maudlin - 2002 - thoroughly muddled mctaggart or how to abuse gauge freedom to generate metaphysical...

Tim Maudlin is Professor of Philosophy at Rutgers University.

Thoroughly MuddledMcTaggart

OrHow to Abuse Gauge Freedom to

Generate Metaphysical Monstrosities

Tim Maudlin

With a Response byJohn Earman

Just when the hubbub over the infamous hole argumentseemed to have died down, and you thought it was safe goback into the placid pool of classical GTR (away from thewhite-water rapids of quantum gravity), John Earman hasconjured up yet another monster to trouble poor old Ein-stein.1 This time, a sprinkling of the magic powder of theconstrained Hamiltonian formalism has been employed toresurrect the decomposing flesh of McTaggart, who intonesthat the GTR (yes, the plain old vanilla GTR) implies that "nogenuine physical magnitude countenanced in GTR changesover time" (p. 6), i.e., that if the GTR is complete, we not onlylive in a block universe, but the block is "frozen", with noreal physical quantities changing. I hope to drive a stakethrough the heart of the undead McTaggart and end thisnew rampage before it has begun.

I choose the image of driving a stake through the heartwith some care. There is much in this paper, especially inthe 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-yond the grasp of most of us, and has done us the great fa-vor of reviewing many technically demanding programsthat are of especial interest to philosophers of physics. Butthe motor of this project is supposed to be a very, very sur-prising feature of the "deep structure" (p. 6) of GTR: namelythat according to the deep structure, nothing physically realchanges. 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.

1 "Thoroughly Modern McTaggart: Or What McTaggart Would Have SaidIf He Had Learned the General Theory of Relativity," Philosophers' ImprintVol. 2, No. 3 (August 2002): http://www.philosophersimprint.org/002003/.

Philosophers' Imprint

<www.philosophersimprint.org/002004/>

Volume 2, No. 4

August 2002© 2002 Tim Maudlin and John Earman

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Tim Maudlin Thoroughly Muddled McTaggart

Let's begin with what we agree on. Earman rejects theoriginal McTaggart's A-series/B-series argument. He agreesthat if one means by a "block universe" only a universewhich can, in its entirety, be modeled by a single 4-dimensional manifold with Lorentz metric, then a block uni-verse can contain real physical change. This is, I think, thecommon sense view. The sort of models of the GTR we aremost familiar with, the solutions of the Einstein Field Equa-tions, represent worlds in which things change: stars col-lapse, perihelions precess, binary star systems radiategravitational waves and increase their rate of spin. The rep-resentation, as a mathematical object, does not change, butthat's just because no mathematical object changes. As Ear-man approvingly paraphrases Savitt: to have a picture ofanimation, one doesn't have to provide an animated picture.But more than that is here conceded. Earman characterizesthe common sense argument for physical change in the GTRas

based on a naively realistic reading of the surface struc-ture of the theory-tensor, vector, and scalar fields onmanifolds. But this naive reading must be radicallymodified if GTR is to count as a deterministic theory,and the modification undercuts the common sense pic-ture of change by freezing the dynamics. [p. 7]

So whatever the new threat to change is supposed to be, itdoes not appear in the "naive reading". The new problem isnot to be founded, for example, on the absence of a preferredfoliation of space-time into instantaneous spaces, since nosuch foliation exists in the "naive reading". Rather, theproblem only appears when the GTR is recast in a waysomewhat different from its usual presentation.

There are two wholly distinct arguments for this new"problem of change" for GTR. One argument involves re-writing the GTR Hamiltonian form, which I will call the"Hamiltonian Argument". The other turns on the proper

definition of an observable in GTR, which I dub the "Ob-servables Argument". I will treat these two arguments sepa-rately, since they rely on different principles.

The Hamiltonian Argument

The Hamiltonian Argument derives from the work of Diracon the "constrained Hamiltonian" formalism for presenting aphysical theory. Earman presents this formalism and its in-terpretation very compactly. I would like to explicate theleading ideas in a somewhat different and, I think, more in-tuitive way. I will not touch on many of the technical detailswhich, I think, play no essential role in understanding the fi-nal result. Let us then approach the Hamiltonian argumentin a series of steps.

First, we are to cast the GTR in Hamiltonian form. Wethen notice that, so cast, the dynamics of the theory appearsto be indeterministic. Next we consider similar cases inwhich apparent indeterminism arises because of a "gaugefreedom" of the theory. We review a standard method forremoving this indeterminism by "quotienting out" the gaugefreedom in the phase space of the theory. Applying thisstandard method to the GTR does indeed restore the deter-minism of the theory—but at a price. The price is that thedynamics of the theory becomes "pure gauge"; that is, statesof the mathematical model which we had originally taken torepresent physically different conditions occurring at differ-ent times are now deemed equivalent since they are relatedby a "gauge transformation". We find that what we took tobe an "earlier" state of the universe is "gauge equivalent" towhat we took to be a "later" state. If gauge equivalent statesare taken to be physically equivalent, it follows that there isno physical difference between the "earlier" and the "later" states:there is no real physical change.

The key step to this argument lies in the technique for

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removing indeterminism by quotienting the phase space. Inorder to understand the nature and prima facie justification ofthis apparatus, it is best to begin on familiar territory: gaugefreedom in classical electromagnetic theory.

If we take the ontology of Maxwellian electrodynamics atface value, we get a picture like this. The complete electro-magnetic state of the universe at some moment is specifiedby the values of the electric and magnetic fields at everypoint of space at that moment, together with their first de-rivatives. (This gives the free field. We may want to add thedistribution of electric charge and its flux to this state, inwhich case there will be constraints between the field valuesand the charge densities, but these complications are not ofthe moment here.) Maxwell's equations then provide thedynamics of this system: they specify how the electric andmagnetic fields evolve through time. Generically, the stateof the universe changes deterministically under this dy-namics.

We can reformulate this theory in Hamiltonian form asfollows. We begin with a Hamiltonian function that,roughly, specifies the total energy of the system. We thenconstruct a phase space of the system. Each point in thephase space specifies the complete global distribution of theelectric and magnetic fields, as well as their conjugate mo-menta. The time evolution of the system is specified byHamilton's equations: the rate of change of any of the ca-nonical variables is given by a partial differential of theHamiltonian. The net result of all this is that the history ofthe entire system is now represented by a trajectory throughphase space, parameterized by time, which solves Hamil-ton's equations. And since the development of the electro-magnetic field in the original (Maxwell) theory was determi-nistic, we expect the dynamics here to be deterministic: eachpoint in phase space will belong to a unique trajectory which

satisfies the equations of motion.Now, it is well known that if a classical electromagnetic

field satisfies Maxwell's equations, then the field can be rep-resented by vector and scalar electromagnetic potentials, Aand Φ, such that

E = -grad Φ - ∂A/c ∂t

and

B = curl A.

It is equally well known that the relation between the poten-tials and the fields is many-one: different scalar and vectorpotentials yield the very same electric and magnetic fields.A mathematical operation changing one pair of potentialsinto another that yields the same fields is a gauge transforma-tion, and the potentials themselves are said to be gaugeequivalent. The justification for this terminology is clear:since the fields are taken to be the fundamental ontology,potentials that are gauge equivalent are taken to representthe very same physical state of affairs. The freedom to chooseamong gauge-equivalent potentials is not a physical degreeof freedom: it rather results from the fact that we have manydistinct mathematical objects all of which represent the samephysical state.

Now suppose we wish to formulate the dynamics of thetheory in terms of the potentials rather than the fields.Without any further ado, we should automatically expectthat (unless something is done), the dynamics in terms of thepotentials ought to be indeterministic. For if the originaldynamics implies that a state of the electromagnetic field E0-B0 will evolve, after a period of time, into E1-B1, then weshould expect the new dynamics only to demand that a pair

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of potentials which yields (by the equations given above) E0-B0 ought to evolve into some pair of potentials which yieldsE1-B1. But since many different pairs of potentials yield E1-B1, we have no reason to expect the dynamics to pick outone of these pairs over any gauge-equivalent pair. That is,the dynamics should now manifest itself as a constraint onthe evolution of the potentials, but a constraint that can bemet by many different trajectories which originate at thesame state.

