1 Introduction There are two traditions in the philosophy of time that, while op- posing one another, are locked in a mutual embrace. The embrace is cemented by two shared assumptions: first, that time presupposes Thoroughly Modern McTaggart change; and, second, that genuine change requires Becoming. Both traditions have ancient roots. One, which takes its inspiration from Parmenides, denies the reality of change and time by rejecting Be- coming; the other, which can be traced to Aristotle, upholds the real- Or ity of change and time by claiming to find Becoming at work in the What McTaggart Would Have Said If He world. What complicates an already complex discussion is that there are at least two distinct senses of Becoming in play. One sense is ex- Had Read the General Theory of Relativity emplified in McTaggart’s (1908, 1927) infamous A-series, in which events 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 John Earman 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 of the incoherency of the A-series less than convincing.1 McTaggart’s brand of Becoming is property-based: that an event becomes present means for him that it loses the (non-relational) Philosophers’ Imprint property of futurity and takes on the (non-relational) property of
ferent times. of change but doesn’t provide a picture of any animation at all. Whether modern McTaggart’s argument proves to be any more This quantitative notion of physical change incorporates qualitative B-series property change as a special case since for any qualitative convincing than his grandfather’s remains to be seen. The main task property one can define a corresponding magnitude which, at any undertaken in the next section is to assess the crucial premise (P30). moment, takes on just two values 0 (“present”) and 1 (“absent”). So far I have just been using common sense, and common sense certainly seems to tell us that there is physical change. (Just look at the world around you where objects are changing in their positions, 4 The frozen dynamics of GTR shapes, etc.) And as the author of a recent learned treatise has noted, GTR as applied to the actual cosmos supplies a precise version of The issues I am going to describe in this section are typically dis- common sense change. But the author of that treatise based his con- clusions on the surface structure of GTR. When the deep structure of cussed under the label of “the problem of time in quantum gravity” the theory is taken into account we see that our first impressions are (see, for example, Isham (1992) and Kucharˇ (1992, 1993)). The badly wrong. For what the deep structure of GTR tells us is that label is at once apt and misleading. It is apt because the problem was brought into prominence by the pursuit of the canonical quan- (P30) No genuine physical magnitude countenanced in GTR changes tization program which aims to marry GTR and quantum theory by over time. quantizing the metric field. But the label is also misleading in two ways. First, the roots of the problem lie in classical GTR, and even 0 0 From (P2 ) and (P3 ) we get if it was decided that it is a mistake to quantize GTR, there would remain the problem of reconciling the frozen dynamics of GTR with (C0) If the set of physical magnitudes countenanced by GTR is the B-series notion of change that is supported not only by common complete, then there is no physical change. sense but by every physical theory prior to GTR. Second, although the aspect of the problem that grabs attention is that of time and 0 0 And from (P1 ) and (C ) we get change, no solution will be forthcoming without tackling the more general issue of what an “observable” of classical GTR is. Some (C00) Physical time as described by GTR is unreal. physicists who work in this area are apt to respond with a “Yes. BUT ...” They will point out that the problem of time and change If you can hear me Grandpop, I hope you are proud to see that only rises to the level of crisis when quantization is attempted and the family is carrying on the tradition! It is a commonplace that in that it is precisely here that the temporal aspect comes to the fore leaving the manifest image for the scientific image we have to leave behind the hankerings of dynamic time theorists for an animated pic- since apparently the frozen dynamics of GTR has to be unfrozen in ture of time in the form of a growing block universe or the like. But order to make quantization possible. I will dispute this wisdom by what I have shown is that we also have to leave behind the famil- claiming that one obtains a key to the problem by getting a handle iar non-dynamic block universe, which allows for the change over time of physical magnitudes, and are forced to the very non-dynamic on what an observable is in classical GTR and that with this key in block universe, which not only doesn’t provide an animated picture hand it may not be necessary to unfreeze in order to quantize. But I 6 John Earman Thoroughly Modern McTaggart am getting ahead of myself. First I have to explain how GTR drives are globally hyperbolic,11 which is another way of restating the as- us out of the ordinary non-dynamic block universe, which is devoid sumption already made above in section 2 that the spacetime can be of Becoming but filled with ordinary B-series change, and into the foliated by Cauchy surfaces. One can then hope that appropriate ini- very non-dynamic block universe, which lacks B-series change. tial data on a Cauchy surface fixes, via Einstein’s gravitational field Because the argument given below is both technical and compli- equations, unique future and past developments. cated, it is important to keep a firm grip on the key point, which A quick insight into why there is a prima facie conflict between is quite simple to state even if the details require an elaborate ex- the surface structure of GTR and Laplacian determinism can be planation. The point is that in a theory with gauge freedom, what gained by deploying the Lagrangian formulation of particle and field is “real” or “objective” is what remains after the gauge freedom is theories used so extensively in modern physics. Consider then the- removed. So, for example, in classical relativistic electromagnetic ories whose equations of motion or field equations follow from an theory, the specification of the electromagnetic potentials contains action principle and, thus, are in the form of (generalized) Euler- a large amount of gauge freedom; what remains after that freedom Lagrange equations. If the action admits as variational symmetries is removed are the electric and magnetic fields or, more property, the elements of a Lie group of transformations that depend on ar- the electromagnetic field tensor–it is this object, and not the values bitrary functions of all the independent variables, then Noether’s of the scalar and vector potentials, that characterizes what is “real” second theorem implies that this is a case of underdetermination– about the electromagnetic state of the world. This much is familiar to a unique solution of the Euler-Lagrange equations is not determined most philosophers with even a nodding acquaintance with physics. by initial data–because the Euler-Lagrange equations must satisfy What is less familiar is that there is a class of gauge theories where a set of mathematical identities and consequently are not indepen- the very dynamics is implemented by a gauge transformation. What dent.12 GTR is a case in point since the relevant action for Einstein’s such a theory describes when the gauge freedom in such theories is gravitational field equations admits the spacetime diffeomorphism killed is a world without B-series change. Einstein’s GTR turns out group as a variational symmetry.13 Hence, taken at face value, GTR to be just such a theory. Now to the details. is indeterministic. The argument in section 2 that classical GTR lends itself to the A way to recoup the fortunes of determinism is to switch from common sense account of physical change was based on a naively the Lagrangian to the Hamiltonian formalism. A case of underdeter- realistic reading of the surface structure of the theory as gleaned mination in the Lagrangian formalism corresponds to the existence from textbook presentations–tensor, vector, and scalar fields on man- ifolds. But this naive reading must be radically modified if GTR is to 11For a definition of this concept, see Wald (1984). count as a deterministic theory, and the modification undercuts the 12I am glossing over various technical conditions needed for the application of common sense picture of change by freezing the dynamics. Prepara- the Noether theorems. See my (2000, 2001b) for an account of the application tory to explaining this startling conclusion, I stipulate that, in order of Noether’s theorems to the issues under discussion. 