Choices in General Relativity

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While it is well-known that general relativity allows degenerate solutions to the field equations, it is believed that this indeterminism does not have experimental consequences. In this article I argue to the contrary: that current ways of thinking about matter distributions miss one important kind of distribution in which objects can swap the events through which they pass. Under the interpretation that an object is an observer, these distributions amount to different experiences for the same observers. I show that current relativity has no way of distinguishing between these empirically distinct distributions and new concepts of matter are needed to distinguish them. To distinguish such indistinguishable alternatives requires choice. The indeterminism of current relativity can be seen as the choice that decides a matter distribution. Gauge freedom can now be interpreted as a new kind of (empirical) indeterminism in which the events in the universe are determined, but who enacts those events is not. Relativity is ‘what’ deterministic but ‘who’ indeterministic. The consequences of this view for our philosophical and physical notions about space-time are explored.

Introduction – The Hole Argument

General relativity (GR) is today formulated in three stages [Norton (2008)]: (1) identify a set of events; also called the manifold of events, (2) distinguish and order the events using a space-time metric; the metric tells us which events are close by and which events are far apart in space or time, (3) distribute matter over the event manifold plus metric structure; the distribution produces world-lines that depict individual moving objects within GR.

Now, the Hole Argument goes as follows. Take a manifold of events and distribute metric and matter fields over it in a smooth way. There are multiple ways of distributing the metric and matter fields over the event manifold (governed by the gauge freedom) in relativity. The Hole Argument is the problem that it is possible to identify a small region in space-time in which matter can be smoothly redistributed in such a way as to preserve everything outside the region. This leads to a problem that matter distribution outside the Hole does not determine distribution in the hole and hence relativity is indeterministic.

Einstein originally formulated the argument (given his penchant for thought experiments) as a justification for why general covariance is impossible but later withdrew the argument [Einstein, 1916] echoing a positivist comment of Kretschmann that observations of science must be presented as interaction between objects. This was later rechristened as the Point Coincidence Argument of Einstein by John Statchel [1980]. The Point Coincidence Argument says that the intersection of world-lines are unchanged by any smooth re-distribution of matter (the intersection of the world-lines is interpreted as the events of measurement by an observer), and such measurements exhaust the empirical content of theory. So, redistribution of matter mathematically appears to be a different material reality but is empirically of no consequence.

Earman and Norton [1987] first used the Point Coincidence Argument to critique space-time substantivalism. The main criticism [Norton, 1988 & 1989] of the Hole Argument can be summarized as follows. GR allows a gauge freedom subject to certain invariants remaining unchanged. It is the contention of those who critique the Hole Argument that these invariants exhaust the empirical content of the world[1]. Invariants include observables such as the space-time interval, the total mass of the universe, the total energy and momentum, etc. It may be said that these invariants together specify the symmetries of the universe and GR allows multiple possible representations of the universe subject to these invariants remaining unchanged. John Norton [2008, Supplement] compares these representations to different ways of modeling the geography of the Earth. His claim is that just as the surface of Earth can be modeled through different cartographic representations, the manifold of events can also be denoted as many space-times. While the distance between two points in one such model might appear to be larger than another, this is only up to the point that we have not accounted for the ‘scale’ that must be applied to that representation. However, as an observer goes from one reference frame to another his space-time is stretched or compressed in a way that the invariants remain unchanged. Because these invariants exhaust the empirical content of the theory, the Hole Argument does not tell us anything new empirically.

Arguing the Hole Argument

I will begin the defense of the Hole Argument with a distinction between two kinds of matter redistributing transforms, only the first of which is overtly considered today in the critiques of the Hole Argument (and GR in general). In the present critiques (shown in the first part of Figure-1), a matter redistributing transform is called a diffeomorphism, which globally changes the coordinate representation of the manifold similar to that used in different cartographic representations of Earth. This coordinate change can alternately be interpreted as matter redistribution (considering equivalence of active and passive diffeomorphisms[2]) and it follows that the equations of GR are compatible with several matter distributions. Note, however, that in a coordinate change, intersections of world-lines are unchanged and this was the fundamental point of Einstein’s argument: the coordinate changes do not tell us anything empirically new about the world.

