Just when the hubbub over the infamous hole argument seemed to have died down, and you thought it was safe go back into the placid pool of classical GTR (away from the white-water rapids of quantum gravity), John Earman has conjured up yet another monster to trouble poor old Ein- stein.1 This time, a sprinkling of the magic powder of the constrained Hamiltonian formalism has been employed to Thoroughly Muddled resurrect the decomposing flesh of McTaggart, who intones that the GTR (yes, the plain old vanilla GTR) implies that "no McTaggart genuine physical magnitude countenanced in GTR changes over time" (p. 6), i.e., that if the GTR is complete, we not only live in a block universe, but the block is "frozen", with no Or real physical quantities changing. I hope to drive a stake through the heart of the undead McTaggart and end this How to Abuse Gauge Freedom to new rampage before it has begun. Generate Metaphysical Monstrosities I choose the image of driving a stake through the heart with some care. There is much in this paper, especially in the latter sections, which will be of considerable use to phi- losophers of physics. Professor Earman has, as usual, mas- tered quite a lot of mathematical physics that would be be- Tim Maudlin yond the grasp of most of us, and has done us the great fa- vor of reviewing many technically demanding programs that are of especial interest to philosophers of physics. But With a Response by the motor of this project is supposed to be a very, very sur- John Earman prising feature of the "deep structure" (p. 6) of GTR: namely that according to the deep structure, nothing physically real changes. This, and only this, is the claim I seek to demolish. Even if I succeed, much of interest may be found in the dis- jecta membra of Professor Earman's paper. Philosophers' Imprint
. But the very same solution can also be foliated as in figure 3:
t'3
t'2
t'1
figure 1
t'0 t4
figure 3
t3 Now in each "instantaneous" state the left-hand clock is ahead of the right-hand clock. This is just as legitimate a way to carve up the model as figure 2. Once we have a foliation, we can begin to apply the t2 Hamiltonian formalism. The points in phase space will rep- resent instantaneous states, i.e., states of Cauchy surfaces. If we use the foliation of figure 2, then the complete four- dimensional solution will be represented by a trajectory t1 through the phase space, and each point on the trajectory will contain clocks that indicate the same time. If we use the foliation of figure 3, then the very same solution will be rep- figure 2 resented by a completely different trajectory, such that each 8 Tim Maudlin Thoroughly Muddled McTaggart point on the trajectory contains clocks that indicate different from the figure 2 slicing at t0 (and, if one likes, at all times times. Obviously, these two trajectories in phase space will prior to t0) but later diverges, wandering over to the region have no points in common. of phase space occupied by the figure 3 slicing. And in gen- But now comes the critical observation. We can also foli- eral, the complete trajectory through phase space up to some ate the solution as in figure 4, with Cauchy surfaces that time does not determine the future trajectory for the simple agree with the figure 2 foliation early but morph into the reason that the foliation of the space-time up to that point does figure 3 slices later on: not determine the foliation later on. And this, in turn, is a con- sequence of the fact that we have arbitrarily chosen the foliation from among the infinitude of ways of splitting the solution into Cauchy slices. Different slicings yield different trajectories through phase space, and slicings that agree to a point and then diverge yield trajectories which agree to a point and then diverge, i.e. , yield dynamical indeterminism. t''3 So before we have written down a single equation, we can make a prediction: casting the GTR into Hamiltonian form will yield a theory with an indeterministic dynamics. t''2 But we also understand the source of the indeterminism: it comes from forcing the GTR into the Procrustean bed of the Hamiltonian formalism. In order to do so, we have to im- port a foliation into our solutions to the EFE's, a foliation t''1 that has no basis in the GTR itself. It is the arbitrary nature of the foliation that makes the resulting trajectory through phase space somewhat arbitrary. But we equally see that t''0 this indeterminism is completely phony: it has nothing to do with any real physical indeterminism. Given the initial state on a Cauchy surface like t0, the GTR admits of a unique figure 4 maximal global solution. Carving up that single solution by different foliations yields different trajectories through phase Again, this slicing will yield a trajectory through the phase space, but all of these trajectories, in their entirety, represent space, and the complete trajectory will correspond to the the same four-dimensional solution. complete four-dimensional solution. But we can immedi- So it should come as no surprise at all that when we put ately see a problem for determinism: the trajectory one gets the GTR into Hamiltonian form we get an apparently inde- from the figure 4 slicing agrees precisely with the trajectory terministic dynamics. How should we deal with this? 9 Tim Maudlin Thoroughly Muddled McTaggart Reviewing the options above, we could, first of all, sim- under those symmetries, but no slices exists that are invari- ply ignore it. The theory in Hamiltonian form represents no ant under them all. So no generic condition can pick out a more physical indeterminism than there is in the EFE's, unique foliation of all of the models of the GTR. namely (given the restriction to maximal globally hyperbolic A more promising approach is this: find a condition that space-times) none. All of the different trajectories through given a single Cauchy surface as data then induces a unique fo- phase space that are solutions to the dynamical equations liation of the space-time. The idea is pretty straightforward: represent the very same complete four-dimensional space- one would expect, for example, that given the slice t0 in fig- time, and the physical magnitudes in the space-time evolve ure 2, the method would generate the complete slicing of deterministically. Furthermore, the magnitudes in the figure 2, and given the slice t'0 of figure 3 it would generate space-time evolve deterministically: the universe may ex- the slicing of figure 3, and no initial Cauchy slice would gener- pand, perihelions may precess, binary star systems may ate the slicing of figure 4. Such a project will be mathemati- speed up their rotations, just as we always thought all along. cally quite non-trivial, but in certain cases one could imagine Changing to the Hamiltonian formalism gives us no new insight how it might go: given one Cauchy surface (call it t0), let the at all into the basic ontology or dynamics of the theory. slice ti (i a positive real) be the locus of all points p in the If one does not want to just ignore the faux indetermin- space-time such that the maximal future-directed time-like curve from t0 to p is of proper relativistic length i, and simi- ism of the Hamiltonian form of the theory (if, for example, it 3 makes quantization more difficult), then one could try to larly for i negative, using a past-directed curve. Or there are eliminate the apparent indeterminism by fixing gauge. In other ways one