, but still isn’t quite the proper explanation. Let us develop some of the details of the twin paradox setup.

Firstly, note that the choice of identical twins is simply done to make the example more colorful: all that matters is the differential ageing that results from the high-speed (relativistic) travel. Let’s name our travellers Angelina (Jolie) and Brad (Pitt). We can simply have them wear twin watches if we wish to, to inspect the difference in seconds of proper time passed. (The time shown on their watches (or in their biological processes, which also function as a clock of sorts) is known as the ‘proper time’ and depends on the state of motion of the clock.) These watches will be synchronized before they split at t = 0 and compared when they meet again. Let us suppose that Angelina travels 4.22 light years away, to Proxima Centauri, at relativistic speed (forget about the biological implications of a human traveling at close to the speed of light). Their respective spacetime trajectories (viewed from Brad’s frame) would look as in figure 4.5. According to this diagram, of course, it looks like Angelina’s journey is by far the longer. However, we need to remember that this is simply a representation of the Lorentzian spacetime interval on a Euclidean page: the longer the spacetime interval, the shorter the journey in special relativity. The crucial element in the twins paradox is the dilation (or ‘gamma’) factor:

Fig. 4.5 Brad (light gray) and Angelina’s (dark) trajectories in spacetime. The motion is plotted from Brad’s perspective who simply ‘stands still’ traveling up the time axis, while Angelina whizzes off to the nearest star, and immediately returns (that is, we have assumed an instantaneous turnaround: we could smooth this vertex off adding additional time to the journey). Note that the 45-degree angle for light implies that it travels one unit of distance per unit of time (formally, we say that c = 1, where c is the velocity of light, since velocity is simply distance divided by time).

(4.2)

This factor gives the ratio for the relative rates of Brad and Angelina’s wristwatches (according to which Angelina’s watch appears to run slow relative to Brad’s). Note that it is entirely velocity dependent, with no sign of rates of change of velocity: the faster one travels, the greater the dilation of one twin’s tick rate relative to the other - but note, in relation to the ‘acceleration solution’ above, that no mention is made in this factor of accelerations: dilations do not care about accelerations! Or they do only inasmuch as accelerations are implicated in speed changes. Note also, that special relativity is perfectly equipped to deal with accelerations, which would simply be represented using curved worldlines on a spacetime diagram.

Suppose we have mastered spaceflight to such an extent that we can instantaneously accelerate Angelina to 80% of the speed of light c: v = 0.80c. Let’s round the distance d to 4 light years for simplicity. Brad would calculate Angelina’s roundtrip to be just twice the distance to Proxima Centauri divided by her speed:

(4.3)

This simply means that Brad will be ten years older when Angelina returns home than when she set off. But taking into account the relativistic speed of Angelina, we need to include the γ-factor, . Recall that v = 0.8c and c itself is just 1 (the speed limit). So we have:

(4.4)

So to find Angelina’s age, Brad’s ten years will be dilated by this specific γ-factor yielding γ × 10 = 0.6 × 10. Angelina will have only aged six years compared to Brad’s ten (or, there are almost two ticks of Brad’s watch for each tick of Angelina’s). Remember, also, that since Angelina is in motion at high speed, her spaceship will be contracted in the direction of motion by the γ-factor, so that from her frame (in which she is at rest, of course) she will have covered only 0.6 × 4 = 2.4 light years, which explains the six-year-long round trip: 2.4/0.8 = 3 (using d/v = t) for each leg - note that during the outward leg, the symmetry of their perspectives is preserved: either could speak of the other as the twin that moved. The 6:10 year ratio is a direct consequence of the dilation factor of 0.6 - as an exercise, try playing around with different values of the dilation factor in order to see how big an age difference one can engineer. This is not just a case of Angelina’s watch showing that six years have elapsed rather than ten: she will have biologically aged six years, unlike Brad and the rest of Earth’s inhabitants who have aged ten.

The impact of frame changing on the age difference can be seen with the aid of a spacetime diagram this time highlighting Angelina’s simultaneity slices (see fig. 4.6):

One can see that the instantaneous switch in Angelina’s direction of motion results in a chunk of Brad’s time (four years’ worth) being leapt over in terms of Angelina’s notion of what is happening now. She will compute startlingly different results for Brad’s age immediately before and immediately after her turnaround because of this frame-change. It should be clear than this is not based in the acceleration felt during the turnaround, and we have mentioned nothing other than plain vanilla special relativity. There is also nothing particularly mysterious about the time being leapt over: it is not ‘missing time,’ and is more of an artefact of the way Angelina’s frame (and so her measurements) must alter with respect to her motion.

Fig. 4.6 As Angelina travels away from Brad, her simultaneity slices (what she considers to be happening right now) tilt relative to Brad’s (which are simply the slices orthogonal to his wordline) in order to preserve the constancy of the speed of light (which, you will recall, we think of as covering one unit of distance in one unit of time) - Angelina’s slices are also orthogonal to her worldline, as determined by the (Lorentzian) inner product associated with Minkowski spacetime. She turns around at the star, there is a sudden switch from one inertial frame to another, which results in the simultaneity slices tilting the other way.

There is a sense in which Angelina has performed a certain kind of time travel into the future: she has slowed down her own ageing (a kind of motion based cryogenics) so that she is out of phase with the ageing of those on Earth. Had she traveled faster and longer she could have returned tens of thousands of years into the Earth’s future (relative to when she left) without ageing much at all - perhaps she finds the Earth scorched thanks to global warming? Of course, this is a one-way journey: there’s no going back to Brad to let him and the rest of Earth know their fate. Any other round trips will only send her further into the Earth’s future.

