Thoroughly Modern McTaggart - quod.lib.umich.edu

Thoroughly Modern McTaggart

OrWhat McTaggart Would Have Said If He

Had Read the General Theory of Relativity

John Earman

Philosophers’ Imprint<http://www.philosophersimprint.org/002003/>

Vol. 2 No. 3August 2002

(c) John Earman 2002

1 IntroductionThere are two traditions in the philosophy of time that, while op-posing one another, are locked in a mutual embrace. The embraceis cemented by two shared assumptions: first, that time presupposeschange; and, second, that genuine change requires Becoming. Bothtraditions have ancient roots. One, which takes its inspiration fromParmenides, denies the reality of change and time by rejecting Be-coming; the other, which can be traced to Aristotle, upholds the real-ity of change and time by claiming to find Becoming at work in theworld. What complicates an already complex discussion is that thereare at least two distinct senses of Becoming in play. One sense is ex-emplified in McTaggart’s (1908, 1927) infamousA-series, in whichevents are ordered as to past, present, and future. In capsule form,McTaggart’s argument for neo-Parmenideanism goes as follows:

(P1) There must be real change if there is to be time.(P2) There must be temporal passage (i.e. a continual change in

events of the non-relational properties of presentness, past-ness, and futurity) if there is to be real change.

(P3) Temporal passage is incoherent.(C) Therefore, time is unreal.

While the majority of philosophers agree with McTaggart’s (P3),there is a significant minority that finds his alleged demonstration ofthe incoherency of theA-series less than convincing.1

McTaggart’s brand of Becoming is property-based: that an eventbecomes present means for him that it loses the (non-relational)property of futurity and takes on the (non-relational) property ofnowness. A non-property-based form of Becoming was articulatedin modern form by C. D. Broad (1923) and has been championed

1See, for example, Savitt (2001a) and the exchange between Smith and Oak-lander, Essays 14-18, in Oaklander and Smith (1994).

John Earman Thoroughly Modern McTaggart

more recently by Michael Tooley (1997). Both Broad and Tooleysubscribe to a form of Aristotle’s doctrine that the future is unrealand/or does not exist and that events become real by coming intoexistence. If we follow convention and call a universe stripped ofitsA-series properties ablock universe, then what Broad and Tooleypresent us with can be called adynamicor growing block universethat continually adds new layers of existence. As Broad put it:

Nothing has happened to the present by becoming past except thatfresh slices of existence have been added to the total history of theworld. The past is thus as real as the present. On the other hand,the essence of a present event is, not that it precedes future events,but that there is quite literally nothing to which it has the relationof precedence. The sum total of existence is always increasing, andit is that which gives the time-series a sense as well as an order. Amomentt is later than a momentt′ if the sum total of existence att includes the sum total of existence att′ together with somethingmore. ... [W]hen an event becomes, it comes into existence; andit was not anything at all until it had become. ... Whatever is hasbecome, and the sum total of existence is continually augmented bybecoming. (1923, 66-69)

Although the text of Gödel’s (1949) essay “A Remark About the Re-lationship Between Relativity Theory and Idealistic Philosophy” isopen to various interpretations, a plausible reading sees Gödel as at-tempting to derive the ideality of time by coupling an acceptance ofBecoming in Broad’s sense as a necessary condition for real changewith the claim that Einstein’s special and general theories of relativ-ity are incompatible with this sense of Becoming.2

The issues surrounding change and Becoming are revisited overand over again in the philosophical literature, with each generationadding new layers of wisdom. Since I do not aim to contribute tothis literature, I will take a cavalier and callous attitude towards

2For various interpretations and evaluations of Gödel’s argument, see Earman(1995, Ch. 6), Yourgrau (1991, 1999), and Belot (2001).

both of the venerable traditions alluded to above: let them remainlocked in their mutual embrace of Becoming and sink from viewinto the metaphysical mire. Becoming, in either McTaggart’s senseor Broad’s sense, is part of the manifest image. The scientific imageknows nothing of either, and yet science does describe a rich androbust sense of change.3 Relinquishing theA-series and eschewingthe metaphor of the piling up of thin slices of existence leaves whathas been called thenon-dynamic block universein which events areordered only by the earlier-than relation (a.k.a. theB-series). Tobe sure, the non-dynamic block universe is itself unanimated; but(to quote Savitt (2001b)) to have a picture of animation, one doesn’thave to provide an animated picture. The animation that is picturedis B-series change–at different moments of time different proper-ties are instantiated, the instantiation of all of which at any singlemoment of time would be contradictory.

Needless to say, the adequacy of theB-series account of changeneeds to be defended against a number of objections, but the de-fense will not be mounted here.4 For present purposes I can as-sume that this account of change is adequate, for my main aim isto call to the attention of philosophers the fact that coupling thisassumption to one of the fundamental theories of modern physics–Einstein’s general theory of relativity (GTR)–revives McTaggart’sworries. For GTR–appropriately interpreted–seems to imply that, iftheB-series account of change is accepted, then there is no physicalchange since–under the appropriate interpretation–GTR implies thatno genuine physical magnitude takes on different values at differ-

3This cavalier attitude glosses over the problem of reconciling the manifestand scientific images; in particular, the problem of how science, if it eschews Be-coming, can give an adequate account of the phenomenology of experience whichdoes involve a transient ‘now’. See Shimony (1993).

4The most thoroughgoing defense of theB-series conception of change is tobe found in Mellor (1981, 1998).

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ent times. This implication naturally raises the question of whetherMcTaggart’s conclusion that time is unreal can be avoided in GTR.

The plan of the paper is as follows. In section 2 I discuss thelogic ofB-series change. The application of this logic to the actualuniverse, as described by textbook versions of GTR, seems to con-firm the common sense conclusion that there is change in the world.In section 3 we meet modern McTaggart who accepts theB-seriesaccount of change but who rejects the common sense conclusion onthe grounds that it rests on taking the surface structure of GTR tooliterally and thatB-series change disappears in the deep structureof the theory. Section 4 describes in detail the considerations thatappear to support modern McTaggart’s claim that in the deep struc-ture of GTR the dynamics is “frozen,” wherein all genuine physicalmagnitudes or “observables” are “constants of the motion.” Sincefamiliar physical quantities do not count as observables in GTR, onemust ask what the observables of the theory are and how they can beused to express the results of observation and measurement. Thesematters are taken up in section 5. Section 6 is devoted to some stocktaking. I indicate why GTR does not imply a flat-out no changeview: it is compatible with an ontology consisting of a time orderedseries of occurrences or events, with different occurrences or eventsoccupying different positions in the series. But GTR does not, Iclaim, sanction an interpretation of thisD-series (as I dub it) thatrestoresB-series or property change. Section 7 contains a digres-sion on the implications of this deep structure interpretation of GTRfor the issue of the status of spacetime points (a.k.a. the issue ofrelationism vs. substantivalism). Sections 8-11 review the pros andcons of various possible reactions to the frozen dynamics of GTR. Iexplain why we ought to take seriously the seemingly radical reac-tion that common senseB-series property change is not to be foundin physical events themselves but only in the mode in which we rep-resent these events to ourselves. At the same time I indicate how the

sting of this reaction can be drawn by showing both that it does notentail McTaggart’s conclusion of the unreality of time and that it iscompatible with preserving much of the common sense talk aboutchange, albeit in an altered form. My conclusions are presented insection 12. I emphasize especially that these issues are not merelyplaythings of academic philosophers since the stance taken on theminfluences the direction of current research in physics.

2 The logic and existence ofB-series changeOn theB-series conception of change, change and the Heracliteanrole of time go hand in hand: the different moments of time sep-arate what would otherwise be contradictories, transforming theminto the temporal alteration that constitutes real change. There aretwo ways to understand how time performs its Heraclitean function–the temporal stage view and the relational view.5 According to thefirst, if Jeremy changes from slim to portly, it is because of theconjunction of three facts: Jeremy is composed of temporal parts,Jeremy-at-t for variablet; Jeremy-at-t1 is slim; and Jeremy-at-t2 isportly, wheret1 < t2. It is crucial, of course, that the temporalstages are stages of the same continuant. But even with this provisoin place, some philosophers remain unsatisfied. Thus, Mellor oncecomplained that “different entities differing in their properties do notamount to change even when ... one is later than the other and bothare parts of something else” (1981, 111). I do not share this qualm,especially when the continuant is the entire universe, in which casethe temporal stages are time slices of the universe and change is a

5A third construal of the logic of change is given bypresentism, the view that theonly things that exist are those that exist now. I will not discuss this view here.For a recent assessment of presentism, the reader is referred to the symposium“The Prospects for Presentism in Spacetime Theories” in Howard (2000).

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change in the state of the universe. Indeed, I would claim that any-thing worthy of being called temporal change entails a change in thestate of the universe.

The competing relational view analyzes the change in Jeremy asconsisting of the facts that Jeremy has the relational properties slim-at-t1 and portly-at-t2, t1 < t2. This view also has its detractors,such as David Lewis who complains that shape is a property, not arelation (1986, 204). I share Lewis’ intuition but do not find it de-finitive. Of more importance to me is the fact that, according to themost fundamental theories of modern physics, the Jeremy and theother enduring physical objects presupposed by the relational viewmust be analyzed in terms of fields. And the most natural way tounderstand the logic of field talk–be it classical or quantum fields–isto take fields to be properties of spacetime points or regions.6 Thismelds naturally with the temporal stage view by construing shape asa property of spacetime regions lying on a time slice (plane of ab-solute simultaneity in the case of Newtonian spacetime or a space-like hypersurface in the case of relativistic spacetimes). Since thediscussion below focuses on GTR it will be helpful to say a bit moreaboutB-series change in the context of general relativistic space-times.