could go about this, using the so-called lapse practice, this means formulating some constraint on the way function and shift vectors. In any case, this sort of "gauge the space-time is foliated, so that diverging foliations that fixing" would help solve the indeterminism problem: since yield diverging trajectories no longer exist. This could be the initial data would be consistent with a unique global fo- done in two ways. liation that satisfies the constraint, one would not get di- verging trajectories through phase space that correspond to One way would be to find some method for canonically foliating a space-time. This would require discovery of some 3The suggestion here, although it would work in some cases, would not condition that exactly one foliation of any space-time can ful- solve the generic problem of fixing a foliation into Cauchy surfaces. It is easy to fill. In certain cases, such a condition might exist (e.g., there see that the suggested method would indeed foliate the space, and that no time- like curve would intersect any of the hypersurfaces of the foliation more than might be a unique foliation in which a background radiation once, but one is not guaranteed that each inextendible timelike curve would in- field is homogeneous and isotropic on every slice), but tersect every hypersurface, so the hypersurfaces need not all be Cauchy surfaces. clearly no such generic condition exists. Minkowski space- In addition, the method is impractical as a means to solving the EFEs, since one time, for example, is a vacuum solution to the EFE's, but is using the metrical structure of the space-time to determine the slicing, but one there can be no condition that picks out a unique foliation of does know the metrical structure until one has solved the EFEs. (Gauge fixing Minkowski space-time on account of its symmetries (e.g., in electrodynamics can make solving the equations easier since the gauge con- straint—e.g., fixing the divergence of A—can be specified before the solution is symmetries under Lorentz boosts, rotations, and transla- known.) So we are here making an abstract point about gauge-fixing, not a tions). Any canonical slices would also have to be invariant practical suggestion. 10 Tim Maudlin Thoroughly Muddled McTaggart the same solution merely because the foliations diverge. Of freedom to foliate the space-time, this solution will be a course, there would still be distinct trajectories through complete disaster. As we have seen, if we are free to foliate, phase space that correspond to the same global solution be- then the state on t0 could evolve into t3 (as in figure 2), but it cause they are generated from different initial Cauchy surfaces could equally well evolve into t''3 (as in figure 4) depending (like t0 and t'0 above), and they might cause problems for, on how we foliate. So the quotienting solution would have say, quantization. But that has nothing to do with indeter- to declare that the state on t3 and the state on t''3 are gauge minism. equivalent, i.e. that they are merely mathematically distinct ways So both ignoring the apparent indeterminism and fixing of expressing the very same physical state. But this is crazy: t3 gauge appear to be viable solutions to the indeterminism has two clocks that indicate the same time, while t''3 has "problem". But if one has fallen in love with the "constrained clocks that indicate different times: these are physically dis- Hamiltonian formalism" and one has solved other faux in- tinct states. Even more damning is this: the left-hand clock determinism problems (as in electromagnetism) by quo- on state t3 indicates a different time than the left-hand clock tienting, then one might be tempted to try this route. But on t''3, so if t3 and t''3 are really the same physical state, then there lies disaster. the physical state of a clock does not really change as it comes to Recall the basic strategy of quotienting. We begin with indicate different times. By similar argumentation, we would apparent dynamical indeterminism: the dynamical equa- come to the conclusion that, no matter how the clock seems to tions permit different solutions with the same initial data so have its hands oriented, it is always in precisely the same that, for example, according to one allowable trajectory ini- physical state: the "change" is merely apparent, not real. tial state S0 evolves into state S1, but according to another These claims are, or course, rather silly—but they are pre- cisely the claims that, according to Earman, the constrained allowable trajectory S0 evolves instead into S'1 and never Hamiltonian formalism reveals about the deep structure of enters S .4 This apparent indeterminism could be removed 1 the GTR. by declaring that S1 and S'1 are really physically identical states: they are gauge equivalent. One then quotients by the It is, in fact, not hard to see that if one is going to restore complete gauge equivalence classes (the gauge orbits), re- determinism by quotienting, then the gauge orbits have to ducing the phase space and restoring determinism. contain every state on every Cauchy surface in a solution to the But if the source of the apparent indeterminism is the EFE's. Only then is one assured that the states along a tra- jectory will belong to the same gauge orbit no matter what 4The problem is most easily stated for a case like electromagnetism done in foliation is used to generate the trajectory. And so what terms of potentials, since there is a common universal time in all models, so one holds for the clock would have to hold for the universe as a can say that according to one trajectory S0 evolves into S1 five minutes later, whole: its physical state never changes, from a millisecond while according to the other trajectory it evolves into S ' five minutes later. The 1 after the Big Bang to a minute before the Big Crunch. In the solution to the apparent indeterminism is then to make S and S ' gauge equiva- 1 1 technical terminology, the dynamics of the theory is pure lent. Since in the GTR there is no such common universal time (introducing a universal time function is equivalent to picking a foliation) things are not so gauge, since all of the states along every trajectory have to simple, as we will see. belong to the same gauge orbit. McTaggart—or more prop- 11 Tim Maudlin Thoroughly Muddled McTaggart erly Parmenides—is vindicated: according to the way of oped the abstract form of the constrained Hamiltonian for- Truth, the universe is ever One and Unchanging, only ac- malism, and more importantly, Dirac suggested that "the cording to the way of Seeming is there change. gauge transformations be identified as the transformations Now there is an attenuated sense in which the state on generated by the first class constraints, where the intended Γ every Cauchy surface in a solution of the EFE's is the same: interpretation is that two points of the phase space which each such state implicitly represents everything that happens are connected by a gauge transformation are to be regarded at all times, since each such state is compatible with exactly as representing the same physical state" (p. 8). It is here that the same maximal globally hyperbolic solution to the EFE's. the method is turned topsy-turvy: instead of starting with an But that does not make them all physically equivalent: oth- understanding of which points in phase space represent the erwise we get an immediate argument from