Figure-1: Two Kinds of Matter Redistributions

Transformations of interest to us in this article instead would be those in which trajectories look different globally but they look identical locally. In the second part of Figure-1, an object ‘A’ that was earlier involved in a trajectory intersection with object ‘P’ is after the transformation involved in an intersection with object ‘Q’. The reverse has happened for object ‘B’. Of course, this change cannot come about by simply a coordinate transformation. It rather involves a matter redistribution. If we simply look at the local intersections of these trajectories, nothing has changed; the intersections remain the same. Given that GR relies on point coincidences, such a matter distribution is eminently allowed. At the point of intersections, the properties of the trajectories (defined by the mass, speed, direction of movement and acceleration of the respective frames of reference) remain unchanged. However, the actual objects inhabiting the frames of reference have changed. It is notable, that the points of intersection can be arbitrarily close to each other, provided there are relevant events at which trajectories are intersecting.

The matter redistributing transforms described here are sometimes called symmetry transforms in mathematics. Let us understand this through an example. Think of an object such as a cube. This object can be described in different coordinate representations—such as spherical and polar coordinates. When I change the coordinate system, I globally apply this change to all objects in the space. All diffeomorphic transformations involve global changes. But, there is also a transformation in which I can change the object without having to change coordinates globally. This is for example when I rotate the cube about an axis perpendicular to one of its faces. If the rotation is any multiple of 900 then the rotated and the original objects are indistinguishable. The various operations that leave an object indistinguishable from its other configurations are its symmetry operations. Swap of location between two objects of identical mass is a symmetry operation in GR. Present GR has considered diffeomorphic transforms but not symmetry operations, as matter distributions.

The kind of symmetry operation described here involves the swapping of objects of the same kind—i.e. with identical properties. For instance, GR cannot distinguish when two masses of 5kg each are swapped between two frames of reference. Indeed, given that we are interested only in Point Coincidences now, a set of objects {X1, X2, X3 … XN} can be constructed from available matter in various ways. Imagine that I have 10 buckets each with 1 liter of water. Suppose also that I am required to transfer this water into another 10 buckets such that each bucket after the transfer has exactly 1 liter of water. The simplest way to do this is to transfer the water of each bucket directly into another bucket. This would correspond to a diffeomorphic transformation (in which each point in space-time is mapped to another point after the transformation). But it is also possible to create 1 liter water buckets by ‘combining’ water from other buckets in many different ways, subject to the conditions that (1) each bucket ultimately has 1 liter of water and (2) the total water in all the buckets put together is 10 liters. The different ways of distributing matter into buckets here is not a diffeomorphic transformation[3] but a symmetry transformation.

Trajectories in classical physics (and GR) are continuous. This means that an object cannot swap its location with another object on another trajectory because it can ‘move’ only infinitesimally forward on a trajectory that is its own designated path. So, symmetric matter distributions are impossible in classical physics, and by implication it might seem that they are forbidden in GR as well. However, I will argue here that GR is compatible with discrete matter swaps (as opposed to continuous motions in classical physics) because the theory only deals with the total energy of the universe, not its distribution. Swapping identical objects preserves the mass, energy, events, and world-lines—the net empirical content of the theory.

In the current way of thinking about world-lines, a world-line evolves in terms of its speed, direction of movement and acceleration. At each point of the world-line, I can attach a frame of reference that defines the speed, direction of movement and acceleration at that point. Now, the world-line can be interpreted in two ways. First, I might say that an object inhabits a particular frame and that frame evolves along with the world-line; so, a world-line is the motion of a frame with an object fixed onto the frame. Second, I might say that each point on a world-line has a frame fixed to it; frames don’t evolve; rather, an object ‘jumps’ from one frame to another. In principle, a jump from one frame to another can be arbitrarily small. In classical physics, I can tend their size of jumps to 0 (through a limit) to achieve the continuity of trajectories. But, in GR, given that I am interested only in point coincidences and point coincidences need not be continuous (indeed, as I will shortly show they cannot be continuous), GR is compatible with the notion that an object actually performs a discrete ‘jump’ from one line to another. The continuity of trajectories (determinism) follows if I tend the jumps to 0, but as I will show, this is not necessary.