Following the model-theoretic view of theories, we can think ofGTR as a class of models of the formM, gab, Tab, whereM is afour-dimensional manifold,gab is a Lorentz signature metric definedon all ofM, andTab is a tensor field that describes the distribu-tion of matter-energy throughout spacetime. The dynamically pos-sible models are the ones that satisfy Einstein’s gravitational fieldequations (which are a set of non-linear partial differential equa-tions linking curvature properties of the spacetime metricgab to the

6However, as will be detailed below in sections 4-7, it is precisely this construalof the ontology and ideology of GTR that leads to a number of perplexing issuesabout the nature of time and observables.

gravitational source termTab) plus various energy conditions that putconstraints on the the stress-energy tensorTab.7 A general relativis-tic spacetimeM, gab is said to betemporally orientablejust in caseit admits a continuous non-vanishing timelike vector field. Choos-ing one of the two possible time orientations provides a globallyconsistent notion oftime directionality.8 With the help of this direc-tionality, a relation¿ of chronological precedencebetween pairsof pointsp, q ∈ M can be defined as follows:p ¿ q iff there isa future directed timelike curve fromp to q. The spacetime is saidto admit aglobal time orderjust in case¿, which is necessarilytransitive, is also irreflexive. Such a spacetime may or may not alsoadmit aglobal time function, i.e. a continuous mapt : M → Rsuch thatt(p) < t(q) wheneverp ¿ q. And even if a spacetimeadmits a global time function, it may not be possible to define atsuch that thet = const time slices areCauchy surfacesin that everytimelike curve without endpoint intersects each of these surfaces ex-actly once. In a spacetime lacking Cauchy surfaces there is no safelaunching pad for the global form of Laplacian determinism whichseeks to determine the entire future and past from appropriate initialdata on an initial time slice.

One of the challenges that GTR poses for philosophers of timeis that the dynamically possible models of GTR contain spacetimesthat lack some or all of the properties on the above wish list. In aspacetime that lacks, say, a globally consistent time order it is hard tosee how to consistently talk about change in even the relatively un-problematicB-series sense. Since the challenge toB-series change Iwant to consider here arises even if the spacetime hasall of the prop-erties on the wish list, no harm is done by restricting attention to such

7For example, the so-called weak energy condition forbids negative energy den-sities. The interested reader may consult Wald (1984) for details.

8How the choice is made is part of the problem of the direction of time, which isis not tackled here.

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nice spacetimes. And since the restriction simplifies the discussionby screening out irrelevant issues, I will impose it in what follows.So consider a general relativistic spacetimeM, gab, a global timefunction t for this spacetime, and a smooth non-vanishing timelikevector fieldV a. Using the integral curves ofV a to identify throughtime the spatial locations on thet = const. slices, we can say thatrelative tot andV a the modelM, gab, Tab does not exhibit any met-ric change if for any two slicest = t1 andt = t2 the values of themetric at the corresponding points of these slices are the same. Ifthe model is dynamically possible and it does not exhibit any metricchange with respect tot andV a then it will not exhibit any changein the matter-energy distribution since the Einstein field equationsimply that Tab inherits symmetries ofgab. This way of approach-ingB-series change raises a worry about relativity of change to thechoice oft andV a. But for present purposes the worries can bebanished with the observation that for general relativistic cosmolog-ical models that stand a fighting chance of representing the actualcosmos, there is no choice oft andV a relative to which there isno metrical and matter-energy change.9 Thus, in the actual cosmosthere isB-series change.

And that’s all there is to it. Philosophers can throw up examplesof possible worlds in which there is a choicet andV a relative towhich there is no change or, even worse, in which there is no globaltime function or not even a global time order. Such brandishings ofpossible worlds can do nothing to undercut the reality ofB-serieschange in the actual world. Other philosophers who are stuck in the

9Technically the point is that realistic cosmological models do not admit evenlocally (i.e. in a finite neighborhood) a timelike vector fieldV a satisfying theKilling equation∇(bV a) = 0, where∇b is covariant differentiation and the roundbrackets denote symmetrization. This is the necessary and sufficient conditionfor there to exist a (local) coordinate systemxµ, µ = 1, 2, 3, 4, with x4 = t,such that∂gµν/∂t = 0.

manifest image can complain that theB-series change which hasbeen exhibited in the actual universe does not deserve to be calledreal change because it lacks Becoming. Those of us who have man-aged to extricate ourselves from the manifest image can only listenwith bemusement.

3 Meet modern McTaggartFor the moment, all seems well. After our labors in defense of com-mon sense we can rest and drift off into a peaceful sleep, dreamingaboutB-series change. Unfortunately this dream is shattered by theloud protests of modern McTaggart10:

Grandpop McTaggart (rest his soul) was unfortunately ahead of histime. If he had waited ten years to write his infamous article onthe unreality of time and had learned Einstein’s GTR, he could havebased his argument on the results of modern science rather than onmetaphysical flim flam. I don’t know whether Grandpop was rightabout theA-series being inconsistent. But it is certainly true thatmodern science gets along without Becoming, conceived either interms of property acquisition (as the fans of theA-series would haveit) or not (as Broad would have it). Thus, there is reason to thinkthat Becoming is not feature of physical events or processes in them-selves. I will therefore reformulate Grandpop’s argument so as toavoid this dubious metaphysics.

(P1′) There must be physical change if there is to be physical time.

What I mean by ‘physical change’ isB-series change in physicalmagnitudes; namely,

(P2′) Physical change occurs only if some genuine physical mag-nitude (a.k.a.“observable”) takes on different values at dif-

10A rather uncouth fellow, having none of the polish and charm of his grand-father.

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ferent times.

This quantitative notion of physical change incorporates qualitativeB-series property change as a special case since for any qualitativeproperty one can define a corresponding magnitude which, at anymoment, takes on just two values 0 (“present”) and 1 (“absent”).

So far I have just been using common sense, and common sensecertainly seems to tell us that there is physical change. (Just look atthe world around you where objects are changing in their positions,shapes, etc.) And as the author of a recent learned treatise has noted,GTR as applied to the actual cosmos supplies a precise version ofcommon sense change. But the author of that treatise based his con-clusions on the surface structure of GTR. When the deep structure ofthe theory is taken into account we see that our first impressions arebadly wrong. For what the deep structure of GTR tells us is that

(P3′) No genuine physical magnitude countenanced in GTR changesover time.

From (P2′) and (P3′) we get

(C′) If the set of physical magnitudes countenanced by GTR iscomplete, then there is no physical change.

And from (P1′) and (C′) we get

(C′′) Physical time as described by GTR is unreal.

If you can hear me Grandpop, I hope you are proud to see thatthe family is carrying on the tradition! It is a commonplace that inleaving the manifest image for the scientific image we have to leavebehind the hankerings of dynamic time theorists for an animated pic-ture of time in the form of a growing block universe or the like. Butwhat I have shown is that we also have to leave behind the famil-iar non-dynamic block universe, which allows for the change overtime of physical magnitudes, and are forced to the very non-dynamicblock universe, which not only doesn’t provide an animated picture

of change but doesn’t provide a picture of any animation at all.

Whether modern McTaggart’s argument proves to be any moreconvincing than his grandfather’s remains to be seen. The main taskundertaken in the next section is to assess the crucial premise (P3′).

4 The frozen dynamics of GTRThe issues I am going to describe in this section are typically dis-cussed under the label of “the problem of time in quantum gravity”(see, for example, Isham (1992) and Kuchar (1992, 1993)). Thelabel is at once apt and misleading. It is apt because the problemwas brought into prominence by the pursuit of the canonical quan-tization program which aims to marry GTR and quantum theory byquantizing the metric field. But the label is also misleading in twoways. First, the roots of the problem lie inclassicalGTR, and evenif it was decided that it is a mistake to quantize GTR, there wouldremain the problem of reconciling the frozen dynamics of GTR withtheB-series notion of change that is supported not only by commonsense but by every physical theory prior to GTR. Second, althoughthe aspect of the problem that grabs attention is that of time andchange, no solution will be forthcoming without tackling the moregeneral issue of what an “observable” of classical GTR is. Somephysicists who work in this area are apt to respond with a “Yes.BUT ...” They will point out that the problem of time and changeonly rises to the level of crisis when quantization is attempted andthat it is precisely here that the temporal aspect comes to the foresince apparently the frozen dynamics of GTR has to be unfrozen inorder to make quantization possible. I will dispute this wisdom byclaiming that one obtains a key to the problem by getting a handleon what an observable is in classical GTR and that with this key inhand it may not be necessary to unfreeze in order to quantize. But I

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am getting ahead of myself. First I have to explain how GTR drivesus out of the ordinary non-dynamic block universe, which is devoidof Becoming but filled with ordinaryB-series change, and into thevery non-dynamic block universe, which lacksB-series change.

Because the argument given below is both technical and compli-cated, it is important to keep a firm grip on the key point, whichis quite simple to state even if the details require an elaborate ex-planation. The point is that in a theory with gauge freedom, whatis “real” or “objective” is what remains after the gauge freedom isremoved. So, for example, in classical relativistic electromagnetictheory, the specification of the electromagnetic potentials containsa large amount of gauge freedom; what remains after that freedomis removed are the electric and magnetic fields or, more property,the electromagnetic field tensor–it is this object, and not the valuesof the scalar and vector potentials, that characterizes what is “real”about the electromagnetic state of the world. This much is familiar tomost philosophers with even a nodding acquaintance with physics.What is less familiar is that there is a class of gauge theories wherethe very dynamics is implemented by a gauge transformation. Whatsuch a theory describes when the gauge freedom in such theories iskilled is a world withoutB-series change. Einstein’s GTR turns outto be just such a theory. Now to the details.

The argument in section 2 that classical GTR lends itself to thecommon sense account of physical change was based on a naivelyrealistic reading of the surface structure of the theory as gleanedfrom textbook presentations–tensor, vector, and scalar fields on man-ifolds. But this naive reading must be radically modified if GTR is tocount as a deterministic theory, and the modification undercuts thecommon sense picture of change by freezing the dynamics. Prepara-tory to explaining this startling conclusion, I stipulate that, in orderthat determinism have the best chance of being true in GTR, atten-tion is to be restricted to general relativistic spacetimesM, gab that

areglobally hyperbolic,11 which is another way of restating the as-sumption already made above in section 2 that the spacetime can befoliated by Cauchy surfaces. One can then hope that appropriate ini-tial data on a Cauchy surface fixes, via Einstein’s gravitational fieldequations, unique future and past developments.