determinism to same state, one rather does the dynamics first and then con- No Real Change. There is real physical change because the cludes from some formal feature of the dynamics that two physical states on the different Cauchy surfaces are different points represent the same physical state. And we can see (i.e., non-isomorphic), even if each surface (together with the why the topsy-turvy method would work for, say, classical EFE's) implies the same global solution. electromagnetic theory: the "gauge transformations" so identified really would be the intuitively correct gauge trans- After all, how could it be otherwise? We know that the formations. But generalizing from this sort of example that GTR is a theory which predicts—and explains—many we should always identify transformations generated by con- changes: the precession of planetary orbits, the collapse of straints in the Hamiltonian as gauge transformations be- stars, the rate of expansion of the universe, the red shift of tween physically equivalent states is a dangerous business, light coming out of a gravitational well. It is, indeed, the ob- as we have seen. servation of precisely such changes that provides our evi- dence for the theory. Any interpretation which claims that It is also only proper to note that criticisms of this use of the deep structure of the theory says that there is no change Dirac's method are not original: as Earman notes, Karel Ku- at all—and that leaves completely mysterious why there charˇ, for example, makes exactly the same point. I only hope seems to be change and why the merely apparent changes are to have made clear why this particular result is to be ex- correctly predicted by the theory—so separates our experi- pected if the GTR is put in Hamiltonian form. ence from physical reality as to render meaningless the evi- Earman seems to feel the force of these objections, since dence that constitutes our grounds for believing the theory. he is at pains to argue that the "frozen time" results are not So the only real question is not that the constrained Hamil- "merely formal tricks or artifacts of the constrained Hamil- tonian formalism (interpreted as Earman suggests) is yield- tonian formalism" (p. 9). To allay these doubts, he proposes ing nonsense in this case, but why it is yielding nonsense. to derive "similar, if not identical, results in the spacetime And the freedom to foliate provides the perfectly compre- setting rather than the (3+1) Hamiltonian formulations" (p. hensible answer. 9). This would be significant, since my argument so far has It is only proper to note that Earman did not construct been that the problems arise from using the (3+1) formula- these arguments for the unreality of change: Dirac devel- tions: it is precisely a foliation that one needs to turn a four- 12 Tim Maudlin Thoroughly Muddled McTaggart dimensional relativistic space-time into a (3+1) dimensional physical change, i.e. no change in his observable quantities, object. And indeed, the arguments that Earman rehearses at least not for those quantities that are constructable in the next do not hinge on the same mistakes that sink the Ham- most straightforward way from the materials at hand" (p. iltonian argument—they hinge rather on a completely dif- 10). One might think that, if true, the moral is to find ob- ferent set of mistakes. To these we now turn. servables—like the position of the perihelion of Mercury relative to the Sun—that are constructable in a less-than- The Observables Argument straightforward-way rather than concluding that there is no real physical change in the world, but having gone through the Earman next takes up the approach suggested by Bergmann looking glass, we are apparently to accept some Alice-in- for guaranteeing determinism in the GTR. Like the quo- Wonderland logic from this point on. tienting technique, Bergmann's approach is certain to yield the result that the GTR is deterministic, at least with respect To fix ideas, let's start with the sort of prediction that we to all observables, because it secures this result by definition: can make using the GTR. It will also help a bit to reflect on in order to be an observable, a quantity must be "unequivo- the way that the GTR can be used in conjunction with other cally predictable from initial data" (p. 9). The only question principles to make predictions, since our focus from here on left is what the observables of the theory are. will be the sorts of predictions that can be made from the We would expect the observables to include: the position GTR neat. So let's start with a "mixed" prediction and then of the perihelion of Mercury after some number of orbits, the try to work back to a more "pure" one. amount by which light from the Sun is redshifted when it The GTR has been used successfully to predict the out- reaches the Earth, the angle at which light from a distant star come of the following experiment: take two synchronized will reach the Earth during an eclipse, and so on. And we atomic clocks, put one on a plane and fly it around the Earth should expect that these observables will change: the peri- on some specified route, then bring the clocks back together helion of Mercury will advance at a predictable rate. So one and compare them. When brought back together, the clocks might expect that we would start with these sorts of observ- will no longer be synchronized, and the amount of dis- ables, the ones that provided evidence for the theory in the agreement can be accurately predicted using the GTR. (Nota first place, the ones that were predicted by the theory, and bene: the GTR can be used to make a deterministic prediction try to discover some generic characterization of observables about how the synchronization of the clocks will change from the that includes these sorts of quantities. But instead we are beginning of the experiment to the end, so a fortiori the GTR can given an abstract characterization of observables that has as a be used to predict that things will change. So the only way consequence that none of the things that were actually ob- to secure McTaggart's conclusion is to argue that the relative served and measured and brought forward as evidence for synchronization of the clocks is not an observable!) How is the theory were observable at all! This isn't just topsy-turvy, this prediction made? it's through-the-looking-glass. One starts with facts about the size and mass of the As Earman puts it: "What may not be familiar to most Earth. These are the sorts of data that can be put on a readers is that Bergmann's proposal implies that there is no Cauchy surface. One then solves the EFE's to get a full four- 13 Tim Maudlin Thoroughly Muddled McTaggart dimensional space-time. All of this is pure GTR. But now physical magnitudes (like clock readings) which are most di- things get a bit tricky. Knowing the flight plan of the air- rectly relevant to laboratory operations are not represented plane, one can then pick out a trajectory through the space- in the purely relativistic mathematics used to make predic- time (by reference to the position of the Earth), representing tions. the trajectory of the flying clock, and one can similarly pick There are two ways to mitigate this problem. One is out the trajectory of the stay-at-home clock. One can calcu- simply to declare that relativistic quantities like the proper late the proper time along these paths, and assuming that the time along a world