‘Jumping’ is also consistent with quantum ideas about matter (think of a quantum particle ‘moving’ in a bubble chamber)—QT does not allow continuous trajectories but allows discrete jumps. If GR has to be reconciled with QT, then the ideas about motion or trajectories must be reconciled prior. Since QT cannot be completed with continuous trajectories (a wavefunction must alternate between measurement ‘collapse’ and deterministic evolution), evolving our notions about motion in GR makes sense. In particular, converting continuous trajectories to discrete ones in GR is one possible approach to ultimate reconciliation. Further, because the next frame to which an object is going to ‘jump’ in can be arbitrarily close to the frame from which it is ‘jumping’ out, the idea of discreteness does not do violence to the general principles of GR. In the present theory, I could think of these points being arbitrarily close by and when I attempt to unify GR with QT, I might then construe that the Planck’s constant ћ places a limit to the ‘size’ of these ‘jumps’. The change from continuity to discreteness is a major change; fixing the size of these jumps later would be only a matter of degree.

The point of introducing the notion of jumps here is that this allows us to think of matter distribution over the event manifold in an entirely new way. Specifically, if an object has to ‘jump’ from one frame to another, then it can jump to several different frames. Redistribution of matter can now be compared to objects playing musical chairs – the chair being the trajectory upon which an object ‘sits’. In the simplest case, two objects very close by, each having the same mass, can swap locations, quite compatible with GR. In a more complex case, multiple objects can re-divide and swap locations subject to the total mass, energy and momentum remaining conserved. (Indeed, if I apply quantum principles to interpreting GR, I might say that there is indeterminism with regard to which frame the object is going to jump into and hence the next frame of the material object cannot be predicted.) In classical mechanics, this is conceivable only in the case of collisions, when trajectories actually intersect, because then the identities of objects can change due to collisions. But, once I conceive of jumps, it is not necessary that the frames of reference of the object playing musical chairs actually intersect. The trajectories could just happen to be nearby.

While the notion of jumps fits neatly with notions of matter in QT, it does not depend upon them. There are justifications within current GR for considering these jumps as valid physical notions. Let us consider these reasons now. Newton’s gravitational theory uses continuous trajectories, each point of which is regarded a fact of measurement because an observer in principle continuously observes a trajectory. When we drop the notion of continuous observation in GR[4], we are left with Point Coincidences—the intersection of trajectories. An object now is observed only at the points at which it intersects with other trajectories. So now, the relevant facts for the theory are not continuous trajectories but the points of trajectory intersections. While GR still uses continuous trajectories, this use is not necessary with the Point Coincidence Argument in place, unless of course, I insist that a trajectory has infinitely many intersections with other trajectories at infinitely short intervals in a continuous fashion.

GR is consistent with the idea that events of intersection can be arbitrarily close without requiring us that this jump tends to 0 (which is needed for a trajectory to be continuous and bars jumps). In fact, the notion of continuous intersections is inconsistent with practical experience because the interval between trajectory intersections tending to 0 amounts to the use of objects of size 0 which is physically impossible (although classical mechanics and GR postulate it). Point particles of size 0 were used in classical mechanics although no physical objects are actually of size 0. Zero size objects don’t create conceptual problems when I have to just draw continuous trajectories but they do create problems when I have to conceive of trajectory intersections. Now, if the intersecting trajectories correspond to finite sized objects, then these intersections must at least leave a gap of the size of the colliding particles for us to be able to identify each intersection as a distinct event. If the events are placed closer than the size of the objects, then events cannot be distinguished. Indeed, I might say that the quantum uncertainty principle has taken us to a logical limit to the divisibility of space-time although within a classical theory such as GR I could merely surmise that there must be a logical limit to divisibility of a trajectory into individual points without actually specifying this limit. In present GR, I am perfectly well off even when I just allow two trajectories to intersect arbitrarily close without tending this distance to 0. In such a case, present-day GR is consistent with the second kind of matter redistributing transformation proposed here.

Giving up continuity of trajectories to retain Point Coincidence only allows us to construe new kinds of matter distribution without violating any physical principles.