A quick insight into why there is aprima facieconflict betweenthe surface structure of GTR and Laplacian determinism can begained by deploying the Lagrangian formulation of particle and fieldtheories used so extensively in modern physics. Consider then the-ories whose equations of motion or field equations follow from anaction principle and, thus, are in the form of (generalized) Euler-Lagrange equations. If the action admits as variational symmetriesthe elements of a Lie group of transformations that depend on ar-bitrary functions of all the independent variables, then Noether’ssecond theorem implies that this is a case of underdetermination–a unique solution of the Euler-Lagrange equations is not determinedby initial data–because the Euler-Lagrange equations must satisfya set of mathematical identities and consequently are not indepen-dent.12 GTR is a case in point since the relevant action for Einstein’sgravitational field equations admits the spacetime diffeomorphismgroup as a variational symmetry.13 Hence, taken at face value, GTRis indeterministic.

A way to recoup the fortunes of determinism is to switch fromthe Lagrangian to the Hamiltonian formalism. A case of underdeter-mination in the Lagrangian formalism corresponds to the existence

11For a definition of this concept, see Wald (1984).12I am glossing over various technical conditions needed for the application of

the Noether theorems. See my (2000, 2001b) for an account of the applicationof Noether’s theorems to the issues under discussion.13A diffeomorphismd :M→M is a one-one mapping of the spacetime mani-

fold M onto itself that preserves the differentiable structure ofM, e.g. ifMis aC∞ manifold, thend must beC∞.

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of constraints in the Hamiltonian formalism: when one Legendretransforms from the Lagrangian velocity phase space(q, q) (whereq stands for the generalized configuration variables andq stands forthe generalized velocity variables) to the Hamiltonian phase spaceΓ(q, p) (wherep stands for the generalized momentum variables),one finds that theps are not independent but must satisfy a set ofidentities called theprimary constraints, which follow from the de-finitions of theps. Other constraints, calledsecondary constraints,may emerge when it is demanded that the primary constraints bepreserved by the motion. For present purposes another way of di-chotomizing constraints is more important: thefirst class constraintsare those that commute (i.e. have weakly vanishing Poisson bracket)with all the other constraints, while thesecond class constraintsfailthis test. P. A. M. Dirac, who was responsible for developing theconstrained Hamiltonian formalism, proposed that thegauge trans-formationsbe identified as the transformations generated by the firstclass constraints, where the intended interpretation is that two pointsof phase spaceΓ which are connected by a gauge transformation areto be regarded as representing the same physical state (see, for ex-ample, Dirac (1964), Henneau and Teitelboim (1992)). The gaugeinvariant quantities, which are referred to asobservablesin the lit-erature on constrained Hamiltonian systems, can be defined in var-ious equivalent ways. LetC ⊂ Γ denote the constraint surface, i.e.the subspace ofΓ where all the constraints hold. An observable isthen defined as a functionF : Γ → R which is constant along thegauge orbits onC or, equivalently, which commutes weakly with allthe first class constraints. Alternatively, if the reduced phase spaceΓ(q, p) is formed by taking the quotient ofC by the gauge orbits,then the observables can be defined as functionsF : Γ → R.14 One

14The mathematically precise way to formulate Hamiltonian mechanics uses asymplectic formΩ, that is, a non-degenerate two form on the phase spaceΓ. ThePoisson bracket for phase functions is defined byf, g := Ω(df, dg). Locally,

payoff of cranking the Dirac algorithm is a dissolution of the threatto determinism since the observables do evolve deterministically. InGTR, however, deterministic dynamics is regained only by freezingthe dynamics!

In GTR the configuration variables (theqs) are Riemann metricshab on a three manifold, which is to be imbedded as a Cauchy sur-face in spacetime, and the momentum variablesπab (theps) are de-fined in terms of the extrinsic curvature tensor, which specifies howthe three-manifold is to be embedded in spacetime. When GTR isrun through the Dirac constraint formalism it is found that there aretwo families of first class constraints, the momentum constraints andthe Hamiltonian constraints.15 Since the latter constraints generatethe motion, all of the observables of GTR, defined as gauge invari-ant quantities, must be constants of the motion. Thus, a “very non-dynamic block universe” is an appropriate appellation for a generalrelativistic world when viewed through the lens of the Dirac con-straint formalism.

It will be helpful to illustrate the above concepts in terms of a toyexample to which I will revert several times in what follows. Startwith the standard treatment of a simple one-dimensional harmonicoscillator. The usual phase space isΓ = Γ(x, px), wherex is theposition of the oscillator andpx is the momentum conjugate tox.The Hamiltonian isH = p2

x

2+ ω2x2

2. There are no constraints and

coordinates(qi, pi), i = 1, 2, ..., N , can be chosen for the2N -dimensionalΓso thatf, g =

∑i(∂f∂qi

∂g∂pi− ∂f

∂pi

∂g∂qi ). In this formalism Hamilton’s equations

can be written asqi = qi,H, pi = −pi,H, whereH is the Hamiltonian.A constraintχ(q, p) is first classjust in case for any other constraintκ(q, p),χ,κ ≈ 0, where “≈ 0” means “vanishes weakly,” i.e. vanishes on the constraintsurfaceC ⊂ Γ. For a detailed treatment of the constrained Hamiltonian formalism,see Henneau and Teitelboim (1992).15“Families” because there is one momentum constraint and one Hamiltonian

constraint for every point of space.

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no gauge freedom, and, of course, both the position and momentumof the oscillator are observables in the sense of Dirac. The chang-ing values of these observables constitutes common sense physicalchange. Now “parametrize” the system by adding the timet as anadditional configuration variable, which necessitates adding the con-jugate momentum variablept. The result is an augmented phasespaceΓ(x, px, t, pt). There is now one constraint, the vanishing ofthe super-HamiltonianH = H + pt. As in GTR the dynamics of theparametrized oscillator is pure gauge so all gauge invariant quan-tities are constants of the motion. Neither the position nor the mo-mentum of the parametrized oscillator are observables. The “frozen”observables can be explicitly constructed as functionsF : Γ → Rwhere the reduced phase spaceΓ consists of points(A,B) ⊂ R xR with A andB the coefficients that appear in an arbitrary solutionx(t) = A cos(ωt) + B sin(ωt) of the equations of motion of theunparametrized oscillator. Notice that within the parametrized treat-ment one can meaningfully say, for example, that eitherA > 0 orB > 0, which in vulgar parlance means that the oscillator is oscillat-ing and doesn’t remain at rest. So it seems meaningful to talk aboutthe oscillator changing. But strictly speaking, if one stays withinthe parametrized formalism, such vulgar talk cannot be cashed intoofficial talk about change in the usual sense of changing values ofobservables, for there aren’t any parametrized-oscillator observableswhose values are different at different times.

Although this toy example serves as a quick and easy illustrationof some complicated concepts, it is potentially misleading. In partic-ular, the unwary reader may be misled into thinking that since in thisexample it is possible to reintroduce change by “deparameterizing”in an obvious way, the same will hold true in non-toy examples. Aswill be seen below in section 8, however, this impression is beliedby the case of GTR. Second, the toy example may give the mis-impression that the counterintuitive results are merely formal tricks

or artifacts of the constrained Hamiltonian formalism. To correctthis impression I will sketch other ways of arriving at similar, if notidentical, conclusions in the spacetime setting rather than the(3+1)Hamiltonian formulation.

The details of the initial value problem in GTR are rather involved(see, for example, Wald (1984)), but for present purposes it is unnec-essary to review those details in order to see why there is an appar-ent failure not only of Laplacian determinism in GTR but even of theweakened version that says that the entire past history of the universedetermines the future. I will illustrate the point by means of a ver-sion of Einstein’s “hole argument.”16 LetM, gab, Tab be any solutionto Einstein’s gravitational field equations, and letd : M → M beany diffeomorphism ofM onto itself. ThenM, d∗g, d∗Tab, whered∗O denotes the object field obtained by “dragging along”O by d,is also a solution. One can choosed to be the identity map on andto the past of some Cauchy surfaceΣo ofM, gab but non-identity tothe future ofΣo. The result is two solutions that agree on the valuesof gab andTab for all p ∈M on or to the past ofΣo but differ on val-ues ofgab andTab for someq ∈ M in the future ofΣo–an apparentviolation of determinism.

To overcome this apparent violation, Bergmann (1961) proposedthat the doctrine of determinism be restricted to quantities–whichhe also dubbed “observables”–whose values are unequivocally pre-dictable from initial data. This proposal for restoring determinismto GTR might seem to have all of the virtues of theft over honesttoil. But despite the apparent circularity involved in Bergmann’sproposal, the doctrine of Laplacian determinism still has content.Notice first that, by the argument just given, GTR will be deter-ministic in Bergmann’s sense only if observables are restricted to

16See Norton (1987) for an account of Einstein’s struggles with the hole ar-gument; and see also section 7 below.

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diffeomorphically invariant quantities. Second, the converse or ‘if’part follows from the existence and uniqueness proofs for the initialvalue problem for GTR, which show that for appropriate initial dataassociated with a three manifoldΣo, there is a uniqueupto diffeo-morphismmaximal development for whichΣo is a Cauchy surface.