13A diffeomorphism d : M → M is a one-one mapping of the spacetime mani- that determinism have the best chance of being true in GTR, atten- fold M onto itself that preserves the differentiable structure of M, e.g. if M tion is to be restricted to general relativistic spacetimes M, gab that is a C∞ manifold, then d must be C∞. 7 John Earman Thoroughly Modern McTaggart of constraints in the Hamiltonian formalism: when one Legendre payoff of cranking the Dirac algorithm is a dissolution of the threat transforms from the Lagrangian velocity phase space (q, q˙) (where to determinism since the observables do evolve deterministically. In q stands for the generalized configuration variables and q˙ stands for GTR, however, deterministic dynamics is regained only by freezing the generalized velocity variables) to the Hamiltonian phase space the dynamics! Γ(q, p) (where p stands for the generalized momentum variables), In GTR the configuration variables (the qs) are Riemann metrics one finds that the ps are not independent but must satisfy a set of hab on a three manifold, which is to be imbedded as a Cauchy sur- identities called the primary constraints, which follow from the de- face in spacetime, and the momentum variables ab (the ps) are de- finitions of the ps. Other constraints, called secondary constraints, fined in terms of the extrinsic curvature tensor, which specifies how may emerge when it is demanded that the primary constraints be the three-manifold is to be embedded in spacetime. When GTR is preserved by the motion. For present purposes another way of di- run through the Dirac constraint formalism it is found that there are chotomizing constraints is more important: the first class constraints two families of first class constraints, the momentum constraints and are those that commute (i.e. have weakly vanishing Poisson bracket) the Hamiltonian constraints.15 Since the latter constraints generate with all the other constraints, while the second class constraints fail the motion, all of the observables of GTR, defined as gauge invari- this test. P. A. M. Dirac, who was responsible for developing the ant quantities, must be constants of the motion. Thus, a “very non- constrained Hamiltonian formalism, proposed that the gauge trans- dynamic block universe” is an appropriate appellation for a general formations be identified as the transformations generated by the first relativistic world when viewed through the lens of the Dirac con- class constraints, where the intended interpretation is that two points straint formalism. of phase space Γ which are connected by a gauge transformation are It will be helpful to illustrate the above concepts in terms of a toy to be regarded as representing the same physical state (see, for ex- example to which I will revert several times in what follows. Start ample, Dirac (1964), Henneau and Teitelboim (1992)). The gauge with the standard treatment of a simple one-dimensional harmonic invariant quantities, which are referred to as observables in the lit- oscillator. The usual phase space is Γ = Γ(x, px), where x is the erature on constrained Hamiltonian systems, can be defined in var- position of the oscillator and px is the momentum conjugate to x. 2 2 2 ious equivalent ways. Let C Γ denote the constraint surface, i.e. px ω x The Hamiltonian is H = 2 + 2 . There are no constraints and the subspace of Γ where all the constraints hold. An observable is then defined as a function F :Γ → R which is constant along the coordinates (qi, p ), i = 1, 2, ..., N, can be chosen for the 2N-dimensional Γ Pi gauge orbits on C or, equivalently, which commutes weakly with all ∂f ∂g ∂f ∂g so that {f, g} = ( i i ). In this formalism Hamilton’s equations i ∂q ∂pi ∂pi ∂q i i the first class constraints. Alternatively, if the reduced phase space can be written as q˙ = {q ,H}, p˙i = {pi,H}, where H is the Hamiltonian. Γ(˜˜ q, p˜) is formed by taking the quotient of C by the gauge orbits, A constraint (q, p) is first class just in case for any other constraint κ(q, p), then the observables can be defined as functions F˜ : Γ˜ → R.14 One { , κ} 0, where “ 0” means “vanishes weakly,” i.e. vanishes on the constraint surface C Γ. For a detailed treatment of the constrained Hamiltonian formalism, 14The mathematically precise way to formulate Hamiltonian mechanics uses a see Henneau and Teitelboim (1992). symplectic form Ω, that is, a non-degenerate two form on the phase space Γ. The 15“Families” because there is one momentum constraint and one Hamiltonian Poisson bracket for phase functions is defined by {f, g} := Ω(df, dg). Locally, constraint for every point of space. 8 John Earman Thoroughly Modern McTaggart no gauge freedom, and, of course, both the position and momentum or artifacts of the constrained Hamiltonian formalism. To correct of the oscillator are observables in the sense of Dirac. The chang- this impression I will sketch other ways of arriving at similar, if not ing values of these observables constitutes common sense physical identical, conclusions in the spacetime setting rather than the (3+1) change. Now “parametrize” the system by adding the time t as an Hamiltonian formulation. additional configuration variable, which necessitates adding the con- The details of the initial value problem in GTR are rather involved jugate momentum variable pt. The result is an augmented phase (see, for example, Wald (1984)), but for present purposes it is unnec- space Γ(x, px, t, pt). There is now one constraint, the vanishing of essary to review those details in order to see why there is an appar- the super-Hamiltonian H = H + pt. As in GTR the dynamics of the ent failure not only of Laplacian determinism in GTR but even of the parametrized oscillator is pure gauge so all gauge invariant quan- weakened version that says that the entire past history of the universe tities are constants of the motion. Neither the position nor the mo- determines the future. I will illustrate the point by means of a ver- 16 mentum of the parametrized oscillator are observables. The “frozen” sion of Einstein’s “hole argument.” Let M, gab,Tab be any solution observables can be explicitly constructed as functions F˜ : Γ˜ → R to Einstein’s gravitational field equations, and let d : M → M be ˜ where the reduced phase space Γ consists of points (A, B) R x any diffeomorphism of M onto itself. Then M, d g, d Tab, where R with A and B the coefficients that appear in an arbitrary solution d O denotes the object field obtained by “dragging along” O by d, x(t) = A cos(ωt) + B sin(ωt) of the equations of motion of the is also a solution. One can choose d to be the identity map on and unparametrized oscillator. Notice that within the parametrized treat- to the past of some Cauchy surface Σo of M, gab but non-identity to ment one can meaningfully say, for example, that either A > 0 or the future of Σo. The result is two solutions that agree on the values B > 0, which in vulgar parlance means that the oscillator is oscillat- of gab and Tab for all p ∈ M on or to the past of Σo but differ on val- ing and doesn’t remain at rest. So it seems meaningful to talk about ues of gab and Tab for some q ∈ M in the future of Σo–an apparent the oscillator changing. But strictly speaking, if one stays within violation of determinism. the parametrized formalism, such vulgar talk cannot be cashed into To overcome this apparent violation, Bergmann (1961) proposed official talk about change in the usual sense of changing values of that the doctrine of determinism be restricted to quantities–which observables, for there aren’t any parametrized-oscillator observables he also dubbed “observables”–whose values are unequivocally pre- whose values are different at different times. dictable from initial data. This proposal for restoring determinism Although this toy example serves as a quick and easy illustration to GTR might seem to have all of the virtues of theft over honest of some complicated concepts, it is potentially misleading. In partic- toil. But despite the apparent circularity involved in Bergmann’s ular, the unwary reader may be misled into thinking that since in this proposal, the doctrine of Laplacian determinism still has content. example it is possible to reintroduce change by “deparameterizing” Notice first that, by the argument just given, GTR will be deter- in an obvious way, the same will hold true in non-toy examples. As ministic in Bergmann’s sense only if observables are restricted to will be seen below in section 8, however, this impression is belied by the case of GTR. Second, the toy example may give the mis- 16See Norton (1987) for an account of Einstein’s struggles with the hole ar- impression that the counterintuitive results are merely formal tricks gument; and see also section 7 below. 