Now we might ask: how is the new kind of matter redistribution empirically any different from the first kind? After all, we still have the same Point Coincidences, and thus the events predicted by the theory remain unchanged. The answer is that although events remain unchanged, the participating objects in those events are different. In principle, any of these objects could be observers. And when observers are present in one frame and not another, there is a tangible difference to their personal experiences.

Frames vs. World-lines

The current space-time perspective in GR is drawn from Special Relativity (SR) in which frames of reference move at constant speeds against each other and an observer essentially looks upon the universe as a bystander who might dispassionately observe the events of the world go by. Because the observer moves at constant speed, and there is no acceleration or gravitation involved, it can be supposed that the observer is indeed mass-less (the mass of the observer has no causal role to play in SR). For, if the observer is massive and he moves through a gravitational field, then, (1) he must accelerate and, in the process, (2) alter the gravitational field in turn. So, the moving frames of SR are mass-less goggles. And because they are mass-less they are causally outside the world. If we apply the perspective of SR to interpret GR, it leads to the premise that frames of reference in GR are different perspectives, quite like those of the moving observers in SR. Thus, an observer uses a frame of reference as he was accelerating although the observer stands outside the manifold of events. This is essentially how GR is constructed today. Thus, in step (2) of the formulation of GR given in the Introduction, I distribute metric fields upon the manifold—a metric that applies globally—from the perspective of an observer that stands outside the manifold. Obviously, if there is a metric, there must be an observer. And if the metric corresponds to an accelerating frame, then the observer too must accelerate. But, the observer in this case is outside the manifold and therefore is also mass-less (his mass has no causal role to play). Two problems can now be raised. First, how can a mass-less observer accelerate? Second, can there even be a mass-less observer?

Figure 2: Observation Frames in SR and GR

The above two questions are rhetorical. In the present theory, answers to both are in the negative and this just goes to show that there cannot be mass-less frames of reference that accelerate. All observers must be massive and therefore observers must be causally inside the world and not outside of it[5]. Now, when an observer is a massive object inside the world, his space-time pertains to a trajectory rather than the global space-time. That is, unlike in SR where space-time included the observation of all objects (because the observer was outside the world and looked upon it) in GR, an observer can observe only the events with which he interacts. These interactions are identical to the observer’s world-line. This requires us to collapse the distinction between a frame of reference and a physical object. The fact that GR allows frames of reference to accelerate introduces the case when real massive observers can observe the world (instead of the mass-less goggles of SR). Now, I can say that the very reason a frame is accelerating is because the observer in it is massive (it requires us to treat every physical object as an observer, and although this is an extreme thought it is compatible with GR). While in SR, frames were 4-dimensional mass-less goggles, these must now be treated as world-lines in GR. Each world-line is a space-time frame as the observer who observes the world from his vantage point.

It follows that a collection of world-lines is not only multiple different objects but also simultaneous different perspectives about the world because each object is now a distinct (accelerating) frame of reference. Because all observers are massive and all observers are within the world, each observer must bring a different space-time view in which the laws of nature are true. Each of these space-times is a simultaneously true representation of the world. Coordinate transforms are not about a hypothetical scenario when observers could accelerate and observe the manifold of events while standing outside the world (like they used to in SR). In GR, to even accelerate, an object must be inside the world as one of the objects. In this view, different coordinate systems are (1) the ability to regard each object in the world as an observer with its own frame of reference and (2) the ability to cut up the manifold of events into different trajectories. Figure-2 shows how a massive observer must exist causally inside the world and not outside it. Now, the number of possible frames of reference at any point in time equals the number of objects in the manifold (each object carries with it its own perspectival space-time). Hence, an observer gains new experiences by being in a different frame and matter distribution over different frames leads to new experiences.