Readers who have already learned GTR will not be surprised bythese moves since variants of them are quite common in the litera-ture. What may not be familiar to most readers is that Bergmann’sproposal implies that there is no physical change, i.e. no change inhis observable quantities, at least not for those quantities that areconstructible in the most straightforward way from the materials athand. To simplify the argument, concentrate on the special case ofvacuum solutions (Tab ≡ 0) to the Einstein field equations; the ar-gument easily generalizes to non-vacuum solutions. Consider firstwhat can be calledlocal field quantitieswhich are constructed fromthe metric and its derivatives upto some finite order and which areevaluated at spacetime point, e.g. the Ricci curvature scalarR. Isthe value of this quantity at some point to the future of an initialvalue hypersurface predictable from initial data on the hypersurface,or even from data on to the the entire past of the hypersurface? Tobe predictable, the value has to be the same in the class of solu-tionsM, gab to the source free Einstein field equations, any twoof which can be related by a diffeomorphism that leaves the initialvalue hypersurfaceΣo ⊂ M and its past fixed. Letp ∈ M be thepoint in the future where we want to predict the valueR(p). Butsince for any otherp′ ∈ M to the future ofΣo a diffeomorphismd can be chosen so as to leaveΣo and its past fixed while map-ping p to p′, predictability leads to the resultR(p) = R(p′); ford∗R(p) := R(d(p)), and the sameness of values ofR in the twosolutionsM, gab andM, d∗gab requires thatR(p) = d∗R(p). Fur-ther, sinceΣo can be chosen wherever one likes,Rmust be constanteverywhere, and a fortiori there is no temporal change to be found in

this quantity, if predictability is to be preserved. Next consider whatcan be calledquasi-local field quantitiesobtained by integrating alocal field quantity over a spacetime region, e.g. the three-volumeintegral, over a sliceΣ, of the Ricci curvature scalarR(3) of thethree-metric induced onΣ by the spacetime metric. Is the value ofthis quantity predictable from initial data onΣo or even from datafrom the entire past ofΣo? Again the answer is negative unless thevalue is the same for all Cauchy surfaces to the future ofΣo; forgiven any solutionM, gab with initial Cauchy surfaceΣo and anypair of Cauchy surfacesΣ andΣ′ to the futureΣo, a diffeomorphismd can be chosen to mapΣ to Σ′ in such a way that inM, d∗gab thevalue of the quasi-local quantity forΣ is the same as the value inM, gab for Σ′. And again sinceΣo can be chosen anywhere onelikes, the value of the quasi-local quantity must be the same for allCauchy slices, and thus there is no temporal change to be found inthis quantity. An analogous conclusion holds for a quasi-local fieldquantity constructed by taking the spacetime volume integral of alocal field quantity over an open regionO ⊂M which has compactclosure and which lies to the future ofΣo. A non-local field quantityobtained by integrating a local field quantity over the entire futureof Σo is predictable; but then such a quantity is incapable of mark-ing any future change. And since again the cut between past andfuture is arbitrary, there is no temporal change to be squeezed fromnon-local field quantities.17

The argument so far has concerned only local and quasi-localfield quantities–that is, quantities that are attached to spacetimepoints and regions–leaving open what happens with quantities thatare not so attached. This issue will be taken up in sections 5 and6. But it is worth remarking in advance that it is not obvious how

17Yet another derivation of the “no change in GTR” conclusion is to be found inSmolin (2000); however, as discussed in section 8 below, Smolin rejects someof the premises used to derive the conclusion.

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these unattached quantities can underwriteB-series change; for suchchange requires a subject, and since spacetime points and regionsare the only obvious candidates for the subject role in GTR, thesepeculiar unattached quantities would seem to remove the subject ofchange from the picture.

Two different senses of the crucial notion of “observable” in GTRhave been put into play–that of Dirac (i.e. real valued functions ofphase space variables that are constant along the Dirac gauge or-bits) and that of Bergmann (diffeomorphically invariant spacetimequantities). Comparing the two is an apparently apples vs. orangesoperation since the former lives in a(3 + 1) phase space formula-tion of GTR while the latter lives in the four-dimensional spacetimesetting. Indeed, establishing a connection is apparently stymied bythe fact there seems to be no natural correspondence between thediffeomorphisms of the spacetime manifold and the Dirac gaugetransformations on phase space. For instance, one can’t even lookfor an isomorphism between the Lie algebra of spacetime diffeo-morphisms and the Poisson algebra of first class constraints sincethe latter isn’t a Lie algebra. However, Isham and Kuchar (1986a,1986b) have shown that if the embedding variables, that describehow a three-manifold is embedded as an initial value hypersurfaceof spacetime, along with their conjugate momentum variables, areadjoined to the phase space of GTR, then there is a natural homo-morphism of the Lie algebra of spacetime diffeomorphisms into thePoisson constraint algebra on the extended phase space. Their ac-count is too technical to discuss here, and for present purposes itwill have to suffice to indicate non-rigorously how spacetime dif-feomorphisms correspond to Dirac gauge transformations. The mo-mentum constraints generate gauge changes in a dynamical variableF (hab, πab) that correspond to the changes wrought by infinitesimaldiffeomorphisms in the initial value hypersurfaceΣo, and the Hamil-tonian constraints generate gauge changes inF (hab, πab) that corre-

spond (at least when the Einstein field equations are satisfied18) tothe changes wrought by infinitesimal diffeomorphisms moving or-thogonal to the initial value hypersurface; and so taken together thegauge transformations in Dirac’s sense may be thought of as gener-ating changes in dynamical variables that correspond to the changeswrought by arbitrary infinitesimal spacetime diffeomorphisms. Onewould expect that this correspondence will lead to a correspondencebetween the class of Dirac observables and the class of Bergmannobservables. But since no precise general account of the latter ex-ists, the subsequent discussion will have to remain sensitive to thedifference in the two senses of observables.

The confidence that the deep structure interpretation of GTR sket-ched above is not an artifact of the(3 + 1) phase space formulationimposed by the Dirac formalism is strengthened by the alternativeapproach of Ashtekar and Bombelli (1991), who show that Hamil-tonian mechanics for GTR does not require a(3+1)-cotangent bun-dle structure. Instead of taking the state space of GTR to be thespace of instantaneous states, they work with the spaceΓ of entirehistories or solutions to the Einstein field equations, which impliesthat dynamics is implemented not by a mapping from one state toanother state in the same solution but as a mapping from one solu-tion to another solution. The spaceΓ has a “presymplectic structure”given by a degenerate two-formΩ. There is no constraint surface, asin the(3 + 1) formulation; rather, the gauge directionsY are givendirectly by the null vectors ofΩ. It turns out that two solutions lie onthe same gauge orbit (i.e. integral curve of the gauge fieldY ) iff theyare diffeomorphically related.19 Thus, in the Ashtekar-Bombelli for-mulation of GTR the Dirac and Bergmann senses “observable” are

18The technical details here are rather tricky; the interested reader may consultUnruh and Wald (1989, 2599)19This conclusion has to be qualified since Ashtekar and Bombelli focus on as-

ymptotically flat spacetimes.

11


united since the gauge invariant quantities are identified as diffeo-morphically invariant functionals of the phase space variables (here,spacetime metrics). However, in what follows I will stick with themore familiar (3+1) phase space formulation since it is this formu-lation of GTR that appears in almost all of the discussions of theproblem of time in quantum gravity.

5 The observables of GTRThe argument sketched in section 4 to show that local and quasi-local field quantities do not provide a basis forB-series change inGTR could equally be construed contrapositively, that is, as showingthat such quantities are not observables since in generic solutions ofEinstein’s field equations these quantities do change with time. Inasymptotically flat solutions, a few such quasi-local quantities–suchas the total ADM mass of the universe–do qualify as observables inDirac’s sense, that is, quantities which are constructed, possibly byintegrating over a Cauchy slice, from the phase space variables andwhich are constant along the gauge orbits. However, in the class ofsource-free solutions to the Einstein field equations having space-times with compact space sections, there are provably no local orquasi-local Dirac observables (see Torre (1993)). Presumably thisno-go result could be turned into a no-go result for Bergmann ob-servables.

This is not a reason for total despair. For by transcendental deduc-tion, there must exist observables for GTR even if they are not to befound among the local or quasi-local field quantities; for, assumingthat the results of observations and measurements must be expressedby the values of observables (hereafter called the Assumption) therewould otherwise be no way to test the theory. For the Dirac senseof observable the Assumption can be justified by appeal to the no-

tion that gauge dependent quantities are not observable or measur-able, while for the Bergmann sense of observable the Assumptionis justified for tests that proceed by deterministic predictions. TheAssumption can, of course, be challenged on a variety of grounds.But it comes into play over and over again in the physics literature,and it will be accepted as a fixed point of the discussion below.

In fact, there are observables for GTR. They are sometimes re-ferred to as “relative” or “relational” quantities, but I find these termsmisleading because the quantities involved are often rather differentfrom the relational quantities familiar from relational accounts ofspace, time, and motion. I prefer the termcoincidence observablesbecause these quantities can be seen as involving a natural gener-alization of the notion of coincidence–the meeting of two “materialpoints”–that Einstein used in 1916 to respond to the threat of inde-terminism posed by the hole argument.20 Einstein wrote:

All our space-time verifications invariably amount to a determinationof space-time coincidences. If, for example, events consisted merelyin the motion of material points, then ultimately nothing would beobservable but the meeting of two or more of these points. Moreover,the results of our measurings are nothing but verifications of suchmeetings of material points of our measuring instruments with othermaterial points, coincidences between the hands of a clock and pointson the clock dial, and observed point-event happenings at the sameplace and the same time. (1916, 117)

(That this passage is part of Einstein’s response to the indeterminismproblem revealed by the hole argument is made clear in Howard(1999).) Although Einstein was on the right track, his notion of thecoincidence of material particles has to be generalized to includefields in order to do justice to the content of his GTR.

20Of course, there are also global observables such as the four-volume integralof the Ricci curvature scalar over all of spacetime, assuming that such integrals arefinite. But is hard to see how such quantities can be used to express the resultsof ordinary measurements.