Topological Twins

Recent work on the twins paradox, and the question of what it is actually showing us about spacetime and relativistic motion, has focused on ingenious topological versions, for closed, non-simply connected spaces (i.e. in which parts of the space are identified, such as gluing the ends of a strip together to create a closed loop). For example, by confining Brad and Angelina to a closed cylindrical universe we can have Angelina complete her round trip without having to accelerate or turn around by simply completing a circuit around the cylinder. (We are assuming that it is only the surfaces that are relevant here, so that the setup would be something like the old-fashioned Asteroids computer game in which one leaves one side of the screen only to emerge on the opposite side. But the example can be generalized to a three-dimensional version, which would simply amount to walking through, e.g. your living room wall and coming out on the opposite side of the interior wall. Identifying one side of the screen with the other results in a non-simply connected topology.) Bear in mind firstly that there is no ‘real’ (intrinsic) curvature in this space: it is locally flat and can be constructed by taking a rectangular section of ordinary flat spacetime (the ‘fundamental domain’) and rolling it into a tube by identifying two sides, just as one can make a pea shooter by rolling up a page of paper. If we identify the remaining open ends then we will have made a torus (see fig. 4.7): simply imagine first creating the cylinder and then gluing the two ends of the cylinder together.

Fig. 4.7 A torus (or ‘doughnut’) characterized by a multiply connected topology. A perfect world for considering the twins scenario without the bother of the traveling twin having to turn around.

Poincaré-disc considerations aside (on which, see the next chapter), one could only verify that one lived in a cylindrical or toroidal universe by determining the global structure of the space, for example by sticking a marker into the ground and traveling in a straight line for long enough until one intersected it again. In this way one finds the ‘loops’ in the space. We can imagine a surveyor letting out a reel of string as they go, eventually coming back to their starting point, and we can imagine them tying a knot in the loop and pulling it tight. This would correspond to one of the ‘cycles’ of the space, of which there are two in this case: one around the handle and one around the hole.10 The question is: will there be the symmetry in this space that led to the original twins paradox, so that each twin views the other’s journey as longer?

Fig. 4.8 The possible motions of quadruplets in a toroidal universe in which they must travel. The left side shows the paths they will take around the space (with 1 staying put), while the right side show their worldlines, beginning and ending at the same spatial location. The worldline 1 represents the stay-at-home quad. Image source: [31], p. 540.

There are various ways one could travel on the surface of this doughnut some of which have no counterpart in flat space. The crucial feature is the hole, responsible for a multiply connected topology involving a pair of non-contractible loops: there are two directions (around the hole and around the handle) in which one is prevented from shrinking loops down to points, as with the reel of string above. This is highly significant in the topological twins scenario. In fact, in a very clear presentation of this example, Jean-Pierre Luminet [31] considers a ‘quadruplet paradox’ instead, with each quad performing a different kind of motion in the space (fig. 4.8).

The quad that travels along the second worldline is simply doing the classic twins journey discussed above: a round trip with turnaround. We know that they will have a longer proper time than the stay-at-home quad and the symmetry between their perspectives about ‘who was really in motion’ will be broken because of the second quad’s switching between different inertial frames. But what of the other two quads: they travel in perfectly straight lines (inertial frames) and at no point do they turnaround or initiate thrusters, availing themselves instead of the wraparound topology. If the symmetry is broken in this case, what breaks it? It cannot be a change of frames since they move inertially throughout their journeys. While not a change of frames as such, there is a difference in their frames caused by the non-simply connected topology.

The difference emerges when we inspect the so-called homotopy classes of the various journeys in the space which encode features of the space’s global topology. We say that a pair of loops belongs to the same homotopy class (or are homotopic) if one can be morphed into the other by continuous deformation (i.e. without snapping or gluing either of them). The first and second quads’ journeys are homotopic since the second quad loop can simply be shrunk to a point without meeting an obstruction (such as a hole or a rolled up dimension, again as happened with the reel of string) - they share the same winding index (0, 0). Their symmetry is broken (and the paradox resolved) in the standard way, by the existence of frame changes. The trajectories of quads 3 and 4, however, do involve the handle and hole of the torus and this means that their loops are mutually non-homotopic: they cannot be morphed into quad 1 and quad 2’s trajectories, nor can they be morphed into each other - they lie in entirely different homotopy classes respectively characterized by the winding numbers (0, 1) and (1, 0).

So here is an asymmetry: there are non-trivial (topological) differences in the quads’ trajectories. This causes differences in their frames and is already enough to dissolve the paradox. However, we are unable to see information about proper time. These windings can be visualized in what is called the universal covering space of the fundamental group, which has the effect of ‘unwrapping’ the loops wrapping around the torus. This simply means that we take the rectangle of flat space that we started with when we built our torus, and rather than rolling it up we simply tile the plane with it, but remembering where the original copy was positioned since this will correspond to our (0, 0) case in which no wrapping around occurs (see fig. 4.9).

One can now use the covering space to retrieve information about the proper times elapsed for each quad since this description includes metrical information. As before, the length of the worldline and the proper time are in inverse proportion: the longer the worldline the shorter the journey. In which case, quad 4 (with winding number (1, 0)) ages the least, followed by quad 3 (with winding number (0, 1)), quad 2 (with winding number (0, 0), but with accelerations), and then the poor stay-at-home quad 1 (also (0, 0), but occupying the same inertial frame throughout).

Fig. 4.9 Unwrapping the loops using the universal covering space for the torus, showing our quads’ paths. Neighboring horizontal and vertical cells correspond to one winding, around the hole and handle respectively. Diagonal cells (1, 1) correspond to single windings around both handle and hole. The starting point and the end point correspond to the same point in space. To consider several windings one would have to move further out to other cells. One can see by direct inspection that the worldlines have different lengths.

Relativistic Reality and the Open Future

The twins paradox was found to be no such thing, no paradox at all: simply a feature of special relativity. It exposes a peculiar yet physically verified aspect of our universe: the faster you move the more slowly you age. Before we leave special relativity, let us consider how the relativity of simultaneity (found to be at the root of the twins paradox) has also been invoked to argue for some very deep metaphysical theses about the nature of reality. Time dilation and spatial contraction play no direct role here, and only the velocity-dependent tilting of worldlines is needed (though of course this implies dilation and contraction).

Recall that the relativity of simultaneity is the idea that the present moment (‘the Now’) is relativized to the state of motion of an observer, so that there is no unique such Now and observers that are in motion relative to one another will identify a different set of events as constituting their present moment - this is a three-dimensional spatial snapshot of the universe (the universe at an instant: a notion that will differ depending on an observer’s state of motion) known in the literature as a spatial hypersurface. In a paper that sparked many responses, philosopher Hilary Putnam [38] argued that since it is possible to find pairs of observers such that present events for one, say Angelina, are to the future of the set of present events for the other, Brad, it must follow that those future events are real for Brad and so pre-determined (given that Brad is real to Angelina). In other words: according to special relativity the future is not open.