In this case, it is obvious that the apparent indeterminismof the dynamics is merely a consequence of the gauge free-dom and does not represent any real physical indetermin-ism. In practice, the apparent indeterminism is removed bygauge-fixing: specifying an additional condition on the scalarand vector potentials such that exactly one member of eachset of gauge-equivalent potentials meets the condition. Oncethe gauge is fixed, there is a one-one correspondence be-tween states of the fields and states of the potentials, and soone expects the determinism of the field dynamics to implydeterminism of the dynamics of the potentials. There aremany such possible conditions for the gauge, which go bynames like Lorentz gauge and Coulomb gauge. One typi-cally picks a gauge so as to make the particular problem athand more mathematically tractable.

All of this is quite uncontroversial and is presented withlittle comment in the standard texts.2 Formulating the dy-

2 An example from a text I pulled off my shelf (Classical ElectromagneticRadiation by J. M. Marion [New York: Academic Press, 1965], p. 113):

Now, it is the field quantities and not the potentials that possessphysical meaningfulness. We therefore say that the field vectors areinvariant to gauge transformations; that is they are gauge invariant.

Because of the arbitrariness in the choice of gauge, we are free toimpose an additional constraint on A. We may state this in other terms:a vector is not completely specified by giving only its curl, but if boththe curl and the divergence of a vector are specified, then the vector is

namics of electromagnetic theory in terms of potentialsrather than fields will yield an indeterministic dynamics,which indeterminism arises solely from the gauge freedomand may be eliminated by fixing gauge. And all of thisholds mutatis mutandis if one were to try construct a Hamil-tonian formulation of classical electromagnetic theory interms of the scalar and vector potentials rather then in termsof the electric and magnetic fields. That is: given a Hamilto-nian, one would begin by constructing a phase space suchthat each point represents a complete global specification ofthe scalar and vector potentials and their conjugate mo-menta. The development of the state of the universe wouldbe represented by a trajectory through this A-Φ phase space.And what one should expect, before writing down or solv-ing a single equation, is that the resulting dynamics will beindeterministic: many trajectories through a given pointshould be solutions to Hamilton's equations. A given initialstate, specified in terms of the potentials, should be able toevolve into any of a set of gauge-equivalent final states, all ofwhich represent the same disposition of the electric andmagnetic fields. And so long as one continues to regard thebasic ontology of the theory as the electric and magneticfields, one will regard this indeterminism as completely un-physical: it arises solely from the freedom to choose differentgauges (i.e. to choose among different gauge-equivalent po-tentials) at any time.

What is one to do about this unphysical indeterminism?There are at least three options:

uniquely determined. Clearly, it is to our advantage to make a choicefor div A that will provide a simplification for the particular problemunder consideration.

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Option 1: Ignore ItThe simplest option of all is just to disregard the inde-

terminism once one has recognized its source. That is, onecould frame the dynamics in terms of the potentials and ad-mit any trajectory that satisfies Hamilton's equations as asolution, recognizing the existence of multiple solutionsfrom the same initial state as due to gauge freedom. Afterall, if one regards the fields as the real ontology, then oneknows that the dynamics of that ontology is deterministic,and one sees the gauge freedom that comes along withswitching to the potentials. One can use the gauge freedomto make particular problems more tractable, but the prima fa-cie indeterminism can just be ignored.

Option 2: Fix GaugeAs discussed above, one could cut down the space of poten-tials by adding an additional gauge condition: one can pick agauge. The phase space will thereby be reduced and—ifthere is a unique pair of potentials that meets the gauge con-dition for each state of the electromagnetic field—one wouldexpect the dynamics on this reduced phase space to again bedeterministic. In the case of electromagnetism, these condi-tions will hold if, for example, the gauge conditions specifythe divergence of A and the value of Φ at spatial infinity.One might wonder why one would go to the trouble of in-flating the phase space by working in terms of the potentialsand then commensurately shrinking the phase space by fix-ing gauge, but the liberty of choosing the gauge could makethe problem at hand more mathematically tractable.

Option 3: QuotientingSuppose one begins with an "inflated" phase space, in whichmultiple points correspond to the same physical state. Andsuppose one has a clear formulation of the conditions in

which a pair of points in the phase space are gauge equiva-lent. Then one can form equivalence classes of points in thephase space, all of which are gauge equivalent and so repre-sent the same physical state. Call these equivalence classesgauge orbits. Finally, one can construct a new phase space,each point of which corresponds to a gauge orbit in theoriginal space. The new phase space is the quotient of theold one by the equivalence classes. And again, intuitively,one expects the dynamics on the new phase space to be de-terministic if the theory one started with was deterministic.

Quotienting has certain formal advantages over gaugefixing as a way to recover a deterministic dynamics. As wehave seen, when one fixes gauge one requires that each dis-tinct physical state have exactly one representation in the"inflated" variables which meets the gauge condition. For iftwo gauge-equivalent points in the inflated phase spacemeet the gauge condition, then the dynamics may not de-termine which of these points the trajectory of the systemwill pass through. And if some physical state has no repre-sentation that meets the gauge condition, then when onefixes gauge one will lose the power to represent some physi-cal possibilities. In contrast, one is automatically guaranteedthat each physical state will correspond to exactly one gaugeorbit, since the orbits by definition contain all the points inthe phase space that represent the state. So if one has a de-terministic theory to begin with, but gets an indeterministicdynamics when casting it into Hamiltonian form, there isgood reason to believe that quotienting will restore the de-terminism.

Of course, one should not overlook that fact that quo-tienting may be a more difficult mathematical matter than,say, fixing gauge. Nor should one overlook the fact thatnone of these formal tricks are really necessary to maintainone's belief in the fundamental determinism of the theory,

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once one has seen that the phase space is inflated so that dif-ferent trajectories can correspond to the same course ofphysical events. Option 1, ignoring the merely apparent in-determinism which inflation creates, is always available.

Notice that our whole discussion up to this point ispredicated on the assumption that one has an antecedent un-derstanding of gauge equivalence and gauge freedom. This is truein classical electrodynamics, where one accepts the basicfield ontology and regards the scalar and vector potentials asmere mathematical conveniences. If one were to becomeconvinced that the potentials were physically real, so thatclassically "gauge equivalent" potentials represent distinctphysical states, then all bets are off. In that case, one mighttake the indeterminism in the dynamics of the potentials toreflect real physical indeterminism. Or more likely, onemight conclude that the dynamics needs to be supplementedto render it deterministic again: after all, the original dy-namics is couched in term of the fields rather than the po-tentials. So one is confident about how to interpret the "in-determinism" in the dynamics that arises for the potentials inelectromagnetic theory (whether that dynamics be in Ham-iltonian form or in standard differential equations) becauseof what one accepts from the outset about the basic ontologyof the theory.

But once the technique of quotienting becomes familiar,there is a temptation to turn this whole process on its head.That is, suppose one is given a phase space and a Hamilto-nian, and suppose that the resulting dynamics is not deter-ministic. (Typically, this manifests itself in certain freelyspecifiable functions that appear when one solves the dy-namical equations.) And suppose that, for some reason orother, one thinks that the true physical dynamics ought to bedeterministic. Then one might well be tempted to render thedynamics deterministic by finding equivalence classes of

phase points such that, when one quotients by them, the re-sulting dynamics is deterministic. That is, one does not beginwith a clear notion of gauge equivalence, but rather postu-lates the "gauge orbits" in such a way as to render the dy-namics deterministic. Having so determined the gauge or-bits, one then concludes that all the states in an given orbitrepresent the same physical state: the "apparent" differencesamong the states arise only from different choices of gauge.

This topsy-turvy use of quotienting contains several dan-gers. One danger arises because, in principle, it is alwayspossible to make it work and so render a theory determinis-tic. One could, for example, assign the whole phase space to asingle gauge orbit, so that the quotiented phase space hasbut a single point. The resulting dynamics is certainly de-terministic, if boring: the universe has only one physicalstate available to it and so always remains in that state. Anyamount of seeming indeterminism in a dynamics can be re-moved by this expedient, but at a heavy price: one wouldhave to abandon both one's belief that the physical state ofthe universe changes, and one's belief that it might havebeen different from what it is. It is hard to imagine whatcould recommend this course of interpretation.