12


I can illustrate the flavor of generalized coincidence observablesin terms of the toy example of the harmonic oscillator. To make thecase a bit more interesting, add a free particle whose positiony ismeasured from the origin of the oscillator. In the non-parametrizedtreatment,y(t) = Ct + D is a good observable which can serveas a clock. The equation of motion of the oscillator can be rewrit-ten using the clock variable asx(y) = A cos

(ω( y

C− D

C))

+ B sin(ω( y

C− D

C)). In the parametrized treatment of the oscillator-plus-

free particle, neitherx nor y is an observable in the Dirac sense.But since any functionF : Γ(A,B,C,D) → R on the reducedphase space is an observable, it follows that for any valuey of y,Xy := A cos

(ω( y

C− D

C))+B sin

(ω( y

C− D

C))

is a good observable.That is, for any valuey of the clock variable, the-position-of-the-oscillator-when-the-value-of-the-clock-variable-is-y is observable inthe Dirac sense; it is also an observable in the spirit of Bergmannsince its value is predictable from appropriate initial data.

Admittedly, however, it remains a bit obscure how the value ofthis coincidence observable is measured. For if the parametrized de-scription is taken seriously, the measuring procedure cannot workby verifying that the coincidence of values described in the equationfor Xy does in fact take place by separately measuring the valuesof the clock variable and the oscillator position and then checkingfor the coincidence. For the positions of the clock and the oscillatorare gauge dependent quantities, and by the Assumption the valuesof these quantities are not fixed by measurement. Rather the mea-surement procedure must be directly responsive to the coincidenceof values itself, even though the coincidence is not a coincidenceof the values of observable quantities. The mystery of how thisis accomplished is underscored by the fact that, because they areconstants of the motion, such coincidence observables in the Diracsense are in principle measurable at any time, e.g. the position-of-the-oscillator-when-the-clock-reads-midnight should be measurable

not just at midnight but at noon and 4:00 PM as well. A theory ofmeasurement for generalized coincidence observables is obviouslyneeded.

The application to GTR of the generalized notion of coincidenceobservable goes back at least as far as Kretchmann (1915, 1917).The basic idea was worked out in some detail four decades laterby Komar (1958). A generic solutionM, gab to Einstein’s vac-uum field equations will not possess non-trivial symmetries, andfor such solutions there will be four independent scalar fieldsφµ,µ = 1, 2, 3, 4, constructed from algebraic combinations of the com-ponents of the Riemann curvature tensor, such that the four-tuples(φ1(p), φ2(p), φ3(p), φ4(p)) and (φ1(p′), φ2(p′), φ3(p′), φ4(p′)) aredifferent wheneverp 6= p′ for any p, p′ ∈ M.21 Thus, the valuesof these fields can be used to coordinatize the spacetime manifold.While these scalars are, of course, not observables in the Bergmannsense, they can be used to support such observables in the followingway. If gαβ are the contravariant components of the metric tensor ina coordinate systemxν, the new components in theφµ systemare given bygµν(φλ) := ∂φµ

∂xα∂φυ

∂xβgαβ. One can speak of the event

of the metric-components-gµν-having-such-and-such-values-in-the-coordinate-system-φµ-at-the-location-where-the-φµ-take-on- val-ues-such-and-so.22 Call such an item aKomar event. That a given

21The construction will not work, for example, in the Friedmann-Walker-Robert-son models used in current cosmological theories because these models employspacetimes that are homogeneous and isotropic. So the attitude behind the con-struction has to be that such models are idealizations and the construction of ob-servables in GTR depends on the fact that the real world is sufficiently com-plex and non-symmetric. See Smolin (2000).22The reader might think that some sleight of hand is involved here because the

‘at’ used in specifying the event seems to be a spatiotemporal at, just as the ‘when’used in specifying the oscillator-clock observable above seems to be a temporalwhen. But as will become clear from the more detailed discussion in section9, no illicit spatiotemporal notions are being smuggled in; all that is going on

13


Komar event occurs (or fails to occur) is an observable matter inBergmann’s sense. But as in the oscillator case, how the occurrenceof a Komar event is to be observed/measured is an unresolved issue.The measurement procedure cannot work by measuring the metricfield, measuring the scalar fieldsφµ, then checking that the coinci-dence of values that constitutes the event does in fact take place, atleast not if the Assumption is granted. Again, the need for a mea-surement theory of coincidence observables is evident.

6 Taking stockWe can now see that modern McTaggart’s diatribe of section 3 wassomewhat hysterical. If we use modern McTaggart’svery non-dyna-mic block universeto denote what results from the gauge interpreta-tion of GTR, then the very non-dynamic block universe does not ac-cord with his conclusions (C′) and (C′′) announcing the unreality oftime and change. One can say in answer to modern McTaggart thatthere is change in the very non-dynamic universe, though not of theusualB-series kind. The occurrence or non-occurrence of a coinci-dence event is an observable matter–at least in the technical senses atissue here–and that one such event occurs earlier than another suchevent is also an observable matter–again in the technical senses atissue. Call this series of coincidence events theD-series (the term‘C-series’ having already been co-opted by McTaggart–see section9). Change now consists in the fact that different positions in theD-series are occupied by different coincidence events.

One can also say thatB-series property change does “exist,” al-

is that some of the quantities are being used to specify a point on a gauge orbit, andonce that specification is made it is easy to define observables in terms of thevalues that other quantities take at the specified point. This is an example ofwhat is called “gauge fixing.”

beit in a representational sense. Using the nomenclature of the previ-ous section, call the functional relationshipgµν(φλ) theKomar state.This state, which floats free of the points ofM, captures the in-trinsic, gauge-independent state of gravitational field. This intrinsicstate is represented or realized by a spacetime modelM, gab, and ifit is represented or realized by one such model it is equally well rep-resented or realized by any model in the diffeomorphic equivalenceclass of the first. It is in the representations that the familiar storyof subjects (spacetime points) and properties of subjects (fields at-tached to spacetime points) can be told, and the familiar story speaksof common senseB-series change. To use Leibnizian terminology,B-series change has been relegated to the status of an appearance,but it is not a mere appearance but a well-founded appearance basedon an objective structure of the physical world.

This story will not be to everyone’s taste. Indeed, the notion thattemporal change in genuine physical magnitudes is not to be foundin the world in itself but only in a representation will be viewed bymany as patently absurd.23 Those who share this view can be di-vided into two categories. The first category consists of those whohold that modern McTaggart’s claim that GTR entails the absence ofB-series change rests on some sort of illicit move or sleight of hand.For example, one way to express modern McTaggart’s claim goeslike this. In a constrained Hamiltonian system the intrinsic dynam-ics, as expressed in terms of the genuine observables, is obtained bypassing to the reduced phase space by quotienting out the gauge or-bits. When this is done for a theory in which motion is pure gauge,there is an “elimination of time” in that the dynamics on the reducedphase space is frozen. In response it is sometimes claimed that this

23For example, Karel Kuchar, one of the leading research workers in canonicalquantum gravity shares this attitude, and as a consequence he explicitly rejectsthe notion that a genuine physical magnitude of GTR must commute with theHamiltonian constraint; see Kuchar (1993).

14


“elimination of time” cannot be carried out in chaotic Hamiltoniansystems since there will not be enough constants of the motion todistinguish the gauge orbits (for in such a system only functions ofthe Hamiltonian are constants of the motion) and since these con-stants will not be smooth functions on the phase space (for in chaoticsystems the trajectory must pass through any open set of the phasespace) (see Smolin (2000)). This objection rests on an equivocationon “constants of the motion”: it could mean either an integral of themotion or a quantity that is constant along the gauge orbits, and thetwo senses are not equivalent. Hájicek (1996a) shows that regardlessof how many integrals of the motion are present in a (finite dimen-sional) Hamiltonian system, parameterizing the system results in aconstrained system for which there exists a set of continuous quanti-ties that are constant along the gauge orbits and which are completein that their values can be used to separate the gauge orbits. To besure, therecan bereal technical problems involved in the “elimina-tion of time” in the guise of constructing the reduced phase space byquotienting out the gauge orbits in a constrained Hamiltonian sys-tem. For instance, while the reduced phase space always exists as atopological space, it may not inherit a manifold structure. However,in the case of GTR the reduced phase space is not fatally ill-behavedsince it is the disjoint union of manifolds.

As a second example of the first category of naysayers to mod-ern McTaggart I would mention Kauffman and Smolin (1997) andSmolin (2000) who classify the problem of time and change in GTRas a pseudo-problem because it is posed in terms of mathematicalstructures that can only be reached by non-constructive methods.Since I think that such structures and methods are needed quite gen-erally in theoretical physics, I do not follow their lead.

In a somewhat different vein, it has been claimed that althoughthe problem of time in GTR is not a pseudo-problem, neither is itintractable since common senseB-series change can be described

in terms of the time independent correlations between gauge depen-dent quantities which change with time. Unruh (1991) has inveighedagainst this proposal:

The problem is that all of our observations must be expressed in termsof the physically measurable quantities of the theory, namely thosecombinations of the dynamic variables which are [gauge invariantand therefore] independent of time. One cannot try to phrase theproblem by saying that one measures the gauge dependent variables,and then looks for time independent correlations between them, sincethe gauge dependent variables are not measurable quantities withinthe context of the theory.24 (266)

I agree with this appraisal. (But I think that a more sympatheticreading of the proposal would lead to the “evolving constants” ideato be discussed below in section 9.) And I agree with Unruh when hegoes on to conclude that “The time independent quantities of generalrelativity alone are simply insufficient to describe time dependentrelations we wish to describe with the theory” (ibid.)–at least I agreeif ‘wish’ refers to our preanalytic desire to find ordinaryB-serieschange described in the theory.

The second category of naysayers to modern McTaggart consistsof those who seek not to find a flaw in his argument but rather toblock or blunt its thrust. One way to do this is to add some addi-tional structure to the theory that has the effect of unfreezing thedynamics by generating additional observables that can change withtime. Examples of this strategy will be considered in the section 8.

A wholly different reaction to modern McTaggart seeks not tofault or blunt his argument but live with its consequences. This reac-tion requires a radical attitude readjustment. It requires us to swal-low the notion that, just as the transient ‘now’ ofA-series and theaccreting layers of existence of Broad’s Becoming have been rele-

24This is an example of the Assumption announced in section 6 coming intoplay.