9 John Earman Thoroughly Modern McTaggart diffeomorphically invariant quantities. Second, the converse or ‘if’ this quantity, if predictability is to be preserved. Next consider what part follows from the existence and uniqueness proofs for the initial can be called quasi-local field quantities obtained by integrating a value problem for GTR, which show that for appropriate initial data local field quantity over a spacetime region, e.g. the three-volume (3) associated with a three manifold Σo, there is a unique upto diffeo- integral, over a slice Σ, of the Ricci curvature scalar R of the morphism maximal development for which Σo is a Cauchy surface. three-metric induced on Σ by the spacetime metric. Is the value of Readers who have already learned GTR will not be surprised by this quantity predictable from initial data on Σo or even from data these moves since variants of them are quite common in the litera- from the entire past of Σo? Again the answer is negative unless the ture. What may not be familiar to most readers is that Bergmann’s value is the same for all Cauchy surfaces to the future of Σo; for proposal implies that there is no physical change, i.e. no change in given any solution M, gab with initial Cauchy surface Σo and any 0 his observable quantities, at least not for those quantities that are pair of Cauchy surfaces Σ and Σ to the future Σo, a diffeomorphism 0 constructible in the most straightforward way from the materials at d can be chosen to map Σ to Σ in such a way that in M, d gab the hand. To simplify the argument, concentrate on the special case of value of the quasi-local quantity for Σ is the same as the value in 0 vacuum solutions (Tab 0) to the Einstein field equations; the ar- M, gab for Σ . And again since Σo can be chosen anywhere one gument easily generalizes to non-vacuum solutions. Consider first likes, the value of the quasi-local quantity must be the same for all what can be called local field quantities which are constructed from Cauchy slices, and thus there is no temporal change to be found in the metric and its derivatives upto some finite order and which are this quantity. An analogous conclusion holds for a quasi-local field evaluated at spacetime point, e.g. the Ricci curvature scalar R. Is quantity constructed by taking the spacetime volume integral of a the value of this quantity at some point to the future of an initial local field quantity over an open region O M which has compact value hypersurface predictable from initial data on the hypersurface, closure and which lies to the future of Σo.A non-local field quantity or even from data on to the the entire past of the hypersurface? To obtained by integrating a local field quantity over the entire future be predictable, the value has to be the same in the class of solu- of Σo is predictable; but then such a quantity is incapable of mark- tions {M, gab} to the source free Einstein field equations, any two ing any future change. And since again the cut between past and of which can be related by a diffeomorphism that leaves the initial future is arbitrary, there is no temporal change to be squeezed from 17 value hypersurface Σo M and its past fixed. Let p ∈ M be the non-local field quantities. point in the future where we want to predict the value R(p). But The argument so far has concerned only local and quasi-local 0 since for any other p ∈ M to the future of Σo a diffeomorphism field quantities–that is, quantities that are attached to spacetime d can be chosen so as to leave Σo and its past fixed while map- points and regions–leaving open what happens with quantities that ping p to p0, predictability leads to the result R(p) = R(p0); for are not so attached. This issue will be taken up in sections 5 and d R(p) := R(d(p)), and the sameness of values of R in the two 6. But it is worth remarking in advance that it is not obvious how solutions M, g and M, d g requires that R(p) = d R(p). Fur- ab ab 17Yet another derivation of the “no change in GTR” conclusion is to be found in ther, since Σo can be chosen wherever one likes, R must be constant Smolin (2000); however, as discussed in section 8 below, Smolin rejects some everywhere, and a fortiori there is no temporal change to be found in of the premises used to derive the conclusion. 10 John Earman Thoroughly Modern McTaggart these unattached quantities can underwrite B-series change; for such spond (at least when the Einstein field equations are satisfied18) to change requires a subject, and since spacetime points and regions the changes wrought by infinitesimal diffeomorphisms moving or- are the only obvious candidates for the subject role in GTR, these thogonal to the initial value hypersurface; and so taken together the peculiar unattached quantities would seem to remove the subject of gauge transformations in Dirac’s sense may be thought of as gener- change from the picture. ating changes in dynamical variables that correspond to the changes Two different senses of the crucial notion of “observable” in GTR wrought by arbitrary infinitesimal spacetime diffeomorphisms. One have been put into play–that of Dirac (i.e. real valued functions of would expect that this correspondence will lead to a correspondence phase space variables that are constant along the Dirac gauge or- between the class of Dirac observables and the class of Bergmann bits) and that of Bergmann (diffeomorphically invariant spacetime observables. But since no precise general account of the latter ex- quantities). Comparing the two is an apparently apples vs. oranges ists, the subsequent discussion will have to remain sensitive to the operation since the former lives in a (3 + 1) phase space formula- difference in the two senses of observables. tion of GTR while the latter lives in the four-dimensional spacetime The confidence that the deep structure interpretation of GTR sket- setting. Indeed, establishing a connection is apparently stymied by ched above is not an artifact of the (3 + 1) phase space formulation the fact there seems to be no natural correspondence between the imposed by the Dirac formalism is strengthened by the alternative diffeomorphisms of the spacetime manifold and the Dirac gauge approach of Ashtekar and Bombelli (1991), who show that Hamil- transformations on phase space. For instance, one can’t even look tonian mechanics for GTR does not require a (3 + 1)-cotangent bun- for an isomorphism between the Lie algebra of spacetime diffeo- dle structure. Instead of taking the state space of GTR to be the morphisms and the Poisson algebra of first class constraints since space of instantaneous states, they work with the space Γˆ of entire the latter isn’t a Lie algebra. However, Isham and Kucharˇ (1986a, histories or solutions to the Einstein field equations, which implies 1986b) have shown that if the embedding variables, that describe that dynamics is implemented not by a mapping from one state to how a three-manifold is embedded as an initial value hypersurface another state in the same solution but as a mapping from one solu- of spacetime, along with their conjugate momentum variables, are tion to another solution. The space Γˆ has a “presymplectic structure” adjoined to the phase space of GTR, then there is a natural homo- given by a degenerate two-form Ωˆ. There is no constraint surface, as morphism of the Lie algebra of spacetime diffeomorphisms into the in the (3 + 1) formulation; rather, the gauge directions Y are given Poisson constraint algebra on the extended phase space. Their ac- directly by the null vectors of Ωˆ. It turns out that two solutions lie on count is too technical to discuss here, and for present purposes it the same gauge orbit (i.e. integral curve of the gauge field Y ) iff they will have to suffice to indicate non-rigorously how spacetime dif- are diffeomorphically related.19 Thus, in the Ashtekar-Bombelli for- feomorphisms correspond to Dirac gauge transformations. The mo- mulation of GTR the Dirac and Bergmann senses “observable” are mentum constraints generate gauge changes in a dynamical variable 18 F (h , ) that correspond to the changes wrought by infinitesimal The technical details here are rather tricky; the interested reader may consult ab ab Unruh and Wald (1989, 2599) diffeomorphisms in the initial value hypersurface Σo, and the Hamil- 19This conclusion has to be qualified since Ashtekar and Bombelli focus on as- tonian constraints generate gauge changes in F (hab, ab) that corre- ymptotically flat spacetimes. 