line, or the gravitational tidal forces at a clocks measure the proper time, one can predict what they point, are observables since we have instruments (clocks or will show when brought back together. water drops) that allow us to observe them, even if we don't Now it is evident that this whole procedure is not simply include those instruments in our models. Furthermore, we a matter of solving the EFE's. In addition to the solution to can just grant that we can know, e.g., how to represent the the EFE's, we have to specify the trajectories of the clocks trajectory of the flying clock, even though we don't solve our and have to deploy a principle about what clocks measure. equations for it. Note that the relevant trajectory is given This second principle could be reduced to a purely relativis- relative to the Earth: the plane flies at a certain altitude above tic one if we used a light-clock (rather than an atomic clock) the Earth for a certain distance before coming back. which can be shown to measure proper time, but even so the The second way to mitigate the problem is to investigate first additional bit of information has to be added from the experimental conditions where gravity is the only important outside. The only way to avoid this would be to include in factor. If we could send our two clocks free-falling along dif- the model a complete physical description of the airplane and of ferent paths through a gravitational lens, then (supposing all physical objects that influence the flight of the airplane and their initial trajectories are part of the Cauchy data) we could then solve for its trajectory, but this is clearly a practical im- solve for their trajectories using the GTR: they will follow possibility. So we need to be a bit careful. What we want to the appropriate time-like geodesics. We can solve for the say is that some quantities, such as the proper time along a point where they will intersect, and predict—from the GTR time-like trajectory, are on the one hand predictable from the alone—how far out of synchronization they will be when GTR and on the other hand observable (by means of clocks), they meet. Or, in a more realistic case, predicting the appar- even though the directly observable instruments (the clocks) ent position of a star during a total eclipse does not demand are not themselves represented in the models of the GTR we significant input from outside the GTR proper. So it is a actually use. This shortcoming cannot be overcome until we plain fact that the GTR makes deterministic predictions have the resources to represent the physics of the clocks in about observable physical magnitudes and about how those the models and would probably be mathematically intracta- magnitudes can change. The only real question before us is ble even then. So when young McTaggart speaks of a how Bergmann's seemingly innocuous definition of an ob- "genuine physical magnitude countenanced by the GTR" (p. servable as any physical magnitude deterministically pre- 6) and when those magnitudes are characterized as "observ- dictable from Cauchy data (irrespective of whether any in- ables", one ought to pause: in many cases, the observable strument can, in the intuitive sense, observe it) can possibly 14 Tim Maudlin Thoroughly Muddled McTaggart get us into trouble. Why don't things like the precession of back the result. So surely this ought to count, on Bergmann's the perihelion of Mercury or the reading of our clocks when criterion, as an observable. they get back together turn out to be observables since they But according to the analysis Earman offers, the Ricci are clearly predictable? The devil, of course, is in the fine curvature at a spacetime point is not observable by Berg- print—and the first bit of fine print occurs in the artfully mann's criterion, i.e., not predictable from initial data, unless wrought statement cited above: Bergmann's criterion implies the Ricci curvature is constant everywhere to the future of that there is no real change, "at least not for those quantities the initial hypersurface. How can that be? that are constructable in the most straightforward way from The trick is how to interpret the phrase "at a spacetime the materials at hand". Let's take a careful look at what ex- point". In my presentation, the relevant spacetime point is actly that means. identified by a definite description: the point where the two The first candidates for Bergmann observables are geodesics meet. In Earman's approach, the point is not so identified. In fact, in Earman's account there is no story at all local field quantities which are constructed from the met- about how the relevant point is identified: it is just somehow ric and its derivatives up to some finite order and which given is not are evaluated at [a] spacetime point, e.g. the Ricci cur- . The point given by a definite description such as the one offered above: the point is not identified by its vature scalar R. Is the value of this quantity at some spacetime relation to the initial hypersurface, or by its point to the future of an initial hypersurface predictable a point ten from initial data on the hypersurface, or even from data spacetime relation to any material object (e.g., as on the entire past of the hypersurface? [p. 10] miles above the Earth), or by the object that occupies it. It is rather just (magically) given as a point in the "bare" manifold, One would initially think that the answer is clearly "yes". To i.e. as a point in the spacetime manifold before any metric has fix ideas, let's take the case discussed above: the initial data been specified for the manifold. include two objects (they may be clocks, but it does not Now, there is, in the first place, no coherent account matter for this example) that are being launched from Earth about how such a point could be identified independently of toward opposite sides of a gravitational lens (such as the the metric and contents of spacetime. And even if there Sun). Can we deterministically predict the value of the Ricci were such an account, there is no account of how the rele- scalar at the space-time point in the future where the two ob- vant point could be observationally identified so that the Ricci jects will meet? curvature there could be checked. So for all intents and pur- Evidently, the answer is "yes". From the initial data and poses, we are in the following situation. We are given the the EFE's, we can construct a complete four-dimensional data on the initial surface, and then someone informs us that solution, identify the point where the relevant geodesics they have a spacetime point somewhere in the future in (which originate at the Earth) meet, and find the Ricci scalar mind, but they provide us with no further information about at that point. And if we wanted to do the experiment, we which point it is, and then they ask whether we can predict could construct rockets to be launched from the Earth that what the Ricci scalar is at that otherwise unidentified point. would measure the curvature when they meet and transmit And of course, we could in such a situation neither predict 15 Tim Maudlin Thoroughly Muddled McTaggart the scalar curvature there (unless, of course, the Ricci scalar where the two rockets collide, we have no problems. We only is the same at every point in the future), nor could we later would have problems if we could per impossible identify empirically determine what the relevant scalar curvature is. points