World-lines vs. Objects

Now, this might appear to be a mysterious issue associated with consciousness, but it is not. This issue rather relates to the fact that when GR draws world-lines, a frame of reference is not explicitly associated with an individual object but only with an object of a certain kind. A frame of reference just defines a role in which a particular object (observer) can step in. Roles are merely objects of a certain kind—for example, I might say that trajectory X is an object of 5 kg moving at 10 meters per second. By cutting up the world into roles I am merely stating which kinds of objects I want in the universe. There can be several objects that have the mass of 5 kg and the frame is compatible with any of them inhabiting it. Now, it is apparent that the fact that an object is capable of inhabiting different frames of reference is just how classical mechanics is designed. Here, an object can have many different velocities and acceleration (the different reference frames). So, how is the problem of consciousness now being introduced? How do I physically decide that an observer goes into one frame and not another? Without going into details I might just suggest that newer ways of distinguishing objects are needed to account for this new problem; these ways must allow us to distinguish between two objects that each weighs 5 kg. We can also suspect that these newer ways might come from QT since all objects (fermions) in QT are distinct. So, if I were to look upon two masses of 5 kg each, quantum theoretically, there would indeed be a distinction between them. The distinction between the experiences of two objects in two different frames is in everyday parlance a fact of consciousness. But this fact requires having a material correlate. GR does not have this correlate, so the correlate must come from elsewhere.

In terms of present GR, I may just draw a distinction between a frame and an object, which I have already made when I said than an object hops through frames. While at this point the ability to hop to one frame rather than another appears as an indeterminism within GR, this indeterminism can be looked upon as opening up the need for newer ways to distinguish objects within GR and hence provides us with a reason to look for a synthesis with QT from within the confines of GR. Such a justification is not present in GR today because the theory by itself appears complete; there are no theoretical reasons for looking to unify GR with QT apart from attaining a unified theory (which is a human reason). The indeterminism of GR can rather be looked upon as the reason why a synthesis with another theory is even required. If the unity with QT does solve the problem of indeterminism, I would say that the quantum and observational reasons for distinguishing between identical masses entering different frames are really the same. Or, that, the differences in the experiences of an object resulting from its presence in a particular frame of reference can be attributed to its quantum theoretic distinction from other objects. In other words, quantum gravity would bring us closer to thinking of material objects as conscious observers. This might be called a quantum gravitational approach to the problem of consciousness. In current similar approaches, Penrose [2004, 2007] proposes the unity as a method by which gravity ‘collapses’ the quantum wavefunction into a definite state. Instead, I propose the unity as a method by which quantum theory distinguishes identical objects in different frames of reference in GR. In Penrose’s approach, GR is deterministic and it overcomes the indeterminism of QT. In our approach, GR itself is indeterministic but in a way that can complement the indeterminism of QT.

Indeterminism in GR

The key point of GR in the context of new kinds of material distributions is the following. When an object accelerates (because of force or because it changed its identity or both), it goes over different events than it would have if it was not accelerating and this modifies the events other observers go over as well. The invariance of properties in GR entails that when an object changes course, it must cause changes in other objects such that some other object now passes over the event that was forsaken by the object that was originally supposed to have passed through it. In other words, GR allows us the freedom to redistribute the matter in the world such that when one object changes course, other objects co-vary with it. The total set of events and their empirical instantiation (the space-time coincidences) remains unchanged. But, the empirical instantiation of events is now enacted by different material objects than before. An event in the world can be multiply realized through various different objects passing over the same events. When different material objects realize the same events, it leads us to a new kind of indeterminism.

Think of two hungry people at a table, with only one ice-cream in front of them. One of the two individuals can eat the ice-cream and if the first one doesn’t, the second would definitely. These two possibilities are identical as far the events are concerned (namely, that someone eats the ice-cream). However, they are not identical with regard to the distribution of matter—who actually eats the ice-cream? Both possibilities are identical in terms of intersection of the world-lines, the so-called empirical content of GR. But this empirical content is global and not individual. While the global empirical content remains unchanged (someone eats the ice-cream) the individual empirical content changes—one of the two individuals eats the ice-cream. In GR, this can be represented by two space-time frames. In the first frame, the observer sees himself eating the ice-cream and in the second frame, the observer sees his partner eating the ice-cream. But which observer is in which frame? Relativity is compatible with either of the observers being in any of the frames and this is the indeterminism. The universe is deterministic in that it defines two roles or frames of reference one of which eats the ice-cream while the other merely observes. But the universe is not deterministic with regard to which object is in which reference frame. It follows that what will happen in the universe is fixed but who will participate in it is not. Figure-3 illustrates this indeterminism.