11 John Earman Thoroughly Modern McTaggart united since the gauge invariant quantities are identified as diffeo- tion that gauge dependent quantities are not observable or measur- morphically invariant functionals of the phase space variables (here, able, while for the Bergmann sense of observable the Assumption spacetime metrics). However, in what follows I will stick with the is justified for tests that proceed by deterministic predictions. The more familiar (3+1) phase space formulation since it is this formu- Assumption can, of course, be challenged on a variety of grounds. lation of GTR that appears in almost all of the discussions of the But it comes into play over and over again in the physics literature, problem of time in quantum gravity. 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 terms 5 The observables of GTR misleading because the quantities involved are often rather different from the relational quantities familiar from relational accounts of The argument sketched in section 4 to show that local and quasi- space, time, and motion. I prefer the term coincidence observables local field quantities do not provide a basis for B-series change in because these quantities can be seen as involving a natural gener- GTR could equally be construed contrapositively, that is, as showing alization of the notion of coincidence–the meeting of two “material that such quantities are not observables since in generic solutions of points”–that Einstein used in 1916 to respond to the threat of inde- Einstein’s field equations these quantities do change with time. In terminism posed by the hole argument.20 Einstein wrote: asymptotically flat solutions, a few such quasi-local quantities–such as the total ADM mass of the universe–do qualify as observables in All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely Dirac’s sense, that is, quantities which are constructed, possibly by in the motion of material points, then ultimately nothing would be integrating over a Cauchy slice, from the phase space variables and observable but the meeting of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such which are constant along the gauge orbits. However, in the class of meetings of material points of our measuring instruments with other source-free solutions to the Einstein field equations having space- material points, coincidences between the hands of a clock and points times with compact space sections, there are provably no local or on the clock dial, and observed point-event happenings at the same quasi-local Dirac observables (see Torre (1993)). Presumably this place and the same time. (1916, 117) no-go result could be turned into a no-go result for Bergmann ob- (That this passage is part of Einstein’s response to the indeterminism servables. problem revealed by the hole argument is made clear in Howard This is not a reason for total despair. For by transcendental deduc- (1999).) Although Einstein was on the right track, his notion of the tion, there must exist observables for GTR even if they are not to be coincidence of material particles has to be generalized to include found among the local or quasi-local field quantities; for, assuming fields in order to do justice to the content of his GTR. that the results of observations and measurements must be expressed by the values of observables (hereafter called the Assumption) there 20Of course, there are also global observables such as the four-volume integral would otherwise be no way to test the theory. For the Dirac sense of the Ricci curvature scalar over all of spacetime, assuming that such integrals are of observable the Assumption can be justified by appeal to the no- finite. But is hard to see how such quantities can be used to express the results of ordinary measurements. 12 John Earman Thoroughly Modern McTaggart
I can illustrate the flavor of generalized coincidence observables not just at midnight but at noon and 4:00 PM as well. A theory of in terms of the toy example of the harmonic oscillator. To make the measurement for generalized coincidence observables is obviously case a bit more interesting, add a free particle whose position y is needed. measured from the origin of the oscillator. In the non-parametrized The application to GTR of the generalized notion of coincidence treatment, y(t) = Ct + D is a good observable which can serve observable goes back at least as far as Kretchmann (1915, 1917). as a clock. The equation of motion of the oscillator can be rewrit- The basic idea was worked out in some detail four decades later ten using the clock variable as x(y) = A cos ω( y D ) + B sin by Komar (1958). A generic solution M, g to Einstein’s vac- C C ab y D uum field equations will not possess non-trivial symmetries, and ω( C C ) . In the parametrized treatment of the oscillator-plus- free particle, neither x nor y is an observable in the Dirac sense. for such solutions there will be four independent scalar fields , But since any function F˜ : Γ(˜ A, B, C, D) → R on the reduced = 1, 2, 3, 4, constructed from algebraic combinations of the com- phase space is an observable, it follows that for any value y¯ of y, ponents of the Riemann curvature tensor, such that the four-tuples y¯ D y¯ D ( 1(p), 2(p), 3(p), 4(p)) and ( 1(p0), 2(p0), 3(p0), 4(p0)) are Xy¯ := A cos ω( C C ) +B sin ω( C C ) is a good observable. That is, for any value y¯ of the clock variable, the-position-of-the- different whenever p 6= p0 for any p, p0 ∈ M.21 Thus, the values oscillator-when-the-value-of-the-clock-variable-is-y¯ is observable in of these fields can be used to coordinatize the spacetime manifold. the Dirac sense; it is also an observable in the spirit of Bergmann While these scalars are, of course, not observables in the Bergmann since its value is predictable from appropriate initial data. sense, they can be used to support such observables in the following Admittedly, however, it remains a bit obscure how the value of way. If g are the contravariant components of the metric tensor in this coincidence observable is measured. For if the parametrized de- a coordinate system {x }, the new components in the { } system ∂ ∂ scription is taken seriously, the measuring procedure cannot work are given by g ( ) := ∂x ∂x g . One can speak of the event by verifying that the coincidence of values described in the equation of the metric-components-g -having-such-and-such-values-in-the- for Xy¯ does in fact take place by separately measuring the values coordinate-system-{ }-at-the-location-where-the- -take-on- val- of the clock variable and the oscillator position and then checking ues-such-and-so.22 Call such an item a Komar event. That a given for the coincidence. For the positions of the clock and the oscillator are gauge dependent quantities, and by the Assumption the values 21The construction will not work, for example, in the Friedmann-Walker-Robert- of these quantities are not fixed by measurement. Rather the mea- son models used in current cosmological theories because these models employ spacetimes that are homogeneous and isotropic. So the attitude behind the con- surement procedure must be directly responsive to the coincidence struction has to be that such models are idealizations and the construction of ob- of values itself, even though the coincidence is not a coincidence servables in GTR depends on the fact that the real world is sufficiently com- of the values of observable quantities. The mystery of how this plex and non-symmetric. See Smolin (2000). is accomplished is underscored by the fact that, because they are 22The reader might think that some sleight of hand is involved here because the constants of the motion, such coincidence observables in the Dirac ‘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 temporal sense are in principle measurable at any time, e.g. the position-of- when. But as will become clear from the more detailed discussion in section the-oscillator-when-the-clock-reads-midnight should be measurable 9, no illicit spatiotemporal notions are being smuggled in; all that is going on 13 John Earman Thoroughly Modern McTaggart