15


gated to the manifest image, so must ordinaryB-series change. Theproblems and prospects of successfully carrying out such a readjust-ment will be discussed in sections 9 and 10. But before turning tothese matters I need to take the reader on a brief digression.

7 Digression: reactions to Einstein’s holeargument

There is an extensive philosophical literature on Einstein’s holeargument, with different authors drawing quite different morals.25

While it would be inappropriate to review this literature here, it willserve a useful function to take up a couple of issues from this litera-ture that link directly to the issues under discussion here.

The first point that needs to be emphasized is that the existingphilosophical literature does not show an appreciation for the factthat the requirement of general covariance comes in two versions,weak and strong. The weak version requires that the laws of a the-ory be written in a form that is valid in all coordinate systems or,equivalently that the laws retain their form under an arbitrary coor-dinate transformation. The strong version requires that the space-time diffeomorphism group be a gauge group of the theory. Theweak version is so called because practically any theory can, withsuitable ingenuity, be massaged so as to fulfill this requirement; forexample, using the tensor calculus Newtonian gravitational theoryand special relativistic theories of motion can be written in gener-ally covariant form. But such massaging does not guarantee thatthe theory satisfies the strong requirement; indeed, as judged by thelight of the constrained Hamiltonian formalism the gauge freedom

25To get a sampling of the variety of reactions, see Maudlin (1990), Hofer andCartwright (1993), Rynasiewicz (1994).

of a weakly generally covariant theory may fall far short of diffeo-morphism invariance, and as a result the observables of the theorymay be much richer than the class of diffeomorphism invariants.26

Thus, it is simply not the case that the problem of change raised bymodern McTaggart in the context of GTR also rears its head in ear-lier theories of physics which generally did not admit the spacetimediffeomorphism group as a gauge symmetry.

The main preoccupation of the philosophical literature on Ein-stein’s hole argument has been on the implications for the status ofspacetime points. Although there is no agreement what these im-plications are, there is a growing realization that an adequate re-sponse to the hole argument is hard to square with either wing ofthe traditional opposition of substantivalism vs. relationism (see,for example, Saunders (2001)). I think that this realization is cor-rect but that it needs to be extended to include the realization thatthe gauge interpretation of diffeomorphism invariance not only un-dercuts the traditional substantivalism vs. relationism distinction butalso calls into question the traditional choices for conceiving the sub-ject vs. attribute distinction. The extremal choices traditionally onoffer consist of taking individuals to be nothing but bundles of prop-erties vs. taking individuals to have a ‘thisness’ (haecceitas) that isnot explained by their properties. The gauge interpretation of GTRdoesn’t provide any grounds forhaecceitasof spacetime points. Nordoes it fit well with taking spacetime points as bundles of propertiessince it denies that the properties that were supposed to make up thebundle are genuine properties. The middle way between thehaec-ceitasview and the bundles-of-properties view takes individuals andproperties to require each other, the slogan being that neither existsindependently of the states of affairs in which individuals instantiatevarious properties. Trying to apply the middle way to GTR runs into

26For details, see my (2001b, 2002a).

16


the same problem as bundles-of-properties view since the gauge in-terpretation implies that the states of affairs composed of spacetimepoints instantiating, say, metrical properties do not capture the lit-eral truth about physical reality; rather, these states of affairs withtheir subject-attribute structure are best seen as representations of areality–perhaps best characterized in terms of coincidence events–that itself does not have this structure.27

I freely admit that what I have just said may prove to be nonsensebecause there is no way to escape the ambit of the traditional posi-tions on subject-attribute relation. All I am sure about here is thatis worth exploring various escape routes and that modern McTag-gart’s argument provides a good occasion for testing rival views ofthe subject-predicate relation.

8 RestoringB-series changeIn section 6 it was suggested that the ontological picture that emergesfrom the deep structure of GTR is aD-series of time ordered coin-cidence events. The occupants of thisD-series can be treated asobjects of predication, and postulating properties of these objectsthat come and go as one moves through the series transforms theD-series into aB-series and restores property change. There is noway to provea priori that such a restoration ofB-series change isnot feasible, but I cannot see anything in or on the horizons of cur-rent physics motivates the idea that theD-series picture of physicalreality is correct as far as it goes but incomplete. If familiar property-

27Alternatively, one could define a “realized at” relation holding between coin-cidence events and spacetime points. Then spacetime points could be thought of asbundles of realization properties. But this seems to me to be just a less per-spicuous way of acknowledging that spacetime points arise as individuals in asubject-attribute representation of a reality from which they are absent.

based change is to be regained, more radical measures are called for.One way to block the thrust of modern McTaggart’s argument is

to break diffeomorphism invariance by adding coordinate conditionsthat privilege a restricted class of coordinate systems. Such a moveis, of course, contrary to the spirit of general relativity. However,diffeomorphism invariance can be restored by treating the privilegedcoordinates as additional scalar fieldsΦK , K = 1, 2, 3, 4, which areexpressed as functionsΦK(xi) of arbitrary label coordinatesxi. Ifone is sufficiently clever, it is often possible to find a way to “pa-rameterize” the action in terms of the new variables so that whatemerges are the old field equations plus the coordinate conditionsexpressed in terms of the new variables. For example, the unimod-ular coordinate condition

√−g = −1, whereg := det(gij), can betreated in this way. When the parameterized action is varied with re-spect to the metric the resulting gravitational field equations are theEinstein equations with an unspecified cosmological constantλ.28

When this unimodular gravitational theory is run through the con-strained Hamiltonian formalism, it is found that ifλ 6= 0 the Hamil-tonian constraints of standard GTR are suspended in favor of weakerconstraints which are compatible with the existence of observablesthat are not constants of the motion (see Kuchar (1991) and Earman(2002b)). Furthermore, unimodular gravitation provides a gauge in-variant time variableτ which, in the case where the spacetime man-ifold M is of the formΣ x R with Σ a compact three-manifold, ismeasured by the four-volume between a fiduciary sliceΦ4 = constandΦK(xi). However, a value ofτ does not correspond to a par-ticular spacelike hypersurface of spacetime, and for this and other

28That is, the “cosmological constant” is constant in that it has the same valuethroughout spacetime but its value can vary from solution to solution. In contrastto the unimodular version of GTR, the action principle for standard GTR assumesthat the cosmological constant is not a dynamical variable in that its value does notvary from solution to solution; see my (2002b).

17


reasons it does not provide the kind of time that is thought to beneeded to solve the problem of time in quantum gravity (see Kuchar(1991) and section 10). Nevertheless, unimodular gravity does serveas a useful example of howB-series change can be restored–albeitin a limited way–and it is given some currency by the fact that recentcosmological observations indicate that our universe is characterizedby a positive value for the cosmological constant (see my (2001a)).

A less sneaky and more direct route to the restoration ofB-serieschange seeks to identifythe, or a, physical time with respect towhich there is change. What this would mean can be illustratedin terms of “deparameterizing” the parametrized harmonic oscil-lator or more generally Hájicek’s (1995, 1996b)reparametrizationinvariant systems(RISs). A RIS captures in geometric terms the no-tion of a (finite dimensional) symplectic system with only first classconstraints and where motion is pure gauge in that each solutioncurve lies within a gauge orbit. A RIS can be thought of as a triple(Γ,Ω, C) where(Γ,Ω) is a (finite dimensional) symplectic systemandC ⊂ Γ corresponds to a first class constraint surface. The pro-jection ΩC of Ω onto C is a degenerate two-form with null vectorfield Y (henceforth called thegauge field) whose integral curvesγare the gauge orbits. This is a very broad class–it includes all fi-nite dimensional constrained Hamiltonian systems–but of course itdoes not include GTR, whose phase space is infinite dimensional.A vector fieldZ on the phase spaceΓ of a RIS(Γ,Ω, C) is called aphysical reference frameif it is of the formZ = ∂/∂T , where theΓsphys := x ∈ Γ : T (x) = s, s ∈ R, are are a family of globaltransversals for the gauge fieldY in that for eachs ∈ R every gaugeorbit γ intersectsΓsphys but is never tangent to it. The restrictionωsof Ω to Γsphys makesΓsphys into a symplectic space. It is requiredthat for anys ands′, the mapsθss′ : Γsphys → Γs

′phys, given by follow-

ing along the integral curves ofZ, are symplectic diffeomorphisms.

For eachγ and eachs let ηγ(s) be the unique point of intersectionof the gauge orbitγ with Γsphys. Drag backηγ(s) to an arbitraryfixed transversal surfaceΓophys by ξγ(s) := θ−1

os (ηγ(s)). The curveξγ : R → Γophys describes the motion relative to thephysical phasespace(Γophys, ωo). It turns out that the physical motion is of Hamil-tonian form.

The payoff of introducing a physical reference frame is that mo-tion in the physical phase spacecan involve common sense change.I say “can” because it could turn out that this motion is frozen in thatthe curveξγ is degenerate, i.e. is just a point inΓophys, but at leastthe possibility is open that motion in the physical phase space is notfrozen. Officially, an observable is a quantity that is constant alongthe gauge orbits or, equivalently, is a phase function on the reducedphase space obtained by quotientingC by the gauge orbits. But theproposal of the deparametrizer is to abandon the official account andrecognize “Z-observables” defined as functions onΓophys. These newobservables need not be constants of the physical motion. To returnto the case of the parametrized simple harmonic oscillator, takingZ to be∂/∂T , whereT := t (with t being “real time”), defines aphysical reference frame, and obviously theZ-observables for thisframe do change with time. More generally a way to construct aphysical reference frame is choose a functionf : Γ → R whoserestrictionfC to the constraint surfaceC is an observable in the offi-cial sense thatfC is constant along the gauge orbits (or equivalently,Y (fC ) = 0). Then, at least locally,Za := Ωab∂bf defines a phys-ical reference frame. In each case one would have to check to seewhether the choice off results in an unfrozen physical dynamics.