With the machinery of spacetime diagrams to hand it is simple to see how this ‘fatalist’ conclusion is supposed to come about - fatalism is the view that all events are predetermined: there is no contingency in what will happen. Consider the following diagram (fig. 4.10), modified from the twins paradox diagram above to show the planes of simultaneity for the observers. Thanks to the relativity of simultaneity, the great spatial distance between Brad and Angelina allows for the possibility of differences in their determinations of temporal separations between events (i.e. what they deem ‘simultaneous,’ ‘before,’ and ‘after’). We don’t need spatial separation to get differences in what events are considered to be simultaneous: one can simply have Brad and Angelina pass each other in opposite directions, perhaps as they manage a quick fleeting kiss, so that their simultaneity surfaces are not parallel. In this case, we can say at the instant of the kiss that they are both real (i.e. both exist in a determinate sense) for each other.

All of this thus far simply reveals just how different spacetime in special relativity is from earlier Newtonian physics. There we had a single Now dividing the events up neatly into past, present, and future. Here we have a more complicated affair, but we can still partition the events in the world according to how they can (or cannot) be causally connected by light signals or signals traveling slower than light. This classification of all of the world’s events involves the spacetime interval built from the separate temporal and spatial intervals using the rule:

Fig. 4.10 As Angelina starts her return journey, her surface of simultaneity can be seen to contain events that are in Brad’s future, as well as events to her past that are also to his future, shown here by the shaded area. This might include such events as Brad’s winning the lottery. Likewise, there are some events on and to the past of Brad’s surface of simultaneity that lie to the future of Angelina’s surface of simultaneity. This will still be the case at the moment of contact if Brad and Angelina intersect as they travel in opposite directions. (Note that the same can be said of Angelina’s outward journey, though with a different set of events that are to the future of Brad’s surface of simultaneity so that what is to the future for Brad on Angelina’s outward journey are to the past for Brad on her return journey (and vice versa).)

(spacetime interval)2 = (time separation)2 − (space separation)2

In Euclidean space, like that used in Newtonian physics, one can only ever speak of positive or zero intervals, since we only ever use sums rather than differences. The revolutionary aspect of special relativity is that we must introduce a third possibility: negative intervals. Hence, we get the following three ways that events can be related:

Timelike: (time separation) > (space separation)

Spacelike: (space separation) > (time separation)

Lightlike: (time separation) = (space separation)

Again, in Euclidean space a zero interval would suggest something uninteresting, pointing to the fact that two events are at the same place. In the context of special relativity we find that vast spatial distances can be linked by null spacetime separation so long as they are linked by light rays. The trick is to modify temporal measurements accordingly so that there is no time interval whatsoever for anything moving at light speed! In terms of the spacetime diagrams, consider how the simultaneity surface must tilt if one travels at light speed: it must lie parallel with the worldline (something enforced by the inner product). A beam of light does not experience events as separated in spacetime since its time and space separation will cancel each other out, hence the expression “null” interval. If it were possible to accelerate a spaceship up to the speed of light and run the twin paradox scenario again, then Angelina’s watch would show that zero seconds have passed - i.e. her proper time would be zero and she would not have aged at all.

One can link this to Putnam’s argument by noting that the events that are ‘simultaneously’ both past and future (past for Angelina; future for Brad) cannot be linked to Angelina by light rays nor any signal traveling more slowly than light: they are spacelike separated from Angelina (though timeline separated for Brad). Let us lay out Putnam’s argument more explicitly, before considering the responses.

The argument invokes a link between what is real and what is in one’s ‘present snapshot’ and also an assumption that this reality is transitive: if we are both real, then what is real to you is also real to me - this latter claim he calls the principle of “no privileged observers.” With this in mind, we have (switching to our characters, Brad and Angelina):

1. Angelina-now is real.

2. At least one other observer is real and can be in motion relative to Angelina (that’s Brad, of course).

3. If all and only things that stand in relation R [simultaneity] to Angelina-now are real, and Brad-now is also real, then all and only things that stand in relation R to Brad-now are real.

From this simple set of premises, Putnam concludes that according to special relativity “future things are already real!” That is, even though Angelina cannot communicate or interact with such events about Brad’s future, the fact that Brad is another member of reality, combined with facts about relative surfaces of simultaneity of those in relative motion (combined with a philosophical assumption about simultaneity grounding what is real), it follows that Brad’s future has ‘already happened’ for Angelina, and so for Brad too (by transitivity).

The ultimate conclusion of this kind of thinking is that the notion of a special three-dimensional surface (a Newtonian Now) carving out and constantly reshaping the past and future11 to generate a four-dimensional universe cannot be sustained given special relativity with its multiplicity of Nows. Instead, it must be replaced with a single spatiotemporal ‘block’ (see fig. 4.11).12 Thinking in terms of space and time as separate entities trips us up, and must be replaced with a fully spatiotemporal picture. But, the argument goes, this must mean that ‘becoming’ (in which reality has a dynamical character) has no place in modern physics.

Fig. 4.11 The block universe picture of the world. All events in the block (the universe as a four-dimensional entity) are taken to have the same ontological status. At the instant of your birth your death is already etched into the block - and indeed, your birth too is etched into the block for all eternity!

The long and short of it is this: if there is no home for a present moment in special relativity, then there is no home for ‘becoming’ (nor becoming real) since that requires a division into ‘past,’ ‘present,’ and ‘future’ and any such division is frame-dependent in special relativity. Reality can’t be frame-dependent: who’s frame? All frames? So becoming must go and, many argue, the block must replace it, and the open future is closed up.

In the case of special relativity the disagreement in measurements only occurs if we take time and space as separate entities. If the focus is on the spacetime interval, there is no disagreement about ‘physical facts.’13 As Minkowski famously said in a public lecture of 1906:

Space by itself and time by itself are doomed to fade away into mere shadows, and only some kind of union between the two can preserve their independent reality.