Komar event occurs (or fails to occur) is an observable matter in beit in a representational sense. Using the nomenclature of the previ- Bergmann’s sense. But as in the oscillator case, how the occurrence ous section, call the functional relationship g ( ) the Komar state. of a Komar event is to be observed/measured is an unresolved issue. This state, which floats free of the points of M, captures the in- The measurement procedure cannot work by measuring the metric trinsic, gauge-independent state of gravitational field. This intrinsic field, measuring the scalar fields , then checking that the coinci- state is represented or realized by a spacetime model M, gab, and if dence of values that constitutes the event does in fact take place, at it is represented or realized by one such model it is equally well rep- least not if the Assumption is granted. Again, the need for a mea- resented or realized by any model in the diffeomorphic equivalence surement theory of coincidence observables is evident. class of the first. It is in the representations that the familiar story of subjects (spacetime points) and properties of subjects (fields at- tached to spacetime points) can be told, and the familiar story speaks 6 Taking stock of common sense B-series change. To use Leibnizian terminology, B-series change has been relegated to the status of an appearance, We can now see that modern McTaggart’s diatribe of section 3 was but it is not a mere appearance but a well-founded appearance based somewhat hysterical. If we use modern McTaggart’s very non-dyna- on an objective structure of the physical world. mic block universe to denote what results from the gauge interpreta- This story will not be to everyone’s taste. Indeed, the notion that tion of GTR, then the very non-dynamic block universe does not ac- temporal change in genuine physical magnitudes is not to be found cord with his conclusions (C0) and (C00) announcing the unreality of in the world in itself but only in a representation will be viewed by time and change. One can say in answer to modern McTaggart that many as patently absurd.23 Those who share this view can be di- there is change in the very non-dynamic universe, though not of the vided into two categories. The first category consists of those who usual B-series kind. The occurrence or non-occurrence of a coinci- hold that modern McTaggart’s claim that GTR entails the absence of dence event is an observable matter–at least in the technical senses at B-series change rests on some sort of illicit move or sleight of hand. issue here–and that one such event occurs earlier than another such For example, one way to express modern McTaggart’s claim goes event is also an observable matter–again in the technical senses at like this. In a constrained Hamiltonian system the intrinsic dynam- D issue. Call this series of coincidence events the -series (the term ics, as expressed in terms of the genuine observables, is obtained by ‘C-series’ having already been co-opted by McTaggart–see section passing to the reduced phase space by quotienting out the gauge or- 9). Change now consists in the fact that different positions in the bits. When this is done for a theory in which motion is pure gauge, D -series are occupied by different coincidence events. there is an “elimination of time” in that the dynamics on the reduced One can also say that B-series property change does “exist,” al- phase space is frozen. In response it is sometimes claimed that this is that some of the quantities are being used to specify a point on a gauge orbit, and 23For example, Karel Kuchar,ˇ one of the leading research workers in canonical once that specification is made it is easy to define observables in terms of the quantum gravity shares this attitude, and as a consequence he explicitly rejects values that other quantities take at the specified point. This is an example of the notion that a genuine physical magnitude of GTR must commute with the what is called “gauge fixing.” Hamiltonian constraint; see Kucharˇ (1993). 14 John Earman Thoroughly Modern McTaggart
“elimination of time” cannot be carried out in chaotic Hamiltonian in terms of the time independent correlations between gauge depen- systems since there will not be enough constants of the motion to dent quantities which change with time. Unruh (1991) has inveighed distinguish the gauge orbits (for in such a system only functions of against this proposal: the Hamiltonian are constants of the motion) and since these con- The problem is that all of our observations must be expressed in terms stants will not be smooth functions on the phase space (for in chaotic of the physically measurable quantities of the theory, namely those systems the trajectory must pass through any open set of the phase combinations of the dynamic variables which are [gauge invariant and therefore] independent of time. One cannot try to phrase the space) (see Smolin (2000)). This objection rests on an equivocation problem by saying that one measures the gauge dependent variables, on “constants of the motion”: it could mean either an integral of the and then looks for time independent correlations between them, since the gauge dependent variables are not measurable quantities within motion or a quantity that is constant along the gauge orbits, and the 24 two senses are not equivalent. Hájicekˇ (1996a) shows that regardless the context of the theory. (266) of how many integrals of the motion are present in a (finite dimen- I agree with this appraisal. (But I think that a more sympathetic sional) Hamiltonian system, parameterizing the system results in a reading of the proposal would lead to the “evolving constants” idea constrained system for which there exists a set of continuous quanti- to be discussed below in section 9.) And I agree with Unruh when he ties that are constant along the gauge orbits and which are complete goes on to conclude that “The time independent quantities of general in that their values can be used to separate the gauge orbits. To be relativity alone are simply insufficient to describe time dependent sure, there can be real technical problems involved in the “elimina- relations we wish to describe with the theory” (ibid.)–at least I agree tion of time” in the guise of constructing the reduced phase space by if ‘wish’ refers to our preanalytic desire to find ordinary B-series quotienting out the gauge orbits in a constrained Hamiltonian sys- change described in the theory. tem. For instance, while the reduced phase space always exists as a The second category of naysayers to modern McTaggart consists topological space, it may not inherit a manifold structure. However, of those who seek not to find a flaw in his argument but rather to in the case of GTR the reduced phase space is not fatally ill-behaved block or blunt its thrust. One way to do this is to add some addi- since it is the disjoint union of manifolds. tional structure to the theory that has the effect of unfreezing the As a second example of the first category of naysayers to mod- dynamics by generating additional observables that can change with ern McTaggart I would mention Kauffman and Smolin (1997) and time. Examples of this strategy will be considered in the section 8. Smolin (2000) who classify the problem of time and change in GTR A wholly different reaction to modern McTaggart seeks not to as a pseudo-problem because it is posed in terms of mathematical fault or blunt his argument but live with its consequences. This reac- structures that can only be reached by non-constructive methods. tion requires a radical attitude readjustment. It requires us to swal- Since I think that such structures and methods are needed quite gen- low the notion that, just as the transient ‘now’ of A-series and the erally in theoretical physics, I do not follow their lead. accreting layers of existence of Broad’s Becoming have been rele- In a somewhat different vein, it has been claimed that although the problem of time in GTR is not a pseudo-problem, neither is it 24This is an example of the Assumption announced in section 6 coming into intractable since common sense B-series change can be described play. 15 John Earman Thoroughly Modern McTaggart gated to the manifest image, so must ordinary B-series change. The of a weakly generally covariant theory may fall far short of diffeo- problems and prospects of successfully carrying out such a readjust- morphism invariance, and as a result the observables of the theory ment will be discussed in sections 9 and 10. But before turning to may be much richer than the class of diffeomorphism invariants.26 these matters I need to take the reader on a brief digression. Thus, it is simply not the case that the problem of change raised by modern McTaggart in the context of GTR also rears its head in ear- lier theories of physics which generally did not admit the spacetime 7 Digression: reactions to Einstein’s hole diffeomorphism group as a gauge symmetry. The main preoccupation of the philosophical literature on Ein- argument stein’s hole argument has been on the implications for the status of There is an extensive philosophical literature on Einstein’s hole spacetime points. Although there is no agreement what these im- argument, with different authors drawing quite different morals.25 plications are, there is a growing realization that an adequate re- While it would be inappropriate to review this literature here, it will sponse to the hole argument is hard to square with either wing of serve a useful function to take up a couple of issues from this litera- the traditional opposition of substantivalism vs. relationism (see, ture that link directly to the issues under discussion here. for example, Saunders (2001)). I think that this realization is cor- The first point that needs to be emphasized is that the existing rect but that it needs to be extended to include the realization that philosophical literature does not show an appreciation for the fact the gauge interpretation of diffeomorphism invariance not only un- that the requirement of general covariance comes in two versions, dercuts the traditional substantivalism vs. relationism distinction but weak and strong. The weak version requires that the laws of a the- also calls into question the traditional choices for conceiving the sub- ory be written in a form that is valid in all coordinate systems or, ject vs. attribute distinction. The extremal choices traditionally on equivalently that the laws retain their form under an arbitrary coor- offer consist of taking individuals to be nothing but bundles of prop- dinate transformation. The strong version requires that the space- erties vs. taking individuals to have a ‘thisness’ (haecceitas) that is time diffeomorphism group be a gauge group of the theory. The not explained by their properties. The gauge interpretation of GTR weak version is so called because practically any theory can, with doesn’t provide any grounds for haecceitas of spacetime points. Nor suitable ingenuity, be massaged so as to fulfill this requirement; for does it fit well with taking spacetime points as bundles of properties example, using the tensor calculus Newtonian gravitational theory since it denies that the properties that were supposed to make up the and special relativistic theories of motion can be written in gener- bundle are genuine properties. The middle way between the haec- ally covariant form. But such massaging does not guarantee that ceitas view and the bundles-of-properties view takes individuals and the theory satisfies the strong requirement; indeed, as judged by the properties to require each other, the slogan being that neither exists light of the constrained Hamiltonian formalism the gauge freedom independently of the states of affairs in which individuals instantiate various properties. Trying to apply the middle way to GTR runs into 25To get a sampling of the variety of reactions, see Maudlin (1990), Hofer and Cartwright (1993), Rynasiewicz (1994). 26For details, see my (2001b, 2002a). 16 John Earman Thoroughly Modern McTaggart the same problem as bundles-of-properties view since the gauge in- based change is to be regained, more radical measures are called for. terpretation implies that the states of affairs composed of spacetime One way to block the thrust of modern McTaggart’s argument is points instantiating, say, metrical properties do not capture the lit- to break diffeomorphism invariance by adding coordinate conditions eral truth about physical reality; rather, these states of affairs with that privilege a restricted class of coordinate systems. Such a move their subject-attribute structure are best seen as representations of a is, of course, contrary to the spirit of general relativity. However, reality–perhaps best characterized in terms of coincidence events– diffeomorphism invariance can be restored by treating the privileged that itself does not have this structure.27 coordinates as additional scalar fields ΦK , K = 1, 2, 3, 4, which are I freely admit that what I have just said may prove to be nonsense expressed as functions ΦK (xi) of arbitrary label coordinates xi. If because there is no way to escape the ambit of the traditional posi- one is sufficiently clever, it is often possible to find a way to “pa- tions on subject-attribute relation. All I am sure about here is that rameterize” the action in terms of the new variables so that what is worth exploring various escape routes and that modern McTag- emerges are the old field equations plus the coordinate conditions gart’s argument provides a good occasion for testing rival views of expressed in terms of the√ new variables. For example, the unimod- the subject-predicate relation. ular coordinate condition g = 1, where g := det(gij), can be treated in this way. When the parameterized action is varied with re- spect to the metric the resulting gravitational field equations are the 8 Restoring B-series change Einstein equations with an unspecified cosmological constant .28 When this unimodular gravitational theory is run through the con- In section 6 it was suggested that the ontological picture that emerges strained Hamiltonian formalism, it is found that if 6= 0 the Hamil- D from the deep structure of GTR is a -series of time ordered coin- tonian constraints of standard GTR are suspended in favor of weaker cidence events. The occupants of this D-series can be treated as constraints which are compatible with the existence of observables objects of predication, and postulating properties of these objects that are not constants of the motion (see Kucharˇ (1991) and Earman that come and go as one moves through the series transforms the (2002b)). Furthermore, unimodular gravitation provides a gauge in- D-series into a B-series and restores property change. There is no variant time variable which, in the case where the spacetime man- B way to prove a priori that such a restoration of -series change is ifold M is of the form Σ x R with Σ a compact three-manifold, is not feasible, but I cannot see anything in or on the horizons of cur- measured by the four-volume between a fiduciary slice Φ4 = const rent physics motivates the idea that the D-series picture of physical and ΦK (xi). However, a value of does not correspond to a par- reality is correct as far as it goes but incomplete. If familiar property- ticular spacelike hypersurface of spacetime, and for this and other
27Alternatively, one could define a “realized at” relation holding between coin- 28That is, the “cosmological constant” is constant in that it has the same value cidence events and spacetime points. Then spacetime points could be thought of as throughout spacetime but its value can vary from solution to solution. In contrast bundles of realization properties. But this seems to me to be just a less per- to the unimodular version of GTR, the action principle for standard GTR assumes spicuous way of acknowledging that spacetime points arise as individuals in a that the cosmological constant is not a dynamical variable in that its value does not subject-attribute representation of a reality from which they are absent. vary from solution to solution; see my (2002b). 17 John Earman Thoroughly Modern McTaggart reasons it does not provide the kind of time that is thought to be For each and each s let (s) be the unique point of intersection s needed to solve the problem of time in quantum gravity (see Kucharˇ of the gauge orbit with Γphys. Drag back (s) to an arbitrary (1991) and section 10). Nevertheless, unimodular gravity does serve o 1 fixed transversal surface Γphys by (s) := os ( (s)). The curve as a useful example of how B-series change can be restored–albeit o : R → Γphys describes the motion relative to the physical phase in a limited way–and it is given some currency by the fact that recent o space (Γphys, ωo). It turns out that the physical motion is of Hamil- cosmological observations indicate that our universe is characterized tonian form. by a positive value for the cosmological constant (see my (2001a)). The payoff of introducing a physical reference frame is that mo- A less sneaky and more direct route to the restoration of B-series tion in the physical phase space can involve common sense change. change seeks to identify the, or a, physical time with respect to I say “can” because it could turn out that this motion is frozen in that which there is change. What this would mean can be illustrated o the curve is degenerate, i.e. is just a point in Γphys, but at least in terms of “deparameterizing” the parametrized harmonic oscil- the possibility is open that motion in the physical phase space is not lator or more generally Hájicek’sˇ (1995, 1996b) reparametrization frozen. Officially, an observable is a quantity that is constant along invariant systems (RISs). A RIS captures in geometric terms the no- the gauge orbits or, equivalently, is a phase function on the reduced tion of a (finite dimensional) symplectic system with only first class phase space obtained by quotienting C by the gauge orbits. But the constraints and where motion is pure gauge in that each solution proposal of the deparametrizer is to abandon the official account and curve lies within a gauge orbit. A RIS can be thought of as a triple o recognize “Z-observables” defined as functions on Γphys. These new (Γ, Ω, C) where (Γ, Ω) is a (finite dimensional) symplectic system observables need not be constants of the physical motion. To return and C Γ corresponds to a first class