As noted by Hájicek (1995, 1996b) three problems arise in seek-ing real change by defining a physical reference frame. The first isexistence: a RIS may not admit any global transversal surfaces. Thesecond isuniqueness: more than one physical reference frame may

18


exist. The third isphysical motivation: what physical grounds jus-tify using a physical reference frame, assuming it exists, to definephysical time and change? I will return to these issues shortly.

Could it be that GTR is a kind of time-parametrized version ofsome more fundamental theory? The initial evidence does not en-courage this notion. In the case of the parametrized oscillator thereare two strong clues for how to deparametrize: first, the super-Hamil-tonian is linear in the momentumpt but quadratic in the other mo-menta, and second, the hypersurfacest = const foliate C and eachof them meets the gauge orbits exactly once. But the Hamiltonianconstraint in GTR is quadratic in all the momenta, and in generalthere is no way to foliate the constraint surface in GTR with hyper-surfaces that meet the gauge orbits exactly once. If there is a truetime hiding in the formalism of GTR, it is well hidden indeed.

Efforts to track down the elusive time in GTR can be labeled as ei-therinternalistor externalistaccording as they rely on factors purelyinternal to GTR or allow external factors to be brought in. The prob-lems that bedevil the internalist attempts have been well documentedin Isham (1992) and Kuchar (1992) and will not be rehearsed here.Instead I will illustrate the externalist approach by reference to CarloRovelli’s (1993) notion of physical time as dependent on a statisticalstate. It is widely accepted that thedirection of time is defined bya statistical state. But here we have the more radical proposal thatphysical time itselfis defined by a statistical state.29 In terms of RISsthe idea is that a stationary statistical state defines a physical refer-ence frame, relative to which there is physical change. There aretechnical problems here connected with the issues of existence anduniqueness mentioned above. What one can hope to get from statis-tical physics is a stationary statistical state in the sense of a function

29See also Connes and Rovelli (1994). Rovelli is now inclined towards theweaker interpretation, discussed below in section 9, on which the statistical statepicks out the “clock variable” for the “evolving constants.”

fC : C → R such thatY (fC) = 0. If one tries to use such a stateto define a reference frameZ by the equationΩC(Z, ·) = dfC(·), theuniqueness problem arises because ifZ satisfies this condition, thenso doesZ ′ = Z + const x Y , whereY is the gauge field. ExtendingfC off the constraint surface to a functionf on all of Γ would allowZ to be uniquely defined as above byΩ(Z, ·) = df(·). But thereare many such extensions and statistical physics seems to provideno reason for preferring one to another since it is concerned onlywith states corresponding to points onC, which represent the physi-cal states. Finally, there is no guarantee any of the physical referenceframes associated with a statistical state defines global transversals,in which case time is at best defined only locally.

The moral to be drawn from this brief discussion of attempts torestoreB-series change is not that the attempts cannot succeed. Themoral is rather that success can only be purchased at the price of newsubstantive physics or new interpretational moves or both. Whetherthe price is worth paying depends in part on what we saddle our-selves with by not paying the price. The next two sections are de-voted to exploring that issue.

9 Learning to live withoutB-series changeThis section and the next explore the consequences of taking se-riously the idea thatB-series change belongs not to the world initself–as described, say, by the reduced phase space of a constrainedHamiltonian system in which motion is pure gauge–but to the mathe-matical and perceptual representations of the world. The idea soundsradical if it is advertised by the slogan thatB-series change is unreal.But radical sound is muted if the fine print of the advertising con-tains the assurances thatB-series change is not a mere illusion andthat what is veridical in our ordinary claims about changing prop-

19


erties and values of physical magnitudes is encoded in the structureof the world in itself. Some of the sought after assurance can begained by studying what Carlo Rovelli (1990, 1991a, 1991b) callsevolvingconstants of the motion, a notion that corresponds to a para-metrized family of the coincidence type observables introduced byKretschmann and Komar (recall section 5). In terms of the con-strained Hamiltonian formalism the general construction of evolvingconstants for the finite dimensional case goes as follows.30 Denotethe original phase space byΓ(qi, pi), i = 1, 2, ..., N . Suppose thatthere areM constraints. Choose coordinates(qa(qi, pi), pa(q

i, pi)),a = 1, 2, ..., D,D = N −M , for the reduced phase spaceΓ (recallthat this is the quotient of the constraint surfaceC ⊂ Γ by the gaugeorbits). The(2N−M)-dimensional constraint surfaceC can then becoordinatized by(qa, pa, tm), for sometm(qi, pi), m = 1, 2, ...,M .The symboltm has been used because these variables, which serveas coordinates of the gauge orbits, are called the “internal time vari-ables” or “clock variables.” In the formal construction the choiceof such variables is somewhat arbitrary, and for some choices thereis no guarantee that the values of these variables will behave likethe reading of anything that one would ordinarily call a clock. Forsake of illustration, assume that the coordinatization(qa, pa, t

m) ofC can be achieved by choosing for thetm a subsetqk, pk consist-ing of M of the original phase space variables. For fixed valuesqk

and pk of these variables, theM equationsqk = qk andpk = pkdetermine, at least locally, a transversal, i.e. a hypersurface ofCthat intersects eachM -dimensional gauge orbit exactly once. Thenfor some other variable, sayq1, its valueQ1 at the intersection canbe writtenQ1

qk,pk= Q1

qk,pk(qa, pa). Each of these quantities is a

genuine Dirac observable; for each defines a function fromC to R

30Here I am following the presentation of Montesinos (2001) and Montesinos andRovelli (2001).

(for (qi, pi) ∈ C, evaluateqa(qi, pi) and pa(qi, pi) and then applyQ1qk,pk

to (qa(qi, pi), pa(qi, pi))) which, by construction is constant

along the gauge orbits). Thek-parameter family of such observ-ables, generated as the chosen clock variables range over their al-lowed values, constitutes an evolving constant of the motion. Forsome fixed intrinsic state(qao , pao), the sub-family that is generatedas the valuesqk, pk of the clock variables vary gives the relative evo-lution Q1

qk,pk:= Q1

qk,pk(qao , pao) of q1 against theM clock variables

qk, pk for the solution picked out by the intrinsic state(qao , pao).The toy example of the parametrized one-dimensional harmonic

oscillator serves as a useful illustration. HereN = 2–there is aretwo configuration variablesx andt–and the original phase space isΓ(x, px, t, pt). M = 1–the one constraint is the super-HamiltonianconstraintH = H + pt which generates the motion. The two-dimensional reduced phase spaceΓ can be coordinatized by(A,B)where the general solution of the unparametrized oscillator isx(t) =A cos(ωt) + B sin(ωt). And one can chooset as the clock variable,giving coordinates(A,B, t) for the three-dimensional constraint sur-faceC. Corresponding to the position variablex–which is not anobservable in the parametrized formulation–there is, for each valuet of t, the Dirac observableXt := A cos(ωt) + B sin(ωt). The one-parameter familyXt : t ∈ R is an evolving constant.

This toy example is somewhat misleading since there is an ob-vious and natural choice for the clock variable–t–which happens tohave the properties we ordinarily associate with time. But in thegeneral case ofM constraints there are many different choices ofMvariables to serve as the clock variables, and once this choice hasbeen made there areM choices of whichM − 1 clock variables tohold fixed while theMth is singled out as the temporal index. Thereis noa priori guarantee that the chosen temporal index will behavelike a good clock variable or bear any direct relationship to our ex-

20


perience of time. Rovelli’s statistical state proposal introduced insection 8 as a way to implement deparametrization can now be rein-terpreted as a way of picking out the variable that is to serve as agood clock variable for an evolving constant rather than as defininga deparametrization.

We can now see how unchanging observables code up informa-tion about what is ordinarily taken forB-series change. For instance,in the oscillator example, any true statement about the motion of theunparametrized oscillator for the solution determined by(Ao, Bo) ∈Γ has a counterpart in a true statement about the familyXt : t ∈ Rof evolving constants. This gives some initial confidence that clas-sical physics can be done without having to unfreeze the dynam-ics and to reintroduce common senseB-series change. Whether allthe information needed to do quantum mechanics and statistical me-chanics can be so encoded in the frozen dynamics will be discussedin the next section.

It remains to explain our perceptions ofB-series change. In thetoy example of the parametrized oscillator the explanation wouldhave to show how the difference(X t2− X t1) in the values of twoconstants of the motion for the solution picked out by the frozenstate(Ao, Bo) ∈ Γ is interpreted by critters such as ourselves asgenuine property change–the change in the position of the oscillatorfrom t = t1 to t = t2. One could go Kantian and simply pos-tulate a human faculty that does the job.31 A real explanation, asopposed to mere postulation, would show how the physics of theobjects and the psychology of critters such as ourselves combinein such a way that we perceive the world as filled withB-serieschange despite the fact that no genuine physical magnitude changesover time. The explanation does not take the form of showing howwe make egregious errors or mistakenly project an illusion onto the

31This was Kant’s theft-over-honest-toil strategy for “solving” tough problems.

world; rather, it takes the form of showing how our perceptual andcognitive apparati represent in a more or less faithful way a structurethat exists independently of us in the world. In this respect the situa-tion is more properly called neo-Hegelian than neo-Kantian,32 and itintersects, though only partially, with (Grandpop) McTaggart’s po-sition. Recall that McTaggart, inspired by his admired Hegel, tookthe world of physical events to be arranged in an intrinsic, observerindependentC-series. But according to McTaggart thisC-series isnon-temporal, and it is by projecting a transient now onto thisC-series that McTaggart thought that we create theB-series and theillusion of change.33

One obvious way to start the sought after explanation of our per-ception ofB-series change is to use the proper time along the worldlines of a suitable system of world lines of observers as the index ofthe family of evolving constants. It is less obvious how to completethe explanation. “World line” is not a primitive notion–my worldline, and yours, is defined in terms of the values of local fields onspacetime, and such values do not constitute genuine observablesin the official sense, though, of course, they may be ingredients ofcoincidence observables. Insofar as a physical explanation must be

32Kant wrote: “I deny to time all claim to absolute reality; that is to say, I denythat it belongs to things absolutely, as their condition or property, independently ofany reference to the form of our sensible intuition ...” (B 52), and “Time is there-fore to be regarded as real, not indeed as an object but as the mode of representa-tion of myself as an object. If without the condition of sensibility I could intuit my-self, or be intuited by another being, the very same determinations which I nowrepresent to ourselves as alternations would yield knowledge into which the repre-sentation of time, and therefore also of alternation, would in no way enter.” (B 54)33For McTaggart theB-series is parasitic on theA-series: “TheB-series, how-

ever, cannot exist except as temporal, since since earlier and later, which are re-lations which connect its terms, are clearly time-relations. So it follows thatthere can be noB-series when here is noA-series, since without theA-seriesthere is no time” (1927, Sec. 312).