However, we must handle Putnam’s argument with some care. We already saw that the events in question lie outside of Angelina’s light cone and, therefore, can be made ‘future’ or ‘past’ by finding the right inertial reference frame moving relative to her (where we are now defining her by the separation point between her future and past light cones). If we have Brad and Angelina intersect as they move in opposite directions, then the events in question are outside the light cones of both, and so could never have anything to do with either. That such events exterior to the light cone are often bundled together and called “the absolute elsewhere” is no accident. There is much conceptual confusion over how to interpret the status of such events. For example, Howard Stein [49] has argued that such events cannot be said to be real with any justification in special relativity; rather, what is real (e.g. when Angelina turns around at Proxima Centauri) is what lies “at points in the topological closure of [her] past [light cone]” and this will be dependent on the origin of the spacetime point in question.14 Stein puts it as follows:

in Einstein-Minkowski space-time an event’s present is constituted by itself alone. In this theory, therefore, the present tense can never be applied correctly to “foreign” objects. This is at bottom a consequence (and a fairly obvious one) of our adopting relativistically invariant language - since, as we know, there is no relativistically invariant notion of simultaneity. ([48], p. 15)

But Putnam considers this to constitute a kind of solipsism: only what has happened in Angelina’s past light cone is real! What about the rest of reality? She was OK in Tomb Raider, but to claim that she is in charge of reality is going a bit far. Stein agrees that it is solipsistic, but it is so in a “pluralistic” way (ibid., p. 18), applying to any point: we must view what events have ‘become’ from the standpoint of some particular event rather than in a global fashion.

Again, and this is part of Stein’s message, special relativity is a theory of space-time, rather than space and time. It is a theory of light cones constraining causal influence.15 The whole setup of Putnam’s argument involves an older way of thinking, in terms of shared present moments. Stein’s ultimate target is with the view that special relativity necessarily involves a conflict with what philosophers call ‘becoming,’ where events are not to be treated as possessing equivalent ontological status, but depend for their reality on whether they are past, present, or future - becoming is, then, the idea that events ‘become more and more past.’ But time is multifaceted in special relativity. As Steven Savitt [42] clearly notes, there is coordinate time and proper (worldline) time, and the latter is perfectly well-suited for linking to a notion of becoming - this view was developed by Rob Clifton and Mark Hogarth [5]. What alters, however, is that becoming is localized to individual worldlines (observers), rather than to a global Newtonian Now. This might be hard to swallow: is this what we mean by the notion of becoming, of a dynamical conception of time? In a different way, Stein also argues that so long as we are willing to make some revisions appropriate to the shift brought about in the transition from Newtonian to relativistic physics, one can make some (limited) sense of becoming in the world. We were perfectly content to align our notions of reality with the notion of a Newtonian present, and we usually (though often reluctantly) let our philosophical views about reality march in step with advances in physics. So, why not let relativity guide us here? If the present moment ceases to be an invariant notion, then, if we are realists about our scientific theories, our conception of reality must shift accordingly.

Though this is really a special relativity section, there is an interesting supplement to this issue that involves the theory of general relativity. An argument due to the great logician (and Einstein’s friend in his later years) Kurt Gödel claims to show that there is (in at least one world that is possible in general relativity) no objective lapse of time [20]. In other words, no becoming: no situation in the world in which it would be true to say “the future is not yet determinate.” What Gödel showed was that there could be physically possible situations in which there was no way to establish a global Now that definitively split the universe up into past, present, and future events. Without such a notion, he argued, one could not speak of objective change either, since change requires the lapse of time (one thing becoming another thing, changing color and so on). For this reason Gödel viewed time (and its related concepts) as entirely subjective or ‘ideal.’ We return to Gödel’s universe again in the final chapter, where we consider its bearing on the possibility of time travel and time machines.

What we should draw from all of this is that the question of whether there is a present (and so becoming) is dependent on the physical conditions of the world: it is a matter for physics rather than philosophy alone to decide. Future advances in quantum gravity, for example, will no doubt serve to refashion (and perhaps reinvigorate) the debate.

4.4 General Relativity and the Hole Argument

Just as special relativity’s symmetry of Lorentz invariance was at the root of the twins paradox, so the characteristic symmetry of general relativity, diffeomorphism invariance, lies at the heart of our next philosophical problem, known as the hole argument. Just as the twins were not really necessary for the twins paradox, so holes are not really necessary for the hole argument (at least not in its modern guise)! Rather surprisingly, the argument was originally developed by Einstein as a way of showing that one could not have a generally covariant theory of gravity since it would clash with Mach’s principle. The details are a little convoluted, but basically the idea is this: Einstein believed that the matter distribution (i.e. the configuration of mass-energy in the universe) should determine a unique metric for spacetime. However, with generally covariant equations we have the freedom to alter the metric in various ways (using diffeomorphisms, which alter the metric smoothly, leaving ‘deeper’ aspects known as topological structure fixed) without leading ourselves from a possible solution (i.e. a matter distribution with a spacetime geometry: ⟨ℳ, g, image⟩) to an impossible solution. That means that we can generate multiple (infinitely many in fact) solutions for the same matter distribution, in violation of Einstein’s understanding of Mach’s principle (as a broadly relational principle involving the idea that the rest of the matter in the universe determines the motion of bodies in even small regions: distant matter has an effect on local motions: inertial motion is governed by the aggregate of masses in the universe as opposed to a Newtonian container). The hole appears since the way the argument was originally set up involved altering the metric only inside a small hole in spacetime (where the hole is defined by the vanishing of matter within it: i.e. T = 0), around which the metric was fixed to a specific value. In this scenario, we can have knowledge of the geometry and matter outside of the hole (and on its boundary), but even with this complete knowledge, we cannot determine uniquely how the metric will develop into the hole: a failure of determination or causality since we can construct a coordinate transformation (a diffeomorphism) that only acts non-trivially within the hole - the modern version of the hole argument to be discussed in a moment simply turns Einstein’s argument violating causality into a temporal one violating determinism, by essentially making the ‘hole’ the entire future to a slice through spacetime and adding the metaphysical component of belief in the reality of spacetime points (substantivalism).16