Or take an only slightly less extreme example. Beginwith a stochastic dynamics for, say, Democritean atoms: theatoms can, from time to time, swerve. Intuitively, a singleinitial state can then evolve in many different ways. If wecast this dynamics on a phase space, we would expect thelaws to admit of solutions that agree for some time and thendiverge. But by clever quotienting, we can remove this in-determinism. Begin with some initial state, then form theclass of all the states that this state could evolve into or fromwhich it could have evolved. Repeat the process for all thestates in this class, and repeat until no more states are added.What we will end up with is the set of all states that have a

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certain number of the various types of atoms, irrespective ofhow those atoms are disposed. Now quotient out with re-spect to these sets of states. The dynamics will again be de-terministic but "frozen": no system ever leaves its "gauge or-bit". There will be alternative possible physical states, in-habited by different numbers or different sorts of particle,but no such state will ever change in time.

All of this is, of course, both formally correct and com-pletely crazy. If we lived in a such a Democritean world, wewould reject the "deterministic" dynamics for the simple rea-son that we see particles moving around and changing posi-tion: the world is not frozen. What possible grounds for be-lieving that the world is deterministic could make rationalthe wholesale rejection of all sense experience?

We must, then, be very judicious in our use of the topsy-turvy method. If we are sure that the dynamics of some the-ory ought to come out deterministic, then we had best keepcareful track of our grounds for that belief and of the pointwhere some faux indeterminism has entered our mathemat-ics. For if we blindly demand determinism from quotient-ing, it can certainly meet our demand, but perhaps in arather nonsensical way.

Let's apply all this to the GTR.We begin with the idea that the GTR is, indeed, determi-

nistic, at least in typical applications. Solutions to the Ein-stein Field Equations are four-dimensional manifolds. Con-ditions for determinism are most easily stated for globallyhyperbolic space-times. Such space-times admit of Cauchysurfaces, i.e. surfaces that every inextendible timelike curveintersects exactly once. If we specify the intrinsic curvatureof such a surface, and the physical state on such a surface,and the way that the surface is embedded in the space-time,then there is typically a unique maximal globally hyperbolic

space-time consistent with that data.Now suppose we want to cast the GTR into a Hamilto-

nian form. (Why would we want to do this? There is somereason to hope that it might help when searching for aquantized version of the theory, but otherwise there is novery compelling reason.) We are immediately faced with adifficulty. Classical mechanics and classical electrodynamicsare formulated in space-times with absolute simultaneity,and so there is a clear-cut notion of the instantaneous state ofthe universe. Points in phase space represent these globalinstantaneous states, and a trajectory though phase spacerepresents the history of the universe as a succession of suchstates. Furthermore, the problem of gauge freedom only in-fects the instantaneous states: the states in a gauge orbit areall representations of the same instantaneous state. Thus adeterministic dynamics over the gauge orbits yields a de-terministic succession of instantaneous states over time,which is the history of the universe.

Now, the fundamental problem when dealing with theGTR is that the four-dimensional solutions to the field equa-tions do not come equipped with anything like absolute si-multaneity, and so there is no unproblematic notion of theinstantaneous state of the universe. We may foliate a globallyhyperbolic space-time by families of Cauchy surfaces, whichcan in many ways serve the role of instantaneous states, butsuch foliations are by no means unique. And in the freedomto foliate lies the key to the Hamiltonian Argument.

Consider a solution to the EFE's that contains two clocks(figure 1). This solution can be "split up" into a stack of in-stantaneous states in various ways. One obvious way is bythe foliation depicted in figure 2. We can now depict thesolution as a succession of global states, in each of which theclocks indicate the same time.

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.

figure 1

t1

t2

t3

t4

figure 2

But the very same solution can also be foliated as in figure 3:

t'0

t'1

t'2

t'3

figure 3

Now in each "instantaneous" state the left-hand clock isahead of the right-hand clock. This is just as legitimate away to carve up the model as figure 2.

Once we have a foliation, we can begin to apply theHamiltonian formalism. The points in phase space will rep-resent instantaneous states, i.e., states of Cauchy surfaces. Ifwe use the foliation of figure 2, then the complete four-dimensional solution will be represented by a trajectorythrough the phase space, and each point on the trajectorywill contain clocks that indicate the same time. If we use thefoliation of figure 3, then the very same solution will be rep-resented by a completely different trajectory, such that each

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point on the trajectory contains clocks that indicate differenttimes. Obviously, these two trajectories in phase space willhave no points in common.

But now comes the critical observation. We can also foli-ate the solution as in figure 4, with Cauchy surfaces thatagree with the figure 2 foliation early but morph into thefigure 3 slices later on:

t''0

t''1

t''3

t''2

figure 4

Again, this slicing will yield a trajectory through the phasespace, and the complete trajectory will correspond to thecomplete four-dimensional solution. But we can immedi-ately see a problem for determinism: the trajectory one getsfrom the figure 4 slicing agrees precisely with the trajectory

from the figure 2 slicing at t0 (and, if one likes, at all timesprior to t0) but later diverges, wandering over to the regionof phase space occupied by the figure 3 slicing. And in gen-eral, the complete trajectory through phase space up to sometime does not determine the future trajectory for the simplereason that the foliation of the space-time up to that point doesnot determine the foliation later on. And this, in turn, is a con-sequence of the fact that we have arbitrarily chosen the foliationfrom among the infinitude of ways of splitting the solution intoCauchy slices. Different slicings yield different trajectoriesthrough phase space, and slicings that agree to a point andthen diverge yield trajectories which agree to a point andthen diverge, i.e. , yield dynamical indeterminism.

So before we have written down a single equation, wecan make a prediction: casting the GTR into Hamiltonianform will yield a theory with an indeterministic dynamics.But we also understand the source of the indeterminism: itcomes from forcing the GTR into the Procrustean bed of theHamiltonian formalism. In order to do so, we have to im-port a foliation into our solutions to the EFE's, a foliationthat has no basis in the GTR itself. It is the arbitrary natureof the foliation that makes the resulting trajectory throughphase space somewhat arbitrary. But we equally see thatthis indeterminism is completely phony: it has nothing to dowith any real physical indeterminism. Given the initial stateon a Cauchy surface like t0, the GTR admits of a uniquemaximal global solution. Carving up that single solution bydifferent foliations yields different trajectories through phasespace, but all of these trajectories, in their entirety, representthe same four-dimensional solution.

So it should come as no surprise at all that when we putthe GTR into Hamiltonian form we get an apparently inde-terministic dynamics. How should we deal with this?

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Reviewing the options above, we could, first of all, sim-ply ignore it. The theory in Hamiltonian form represents nomore physical indeterminism than there is in the EFE's,namely (given the restriction to maximal globally hyperbolicspace-times) none. All of the different trajectories throughphase space that are solutions to the dynamical equationsrepresent the very same complete four-dimensional space-time, and the physical magnitudes in the space-time evolvedeterministically. Furthermore, the magnitudes in thespace-time evolve deterministically: the universe may ex-pand, perihelions may precess, binary star systems mayspeed up their rotations, just as we always thought all along.Changing to the Hamiltonian formalism gives us no new insightat all into the basic ontology or dynamics of the theory.

If one does not want to just ignore the faux indetermin-ism of the Hamiltonian form of the theory (if, for example, itmakes quantization more difficult), then one could try toeliminate the apparent indeterminism by fixing gauge. Inpractice, this means formulating some constraint on the waythe space-time is foliated, so that diverging foliations thatyield diverging trajectories no longer exist. This could bedone in two ways.

One way would be to find some method for canonicallyfoliating a space-time. This would require discovery of somecondition that exactly one foliation of any space-time can ful-fill. In certain cases, such a condition might exist (e.g., theremight be a unique foliation in which a background radiationfield is homogeneous and isotropic on every slice), butclearly no such generic condition exists. Minkowski space-time, for example, is a vacuum solution to the EFE's, butthere can be no condition that picks out a unique foliation ofMinkowski space-time on account of its symmetries (e.g.,symmetries under Lorentz boosts, rotations, and transla-tions). Any canonical slices would also have to be invariant

under those symmetries, but no slices exists that are invari-ant under them all. So no generic condition can pick out aunique foliation of all of the models of the GTR.