21


couched entirely in terms of genuine observables, the sought afterexplanation cannot be purely physical. This is not to say that the ex-planation must be mentalistic in some sense that rests on a mental-physical dualism. But it is to say that it will have to involve rep-resentations in terms of quantities that are not part of the intrinsicphysical description of the world.

10 Reconciling the absence ofB-series changewith quantum mechanics and statistical

mechanics.In the preceding section I tried indicate why the idea thatB-serieschange exists not in the world in itself but only in representationsof the world is not as radical as it sounds. But apparently the ideacomes unstuck when one tries to apply it to quantum mechanics andstatistical mechanics because, it has been contended, both of thesetypes of mechanics require an unfrozen dynamics. I will explainsome of the considerations behind this contention, and at the sametime indicate why I think that the matter is far from being settled.

Consider the program of Dirac quantization for constrained Ham-iltonian systems (see Dirac (1964) and Henneau and Teitelboim(1992)). In barest outline form, the constraints are promoted to op-erators on some appropriate Hilbert space, and the physical sectorof this space is spanned by state vectors that are annihilated by theoperator constraints. When applied to the Hamiltonian constraint ofGTR the requirement that the Hamiltonian operator constraint (foran appropriate choice of operator orderings) annihilate the physicalstate vectors produces what is called the Wheeler-DeWitt equation.This equation is a kind of degenerate Schrödinger equation in whichthere is no time dependence, a feature that is thought to pose in-

superable technical and interpretational problems for the quantumformalism (see Kuchar (1992)). For our purposes the key questionis whether quantization is stymied until the dynamics is unfrozenby deparameterization or some other means. To keep the discussionmanageable I will abandon the formidable case of quantum gravityin favor of my running toy example.

Consider what would happen if one tried to quantize à la Dirac theparametrized simple harmonic oscillator without decoding the con-stants of the motion in such a way as to reintroduceB-series change.Instead of the single configuration variablex one now has a pairof configuration variables,x, t. And instead of building a Hilbertspace ofL2 complex valued functionsψ(x), the Hilbert space wouldhave to be constructed fromL2 functionsΨ(x, t) of both x and t.Following the program of Dirac constraint quantization, the super-Hamiltonian constraint is applied by turningH = H + pt into anoperatorH = H + pt and requiring that the physical states be anni-hilated by this operator:

HΨ = HΨ + ptΨ = 0 (1)

Equation (1) is the Schrödinger equation forΨ(x, t) if t is identifiedas time. But as Unruh and Wald (1989) note, there seems to be anobvious problem here; namely, it doesn’t seem possible to build aHilbert space out of solutions of (1) since presumably there aren’tany solutions that are square integrable with respect tox and t, atleast not with respect to the naive measuredxdt. They concludethat in order to quantize the parametrized oscillator it is necessaryto identify a time variable that “sets the conditions” for a measure-ment. This is a little too quick, as shown by Ashtekar and Tate(1994). The family of operatorsXt = Xo cos(ωt) + Po

ωsin(ωt)–

which are the quantum operator analogues of the position evolv-ing constants–andPt = −ωXo sin(ωt) + Po cos(ωt)–which are the

22


corresponding family of momentum operator evolving constants–commute with the quantum super-Hamiltonian constraint. It turnsout that requiring that this family of operators be self-adjoint fixesthe measureµ(x, t)dxdt for the inner product of solutions to (1) tobe of the formµ(t)dxdt, whereµ(t) is unconstrained. One is there-fore free to chooseµ(t) to get the desired normalization. Thus, asAshtekar and Tate conclude, “it is not necessary to isolate time inorder to construct the Hilbert space of physical states” (1994, 6468).

Montesinos, Rovelli, and Thiemann (1999) have studied the Diracquantization of another toy model which more closely mimics theconstraint structure of GTR in that it has two non-commuting Hamil-tonian constraints. (Recall that GTR has a different Hamiltonianconstraint for each point of space.) It was found that the require-ment that the quantum operator analogues of a family of classicalgauge invariant evolving constants be self-adjoint determines theinner product for an apparently sensible Hilbert space of physicalstates. So again quantization does not imply that it is necessary tofind a time variable with respect to which the dynamics is unfrozen.34

Some skepticism has been expressed about whether the evolvingconstants scheme can lead to an adequate quantization of GTR viathe Dirac canonical quantization scheme (see Hájicek (1991)–andRovelli’s (1991b) reply–and also Kuchar (1992, 1993)). It is tooearly to make predictions about the outcome of this highly technical

34The idea of evolving constants is not supposed to be a panacea for the prob-lem of quantizing parametrized systems. For example, it may well be the case thatquantizing using different families of evolving constants leads to different quan-tizations that are not equivalent even up to the usual operator ordering ambi-guities. However, this should not obscure the conceptual point that the evolv-ing constants idea shows how quantization is possible without unfreezing the dy-namics. This conceptual point would be undercut only if it turned out that the iden-tification of the route to the construction of the “correct” Hilbert space requires theprior isolation of time.

issue. A positive verdict in favor of evolving constants would addsupport to the line I sketched in the preceding section. By the sametoken a negative verdict would indicate thateither the dynamics ofGTR must be unfrozen if gravity is to be quantizedor elsethat grav-ity cannot be quantized by the canonical Hamiltonian procedure.

Turning now to statistical mechanics, it might seem even moreobvious than in quantum mechanics that unfreezing the dynamics isnecessary since without identifying an unfrozen time variable and acorresponding non-zero Hamiltonian, ordinary statistical mechan-ics is not possible: Without time and change, how can thermal-ization occur? What replaces the familiar classical Gibbs equilib-rium stateρ ∼ exp(−βH), whereH is the Hamiltonian andβis the inverse temperature? Nevertheless, Montesinos and Rovelli(2001) have argued that a Boltzmann-like approach is possible with-out deparametrization. The idea is that even though the dynamicsof a parametrized system is frozen, an appropriate analogue of ther-malization can be obtained by taking a large ensemble of identicalparametrized systems and turning on a “small interaction” amongthem. One would expect that with the interaction turned on, only asmall number of global macroscopic quantities will be conserved.In place of the usual equilibrium statistical state one will have astateρ determined by the requirement that it maximize the entropyS := −k ∫

Γρ ln ρdz, where the integration is over the phase space

Γ = Γ1 x Γ2 x ... of the ensemble of identical parametrized sys-tems, subject to the requirements thatρ is normalized and that it re-turn the observed expectation values〈Ok〉 for the conserved macro-scopic quantitiesOk. ρ then has the formZ−1 exp(−∑k αk〈Ok〉),whereZ :=

∫Γρdz. The application of this scheme to the example

of Montesinos, Rovelli, and Thiemann (1999) has been worked outin Montesinos and Rovelli (2001). The parametersαk that charac-terize the equilibrium state generally do not include temperature as

23


ordinarily conceived, with the upshot that from the point of view ofthe proposed approach “temperature plays no fundamental role inthe statistical analysis. It may not be defined, or, if it is defined atall, temperature is just one of the intensive macroscopic parameterscharacterizing the equilibrium configuration of the system” (Mon-tesinos and Rovelli (2001, 567)). Whether or not these radical ideaswill yield fruit when applied to statistical mechanics of black holes,of matter interacting with gravity, etc. remains to be seen. But theexploration of the implications for statistical physics of modern Mc-Taggartism seem well worth pursuing.

11 Conclusion

There is indeed a real problem about change. But it is not the oneposed by Grandpop McTaggart and debated endlessly in the philo-sophical literature. The real problem is not concerned with McTag-gart’sA-series, Broad’s Becoming, or any of the other metaphysicalextravagances that dot the philosophical literature on time. The realproblem about change is the result of substantive discoveries in mod-ern physics–in particular, Einstein’s GTR–which seem to imply thatthere is no temporal change in any genuine physical magnitude, i.e.there is noB-series or property change. This is not to say that in thepicture of physical reality that emerges from these discoveries thatthere is no changetout court; for there is still a temporally orderedseries of coincidence events (theD-series), and different events oc-cupy different places in the series. But the events in this series can-not be construed as arising from the occurrence of ordinary propertychange, and physics provides no motivation for positing changingproperties of the occupants of the series.

Understanding the real problem of change requires understand-ing the foundations of classical GTR and the issues involved in con-

structing a quantum theory of gravity. Its solution will most likelyrequire the contributions of both philosophers and physicists. Whiledismissing or maneuvering around the problem remains an option, itis not an option that is easy to implement. The alternative of takingthe problem seriously and working out the implications for physics,for the philosophy of science, and for the philosophy of mind is thecourse I recommend.35 It is a course fraught with many difficultiesand unpleasant surprises. But the potential payoff seems to me tomake the effort worthwhile.

35An even more radical solution to the problem of time in classical GTR andquantum gravity is given by Barbour’s (2000) version of presentism. See But-terfield (2002) for a critical review.

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Acknowledgment: I am grateful to Gordon Belot, Craig Callen-der, John Norton, Carlo Rovelli, Steve Savitt, and Bill Unruh bothfor encouragement and a number of helpful suggestions. Specialthanks are due to Jeremy Butterfield, Tim Maudlin, and an anony-mous referee for forcing me to make a number of crucial clarifica-tions. It is fair to say, however, that each of these (otherwise rational)people disagrees with some aspect of my position.

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