What Einstein wanted from his theory was that the geometrical features of spacetime were uniquely determined by the distribution of matter and energy. Before we get to this, and the hole argument itself, we should first briefly explain what general relativity is and how it works. The theory is rooted in the idea that spacetime (the history of the universe) is modeled by a four-dimensional manifold (think of this as a space that can be labeled by coordinates) equipped with a (Lorentzian) metric that specifies distances and angles between points of spacetime (i.e. events). The crucial difference with respect to all other spacetime theories that came before is that this metric obeys equations of motion (Einstein’s field equations): it is a dynamical actor in the theory that couples to the state of matter and energy. The metric in general relativity multitasks, representing both gravitational as well as the above spatiotemporal features. This means that if mass and energy can act as a source of gravity (which they can of course), then they can also act as a source of warping of the geometry of spacetime - this was Einstein’s understanding of the equivalence principle, which sits at the heart of general relativity: in terms of observable properties, a gravitational field applied to a reference frame is identical to an acceleration of the reference frame in the opposite direction. This was argued for using the famous ‘elevator experiment’ (a thought experiment, very similar to Galileo’s ship example: I expect that Einstein had this in mind). Suppose you are confined in an elevator (with no way of seeing out) on the surface of the Earth, which has a mass that induces a gravitational acceleration on objects of 9.81 meters per second squared. Now suppose that an evil scientist floods the elevator with a gas that sends you to sleep, and then shifts the elevator into deep space, but straps a rocket onto the underside that accelerates you at 9.81 meters per second squared: you would not be able to tell by performing experiments located within the elevator that you had been moved at all. But, light beams allowed to stream through the elevator would appear to have a slight curve due to the motion of the elevator through space. This feature allowed Einstein to make the prediction that the gravitational field of a massive body must cause the light to curve in an identical way: this was tested (and confirmed) by Sir Arthur Eddington, who measured the deflection of light by the Sun during an eclipse. Since light travels along geodesics (shortest time/energy paths), it must be the spacetime geometry that is being ‘bent’ by the Sun (the gravitational source).

General relativity demanded a very high degree of symmetry (diffeomorphism symmetry) to perform its function.17 In a Galilean invariant theory one mustn’t be able to detect operations that translate, rotate, or give uniform boosts to the reference frame you are conducting your experiments in. Confined to your ship’s cabin, with only Newton’s laws, you shouldn’t be able to figure out how the boat is moving and where and when it is. In the case of general relativity, the operations are generalized to any motions, including accelerated ones: in your spaceship’s cabin, armed with the laws of general relativity, you can’t tell whether you are accelerating or sitting on a planet by measurements using rods and clocks - in other words: there is no way of telling that an isometry (a spacetime distance preserving map) has been applied.

In more visual terms, this is the meaning of diffeomorphism invariance: warp your spacetime geometry from a perfect sphere into a teddy bear shape and the laws of the theory won’t bat an eyelid. They won’t notice since from their perspective all that matters are the topological properties (the invariants) and these don’t care about stretching and squishing, so long as one doesn’t tear the spacetime or glue pieces together (as with the cylinder becoming a torus above). It is this same feature that leads mathematicians to identify coffee cups with doughnuts: they are topologically identical, each having only one hole or handle. So the predictions of general relativity in a coffee cup universe are identical to those in a doughnut universe so long as the metric field is transformed in the same way as any matter fields by the diffeomorphism that brings about this shape shifting. Likewise, the predictions will be identical regardless of whether the world is shaped like a ball or a bowl.

The Einstein Shift

This idea in general relativity that if one replaces the spacetime manifold with a topologically equivalent (i.e. homeomorphic) manifold, then the physics ‘stays the same’ is at the root of the hole argument - originally, it was the ability to use any coordinate system to describe some physical situation (i.e. the passive understanding of the symmetry). Since homeomorphic manifolds do not differ in their topological properties, and these are what matters, the observable content of the theory is unaltered by the action of a diffeomorphism. This feature has forced many to give up on the idea of a spacetime ‘sitting under’ physical events, and the hole argument plays a large part in this. So let’s present a simplified version.

We can run something like the Galileo ship argument in this case too. Now imagine again that a rather more powerful being, such as the Flying Spaghetti Monster, wants to fool you. First, you make a bunch of measurements using all the machinery we have, to determine a model of the universe, with some spreading of the fields onto the manifold. The Spaghetti Monster then puts you to sleep, does some reshuffling of the points of the manifold (or smearing of the fields over those points) and then wakes you up. If the points are real then the Spaghetti Monster has generated a physically distinct situation. But you won’t be able to tell what has happened, since all observables (those physical quantities that satisfy the laws of general relativity) will be the same, since all the monster did was apply an operation relative to which the laws are insensitive - shifting matter and metric by the same transformation - in which case so are the observables. The formal foundation of this lies in the fact that we are now moving structure that was fixed in the context of the Leibniz shift argument (the metrical structure) together with the matter fields. This means that what look like very significant changes, that warp and bend things out of shape, are not detectable, much as doubling the rate of everything (includings one’s means for checking on the rates of change of processes: clocks, pulses, orbits, etc.) would leave things looking just as they did before.

A little more technically now: we like to think of spacetime in relativistic theories as a four-dimensional block, but if we want to look at dynamical features, it’s useful to carve this block up into three-dimensional slices. We can do this in general relativity, so long as we realize that our slicing has no real physical significance and many such slicings (which would lead to the same four-dimensional block) are possible - this is known as ‘foliation invariance.’ Suppose that we know the matter distribution and the geometry with absolute precision up to and including some slice image. The freedom to perform a diffeomorphism means that even though we have specified everything up to and on our slice (roughly representing our ‘Now’), infinitely many possible developments of the fields off that slice are possible.

As we will see in §5.3, having multiple possible futures from some initial conditions amounts to indeterminism, and so it appears as though general relativity itself is indeterministic. For example, by running the initial state through the equations of general relativity (the laws), we might generate the world represented by the model ⟨ℳ, g, T⟩ or we might generate the world ⟨ℳ, ϕ*g, ϕ*T⟩ (where ϕ is a diffeomorphism, of which we have infinitely many to choose from, and ϕ* an operation that drags fields, such as g and T, around over the points of the manifold).