A more promising approach is this: find a condition thatgiven a single Cauchy surface as data then induces a unique fo-liation of the space-time. The idea is pretty straightforward:one would expect, for example, that given the slice t0 in fig-ure 2, the method would generate the complete slicing offigure 2, and given the slice t'0 of figure 3 it would generatethe slicing of figure 3, and no initial Cauchy slice would gener-ate the slicing of figure 4. Such a project will be mathemati-cally quite non-trivial, but in certain cases one could imaginehow it might go: given one Cauchy surface (call it t0), let theslice ti (i a positive real) be the locus of all points p in thespace-time such that the maximal future-directed time-likecurve from t0 to p is of proper relativistic length i, and simi-larly for i negative, using a past-directed curve.3 Or there areother ways one could go about this, using the so-called lapsefunction and shift vectors. In any case, this sort of "gaugefixing" would help solve the indeterminism problem: sincethe initial data would be consistent with a unique global fo-liation that satisfies the constraint, one would not get di-verging trajectories through phase space that correspond to

3The suggestion here, although it would work in some cases, would notsolve the generic problem of fixing a foliation into Cauchy surfaces. It is easy tosee 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 thanonce, but one is not guaranteed that each inextendible timelike curve would in-tersect every hypersurface, so the hypersurfaces need not all be Cauchy surfaces.In addition, the method is impractical as a means to solving the EFEs, since oneis using the metrical structure of the space-time to determine the slicing, but onedoes know the metrical structure until one has solved the EFEs. (Gauge fixingin 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 isknown.) So we are here making an abstract point about gauge-fixing, not apractical suggestion.

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the same solution merely because the foliations diverge. Ofcourse, there would still be distinct trajectories throughphase space that correspond to the same global solution be-cause they are generated from different initial Cauchy surfaces(like t0 and t'0 above), and they might cause problems for,say, quantization. But that has nothing to do with indeter-minism.

So both ignoring the apparent indeterminism and fixinggauge appear to be viable solutions to the indeterminism"problem". But if one has fallen in love with the "constrainedHamiltonian formalism" and one has solved other faux in-determinism problems (as in electromagnetism) by quo-tienting, then one might be tempted to try this route. Butthere lies disaster.

Recall the basic strategy of quotienting. We begin withapparent dynamical indeterminism: the dynamical equa-tions permit different solutions with the same initial data sothat, for example, according to one allowable trajectory ini-tial state S0 evolves into state S1, but according to anotherallowable trajectory S0 evolves instead into S'1 and neverenters S1.4 This apparent indeterminism could be removedby declaring that S1 and S'1 are really physically identicalstates: they are gauge equivalent. One then quotients by thecomplete gauge equivalence classes (the gauge orbits), re-ducing the phase space and restoring determinism.

But if the source of the apparent indeterminism is the

4The problem is most easily stated for a case like electromagnetism done interms of potentials, since there is a common universal time in all models, so onecan say that according to one trajectory S0 evolves into S1 five minutes later,

while according to the other trajectory it evolves into S1' five minutes later. The

solution to the apparent indeterminism is then to make S1 and S1' gauge equiva-lent. Since in the GTR there is no such common universal time (introducing auniversal time function is equivalent to picking a foliation) things are not sosimple, as we will see.

freedom to foliate the space-time, this solution will be acomplete disaster. As we have seen, if we are free to foliate,then the state on t0 could evolve into t3 (as in figure 2), but itcould equally well evolve into t''3 (as in figure 4) dependingon how we foliate. So the quotienting solution would haveto declare that the state on t3 and the state on t''3 are gaugeequivalent, i.e. that they are merely mathematically distinct waysof expressing the very same physical state. But this is crazy: t3has two clocks that indicate the same time, while t''3 hasclocks that indicate different times: these are physically dis-tinct states. Even more damning is this: the left-hand clockon state t3 indicates a different time than the left-hand clockon t''3, so if t3 and t''3 are really the same physical state, thenthe physical state of a clock does not really change as it comes toindicate different times. By similar argumentation, we wouldcome to the conclusion that, no matter how the clock seems tohave its hands oriented, it is always in precisely the samephysical state: the "change" is merely apparent, not real.These claims are, or course, rather silly—but they are pre-cisely the claims that, according to Earman, the constrainedHamiltonian formalism reveals about the deep structure ofthe GTR.

It is, in fact, not hard to see that if one is going to restoredeterminism by quotienting, then the gauge orbits have tocontain every state on every Cauchy surface in a solution to theEFE's. Only then is one assured that the states along a tra-jectory will belong to the same gauge orbit no matter whatfoliation is used to generate the trajectory. And so whatholds for the clock would have to hold for the universe as awhole: its physical state never changes, from a millisecondafter the Big Bang to a minute before the Big Crunch. In thetechnical terminology, the dynamics of the theory is puregauge, since all of the states along every trajectory have tobelong to the same gauge orbit. McTaggart—or more prop-

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erly Parmenides—is vindicated: according to the way ofTruth, the universe is ever One and Unchanging, only ac-cording to the way of Seeming is there change.

Now there is an attenuated sense in which the state onevery Cauchy surface in a solution of the EFE's is the same:each such state implicitly represents everything that happensat all times, since each such state is compatible with exactlythe same maximal globally hyperbolic solution to the EFE's.But that does not make them all physically equivalent: oth-erwise we get an immediate argument from determinism toNo Real Change. There is real physical change because thephysical states on the different Cauchy surfaces are different(i.e., non-isomorphic), even if each surface (together with theEFE's) implies the same global solution.

After all, how could it be otherwise? We know that theGTR is a theory which predicts—and explains—manychanges: the precession of planetary orbits, the collapse ofstars, the rate of expansion of the universe, the red shift oflight coming out of a gravitational well. It is, indeed, the ob-servation of precisely such changes that provides our evi-dence for the theory. Any interpretation which claims thatthe deep structure of the theory says that there is no changeat all—and that leaves completely mysterious why thereseems to be change and why the merely apparent changes arecorrectly predicted by the theory—so separates our experi-ence from physical reality as to render meaningless the evi-dence that constitutes our grounds for believing the theory.So the only real question is not that the constrained Hamil-tonian formalism (interpreted as Earman suggests) is yield-ing nonsense in this case, but why it is yielding nonsense.And the freedom to foliate provides the perfectly compre-hensible answer.

It is only proper to note that Earman did not constructthese arguments for the unreality of change: Dirac devel-

oped the abstract form of the constrained Hamiltonian for-malism, and more importantly, Dirac suggested that "thegauge transformations be identified as the transformationsgenerated by the first class constraints, where the intendedinterpretation is that two points of the phase space Γ whichare connected by a gauge transformation are to be regardedas representing the same physical state" (p. 8). It is here thatthe method is turned topsy-turvy: instead of starting with anunderstanding of which points in phase space represent thesame state, one rather does the dynamics first and then con-cludes from some formal feature of the dynamics that twopoints represent the same physical state. And we can seewhy the topsy-turvy method would work for, say, classicalelectromagnetic theory: the "gauge transformations" soidentified really would be the intuitively correct gauge trans-formations. But generalizing from this sort of example thatwe should always identify transformations generated by con-straints in the Hamiltonian as gauge transformations be-tween physically equivalent states is a dangerous business,as we have seen.

It is also only proper to note that criticisms of this use ofDirac's method are not original: as Earman notes, Karel Ku-char, for example, makes exactly the same point. I only hopeto have made clear why this particular result is to be ex-pected if the GTR is put in Hamiltonian form.

Earman seems to feel the force of these objections, sincehe is at pains to argue that the "frozen time" results are not"merely formal tricks or artifacts of the constrained Hamil-tonian formalism" (p. 9). To allay these doubts, he proposesto derive "similar, if not identical, results in the spacetimesetting rather than the (3+1) Hamiltonian formulations" (p.9). This would be significant, since my argument so far hasbeen that the problems arise from using the (3+1) formula-tions: it is precisely a foliation that one needs to turn a four-

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dimensional relativistic space-time into a (3+1) dimensionalobject. And indeed, the arguments that Earman rehearsesnext do not hinge on the same mistakes that sink the Ham-iltonian argument—they hinge rather on a completely dif-ferent set of mistakes. To these we now turn.

The Observables Argument

Earman next takes up the approach suggested by Bergmannfor guaranteeing determinism in the GTR. Like the quo-tienting technique, Bergmann's approach is certain to yieldthe result that the GTR is deterministic, at least with respectto all observables, because it secures this result by definition:in order to be an observable, a quantity must be "unequivo-cally predictable from initial data" (p. 9). The only questionleft is what the observables of the theory are.