on the bare manifold as such. So if the spacetime point is only given to us as a point on the At this point, the attentive reader will begin to feel bare manifold, the Ricci scalar at that point is not predictable queasy. For what we have is just another incarnation of the and so, by Bergmann's criterion, not observable. (Nota bene: notorious "hole" argument. That argument, recall, was sup- the failure of observability has nothing whatever to do with posed to establish that the GTR is indeterministic because the a lack of instrumentation by which one can empirically de- EFE's only determine a solution from initial data up to a dif- termine the Ricci scalar; it has to do with the lack of instru- feomorphism. Now, whatever one thinks of that argument mentation by which one can determine the identity of the (my own views have been expressed ad nauseum elsewhere5), relevant point.) all hands in that debate agreed that if there is any indeter- Earman makes this argument using the technical ma- minism, it is an unobservable indeterminism, since it concerns chinery of diffeomorphism invariance. Take a solution to what happens at particular points of the bare manifold, but the EFE's, and choose a diffeomorphism that is the identity particular points of the bare manifold are not, per se, observ- map on the initial data surface but not on the point p in the able. Now that argument is being used, in conjunction with bare manifold. In particular, suppose that the Ricci scalar to Bergmann's criterion, to a perfectly risible conclusion. The the future of the initial surface is not everywhere constant. proposed logical form of a "local field quantity" is a quantity Then there is a point q to the future whose Ricci scalar differs attached to a point on the bare manifold, and then the indeter- from that of p in the solution. Choose a diffeomorphism that minism is used to argue that these quantities are not observ- is the identity on the initial hypersurface and that maps q to ables (unless, of course, the quantity is the same everywhere, p. Now "drag" the original solution along the diffeomor- and so predictable). But even if the values of quantities at- phism to get a new solution to the EFE's. In this new solu- tached to points of the bare manifold were observable in tion, the Ricci scalar at p will be different from the original: it Bergmann's sense, they would not be observable in the nor- will now be the value that was formerly at q. But since this mal sense, since we can't identify the relevant points by ob- is also a solution of the EFE's from the same initial data, that servation. What we can identify by observation are the data and the EFE's cannot predict the Ricci scalar at p. points that satisfy definite descriptions such as "the point where these geodesics which originate here meet", and Of course, if p happens to be the point where the two against these sorts of quantities Earman's diffeomorphism geodesics intersect in the first solution, it will not be the argument has exactly zero force. point where they intersect in the new solution. The geodes- So if one were to start by reflecting on the logical form of ics will be "dragged along" by the diffeomorphism, and they the predictions we actually make using the GTR, or if one will now intersect at a new point on the "bare" manifold, a point whose Ricci scalar is identical to that at p in the origi- 5 E.g,. in "Substances and Space-Time: What Aristotle Would Have Said to nal solution. And so if we identify the relevant point by the Einstein", Studies in the History and Philosophy of Science Vol. 21, No. 4, pp. definite description and empirically identify the point by 531-561. 16 Tim Maudlin Thoroughly Muddled McTaggart were to start by reflecting on what sorts of things we take to times at different places in the space-time, and (b) the differ- be actually observable in spacetime, then one would not be- ent places are timelike related to each other, so one can gin with the logical form of a quantity attached to a point of speak of which of a pair of indications is earlier and which the bare manifold. Earman's strategy instead is to start with later. Notice that the subject of the change is a material ob- quantities attached to bare points because they "are con- ject—a clock—which is represented by a spacetime worm structable in the most straightforward way from the materi- that has different features at different points. Nothing at a als at hand", and then to show, to no great surprise, that spacetime point can change, for the simple reason that a these are not Bergmann observables. There is nothing tech- spacetime point has no temporal extension: individual nically wrong with Earman's argument; it just seems like a spacetime points are not even candidates for "subjects of rather senseless way to proceed once one reflects on the change", i.e., for the things that change. Subjects of change source of the difficulties it encounters. must persist through time so they can have different prop- Earman does not claim that the examination of "local erties at different times. And the things that persist through field quantities" (or the "quasi-local field quantities" one gets time are typically material objects like clocks or stars or gal- by integrating the local ones over patches of the bare mani- axies. These are represented by spacetime worms that are fold, which obviously inherit the same problems) is a deci- identified by their material contents: the spacetime worm that is sive argument that there are no Bergmann observables that the clock is the collection of spacetime points that are occupied are different at different places in spacetime. But he casts a by the material of which the clock is made. And in this sense, the skeptical eye on the utility of any quantities which are not subjects of change are not "attached to" bare spacetime "attached" to points of the bare manifold (like the value of the points: under an active diffeomorphism, the subject of Ricci scalar where the geodesics meet): "[I]t is worth remarking change (e.g. the clock) will be "dragged along" to a new part that it is not obvious how these unattached quantities could of the bare manifold. underwrite B-series change; for such change requires a sub- So even if, in some sense, spacetime points are the ultimate ject, and since spacetime points and regions are the only ob- "subjects of predication" in the GTR, it does not at all follow vious candidates for the subject role in GTR, these peculiar that they are the subjects of change, i.e., the things that unattached quantities would seem to remove the subject of change. This is just a confusion of two rather unrelated uses change from the picture" (pp. 10-11). So let's answer this of the term 'subject'. concern before going forward. In succeeding sections of the paper, Earman takes up the At the beginning of our discussion, it was agreed that ac- project of identifying some other "observables" beside his lo- cording to the "naive" reading of the GTR, the GTR allows cal and quasi-local field quantities. Unfortunately, he never for B-series change. For the models of the GTR are four- considers anything as simple as "the value of the Ricci scalar dimensional manifolds with a relativistic metric, and that where two given (i.e., given in the initial data) geodesics metric allows one to define certain timelike relations among meet". He has some remarks about so-called "coincidence events in the