The catch is that the various apparently possible futures only differ with respect to which points are sitting under which field-values. So we want to know whether g = x or ϕ*g = y sits at some specific point p. As mentioned, no observable facts are affected by this indeterminism. So why on Earth should we be concerned? In the philosopher’s version of the hole argument, due to John Earman and John Norton [8], spacetime substantivalists ought to be concerned, since they believe in the reality of the spacetime manifold and its points. If we follow this view then there should be a fact of the matter as to which point the Spaghetti Monster shifted some field value to, even though it is opaque to experiments. This is, of course, extremely close to the Leibniz shift argument, only with the more general diffeomorphism group taking the place of the Galilean group. Just as Leibniz thought that the proliferation of possibilities that realism about space and time generated in Newton’s world amounted to a demonstration of its absurdity, so Earman and Norton argue that substantivalism must be rejected, for reasons of physics: general relativity is not indeterministic in any sense that matters, and an interpretation that says it is should be rejected. In their own words:

Determinism may fail, but if it fails, it should fail for a reason of physics, not because of commitment to substantival properties which can be eradicated without affecting the empirical consequences of the theory. ([8], p. 524)

The alternative is associated with Leibnizian relationalism: view all of the diffeomorphic futures as representing one and the same physical possibility - they call this “Leibniz equivalence.” The Spaghetti Monster was fooled into thinking she was doing some non-trivial operation by the mathematical machinery we use to talk about the world according to general relativity! In reality, so this Leibniz equivalence option goes, the mathematics of such transformations is a piece of representation that, while helpful in many ways, does not map onto the world: the world is best represented by the (intrinsic) structure that is invariant under such transformations - this equivalence class of diffeomorphic metrics is called the geometry by physicists, by contrast with the metric.

Earman and Norton wrote their paper in 1987, and it sparked an explosion of papers and alternative views, some seeking to defend substantivalism, some accepting the relationalist thrust. We will sample a few of these here, but I leave it to you to decide which makes best sense.

Getting out of the Hole

Firstly, we need to say something about the extent to which endorsing Leibniz equivalence (i.e. the idea that general relativity is about diffeomorphism equivalence classes of metrics) is in fact relationalist. Historically, of course, accepting the idea that a bunch of symmetric possibilities represent the same state is associated with relationalism, as we saw in the Leibniz-Newton debate. But one immediate problem with this view is that general relativity in fact allows for so-called ‘vacuum solutions’ to the field equations, meaning that there is just pure gravity in such worlds (nothing we would ordinarily call matter). So we must ask ourselves, how can there be a relationalist interpretation of entirely empty space? Indeed, how do we contend with the fact that general relativity is a theory of spacetime, and so is presumably committed to its existence?

The catch here is that, as we suggested earlier, spacetime in general relativity (the metric field) is a rather different beast to spacetime in all prior theories, and this has to do with the multitasking feature (spacetime doubles as the gravitational field), and as the quantum gravity theorist Carlo Rovelli likes to put it, with a strong enough gravitational wave, you could smash a rock to pieces. Since the metric field is everywhere defined, if it is as substantial as it seems, then the relationalist has something defined all over and needn’t worry about empty space. But, the substantial entity is spacetime, so why is this the property of the relationist rather than the substantivalist? This confusion (among others) has led the philosopher Robert Rynasiewicz [41] to dismiss the substantivalist versus relationalist debate as “outmoded” in the context of general relativity since the categories used in the original formulation no longer make sense. This makes sense: if both sides are claiming that the self-same object is real then what are they fighting about?

However, it is possible to restructure the debate in the light of the new developments so that we can still have a meaningful debate about how general relativity maps to the world. Carl Hoefer [24] has argued that a version of substantivalism fit for general relativity can be constructed and, given that relationalism is just the denial of substantivalism, so can an account of relationalism in general relativity: it is just the rejection of the idea that spacetime is part of the theory’s fundamental ontology - though, he argues, the latter is not necessarily as well supported as the former and it might well be that the debate is more or less settled in favor of substantivalism. The idea is simply that the metric field plays the same role in this new context that the Newtonian container played in the classical debate: the difference is that this new spacetime is not inert, but influences matter and is backreacted on by that matter.18 This change does not affect the fact that the metric provides us with our basic spatiotemporal facts. Either these spatiotemporal facts are grounded in a real substantival spacetime as modeled by the metric field, or they are grounded in something else.

However, we must not forget that this can be no direct mapping from metric field to spacetime: a lesson of the hole argument is that the metric at a point is itself not physical, since we can smear it arbitrarily over the manifold without changing the physical possibility described. What is physical is, instead, the equivalence class of such smeared metrics (the geometry). But this move as we saw was associated with relationalism. A reaction from substantivalists has been that since such an equivalence class would simply encode the intrinsic physical structure of spacetime, there is no reason why they too shouldn’t help themselves to it - this is known as ‘sophisticated substantivalism’! But, if both relationalists and substantivalists are invoking the same structure then where does the difference lie? Hasn’t the distinction between relationalism and substantivalism simply collapsed? Not quite. On the surface, the substantivalist has more of a case for claiming that the structure corresponds to a truly existing spacetime than the relationalist has for saying that it is in some sense generated by relations. But it is not clear cut. New work, originating in research on quantum gravity, argues that the observables (invariants) of general relativity are necessarily relational (taking the form of correlations between field values).