We would expect the observables to include: the positionof the perihelion of Mercury after some number of orbits, theamount by which light from the Sun is redshifted when itreaches the Earth, the angle at which light from a distant starwill reach the Earth during an eclipse, and so on. And weshould expect that these observables will change: the peri-helion of Mercury will advance at a predictable rate. So onemight expect that we would start with these sorts of observ-ables, the ones that provided evidence for the theory in thefirst place, the ones that were predicted by the theory, andtry to discover some generic characterization of observablesthat includes these sorts of quantities. But instead we aregiven an abstract characterization of observables that has as aconsequence that none of the things that were actually ob-served and measured and brought forward as evidence forthe theory were observable at all! This isn't just topsy-turvy,it's through-the-looking-glass.

As Earman puts it: "What may not be familiar to mostreaders is that Bergmann's proposal implies that there is no

physical change, i.e. no change in his observable quantities,at least not for those quantities that are constructable in themost straightforward way from the materials at hand" (p.10). One might think that, if true, the moral is to find ob-servables—like the position of the perihelion of Mercuryrelative to the Sun—that are constructable in a less-than-straightforward-way rather than concluding that there is noreal physical change in the world, but having gone through thelooking glass, we are apparently to accept some Alice-in-Wonderland logic from this point on.

To fix ideas, let's start with the sort of prediction that wecan make using the GTR. It will also help a bit to reflect onthe way that the GTR can be used in conjunction with otherprinciples to make predictions, since our focus from here onwill be the sorts of predictions that can be made from theGTR neat. So let's start with a "mixed" prediction and thentry to work back to a more "pure" one.

The GTR has been used successfully to predict the out-come of the following experiment: take two synchronizedatomic clocks, put one on a plane and fly it around the Earthon some specified route, then bring the clocks back togetherand compare them. When brought back together, the clockswill no longer be synchronized, and the amount of dis-agreement can be accurately predicted using the GTR. (Notabene: the GTR can be used to make a deterministic predictionabout how the synchronization of the clocks will change from thebeginning of the experiment to the end, so a fortiori the GTR canbe used to predict that things will change. So the only wayto secure McTaggart's conclusion is to argue that the relativesynchronization of the clocks is not an observable!) How isthis prediction made?

One starts with facts about the size and mass of theEarth. These are the sorts of data that can be put on aCauchy surface. One then solves the EFE's to get a full four-

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dimensional space-time. All of this is pure GTR. But nowthings get a bit tricky. Knowing the flight plan of the air-plane, one can then pick out a trajectory through the space-time (by reference to the position of the Earth), representingthe trajectory of the flying clock, and one can similarly pickout the trajectory of the stay-at-home clock. One can calcu-late the proper time along these paths, and assuming that theclocks measure the proper time, one can predict what theywill show when brought back together.

Now it is evident that this whole procedure is not simplya matter of solving the EFE's. In addition to the solution tothe EFE's, we have to specify the trajectories of the clocksand have to deploy a principle about what clocks measure.This second principle could be reduced to a purely relativis-tic one if we used a light-clock (rather than an atomic clock)which can be shown to measure proper time, but even so thefirst additional bit of information has to be added from theoutside. The only way to avoid this would be to include inthe model a complete physical description of the airplane and ofall physical objects that influence the flight of the airplane andthen solve for its trajectory, but this is clearly a practical im-possibility. So we need to be a bit careful. What we want tosay is that some quantities, such as the proper time along atime-like trajectory, are on the one hand predictable from theGTR and on the other hand observable (by means of clocks),even though the directly observable instruments (the clocks)are not themselves represented in the models of the GTR weactually use. This shortcoming cannot be overcome until wehave the resources to represent the physics of the clocks inthe models and would probably be mathematically intracta-ble even then. So when young McTaggart speaks of a"genuine physical magnitude countenanced by the GTR" (p.6) and when those magnitudes are characterized as "observ-ables", one ought to pause: in many cases, the observable

physical magnitudes (like clock readings) which are most di-rectly relevant to laboratory operations are not representedin the purely relativistic mathematics used to make predic-tions.

There are two ways to mitigate this problem. One issimply to declare that relativistic quantities like the propertime along a world line, or the gravitational tidal forces at apoint, are observables since we have instruments (clocks orwater drops) that allow us to observe them, even if we don'tinclude those instruments in our models. Furthermore, wecan just grant that we can know, e.g., how to represent thetrajectory of the flying clock, even though we don't solve ourequations for it. Note that the relevant trajectory is givenrelative to the Earth: the plane flies at a certain altitude abovethe Earth for a certain distance before coming back.

The second way to mitigate the problem is to investigateexperimental conditions where gravity is the only importantfactor. If we could send our two clocks free-falling along dif-ferent paths through a gravitational lens, then (supposingtheir initial trajectories are part of the Cauchy data) we couldsolve for their trajectories using the GTR: they will followthe appropriate time-like geodesics. We can solve for thepoint where they will intersect, and predict—from the GTRalone—how far out of synchronization they will be whenthey meet. Or, in a more realistic case, predicting the appar-ent position of a star during a total eclipse does not demandsignificant input from outside the GTR proper. So it is aplain fact that the GTR makes deterministic predictionsabout observable physical magnitudes and about how thosemagnitudes can change. The only real question before us ishow Bergmann's seemingly innocuous definition of an ob-servable as any physical magnitude deterministically pre-dictable from Cauchy data (irrespective of whether any in-strument can, in the intuitive sense, observe it) can possibly

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get us into trouble. Why don't things like the precession ofthe perihelion of Mercury or the reading of our clocks whenthey get back together turn out to be observables since theyare clearly predictable? The devil, of course, is in the fineprint—and the first bit of fine print occurs in the artfullywrought statement cited above: Bergmann's criterion impliesthat there is no real change, "at least not for those quantitiesthat are constructable in the most straightforward way fromthe materials at hand". Let's take a careful look at what ex-actly that means.

The first candidates for Bergmann observables are

local field quantities which are constructed from the met-ric and its derivatives up to some finite order and whichare evaluated at [a] spacetime point, e.g. the Ricci cur-

vature scalar R . Is the value of this quantity at some

point to the future of an initial hypersurface predictablefrom initial data on the hypersurface, or even from dataon the entire past of the hypersurface? [p. 10]

One would initially think that the answer is clearly "yes". Tofix ideas, let's take the case discussed above: the initial datainclude two objects (they may be clocks, but it does notmatter for this example) that are being launched from Earthtoward opposite sides of a gravitational lens (such as theSun). Can we deterministically predict the value of the Ricciscalar at the space-time point in the future where the two ob-jects will meet?

Evidently, the answer is "yes". From the initial data andthe EFE's, we can construct a complete four-dimensionalsolution, identify the point where the relevant geodesics(which originate at the Earth) meet, and find the Ricci scalarat that point. And if we wanted to do the experiment, wecould construct rockets to be launched from the Earth thatwould measure the curvature when they meet and transmit

back the result. So surely this ought to count, on Bergmann'scriterion, as an observable.

But according to the analysis Earman offers, the Riccicurvature at a spacetime point is not observable by Berg-mann's criterion, i.e., not predictable from initial data, unlessthe Ricci curvature is constant everywhere to the future ofthe initial hypersurface. How can that be?

The trick is how to interpret the phrase "at a spacetimepoint". In my presentation, the relevant spacetime point isidentified by a definite description: the point where the twogeodesics meet. In Earman's approach, the point is not soidentified. In fact, in Earman's account there is no story at allabout how the relevant point is identified: it is just somehowgiven. The point is not given by a definite description such asthe one offered above: the point is not identified by itsspacetime relation to the initial hypersurface, or by itsspacetime relation to any material object (e.g., as a point tenmiles above the Earth), or by the object that occupies it. It israther just (magically) given as a point in the "bare" manifold,i.e. as a point in the spacetime manifold before any metric hasbeen specified for the manifold.