constraint surface. The pro- to the case of the parametrized simple harmonic oscillator, taking jection ΩC of Ω onto C is a degenerate two-form with null vector Z to be ∂/∂T , where T := t (with t being “real time”), defines a field Y (henceforth called the gauge field) whose integral curves physical reference frame, and obviously the Z-observables for this are the gauge orbits. This is a very broad class–it includes all fi- frame do change with time. More generally a way to construct a nite dimensional constrained Hamiltonian systems–but of course it physical reference frame is choose a function f :Γ → R whose does not include GTR, whose phase space is infinite dimensional. restriction fC to the constraint surface C is an observable in the offi- A vector field Z on the phase space Γ of a RIS (Γ, Ω, C) is called a cial sense that fC is constant along the gauge orbits (or equivalently, a ab physical reference frame if it is of the form Z = ∂/∂T , where the Y (fC ) = 0). Then, at least locally, Z := Ω ∂bf defines a phys- s Γphys := {x ∈ Γ: T (x) = s}, s ∈ R, are are a family of global ical reference frame. In each case one would have to check to see transversals for the gauge field Y in that for each s ∈ R every gauge whether the choice of f results in an unfrozen physical dynamics. s orbit intersects Γphys but is never tangent to it. The restriction ωs As noted by Hájicekˇ (1995, 1996b) three problems arise in seek- s s of Ω to Γphys makes Γphys into a symplectic space. It is required ing real change by defining a physical reference frame. The first is 0 s s0 that for any s and s , the maps ss0 :Γphys → Γphys, given by follow- existence: a RIS may not admit any global transversal surfaces. The ing along the integral curves of Z, are symplectic diffeomorphisms. second is uniqueness: more than one physical reference frame may
18 John Earman Thoroughly Modern McTaggart exist. The third is physical motivation: what physical grounds jus- fC : C → R such that Y (fC) = 0. If one tries to use such a state tify using a physical reference frame, assuming it exists, to define to define a reference frame Z by the equation ΩC(Z, ) = dfC( ), the physical time and change? I will return to these issues shortly. uniqueness problem arises because if Z satisfies this condition, then Could it be that GTR is a kind of time-parametrized version of so does Z0 = Z + const x Y , where Y is the gauge field. Extending some more fundamental theory? The initial evidence does not en- fC off the constraint surface to a function f on all of Γ would allow courage this notion. In the case of the parametrized oscillator there Z to be uniquely defined as above by Ω(Z, ) = df( ). But there are two strong clues for how to deparametrize: first, the super-Hamil- are many such extensions and statistical physics seems to provide tonian is linear in the momentum pt but quadratic in the other mo- no reason for preferring one to another since it is concerned only menta, and second, the hypersurfaces t = const foliate C and each with states corresponding to points on C, which represent the physi- of them meets the gauge orbits exactly once. But the Hamiltonian cal states. Finally, there is no guarantee any of the physical reference constraint in GTR is quadratic in all the momenta, and in general frames associated with a statistical state defines global transversals, there is no way to foliate the constraint surface in GTR with hyper- in which case time is at best defined only locally. surfaces that meet the gauge orbits exactly once. If there is a true The moral to be drawn from this brief discussion of attempts to time hiding in the formalism of GTR, it is well hidden indeed. restore B-series change is not that the attempts cannot succeed. The Efforts to track down the elusive time in GTR can be labeled as ei- moral is rather that success can only be purchased at the price of new ther internalist or externalist according as they rely on factors purely substantive physics or new interpretational moves or both. Whether internal to GTR or allow external factors to be brought in. The prob- the price is worth paying depends in part on what we saddle our- lems that bedevil the internalist attempts have been well documented selves with by not paying the price. The next two sections are de- in Isham (1992) and Kucharˇ (1992) and will not be rehearsed here. voted to exploring that issue. Instead I will illustrate the externalist approach by reference to Carlo Rovelli’s (1993) notion of physical time as dependent on a statistical state. It is widely accepted that the direction of time is defined by 9 Learning to live without B-series change a statistical state. But here we have the more radical proposal that physical time itself is defined by a statistical state.29 In terms of RISs This section and the next explore the consequences of taking se- riously the idea that B-series change belongs not to the world in the idea is that a stationary statistical state defines a physical refer- itself–as described, say, by the reduced phase space of a constrained ence frame, relative to which there is physical change. There are technical problems here connected with the issues of existence and Hamiltonian system in which motion is pure gauge–but to the mathe- matical and perceptual representations of the world. The idea sounds uniqueness mentioned above. What one can hope to get from statis- B tical physics is a stationary statistical state in the sense of a function radical if it is advertised by the slogan that -series change is unreal. But radical sound is muted if the fine print of the advertising con- 29See also Connes and Rovelli (1994). Rovelli is now inclined towards the tains the assurances that B-series change is not a mere illusion and weaker interpretation, discussed below in section 9, on which the statistical state that what is veridical in our ordinary claims about changing prop- picks out the “clock variable” for the “evolving constants.” 19 John Earman Thoroughly Modern McTaggart
i a i i erties and values of physical magnitudes is encoded in the structure (for (q , pi) ∈ C, evaluate q˜ (q , pi) and p˜a(q , pi) and then apply 1 a i i of the world in itself. Some of the sought after assurance can be Q k to (˜q (q , pi), p˜a(q , pi))) which, by construction is constant q¯ ,p¯k gained by studying what Carlo Rovelli (1990, 1991a, 1991b) calls along the gauge orbits). The k-parameter family of such observ- evolving constants of the motion, a notion that corresponds to a para- ables, generated as the chosen clock variables range over their al- metrized family of the coincidence type observables introduced by lowed values, constitutes an evolving constant of the motion. For a Kretschmann and Komar (recall section 5). In terms of the con- some fixed intrinsic state (˜qo , p˜ao), the sub-family that is generated k strained Hamiltonian formalism the general construction of evolving as the values q¯ , p¯k of the clock variables vary gives the relative evo- 30 ˚ 1 ˚ 1 a 1 constants for the finite dimensional case goes as follows. Denote lution Q k := Q k (˜q , p˜ao) of q against the M clock variables q¯ ,p¯k q¯ ,p¯k o the original phase space by Γ(qi, p ), i = 1, 2, ..., N. Suppose that k a i {q , pk} for the solution picked out by the intrinsic state (˜q , p˜ao). a i i o there are M constraints. Choose coordinates (˜q (q , pi), p˜a(q , pi)), The toy example of the parametrized one-dimensional harmonic ˜ a = 1, 2, ..., D, D = N M, for the reduced phase space Γ (recall oscillator serves as a useful illustration. Here N = 2–there is are that this is the quotient of the constraint surface C Γ by the gauge two configuration variables x and t–and the original phase space is orbits). The (2N M)-dimensional constraint surface C can then be Γ(x, px, t, pt). M = 1–the one constraint is the super-Hamiltonian coordinatized by (˜qa, p˜ , tm), for some tm(qi, p ), m = 1, 2, ..., M. a i constraint H = H + pt which generates the motion. The two- m The symbol t has been used because these variables, which serve dimensional reduced phase space Γ˜ can be coordinatized by (A, B) as coordinates of the gauge orbits, are called the “internal time vari- where the general solution of the unparametrized oscillator is x(t) = ables” or “clock variables.” In the formal construction the choice A cos(ωt) + B sin(ωt). And one can choose t as the clock variable, of such variables is somewhat arbitrary, and for some choices there giving coordinates (A, B, t) for the three-dimensional constraint sur- is no guarantee that the values of these variables will behave like face C. Corresponding to the position variable x–which is not an the reading of anything that one would ordinarily call a clock. For observable in the parametrized formulation–there is, for each value sake