Earlier responses to the hole argument attempted to prop up substantivalism by consideration of modal metaphysics (having to do with possibilities and possible worlds), some of it quite arcane. To see how these work, bear in mind that the distinct models (or worlds, if you prefer) that are generated by diffeomorphisms differ only with respect to which (invisible) manifold point plays host to which (visible) feature. In one world the point p might be host to the location of maximum curvature, while in a diffeomorphic version that role is played by the point q. This kind of non-qualitative difference (amounting to an invisible role-swapping) is known as a ‘haecceitistic’ difference: the same individuals (points and fields) are present in both worlds, and exactly the same observable relations are realized, but by different individuals in each case. One approach, due to Tim Maudlin [32], to saving substantivalism suggested that points might wear their metric field values as essential properties, so that a world in which they don’t have those selfsame properties is simply not a genuine physical possibility - this only works in situations where the diffeomorphisms do not preserve the points’ metric properties (i.e. where they are not ‘isometries’), so that a world with symmetries is excluded. There is a simple and cogent objection to this view, which is that, while we might agree that metrical properties of some kind are essential to spacetime points, to rigidly attach just those metrical properties a point happens to have as a matter of fact (in our world) seems to be too strong. For example, we can’t talk about fairly innocuous counterfactuals that involve the point having different properties, such as ‘if I hadn’t made a cup of tea five minutes ago, the curvature around my desk would have been different’ - this seems to commit us to the necessity of my teacup being on the desk since the points within the desk and cup would take on different metrical properties!

Maudlin’s response is that we can help ourselves to modal talk of this kind, but without invoking the same points clothed in different properties, by using a tool associated with modal logic, known as counterpart theory (due to David Lewis). This says that the statement ‘if I hadn’t made a cup of tea five minutes ago, the curvature around my desk would have been different’ is true because there is a counterpart desk, cup, and point with these properties. Jeremy Butterfield [3] simply bypasses the metrical essentialist component and uses counterpart alone to motivate a defence of substantivalism: there is only one world with my desk, cup, and the points they occupy. Again, we can consider modal facts about them, but this need not involve those objects being the same in the possible worlds considered: the points in a spacetime and a diffeomorphic version are not the same since the counterpart relation is different from the identity relation. Indeed, choosing a good counterpart relation would involve choosing the closest match for some point in the other scenario, and that would be the one to which the fields were dragged by the diffeomorphism.

Other responses work by similarly denying that there is ‘transworld identity’ linking the points in the different solutions (in a non-qualitative way), but without the additional modal gymnastics. If we simply deny that points have some kind of primitive identity that transcends their qualitative properties, then we end up achieving Leibniz equivalence through the back door. The points of the manifold aren’t transported from world to world, forming an absolute background: if we want to know what points are the same across worlds we look at their qualitative properties. This is the basis for the sophisticated substantivalism mentioned above. Simply put: haecceitism need not be viewed as part and parcel of substantivalism as Earman and Norton had suggested. The problem is, however, that this leaves us very little room to distinguish relationalism and substantivalism, as before. It is possible that a view that simply merges these positions might be more favorable.

4.5 Further Readings

There are a great many books on both the physics and the philosophy of space, time, and spacetime. Many of the latter can often depart from the physics, and lie more within metaphysics than philosophy of physics.

Fun

· Edwin Taylor and John Wheeler (1992) Spacetime Physics: Introduction to Special Relativity (2nd edn). W. H. Freeman and Company.
-This remains one of the best textbooks for beginners to gain some actual computational feeling for special relativity in a light-hearted way - it helps that John Wheeler was one of the great physicists.

· Nick Huggett, ed. (1999) Space from Zeno to Einstein: Classic Readings with a Contemporary Commentary. MIT Press.
-Very useful collection of many of the ancestral voices of contemporary philosophy of spacetime physics, including some of the original papers corresponding to topics discussed in this (and the next) chapter (by Leibniz, Newton, and Kant, for example).

Serious

· John Earman (1989) World Enough and Space-Time Absolute vs. Relational Theories of Space and Time. MIT Press.
- The classic treatment of the debate between substantivalists and relationalists. Exceptionally clear, full of good sense, and still relevant.

Connoisseurs

· Jeremy Butterfield, Mark Hogarth, and Gordon Belot, eds. (1996) Spacetime. Dartmouth.
- It will cost you an arm and a leg to buy, but this is truly a dream collection of pivotal papers on themes discussed in this (and the next) chapter. One can gain a very good feel for the field of philosophy of spacetime physics from this one text.