Now, there is, in the first place, no coherent accountabout how such a point could be identified independently ofthe metric and contents of spacetime. And even if therewere such an account, there is no account of how the rele-vant point could be observationally identified so that the Riccicurvature there could be checked. So for all intents and pur-poses, we are in the following situation. We are given thedata on the initial surface, and then someone informs us thatthey have a spacetime point somewhere in the future inmind, but they provide us with no further information aboutwhich point it is, and then they ask whether we can predictwhat the Ricci scalar is at that otherwise unidentified point.And of course, we could in such a situation neither predict

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the scalar curvature there (unless, of course, the Ricci scalaris the same at every point in the future), nor could we laterempirically determine what the relevant scalar curvature is.So if the spacetime point is only given to us as a point on thebare manifold, the Ricci scalar at that point is not predictableand so, by Bergmann's criterion, not observable. (Nota bene:the failure of observability has nothing whatever to do witha lack of instrumentation by which one can empirically de-termine the Ricci scalar; it has to do with the lack of instru-mentation by which one can determine the identity of therelevant point.)

Earman makes this argument using the technical ma-chinery of diffeomorphism invariance. Take a solution tothe EFE's, and choose a diffeomorphism that is the identitymap on the initial data surface but not on the point p in thebare manifold. In particular, suppose that the Ricci scalar tothe future of the initial surface is not everywhere constant.Then there is a point q to the future whose Ricci scalar differsfrom that of p in the solution. Choose a diffeomorphism thatis the identity on the initial hypersurface and that maps q top. Now "drag" the original solution along the diffeomor-phism to get a new solution to the EFE's. In this new solu-tion, the Ricci scalar at p will be different from the original: itwill now be the value that was formerly at q. But since thisis also a solution of the EFE's from the same initial data, thatdata and the EFE's cannot predict the Ricci scalar at p.

Of course, if p happens to be the point where the twogeodesics intersect in the first solution, it will not be thepoint where they intersect in the new solution. The geodes-ics will be "dragged along" by the diffeomorphism, and theywill now intersect at a new point on the "bare" manifold, apoint whose Ricci scalar is identical to that at p in the origi-nal solution. And so if we identify the relevant point by thedefinite description and empirically identify the point by

where the two rockets collide, we have no problems. We onlywould have problems if we could per impossible identifypoints on the bare manifold as such.

At this point, the attentive reader will begin to feelqueasy. For what we have is just another incarnation of thenotorious "hole" argument. That argument, recall, was sup-posed to establish that the GTR is indeterministic because theEFE's only determine a solution from initial data up to a dif-feomorphism. Now, whatever one thinks of that argument(my own views have been expressed ad nauseum elsewhere5),all hands in that debate agreed that if there is any indeter-minism, it is an unobservable indeterminism, since it concernswhat happens at particular points of the bare manifold, butparticular points of the bare manifold are not, per se, observ-able. Now that argument is being used, in conjunction withBergmann's criterion, to a perfectly risible conclusion. Theproposed logical form of a "local field quantity" is a quantityattached to a point on the bare manifold, and then the indeter-minism is used to argue that these quantities are not observ-ables (unless, of course, the quantity is the same everywhere,and so predictable). But even if the values of quantities at-tached to points of the bare manifold were observable inBergmann's sense, they would not be observable in the nor-mal sense, since we can't identify the relevant points by ob-servation. What we can identify by observation are thepoints that satisfy definite descriptions such as "the pointwhere these geodesics which originate here meet", andagainst these sorts of quantities Earman's diffeomorphismargument has exactly zero force.

So if one were to start by reflecting on the logical form ofthe predictions we actually make using the GTR, or if one

5 E.g,. in "Substances and Space-Time: What Aristotle Would Have Said toEinstein", Studies in the History and Philosophy of Science Vol. 21, No. 4, pp.531-561.

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were to start by reflecting on what sorts of things we take tobe actually observable in spacetime, then one would not be-gin with the logical form of a quantity attached to a point ofthe bare manifold. Earman's strategy instead is to start withquantities attached to bare points because they "are con-structable in the most straightforward way from the materi-als at hand", and then to show, to no great surprise, thatthese are not Bergmann observables. There is nothing tech-nically wrong with Earman's argument; it just seems like arather senseless way to proceed once one reflects on thesource of the difficulties it encounters.

Earman does not claim that the examination of "localfield quantities" (or the "quasi-local field quantities" one getsby integrating the local ones over patches of the bare mani-fold, which obviously inherit the same problems) is a deci-sive argument that there are no Bergmann observables thatare different at different places in spacetime. But he casts askeptical eye on the utility of any quantities which are not"attached" to points of the bare manifold (like the value of theRicci scalar where the geodesics meet): "[I]t is worth remarkingthat it is not obvious how these unattached quantities couldunderwrite B-series change; for such change requires a sub-ject, and since spacetime points and regions are the only ob-vious candidates for the subject role in GTR, these peculiarunattached quantities would seem to remove the subject ofchange from the picture" (pp. 10-11). So let's answer thisconcern before going forward.

At the beginning of our discussion, it was agreed that ac-cording to the "naive" reading of the GTR, the GTR allowsfor B-series change. For the models of the GTR are four-dimensional manifolds with a relativistic metric, and thatmetric allows one to define certain timelike relations amongevents in the space-time. Thus, a clock in the GTR canchange the time it indicates because (a) it indicates different

times at different places in the space-time, and (b) the differ-ent places are timelike related to each other, so one canspeak of which of a pair of indications is earlier and whichlater. Notice that the subject of the change is a material ob-ject—a clock—which is represented by a spacetime wormthat has different features at different points. Nothing at aspacetime point can change, for the simple reason that aspacetime point has no temporal extension: individualspacetime points are not even candidates for "subjects ofchange", i.e., for the things that change. Subjects of changemust persist through time so they can have different prop-erties at different times. And the things that persist throughtime are typically material objects like clocks or stars or gal-axies. These are represented by spacetime worms that areidentified by their material contents: the spacetime worm that isthe clock is the collection of spacetime points that are occupiedby the material of which the clock is made. And in this sense, thesubjects of change are not "attached to" bare spacetimepoints: under an active diffeomorphism, the subject ofchange (e.g. the clock) will be "dragged along" to a new partof the bare manifold.

So even if, in some sense, spacetime points are the ultimate"subjects of predication" in the GTR, it does not at all followthat they are the subjects of change, i.e., the things thatchange. This is just a confusion of two rather unrelated usesof the term 'subject'.

In succeeding sections of the paper, Earman takes up theproject of identifying some other "observables" beside his lo-cal and quasi-local field quantities. Unfortunately, he neverconsiders anything as simple as "the value of the Ricci scalarwhere two given (i.e., given in the initial data) geodesicsmeet". He has some remarks about so-called "coincidenceobservables", and this looks promising: we may ask after thevalue of a quantity where the two geodesics coincide. But

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even here things strike him as problematic. Adjusting hisremarks to fit the case of the intersecting geodesics, onewould get this:

Admittedly, however, it remains a bit obscure how thevalue of this coincidence observable is measured. For ...the measuring procedure cannot work by verifying thatthe coincidence of values ... does in fact take place byseparately measuring the [bare manifold position of apoint on one geodesic] and the [bare manifold positionof a point on the other geodesic], and then checking forthe coincidence. For [the positions on the bare manifold]are gauge dependent quantities, and by the Assumptionthese quantities are not fixed by measurement ... . [p. 13]

Of course, one would not tell where the geodesics coincidein anything like this way: one would tell by sending a rocketalong each path and making the measurement when theycollide. Nothing in any of Earman's arguments suggests anydifficulty about this procedure.

So the Observables Argument gets any traction only byconsidering candidates for observables (values at points ofthe bare manifold) which are neither the sorts of things oneactually uses the GTR to predict nor the sorts of things onewould expect—quite apart from diffeomorphism invari-ance—to be observable. And as soon as one tries the argu-ments out on something that one would predict or observe,they fail. The Observables Argument runs on completelydifferent principles from the Hamiltonian Argument, but itis equally broken-backed.

The critical Section 4 ends with yet a third observation,drawn from an entirely different source: the "alternative ap-proach of Ashtekar and Bombelli" (p 15). I can only admitthat I have no first-hand knowledge of the theory and cannotmake head nor tail out of Earman's description. He says thetheory is defined on the space "of entire histories or solutions

to the Einstein field equations, which implies that the dy-namics is implemented not by a mapping from one state toanother state in the same solution, but from one solution toanother solution". But the whole argument to date has beenthat complete solutions to the EFE's—complete relativistichistories of the world—can represent worlds in which thingschange. Why a mapping from one complete solution to an-other—from one complete possible world to an-other—should even be called "dynamics", or what it has todo with the physics of the one world we live in, is com-pletely obscure.