Figure-3 Dilemma of the Ice-Cream Baiters

There are two ways in which to look at indeterminism. The first way is about how I draw trajectories in the manifold of events. This involves connecting various events in the manifold and there are several distinct ways of doing it. Notably, these event connections may not necessarily indicate the evolution of the same object—like they used to in classical mechanics (in which the identity of an object did not change while moving). When matter is redistributed, an object’s identity (mass in the case of GR) might change as it goes through different events. In the ice-cream case, it corresponds to whether the event of ice-cream eating is enacted by the customers or by the ice-cream vendor himself. In one case, the eating of ice-cream is preceded by people driving to the vendor and being served on tables. In the second case, the eating of ice-cream might be preceded with ice-cream vendor type of activities such as serving a customer ice-cream or collecting their payments. Different kinds of trajectories can therefore pass over the same event. The second way is about which actual instance of identically looking ice-cream cones is actually served to a particular customer. This difference is not seen if I just look at the different frames (the possibilities of whether or not a person eats the ice-cream or which particular cone he eats) because different frames are different perspectives about the world (the different objects in the world). The difference is rather evidenced by deciding upon which object inherits which frame of reference because then the other objects must inherit other frames of references. The two ways of seeing indeterminism are the same—namely, which object passes over an event. In the present case, there are two objects passing—the ice-cream and the ice-cream eater—and both are subject to the same indeterminism.

Now, one might ask: if GR is indeterministic why hasn’t this been empirically discovered? And the reason is that current relativity does not a priori specify the manifold of events. So, when I try to confirm the theory using experiments, a variation in some individual observations can be attributed either to indeterminism in the theory or to a different manifold of events. In the case of current GR, I would do the latter. So, I would say that the universe passes through the observed events because the universe must deterministically pass through these events (which, however, could not have been known a priori because I don’t know the exact set of events in the world). However, as I have argued in this article, the actual flow of matter over these events is based upon choice (theoretically it is a symmetry between different distributions) and the observations reaped depend upon the choices made. It is also possible to argue that choices are not apparent, because changes to world-lines are caused by the gravitational force, which is quite a weak force. So, it might be concluded that large galaxies have a tangible freedom but I as an individual do not. Although it is outside the scope of this article, I can speculate, that this is where the reconciliation of relativity with quantum theory becomes important. As is known, quantum theory deals with the electromagnetic, weak and strong forces while relativity deals with gravity. It is the forces under quantum theory that govern most of everyday objects (almost nothing in modern technology has anything to do with relativity). It has often been supposed that relativity being a deterministic classical theory is contrary to quantum theory, owing to the nature of indeterminism faced in quantum theory. It might instead be said that both theories are indeterministic and this makes their reconciliation that much easier. Of course, this means that I would need to introduce new causal mechanisms by which indeterminism in the joint theory can be overcome but that is a step that I might take for achieving the unity. Before that, it would also be necessary to reformulate GR along the lines of the new kind of construction proposed in this article (equating frames of reference with world-lines; applying not a single metric but multiple metrics simultaneously to the manifold of events).

Reinterpreting Relativity

When currently GR distributes metric and matter fields over the manifold it does create world-lines but all these world-lines are in a single space-time. We don’t put each of the objects in different space-times because I don’t think of them as observers. But, the space-time in which all the world-lines exist is drawn from the perspective of an observer who is mass-less and not part of the observation. This is the classic mind-matter divide in which the masses are not observers and observers don’t have a mass. When an observer positioned outside the world performs a frame transform, he would only change the coordinate representation of the world-lines without changing their identities. This can lead to someone (correctly) arguing that the Hole Argument does not entail new experiences. Such changes are indeed akin to various cartographic representations of the Earth’s geography [Norton, 2008]. This is because new experiences only come about when someone performs a frame transform being inside the world. In this case, an object’s changing frame of reference can mean that an object is now jumping through a different set of frames. A change in the association between matter and frames of reference is a symmetry operation that redistributed matter in a way that GR cannot discriminate.

The empiricism of events does not exhaust the total empirical content of the world; there are facts such as which world-lines pass over which events, and which objects inhabit which world-lines which are not explained by GR. GR is in fact totally compatible with different distributions of matter, each of which results in a different history of the universe, and this is the indeterminism. The history of the universe is not just the events that occur in it but also which material objects pass over those events. There is a clear empirical difference between various alternatives in which a particular object passes over an event and when it does not pass over that event because the object passing over the events can itself be the observer.