Notes

1 See §5.1 of his Space, Time, and Stuff(Oxford University Press, 2012).2 The details are a little complicated, but philosophically very interesting. The theory is, in a sense, timeless, making do with the instantaneous configurations and intrinsic differences between them. Space and time are given over to configuration space (usually viewed as an abstract framework for talking about things in space and time), though with the usual symmetries inherited from Newtonian space and time removed, so that each point in the configuration space represents all of the configurations of Newtonian space and time that are isometric. A relationalist theory would then take place relative to this space instead. I refer the reader to Julian Barbour’s popular book on the subject, The End of Time (Oxford University Press, 2000).3 This is very much a ‘Leibniz for dummies’ approach. His true position is extraordinarily complex, and amounts to a non-relationist position involving fundamentally spaceless objects known as monads. We will not delve into these issues, but, for a good place to start, the interested reader is directed to John Earman’s article “Was Leibniz a Relationist?” (in P. A. French et al. (eds.) Studies in Metaphysics, Volume 4, University of Minnesota Press, 1979: pp. 263-276).4 Readers wanting more detail here are advised to consult chapter 17 of Roger Penrose’s Road to Reality (Alfred A. Knopf, 2004).5 This strategy, of making what naively seems to be a relational property a monadic one (internal to the object possessing it) is known as Sklar’s Manoeuvre. On the surface it sounds like a workable proposal, however, it has been rather controversial: see Brad Skow’s “Sklar’s Maneuver” (The British Journal for the Philosophy of Science 58(4), 2007: 777-786) for a contrary voice.6 For a clear-headed philosophical analysis of these issues and more, see Sklar’s Philosophy and the Foundations of Dynamics (Cambridge University Press, 2013).7 Martin Gardner describes a ‘real world’ version of this thought experiment in “The Ozma Problem and the Fall of Parity” (in J. Van Cleve and R. E. Frederick (eds.), The Philosophy of Right and Left: Incongruent Counterparts and the Nature of Space, Springer, 1991: pp. 76-77). The Ozma project was an early attempt to communicate with other planets - ‘Ozma’ refers to the ruler of Oz in the Wizard of Oz. The problem was to design a language that could operate across cosmic boundaries (which would, of course, be characterized by their own idiosyncratic conventions). One such attempt involved a method of transmitting pictures by using a kind of ‘data matrix’ method in which binary code is sent to indicate whether a cell is dark or light. One might send instructions on how to build a piece of technology for example - recall how in the movie Contact aliens sent us instructions for building a wormhole generator. The problem is: how do we transmit information about whether to use, e.g. left or right-handed screws? They might well print their matrix with the instructions entirely the wrong way around relative to the instructions we sent. A possible solution is provided by having them utilize universal laws of physics that violate mirror symmetry.8 Carl Hoefer (2000) has reconstructed Kant’s argument in terms of ascribing ‘primitive identities’ (i.e. brute, non-qualitative facts that allow for comparisons across counterfactual situations: different possible worlds) to the points of space. This is needed to make sense of performing a reflection on the ‘lone hand’ world in such a way that it would generate a new, different possibility - he takes primitive identities to be too great a price to pay given that, he argues, ultimately the relationalist can also explain any facts that need to be explained (“Kant’s Hands and Earman’s Pions: Chirality Arguments for Substantival Space,” International Studies in the Philosophy of Science 14(3): 237-256).9 I refer the reader wishing to have a more technical account of this problem to Oliver Pooley’s “Handedness, Parity Violation, and the Reality of Space” (in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections (pp. 250-280). Cambridge University Press.).10 We can classify these topological invariants by invoking winding numbers that count the number of times a loop (in this case a trajectory of one of our travellers) wraps around the rolled up dimension. So in the case of the torus (m, n) refers to m windings around the hole and n windings around the handle. No winding at all around either would simply be represented by (0, 0). If you have a taste for interesting mathematics like this, then I urge you to read Richard Evan Schwartz’s Mostly Surfaces (AMS, 2011).11 This reshaping can be understood in a variety of ways. For example, we might think of the Now as advancing forward as part of a ‘growing block,’ so that the future is not yet fixed though the past is. Or we might think of this in reverse as a ‘shrinking block’ so that future is ‘eaten away’ by the ever-advancing present. Or, as presentists argue, we might deny reality to anything but the distinguished present moment. Even independently of special relativistic considerations, it has been argued that the notions of ‘Now’ and ‘present’ are anthropocentric, amounting to nothing more than “simultaneous with this utterance” - see, e.g. J. J. C. Smart, Philosophy and Scientific Realism (Routledge and Kegan Paul, 1963: p. 137). Special relativity provides a means of extending this kind of reasoning, linking the Now to a frame of reference rather than anything specifically anthropcentric.12 Nicholas Maxwell has argued that this conclusion (that the world’s events are ontologically fixed) should lead us to reject special relativity because it conflicts with the ‘probabilism’ of quantum mechanics - “Are Probabilism and Special Relativity Incompatible?” (Philosophy of Science 52, 1985: 23-43). This would make a good paper to ‘cut your critical teeth on’ in the light of the discussion in this section and Chapter 7. David Albert also argued that Minkowski spacetime is a hard place to become (or “unfold”) if you’re a quantum mechanical state, though he invokes more aspects than the probabilistic evolution of Maxwell - “Special Relativity as an Open Question” (in H.-P. Breuer and F. Petruccione (eds.), Relativistic Quantum Measurement and Decoherence, Springer, 1999: pp. 1-13).13 Interestingly, since the (three-dimensional) shapes of ordinary objects involve spatial extension (parts separated by space), they too are relativized to frames of reference. It has been argued, therefore, that intrinsic shapes, if they are to exist, should be transfigured into four-dimensional, relativistically invariant properties - for more on this point, see Yuri Balahov’s Persistence and Spacetime (Oxford University Press, 2010).14 Mark Hinchliff has defended the view (called ‘cone presentism’) that identifies the present moment with the surface of the past light cone, so that any light signals that have reached a point constitute that event’s present - “A Defense of Presentism in a Relativistic Setting” (Philosophy of Science 67, Supplement, 2000: S575-S586). This is very hard to swallow for all sorts of reasons. Firstly, it relativizes things to points, so that there are as many presents as spacetime points. Secondly, if we manage to capture light from the first photon created after the Big Bang, then that qualifies as present. Revisions are necessary, but this might be a step too far.15 Stein prefers to call events that related outside of the light cone of a point “causally alien.” This perhaps better captures the issues Stein has with Putnam’s argument, and strikes me as more appropriate than any of the alternatives, but alas the terminology never took hold - “A Note on Time and Relativity Theory” (The Journal of Philosophy 67(9), 1970: 289-294).16 For an excellent philosophical-historical discussion of this episode, see John Stachel’s “The Hole Argument and Some Physical and Philosophical Implications” (Living Reviews in Relativity 17 (2014): http://relativity.livingreviews.org/Articles/lrr-2014-1/download/lrr-2014-1BW.pdf).17 Diffeomorphisms are somewhat difficult to explain properly in an elementary manner, but fortunately we don’t really need the details for this example. They are, more or less, transformations (isomorphisms) along the same lines as translations mapping one point or region to another (i.e. they are maps ϕ from the manifold to itself or to some other manifold, ϕ: ℳ → ℳ′), but that satisfy properties having to do with continuity. In the case of the hole argument we use the action of such maps on fields, so that they have the effect of dragging, e.g. the metric field from one point of the manifold onto another (this action is distinguished by an asterisk, ϕ*). So given a field (with a physical interpretation), which might be defined at a point p, say, the diffeomorphism gives us ϕ* defined at another point q. The value of ϕ* at q is the same as the value of at p (because q = ϕ(p)), but the value of ϕ* at p is not the same as the value of at p. This highlights the way in which the points p and q of the manifold play an important role in comparing the diffeomorphic fields: if the points are real then the field that we have dragged around is truly different in the two cases. If we have some dynamical equations that cannot tell us which is realized in the world, for some complete specification of initial facts, then we will have a case of indeterminism.18 A fuller statement of the view Hoefer calls “metric field substantivalism” can be found in his “The Metaphysics of Space-Time Substantivalism” (Journal of Philosophy 93(1), 1996: 5-27). In this paper he eliminates much of the metaphysical baggage that bloated earlier responses to the hole argument (specifically, the notion of ‘primitive identity’ for spacetime points, that we discuss below).