space-time. Thus, a clock in the GTR can observables", and this looks promising: we may ask after the change the time it indicates because (a) it indicates different value of a quantity where the two geodesics coincide. But 17 Tim Maudlin Thoroughly Muddled McTaggart even here things strike him as problematic. Adjusting his to the Einstein field equations, which implies that the dy- remarks to fit the case of the intersecting geodesics, one namics is implemented not by a mapping from one state to would get this: another state in the same solution, but from one solution to another solution". But the whole argument to date has been Admittedly, however, it remains a bit obscure how the that complete solutions to the EFE's—complete relativistic value of this coincidence observable is measured. For ... histories of the world—can represent worlds in which things the measuring procedure cannot work by verifying that change. Why a mapping from one complete solution to an- the coincidence of values ... does in fact take place by separately measuring the [bare manifold position of a other—from one complete possible world to an- point on one geodesic] and the [bare manifold position other—should even be called "dynamics", or what it has to of a point on the other geodesic], and then checking for do with the physics of the one world we live in, is com- the coincidence. For [the positions on the bare manifold] pletely obscure. are gauge dependent quantities, and by the Assumption these quantities are not fixed by measurement ... . [p. 13] So in the end, we have three arguments against change in the GTR, two demonstrably inadequate and the third in- Of course, one would not tell where the geodesics coincide comprehensible (at least to me). One might wonder why in anything like this way: one would tell by sending a rocket Earman would bother with three arguments if he thought along each path and making the measurement when they that any one of them sufficed to establish the conclusion. collide. Nothing in any of Earman's arguments suggests any The principle he seems to be following is: where there's difficulty about this procedure. smoke, there's fire. But sometimes where there's smoke, So the Observables Argument gets any traction only by there's mirrors. The apparent difficulty for change—and considering candidates for observables (values at points of therefore for time itself—which Earman’s McTaggart dis- the bare manifold) which are neither the sorts of things one cerns in the GTR is only an artifact of a bad choice of for- actually uses the GTR to predict nor the sorts of things one malism or a bad choice for the logical form of an observable, would expect—quite apart from diffeomorphism invari- not because of any intrinsic problem in the theory. ance—to be observable. And as soon as one tries the argu- Has our encounter with McTaggart yielded any positive ments out on something that one would predict or observe, results? There is, if not a moral, at least an intriguing sug- they fail. The Observables Argument runs on completely gestion of the possibility of a moral to our story. It is now a different principles from the Hamiltonian Argument, but it commonplace that there is a deep problem of time that arises is equally broken-backed. when one tries to quantize the GTR, and that this implies a The critical Section 4 ends with yet a third observation, fundamental incompatibility between the GTR and quantum drawn from an entirely different source: the "alternative ap- theory. McTaggart’s puzzling claim was that he had found a proach of Ashtekar and Bombelli" (p 15). I can only admit problem of time in the purely classical theory. We have that I have no first-hand knowledge of the theory and cannot shown McTaggart’s worries to be unfounded—but are left make head nor tail out of Earman's description. He says the with the intriguing possibility that the “problem of time” in theory is defined on the space "of entire histories or solutions quantum gravity is equally chimerical. If one casts the GTR 18 Tim Maudlin Thoroughly Muddled McTaggart in Hamiltonian form—as is commonly done—in order to (1) At the outset I want to emphasize a point generally quantize it, then the interpretative problems attending that accepted in the physics community but largely unappreci- form of the classical theory will likely arise again in the ated in the philosophy of science community: There is a uni- quantum version. And if tying "observables" to particular form method for getting a fix on gauge that applies to any points in the bare manifold makes trouble in the classical theory in mathematical physics whose equations of mo- version, then we should anticipate difficulties in defining tion/field equations are derivable from an action principle. I observables in the quantum version. So McTaggart may emphasize that any particle theories and field theories, have done us a favor after all, by redirecting our attention Newtonian theories and relativistic theories, etc., all fall- away from quantization back to the original theory as a within the scope of the method.6 source of various technical problems that might arise. The first step in employing the method is to convert from But however things work out for quantum gravity, there the Lagrangian form of the theory to the Hamiltonian form. is, after all, no problem of time or change in the GTR. Let's If non-trivial gauge freedom is involved in the theory, it re- return McTaggart to his final resting place, and let him veals itself in the existence of Hamiltonian constraints. One molder there in peace. then proceeds to identify the first class constraints and, fol- lowing Dirac, these constraints are taken to generate the Response by John Earman gauge transformation on the Hamiltonian phase space. Fi- nally, the genuine physical magnitudes or "observables" are I am grateful to Tim for posing his disagreements with me in identified as the gauge invariant quantities. a form that more than matches my attempt to state the issues Now apply this method to some theory of physics. Sup- in a provocative way. Tim does a brilliant job of explaining pose that the result offends your (or Tim's) intuitions. This the guts of some difficult technical issues. He takes his ex- might indicate that the method, despite its universally ac- planation to show that the sorts of considerations I adduced knowledged success in providing a precise and systematic in favor of modern McTaggartism lead to a precipice below explication of the gauge concept across a vast range of cases, which lies absurdity. I see no precipice but rather a series of breaks down in the case at hand. But in the absence of any steps that lead to an understanding of the motivation and competing method for getting a fix on gauge—and I haven't content of contemporary main-line research in the founda- heard a definite competing proposal—you (or Tim) should tions of classical general relativity theory (GTR) and quan- seriously consider the possibility that your intuitions have to tum gravity. This research may lead only to a dead end, but be retrained. there is no a priori way to know this, much less that the re- Tim doesn't want a competing method, but he does want search is based on absurd ideas. In what follows I will con- to be able to cherry-pick the results of the constraint formal- fine my comments to four points that lie at the heart of our ism. The motivation behind the technical apparatus is to disagreements.