So in the end, we have three arguments against change inthe GTR, two demonstrably inadequate and the third in-comprehensible (at least to me). One might wonder whyEarman would bother with three arguments if he thoughtthat any one of them sufficed to establish the conclusion.The principle he seems to be following is: where there'ssmoke, there's fire. But sometimes where there's smoke,there's mirrors. The apparent difficulty for change—andtherefore for time itself—which Earman’s McTaggart dis-cerns in the GTR is only an artifact of a bad choice of for-malism or a bad choice for the logical form of an observable,not because of any intrinsic problem in the theory.

Has our encounter with McTaggart yielded any positiveresults? There is, if not a moral, at least an intriguing sug-gestion of the possibility of a moral to our story. It is now acommonplace that there is a deep problem of time that ariseswhen one tries to quantize the GTR, and that this implies afundamental incompatibility between the GTR and quantumtheory. McTaggart’s puzzling claim was that he had found aproblem of time in the purely classical theory. We haveshown McTaggart’s worries to be unfounded—but are leftwith the intriguing possibility that the “problem of time” inquantum gravity is equally chimerical. If one casts the GTR

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in Hamiltonian form—as is commonly done—in order toquantize it, then the interpretative problems attending thatform of the classical theory will likely arise again in thequantum version. And if tying "observables" to particularpoints in the bare manifold makes trouble in the classicalversion, then we should anticipate difficulties in definingobservables in the quantum version. So McTaggart mayhave done us a favor after all, by redirecting our attentionaway from quantization back to the original theory as asource of various technical problems that might arise.

But however things work out for quantum gravity, thereis, after all, no problem of time or change in the GTR. Let'sreturn McTaggart to his final resting place, and let himmolder there in peace.

Response by John EarmanI am grateful to Tim for posing his disagreements with me ina form that more than matches my attempt to state the issuesin a provocative way. Tim does a brilliant job of explainingthe guts of some difficult technical issues. He takes his ex-planation to show that the sorts of considerations I adducedin favor of modern McTaggartism lead to a precipice belowwhich lies absurdity. I see no precipice but rather a series ofsteps that lead to an understanding of the motivation andcontent of contemporary main-line research in the founda-tions of classical general relativity theory (GTR) and quan-tum gravity. This research may lead only to a dead end, butthere is no a priori way to know this, much less that the re-search is based on absurd ideas. In what follows I will con-fine my comments to four points that lie at the heart of ourdisagreements.

(1) At the outset I want to emphasize a point generallyaccepted in the physics community but largely unappreci-ated in the philosophy of science community: There is a uni-form method for getting a fix on gauge that applies to anytheory in mathematical physics whose equations of mo-tion/field equations are derivable from an action principle. Iemphasize that any particle theories and field theories,Newtonian theories and relativistic theories, etc., all fall-within the scope of the method.6

The first step in employing the method is to convert fromthe Lagrangian form of the theory to the Hamiltonian form.If non-trivial gauge freedom is involved in the theory, it re-veals itself in the existence of Hamiltonian constraints. Onethen proceeds to identify the first class constraints and, fol-lowing Dirac, these constraints are taken to generate thegauge transformation on the Hamiltonian phase space. Fi-nally, the genuine physical magnitudes or "observables" areidentified as the gauge invariant quantities.

Now apply this method to some theory of physics. Sup-pose that the result offends your (or Tim's) intuitions. Thismight indicate that the method, despite its universally ac-knowledged success in providing a precise and systematicexplication of the gauge concept across a vast range of cases,breaks down in the case at hand. But in the absence of anycompeting method for getting a fix on gauge—and I haven'theard a definite competing proposal—you (or Tim) shouldseriously consider the possibility that your intuitions have tobe retrained.

Tim doesn't want a competing method, but he does wantto be able to cherry-pick the results of the constraint formal-ism. The motivation behind the technical apparatus is to

6Here is one explicit expression of faith in the generality of the method: "Itis well known that all the theories containing gauge transformations are de-scribed by constrained systems" (Gomis, Henneaux, and Pons 1990, p. 1089).

John Earman is University Professor of the History and Philoso-

phy of Science at the University of Pittsburgh

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detect when an apparent violation of determinism is merelya faux violation. Tim's intuitions tell him that someviolations of determinism are tolerable, and in these cases hesees no need to save determinism by appeal to gauge free-dom. But all the potential violations of determinism coveredby the constraint apparatus are of a piece; namely, arbitraryfunctions of the independent variables show up in solutionsto the equations of motion. Since time is normally the or oneof the independent variables, what this means is not just thatsolutions to the equations of motion can agree on initial datawhile disagreeing at a later time but also that given any al-lowed initial value of a dependent variable, there is a solu-tion of the equations of motion which has the prescribed ini-tial value of the dependent variable but which gives to thedependent variable any value you like at any future timeyou choose. If this isn't a violation of determinism, it is hardto know what one would be. And it is hard to see any prin-cipled way to distinguished tolerable vs. intolerable viola-tions of this kind. It is sometimes said that the kind of inde-terminism that threatens GTR is uninteresting because it isunobservable. But this response buys into the gauge inter-pretation of the theory: the observables of the theory, in theguise of gauge independent quantities, do evolve determi-nistically. Furthermore, this response commits one to anontology and ideology that is quite different from the extantproposals in the philosophical literature. More on this in (3)and (4) below.

(2) In a sense there is an alternative to the Dirac con-strained Hamiltonian formalism, but what this alternativegives isn't so much a rival account of gauge as a differentnomenclature. The alternative works on the Lagrangianformulation of the theory, and it sees gauge freedom at workwhen Noether's second theorem applies, that is, when theaction is invariant under an infinite dimensional Lie group

of transformations which depend on arbitrary functions ofthe independent variables. Noether's second theorem im-plies that the Euler-Lagrange equations of motion are un-derdetermined—i.e., there is an apparent breakdown of de-terminism. The underdetermination is overcome if the ele-ments of the invariance group are seen as gauge transforma-tions—as relating different descriptions of the same physicalsituation. One now has to face the issue of how these La-grangian gauge transformations—which act on the inde-pendent and dependent variables of the Lagrangian andwhich map solutions of the Euler-Lagrange equations ontosolutions—are related to the Dirac-Hamiltonian gauge trans-formations, which are point transformations on the Hamil-tonian phase space and which map solutions of the Hamil-ton-Dirac equations onto solutions. In some cases the rela-tion between the two concepts of gauge is transparent; inother cases, such as Einstein GTR, the relation is opaque andrequires special effort to discern.7 I will return to the alter-native Lagrangian approach below, but now I want to takeup Tim's question of why one would want to put GTR inHamiltonian form.

The reason that physicists use the Dirac formalism to geta fix on gauge is that they always have one eye cocked to-wards quantization and because the standard route to quan-tization goes through the Hamiltonian formulation of a the-ory. Trying to travel this route with respect to GTR in orderto produce a quantum theory of gravity is known as the ca-nonical quantization program. To my knowledge, all of theleading research workers in this program (with the one no-table exception of Karel Kuchar) accept the consequences ofapplying the Dirac formalism to GTR—in particular, theconsequence that the observables of GTR are "constants ofthe motion", a consequence that Tim labels as absurd and

7 See my (2002) for a discussion of this issue.

John Earman Response to Maudlin

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disastrous. The popular press would have you believe thatthe only viable approach to quantum gravity is via stringtheory or M-theory as it is now called. But the loop formu-lation of quantum gravity—which falls within the canonicalquantization program—is currently an active research pro-gram. In contrast to M(ystery)-theory, loop quantum gravityis a definite theory rather than a wannabe theory, and it is atheory with notable theoretical success in the form of an ex-planation of black hole entropy and the prediction of areaand volume quantization of space.8

I am not giving an argument from authority, although Ido think that philosophers are on dangerous ground whenthey are dismissive of the prevailing opinions of physicistson matter of interpretation. Rather, my point is that inter-pretations of scientific theories are subject to empirical tests,albeit of an indirect sort. If the loop formulation of quantumgravity continues to make theoretical progress and, eventu-ally, passes experimental checks, then I would take thesesuccesses to be confirmation of the gauge interpretation ofGTR dictated by the Dirac constraint formalism.