The difference between SR and GR can be obvious to a super-observer[6] who sits in the manifold of events (and not on a particular world-line) and can see the various distributions of world-lines over the events. Such an observer would be a priori unable to decide which course the universe would take. The indeterminism of relativity isn’t a hypothetical one, and this is apparent from the position of a super-observer. GR is compatible with the universe following different courses of matter distribution, while preserving the events. Whereas SR is about possibilities of observing the world differently from different individual perspectives, GR is about various possibilities in the perspective of the super-observer himself.

Space-Time Substantivalism

Prior the advent of SR, the 4-dimensional Euclidean space-time was regarded to be the universal container in which the world exists. But, with the advent of SR itself, it became difficult to hold this viewpoint. If there is a unique space-time, then why do different observers observe different lengths and durations from moving frames? Indeed, the notion of the constant speed of light is generally taken to be the refutation of the existence of any plenum in which things exist. But, now, GR throws up new possibilities. In particular, given that all individual space-time perspectives are derived by overlaying a metric and matter structure over the event manifold, I might be tempted to think of the event manifold as the universal space-time [Earman & Norton, 1987]. Because the metric carries the matter and energy, this metric can be considered the contents as opposed to the container of the space-time events.

But, keep in mind that the metric distributed over the manifold of events (the steps 2 & 3 in the construction of GR) are carried out from the perspective of an observer outside the manifold and these therefore don’t pertain to the observed intervals. The observed intervals are the distances between events along the world-lines, since every observation event is on the world-line itself. How then do I determine the distances between different world-lines? The answer is that nothing in the perspectival space-time can give us that distance. Perspectival space-time is created at the time of an observation and only gives us the distances between events on the same world-line; it does not give us the interval between the events on different world-lines. The a priori ‘distance’ between events in the manifold (before matter is distributed on these events) must therefore be defined in some other yet unknown way. Because to even redistribute matter over the manifold of events I need a metric and because the perspectival metric is defined only after matter has been distributed (at least objects of a certain kind have been identified), there must be another way of distinguishing events in the manifold before matter is distributed upon these events. We can call this new metric the noumenal metric as opposed to the phenomenal metric of the perspectival space-time which describes the observed intervals.

Because the event manifold is fixed and objects of the world cover this event manifold, the noumenal metric is a fixed set of intervals between the events. Of course, this is not necessarily how an individual observer sees these intervals since in principle, the trajectory of an object is not fixed, but determined run-time when the object passes over certain events. It is evident to us from everyday experience that when multiple objects pass through different events, they create a distinction between events. The distinction between different events—at any instance in time—is precisely the distinction between objects. And since different material objects can pass over the same events, we might say that the spatial distinction between events is given at the point of matter distribution over events. The noumenal metric—it can be surmised—is purely a time-like metric between events. We might look upon the manifold of events plus this time-like metric as a fixed ‘script’ through which the universe evolves. The script describes events but not the actors. And this is precisely the indeterminism of relativity theory.

The manifold of events and its associated metric can be viewed as an ontological reality, which is translated into phenomena. In an actual history, this ontological reality is known, and the knowledge of reality could vary from one history to another depending upon which objects pass through which events. So, the event ontology does not fix the history, but only constrains it. When an event is enacted by different objects, knowledge of which object has enacted an event changes but the event itself remains unchanged. The perspectival space-time is a moving world-line and that moving observer views (parts of) the event manifold from his perspective; it is created at the point of measurement and this space-time which goes with the observer. The perspectival space-time is relative but the manifold of events and noumenal metric are universal.

I don’t intend here to suggest that the space-time manifold is a plenum, thing or object (the conventional ways of thinking of ontology). The manifold is rather like the script of a drama that doesn’t yet have any actors. New ways of conceiving ontology are needed to accommodate this view. In particular, we are talking about the script of the universe independent of the actors in it. The actors can see the script and then choose to participate in it.