John Earman is University Professor of the History and Philoso- 6Here is one explicit expression of faith in the generality of the method: "It is well known that all the theories containing gauge transformations are de- phy of Science at the University of Pittsburgh scribed by constrained systems" (Gomis, Henneaux, and Pons 1990, p. 1089). 19 TimJohn Maudlin Earman ThoroughlyResponse Muddled to Maudlin McTaggart detect when an apparent violation of determinism is merely of transformations which depend on arbitrary functions of a faux violation. Tim's intuitions tell him that some the independent variables. Noether's second theorem im- violations of determinism are tolerable, and in these cases he plies that the Euler-Lagrange equations of motion are un- sees no need to save determinism by appeal to gauge free- derdetermined—i.e., there is an apparent breakdown of de- dom. But all the potential violations of determinism covered terminism. The underdetermination is overcome if the ele- by the constraint apparatus are of a piece; namely, arbitrary ments of the invariance group are seen as gauge transforma- functions of the independent variables show up in solutions tions—as relating different descriptions of the same physical to the equations of motion. Since time is normally the or one situation. One now has to face the issue of how these La- of the independent variables, what this means is not just that grangian gauge transformations—which act on the inde- solutions to the equations of motion can agree on initial data pendent and dependent variables of the Lagrangian and while disagreeing at a later time but also that given any al- which map solutions of the Euler-Lagrange equations onto lowed initial value of a dependent variable, there is a solu- solutions—are related to the Dirac-Hamiltonian gauge trans- tion of the equations of motion which has the prescribed ini- formations, which are point transformations on the Hamil- tial value of the dependent variable but which gives to the tonian phase space and which map solutions of the Hamil- dependent variable any value you like at any future time ton-Dirac equations onto solutions. In some cases the rela- you choose. If this isn't a violation of determinism, it is hard tion between the two concepts of gauge is transparent; in to know what one would be. And it is hard to see any prin- other cases, such as Einstein GTR, the relation is opaque and cipled way to distinguished tolerable vs. intolerable viola- requires special effort to discern.7 I will return to the alter- tions of this kind. It is sometimes said that the kind of inde- native Lagrangian approach below, but now I want to take terminism that threatens GTR is uninteresting because it is up Tim's question of why one would want to put GTR in unobservable. But this response buys into the gauge inter- Hamiltonian form. pretation of the theory: the observables of the theory, in the The reason that physicists use the Dirac formalism to get guise of gauge independent quantities, do evolve determi- a fix on gauge is that they always have one eye cocked to- nistically. Furthermore, this response commits one to an wards quantization and because the standard route to quan- ontology and ideology that is quite different from the extant tization goes through the Hamiltonian formulation of a the- proposals in the philosophical literature. More on this in (3) ory. Trying to travel this route with respect to GTR in order and (4) below. to produce a quantum theory of gravity is known as the ca- nonical quantization program. To my knowledge, all of the (2) In a sense there is an alternative to the Dirac con- leading research workers in this program (with the one no- strained Hamiltonian formalism, but what this alternative table exception of Karel Kucharˇ) accept the consequences of gives isn't so much a rival account of gauge as a different applying the Dirac formalism to GTR—in particular, the nomenclature. The alternative works on the Lagrangian consequence that the observables of GTR are "constants of formulation of the theory, and it sees gauge freedom at work the motion", a consequence that Tim labels as absurd and when Noether's second theorem applies, that is, when the action is invariant under an infinite dimensional Lie group 7 See my (2002) for a discussion of this issue. 20 JohnTim Maudlin Earman ThoroughlyResponse Muddled to Maudlin McTaggart disastrous. The popular press would have you believe that phase space where all of the Hamiltonian constraints are the only viable approach to quantum gravity is via string satisfied); namely, the gauge condition must define a trans- theory or M-theory as it is now called. But the loop formu- versal in the constraint surface, i.e., a lower dimensional sur- lation of quantum gravity—which falls within the canonical face that intercepts each of the gauge orbits (as generated by quantization program—is currently an active research pro- the first class constraints) exactly once. In some cases, such gram. In contrast to M(ystery)-theory, loop quantum gravity as Yang-Mills theories, familiar gauge conditions fail to de- is a definite theory rather than a wannabe theory, and it is a fine a transversal and, thus, the brute force attempt at quan- theory with notable theoretical success in the form of an ex- tization is defective. But what is more important is that planation of black hole entropy and the prediction of area when a gauge condition does succeed in fixing a global and volume quantization of space.8 transversal, what one is getting on the cheap, so to speak, is I am not giving an argument from authority, although I an isomorphic copy of the reduced phase space which re- do think that philosophers are on dangerous ground when sults when the Dirac gauge orbits are quotiented out. What they are dismissive of the prevailing opinions of physicists this strongly suggests is that the theoretically desirable tech- on matter of interpretation. Rather, my point is that inter- nique of quantization of a gauge theory would be to pass to pretations of scientific theories are subject to empirical tests, the reduced phase space (where the new phase variables are albeit of an indirect sort. If the loop formulation of quantum gauge invariant quantities) and then to perform normal gravity continues to make theoretical progress and, eventu- quantization on the resulting unconstrained system. Un- ally, passes experimental checks, then I would take these fortunately, various technical obstructions can block the pas- successes to be confirmation of the gauge interpretation of sage to the reduced phase space, and even if the passage is GTR dictated by the Dirac constraint formalism. not blocked there remains the fact that the constraints may Before leaving the issue of quantization of gauge theories be too difficult for physicists to solve. This is why Dirac in- I want to emphasize that it is hard to see how the primacy of vented a short-cut method referred to as constraint quanti- the Dirac account of gauge can be abandoned.9 As Tim zation, which consists in promoting the first class constraints mentions, one approach to quantizing a gauge theory is the to operators on a suitable Hilbert space and then identifying brute-force method: impose a gauge condition to kill off the the physical sector of this space in terms of the state vectors gauge freedom and then quantize in that gauge. What fixing that are annihilated by the operator constraints. But whether a gauge means is explained in terms of the geometry of the one is performing reduced phase space quantization or Dirac Dirac constraint surface (the subspace of the Hamiltonian constraint quantization, the philosophy is the same: only Dirac observables (= quantities which are constant along the 8 For a review of the loop formulation of quantum gravity, see Rovelli Dirac gauge orbits or, equivalently, phase functions on the (1998). This approach to quantum gravity takes advantage of a reformulation of reduced phase space) get associated with quantum observ- classical GTR in terms of a new set of variables (due independently to Abhay Ashtekar and Amitaba Sen) that makes the Hamiltonian constraints easier to ables in the form of self-adjoint operators. handle. 