Before leaving the issue of quantization of gauge theoriesI want to emphasize that it is hard to see how the primacy ofthe Dirac account of gauge can be abandoned.9 As Timmentions, one approach to quantizing a gauge theory is thebrute-force method: impose a gauge condition to kill off thegauge freedom and then quantize in that gauge. What fixinga gauge means is explained in terms of the geometry of theDirac constraint surface (the subspace of the Hamiltonian

8For a review of the loop formulation of quantum gravity, see Rovelli(1998). This approach to quantum gravity takes advantage of a reformulation ofclassical GTR in terms of a new set of variables (due independently to AbhayAshtekar and Amitaba Sen) that makes the Hamiltonian constraints easier tohandle.

9The standard reference on quantization of gauge theories is Henneaux andTeitelboim (1992).

phase space where all of the Hamiltonian constraints aresatisfied); namely, the gauge condition must define a trans-versal in the constraint surface, i.e., a lower dimensional sur-face that intercepts each of the gauge orbits (as generated bythe first class constraints) exactly once. In some cases, suchas Yang-Mills theories, familiar gauge conditions fail to de-fine a transversal and, thus, the brute force attempt at quan-tization is defective. But what is more important is thatwhen a gauge condition does succeed in fixing a globaltransversal, what one is getting on the cheap, so to speak, isan isomorphic copy of the reduced phase space which re-sults when the Dirac gauge orbits are quotiented out. Whatthis strongly suggests is that the theoretically desirable tech-nique of quantization of a gauge theory would be to pass tothe reduced phase space (where the new phase variables aregauge invariant quantities) and then to perform normalquantization on the resulting unconstrained system. Un-fortunately, various technical obstructions can block the pas-sage to the reduced phase space, and even if the passage isnot blocked there remains the fact that the constraints maybe too difficult for physicists to solve. This is why Dirac in-vented a short-cut method referred to as constraint quanti-zation, which consists in promoting the first class constraintsto operators on a suitable Hilbert space and then identifyingthe physical sector of this space in terms of the state vectorsthat are annihilated by the operator constraints. But whetherone is performing reduced phase space quantization or Diracconstraint quantization, the philosophy is the same: onlyDirac observables (= quantities which are constant along theDirac gauge orbits or, equivalently, phase functions on thereduced phase space) get associated with quantum observ-ables in the form of self-adjoint operators.

(3) Leaving now the issues of quantization and eschew-ing the (3+1)-dimensional Hamiltonian approach in favor of


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the 4-dimensional Lagrangian approach, the nomenclaturechanges but the essential conclusions remain the same; inparticular, applying the considerations outlined at the be-ginning of (2) to GTR leads to the result that the Euler-Lagrange equations of GTR are underdetermined and, thus,that GTR is apparently an indeterministic theory. Tim isright that this is just Einstein's notorious "hole argument"dressed in a new guise. Tim feels that this argument shouldhave been put to rest long ago. I agree, but for different rea-sons. The reactions in the philosophical literature to the holeargument are amazing in terms of their ingenuity and theextravagances they employ, and generally they have beenskewed because philosophers want to seize the opportunityto ride a favorite hobby horse—a favorite account of identityacross possible worlds, a favorite account of how languageand reference work, a favorite account of essential proper-ties, etc. I ask them to pause for a moment and consider thefact there is an almost universally uniform reaction amongpracticing general relativists; namely, the lesson of the holeargument is (as the Lagrangian approach to gauge tells us)that the spacetime diffeomorphism group is a gauge groupof GTR.10

Some philosophers mouth these words but they gener-ally fail to work out the implications of their words, the mostimmediate of which is that the gauge invariant quantities ofthe theory must be diffeomorphic invariants. What are suchquantities? Generally philosophers don't have an answerbecause they haven't bothered to ask the question. Generalrelativists have asked, and the negative part of the answerthey find is that the gauge invariants of GTR do not include

10Again I would emphasize that the loop formulation of quantum gravity is aself-conscious attempt to accommodate the diffeomorphism invariance of classi-cal GTR as a gauge symmetry. String theorist, in conversation if not in print,say that they hope that M-theory will display this accommodation.

local field quantities (whether scalar, vector, or tensor). Thepositive part of their answer is that the gauge invariants in-clude (at least) two different sorts of quantities: first, highlynon-local quantities, such as volume integrals of local fieldquantities over all of spacetime; and, second, what I calledcoincidence events, which consist of a special kind of coinci-dence of values of two gauge-dependent quantities that gotogether to form a gauge independent one. Even apart fromthe issue of change and McTaggartism, this answer is inter-esting because the ontological picture that emerges from itlies outside the ambit of the normal discussion in the phi-losophy of space and time. Of course, the answer may bewrong. But it is remiss of philosophers to refuse to exploreits ramifications on the grounds that their philosophy tellthem it must be wrong.

(4) Coming now to the main issue of modern McTaggar-tism, I am unrepentant in agreeing with Carlo Rovelli that, atbase, what GTR describes is not evolution of the familiarkind, i.e., the change over time of observables in the sense ofgenuine physical magnitudes. Carlo tries to draw some ofthe sting of this position by saying that what the theory de-scribes is relative change, the change of "partial observables"with respect to one another. I am not fond of this way ofputting the matter since these partial observables are notgauge independent quantities and, thus, are not the kind ofthings whose values, or change of values, could be experi-mentally detected.

My alternative for drawing the sting of the no-evolutionview is threefold. First, I note that the form of McTaggar-tism that emerges from GTR does not support McTaggart'sultimate conclusion that time is unreal. Second, I point outthat the gauge interpretation of GTR is compatible with anattenuated kind of change, for it is compatible with takingthe history of the universe to be what I dub a D-series, i.e., a


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time ordered series of coincidence events with differentevents occupying different places in the series. This does notgo very far towards restoring normal change since the coin-cidence events do not consist of the occurrence of a changein a genuine physical quantity. Third, I recommend that wedon't try to restore normal change and evolution either bychanging the theory or by some clever interpretational ploythat rejects or bypasses the gauge interpretation of GTR.Rather, I recommend that if normal change and temporalevolution are wanted, we should seek them not in the intrin-sic physics of classical GTR or quantum gravity but in therepresentations of the gauge invariant content of solutions toEinstein's field equations in terms of the standard textbookmodels of fields evolving on manifolds. Such a representa-tional stance is nothing new; indeed, it should be familiarfrom the history of the debates over absolute vs relationalaccounts of space and time. For example, a good construalof Leibniz's relational account of space is to take him assaying that the Newtonians are welcome to talk about bodiesbeing contained in and moving through space as long assuch talk is taken not literally but rather as a way repre-senting the actual and possible relative configurations ofbodies. What I am suggesting is that a similar representa-tional account be applied to GTR, and that if it is done in theproper way we can have our cake and eat it too: ordinarytalk about change is accommodated in the representationswhile the gauge interpretation of the theory is respected byrecognizing that what these representations are representa-tions of is not of evolution in any ordinary sense.

In conclusion, my major disappointment with Tim's re-sponse is really a disappointment with my presentation. Forwhat his response reveals is that I failed to convey how theissues surrounding modern McTaggartism are not mereshuttlecocks to be batted back and forth in a philosopher's

game of badminton. These issues connect directly to deci-sions that physicists working on the frontiers of researchhave to make, for example, in searching for a way to marrygeneral relativity and quantum mechanics. If I could makeTim feel the excitement I experience when I see how phi-losophical concerns about time and change intertwine withcontemporary research in physics, he might be willing tojoin me on the precipice—a precipice not of absurdity but ofa new understanding of old issues.

References

Earman, J. 2002. "Getting a Fix on Gauge: An Ode to theConstrained Hamiltonian Formalism," to appear in K.Brading and E. Castellani (eds.), Symmetries in Physics.Cambridge: Cambridge University Press.

Gomis, J., M. Henneaux, and J. M. Pons (1990). "ExistenceTheorem for Gauge Symmetries in Hamiltonian Con-strained Systems', Classical and Quantum Gravity 7, 1089-1096.

Henneaux, M., and C. Teitelboim (1992). Quantization ofGauge Systems (Princeton, NJ: Princeton UniversityPress).

Rovelli, C. 1998. " Loop Quantum Gravity," Living Reviews inRelativity. http://www.livingreviews.org


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