When someone looks at a desert and envisions a grand city in its place, he is ‘seeing’ although not experiencing in the phenomenal sense of having sensations. We constantly ‘see’ the world as possibilities for the future. In that sense, the manifold of events is not a sensation (until matter is distributed over events) and yet it is known as an imminent possibility. The manifold of events becomes a sensation at the point of observation. This comes very close to views about matter in QT in which reality is said to be ‘created’ at the point of experience, by the act of measurement, and this could prove helpful when conceiving of a unified theory of quantum gravity. That GR enables a similar viewpoint as QT might be the element of surprise for many. And yet, with the foregoing discussion it is possible to see GR as dealing in the script of a drama while QT deals with the actors which participate in the script. The separation between an actor and a role gives us the intuitive basis to reconcile two formidable theories—QT and GR. But it also introduces a new type of ontology in which the universe has a destiny or script quite separate from the freedoms of choices each individual enjoys.


  • Earman, John and Norton, John (1987) ‘What Price Spacetime Substantivalism? The Hole Story’, J. Phil. Set. 38 (1987), 515-525
  • Einstein, Albert (1916), ‘The Foundation of the General Theory of Relativity’ in H.A. Lorentz et al., The Principle of Relativity, New York: Dover, 1952, pp. 111-164.
  • Norton, John, ‘The Hole Argument’, The Stanford Encyclopedia of Philosophy (Spring 2008 Edition), Edward N. Zalta (ed.), URL = <>.
  • Norton, John D. (1987), ‘Einstein, the Hole Argument and the Reality of Space’, in John Forge (ed.), Measurement, Realism and Objectivity, Dordrecht: Reidel, pp. 153-188.
  • Norton, John D. (1988), ‘The Hole Argument’, in A. Fine and J. Leplin (eds.), PSA 1988, Volume 2, pp. 56-64.
  • Penrose, R. (2004), Shadows of the Mind: A Search for the Missing Science of Consciousness, (Oxford University Press).
  • Penrose, R. (2007), The Road to Reality: A Complete Guide to the Laws of Nature, (Vintage Books).
  • Stachel, John (1980), ‘Einstein’s Search for General Covariance’, in Don Howard and John Stachel (eds.), Einstein and the History of General Relativity (Einstein Studies, Volume 1), Boston: Birkhäuser, 1989, pp. 63-100.


[1] I will later dispute this contention. I will argue that invariants exhaust the empirical content of present GR although not of the world when material objects are also observers.

[2] The difference between active and passive diffeomorphisms can be understood by the following example. Imagine that all continents on the surface of Earth are clockwise rotated without rotating the Earth itself and without changing the distances between continents. This is Active Diffeomorphism. Contrast it to the case when the Earth itself is rotated counter-clockwise, while leaving the continents intact. This is Passive Diffeomorphism. It is easy to see how Active and Passive Diffeomorphisms are equivalent. In fact, given their equivalence, it might be hard to see why they should be distinguished. The picture gets slightly more complicated when the coordinate transformation also involves time because then, in the Passive case, the continents are at a new location at time T+1 whereas in Active case they are at the new location at time T itself.

[3] Diffeomorphic transformations are bijective (one-to-one) and differentiable, which the distribution of water is not since the water distribution can be one-to-one, many-to-one or one-to-many.

[4] Now the observer itself must be an object and must have its own trajectory. When an observer observes, trajectories intersect. To suggest that an observer continuously observes an object (which is the premise in classical physics) two trajectories must continuously intersect – or practically speaking – must be congruous all over. The notion of continuous observation can therefore be replaced with at-an-instance observation without losing empirical validity. Now, at the times when an observer observes, his trajectory intersects with the observed world.

[5] This point is further important because Gauge invariance pertains to local symmetries and not global ones. Global invariances correspond to an observer outside the world who observes the world but does not itself have physical properties – e.g. mass. This idealization is misconstrued because all observation involves a physical interaction, and for that to happen the observer must also have physical properties, quite like the objects he observes. This essentially reduces an observer to a physical object (at least from the standpoint of empirical measurements) and the interaction between the observer and the objects is a local interaction based on transfer of energy. The observer can’t know the world globally unless the real-world specifies an energy transfer from every object in the world into a single observer-object.

[6] Such a super-observer is a hypothesis, introduced merely to illustrate the distinction between SR and GR.