9The standard reference on quantization of gauge theories is Henneaux and (3) Leaving now the issues of quantization and eschew- Teitelboim (1992). ing the (3+1)-dimensional Hamiltonian approach in favor of 21 JohnTim MaudlinEarman ThoroughlyResponse Muddled to Maudlin McTaggart the 4-dimensional Lagrangian approach, the nomenclature local field quantities (whether scalar, vector, or tensor). The changes but the essential conclusions remain the same; in positive part of their answer is that the gauge invariants in- particular, applying the considerations outlined at the be- clude (at least) two different sorts of quantities: first, highly ginning of (2) to GTR leads to the result that the Euler- non-local quantities, such as volume integrals of local field Lagrange equations of GTR are underdetermined and, thus, quantities over all of spacetime; and, second, what I called that GTR is apparently an indeterministic theory. Tim is coincidence events, which consist of a special kind of coinci- right that this is just Einstein's notorious "hole argument" dence of values of two gauge-dependent quantities that go dressed in a new guise. Tim feels that this argument should together to form a gauge independent one. Even apart from have been put to rest long ago. I agree, but for different rea- the issue of change and McTaggartism, this answer is inter- sons. The reactions in the philosophical literature to the hole esting because the ontological picture that emerges from it argument are amazing in terms of their ingenuity and the lies outside the ambit of the normal discussion in the phi- extravagances they employ, and generally they have been losophy of space and time. Of course, the answer may be skewed because philosophers want to seize the opportunity wrong. But it is remiss of philosophers to refuse to explore to ride a favorite hobby horse—a favorite account of identity its ramifications on the grounds that their philosophy tell across possible worlds, a favorite account of how language them it must be wrong. and reference work, a favorite account of essential proper- (4) Coming now to the main issue of modern McTaggar- ties, etc. I ask them to pause for a moment and consider the tism, I am unrepentant in agreeing with Carlo Rovelli that, at fact there is an almost universally uniform reaction among base, what GTR describes is not evolution of the familiar practicing general relativists; namely, the lesson of the hole kind, i.e., the change over time of observables in the sense of argument is (as the Lagrangian approach to gauge tells us) genuine physical magnitudes. Carlo tries to draw some of that the spacetime diffeomorphism group is a gauge group 10 the sting of this position by saying that what the theory de- of GTR. scribes is relative change, the change of "partial observables" Some philosophers mouth these words but they gener- with respect to one another. I am not fond of this way of ally fail to work out the implications of their words, the most putting the matter since these partial observables are not immediate of which is that the gauge invariant quantities of gauge independent quantities and, thus, are not the kind of the theory must be diffeomorphic invariants. What are such things whose values, or change of values, could be experi- quantities? Generally philosophers don't have an answer mentally detected. because they haven't bothered to ask the question. General relativists have asked, and the negative part of the answer My alternative for drawing the sting of the no-evolution they find is that the gauge invariants of GTR do not include view is threefold. First, I note that the form of McTaggar- tism that emerges from GTR does not support McTaggart's
10 ultimate conclusion that time is unreal. Second, I point out Again I would emphasize that the loop formulation of quantum gravity is a that the gauge interpretation of GTR is compatible with an self-conscious attempt to accommodate the diffeomorphism invariance of classi- cal GTR as a gauge symmetry. String theorist, in conversation if not in print, attenuated kind of change, for it is compatible with taking say that they hope that M-theory will display this accommodation. the history of the universe to be what I dub a D-series, i.e., a 22 JohnTim MaudlinEarman ThoroughlyResponse Muddled to Maudlin McTaggart time ordered series of coincidence events with different game of badminton. These issues connect directly to deci- events occupying different places in the series. This does not sions that physicists working on the frontiers of research go very far towards restoring normal change since the coin- have to make, for example, in searching for a way to marry cidence events do not consist of the occurrence of a change general relativity and quantum mechanics. If I could make in a genuine physical quantity. Third, I recommend that we Tim feel the excitement I experience when I see how phi- don't try to restore normal change and evolution either by losophical concerns about time and change intertwine with changing the theory or by some clever interpretational ploy contemporary research in physics, he might be willing to that rejects or bypasses the gauge interpretation of GTR. join me on the precipice—a precipice not of absurdity but of Rather, I recommend that if normal change and temporal a new understanding of old issues. evolution are wanted, we should seek them not in the intrin- sic physics of classical GTR or quantum gravity but in the representations of the gauge invariant content of solutions to References Einstein's field equations in terms of the standard textbook models of fields evolving on manifolds. Such a representa- Earman, J. 2002. "Getting a Fix on Gauge: An Ode to the tional stance is nothing new; indeed, it should be familiar Constrained Hamiltonian Formalism," to appear in K. from the history of the debates over absolute vs relational Brading and E. Castellani (eds.), Symmetries in Physics. accounts of space and time. For example, a good construal Cambridge: Cambridge University Press. of Leibniz's relational account of space is to take him as saying that the Newtonians are welcome to talk about bodies being contained in and moving through space as long as Gomis, J., M. Henneaux, and J. M. Pons (1990). "Existence such talk is taken not literally but rather as a way repre- Theorem for Gauge Symmetries in Hamiltonian Con- senting the actual and possible relative configurations of strained Systems', Classical and Quantum Gravity 7, 1089- bodies. What I am suggesting is that a similar representa- 1096. tional account be applied to GTR, and that if it is done in the proper way we can have our cake and eat it too: ordinary Henneaux, M., and C. Teitelboim (1992). Quantization of talk about change is accommodated in the representations Gauge Systems (Princeton, NJ: Princeton University while the gauge interpretation of the theory is respected by Press). recognizing that what these representations are representa- tions of is not of evolution in any ordinary sense. Rovelli, C. 1998. " Loop Quantum Gravity," Living Reviews in Relativity. http://www.livingreviews.org In conclusion, my major disappointment with Tim's re- sponse is really a disappointment with my presentation. For what his response reveals is that I failed to convey how the issues surrounding modern McTaggartism are not mere shuttlecocks to be batted back and forth in a philosopher's 23