of illustration, assume that the coordinatization (˜qa, p˜ , tm) of a t¯of t, the Dirac observable Xt¯ := A cos(ωt¯) + B sin(ωt¯). The one- C can be achieved by choosing for the tm a subset {qk, p } consist- k parameter family {Xt¯ : t¯∈ R} is an evolving constant. k ing of M of the original phase space variables. For fixed values q¯ This toy example is somewhat misleading since there is an ob- k k and p¯k of these variables, the M equations q =q ¯ and pk =p ¯k vious and natural choice for the clock variable–t–which happens to determine, at least locally, a transversal, i.e. a hypersurface of C have the properties we ordinarily associate with time. But in the that intersects each M-dimensional gauge orbit exactly once. Then general case of M constraints there are many different choices of M 1 1 for some other variable, say q , its value Q at the intersection can variables to serve as the clock variables, and once this choice has 1 1 a be written Q k = Q k (˜q , p˜a). Each of these quantities is a q¯ ,p¯k q¯ ,p¯k been made there are M choices of which M 1 clock variables to genuine Dirac observable; for each defines a function from C to R hold fixed while the Mth is singled out as the temporal index. There is no a priori guarantee that the chosen temporal index will behave 30Here I am following the presentation of Montesinos (2001) and Montesinos and like a good clock variable or bear any direct relationship to our ex- Rovelli (2001). 20 John Earman Thoroughly Modern McTaggart perience of time. Rovelli’s statistical state proposal introduced in world; rather, it takes the form of showing how our perceptual and section 8 as a way to implement deparametrization can now be rein- cognitive apparati represent in a more or less faithful way a structure terpreted as a way of picking out the variable that is to serve as a that exists independently of us in the world. In this respect the situa- good clock variable for an evolving constant rather than as defining tion is more properly called neo-Hegelian than neo-Kantian,32 and it a deparametrization. intersects, though only partially, with (Grandpop) McTaggart’s po- We can now see how unchanging observables code up informa- sition. Recall that McTaggart, inspired by his admired Hegel, took tion about what is ordinarily taken for B-series change. For instance, the world of physical events to be arranged in an intrinsic, observer in the oscillator example, any true statement about the motion of the independent C-series. But according to McTaggart this C-series is unparametrized oscillator for the solution determined by (Ao,Bo) ∈ non-temporal, and it is by projecting a transient now onto this C- ˜ ˚ Γ has a counterpart in a true statement about the family {Xt¯ : t ∈ R} series that McTaggart thought that we create the B-series and the of evolving constants. This gives some initial confidence that clas- illusion of change.33 sical physics can be done without having to unfreeze the dynam- One obvious way to start the sought after explanation of our per- ics and to reintroduce common sense B-series change. Whether all ception of B-series change is to use the proper time along the world the information needed to do quantum mechanics and statistical me- lines of a suitable system of world lines of observers as the index of chanics can be so encoded in the frozen dynamics will be discussed the family of evolving constants. It is less obvious how to complete in the next section. the explanation. “World line” is not a primitive notion–my world It remains to explain our perceptions of B-series change. In the line, and yours, is defined in terms of the values of local fields on toy example of the parametrized oscillator the explanation would spacetime, and such values do not constitute genuine observables ˚ ˚ in the official sense, though, of course, they may be ingredients of have to show how the difference (Xt¯2 Xt¯1 ) in the values of two constants of the motion for the solution picked out by the frozen coincidence observables. Insofar as a physical explanation must be ˜ state (Ao,Bo) ∈ Γ is interpreted by critters such as ourselves as genuine property change–the change in the position of the oscillator 32Kant wrote: “I deny to time all claim to absolute reality; that is to say, I deny from t = t¯1 to t = t¯2. One could go Kantian and simply pos- that it belongs to things absolutely, as their condition or property, independently of tulate a human faculty that does the job.31 A real explanation, as any reference to the form of our sensible intuition ...” (B 52), and “Time is there- opposed to mere postulation, would show how the physics of the 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- objects and the psychology of critters such as ourselves combine self, or be intuited by another being, the very same determinations which I now in such a way that we perceive the world as filled with B-series represent to ourselves as alternations would yield knowledge into which the repre- change despite the fact that no genuine physical magnitude changes sentation of time, and therefore also of alternation, would in no way enter.” (B 54) over time. The explanation does not take the form of showing how 33For McTaggart the B-series is parasitic on the A-series: “The B-series, how- we make egregious errors or mistakenly project an illusion onto the 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 that there can be no B-series when here is no A-series, since without the A-series 31This was Kant’s theft-over-honest-toil strategy for “solving” tough problems. there is no time” (1927, Sec. 312). 21 John Earman Thoroughly Modern McTaggart couched entirely in terms of genuine observables, the sought after superable technical and interpretational problems for the quantum explanation cannot be purely physical. This is not to say that the ex- formalism (see Kucharˇ (1992)). For our purposes the key question planation must be mentalistic in some sense that rests on a mental- is whether quantization is stymied until the dynamics is unfrozen physical dualism. But it is to say that it will have to involve rep- by deparameterization or some other means. To keep the discussion resentations in terms of quantities that are not part of the intrinsic manageable I will abandon the formidable case of quantum gravity physical description of the world. in favor of my running toy example. Consider what would happen if one tried to quantize à la Dirac the parametrized simple harmonic oscillator without decoding the con- 10 Reconciling the absence of B-series change stants of the motion in such a way as to reintroduce B-series change. Instead of the single configuration variable x one now has a pair with quantum mechanics and statistical of configuration variables, x, t. And instead of building a Hilbert mechanics. space of L2 complex valued functions (x), the Hilbert space would have to be constructed from L2 functions Ψ(x, t) of both x and t. In the preceding section I tried indicate why the idea that B-series Following the program of Dirac constraint quantization, the super- change exists not in the world in itself but only in representations H = H + p of the world is not as radical as it sounds. But apparently the idea Hamiltonian constraint is applied by turning t into an operator Hˆ = Hˆ +p ˆ and requiring that the physical states be anni- comes unstuck when one tries to apply it to quantum mechanics and t statistical mechanics because, it has been contended, both of these hilated by this operator: types of mechanics require an unfrozen dynamics. I will explain ˆ ˆ some of the considerations behind this contention, and at the same HΨ = HΨ +p ˆtΨ = 0 (1) time indicate why I think that the matter is far from being settled. Equation (1) is the Schrödinger equation for Ψ(x, t) if t is identified Consider the program of Dirac quantization for constrained Ham- as time. But as Unruh and Wald (1989) note, there seems to be an iltonian systems (see Dirac (1964) and Henneau and Teitelboim obvious problem here; namely, it doesn’t seem possible to build a (1992)). In barest outline form, the constraints are promoted to op- Hilbert space out of solutions of (1) since presumably there aren’t erators on some appropriate Hilbert space, and the physical sector any solutions that are square integrable with respect to x and t, at of this space is spanned by state vectors that are annihilated by the least not with respect to the naive measure dxdt. They conclude operator constraints. When applied to the Hamiltonian constraint of that in order to quantize the parametrized oscillator it is necessary GTR the requirement that the Hamiltonian operator constraint (for to identify a time variable that “sets the conditions” for a measure- an appropriate choice of operator orderings) annihilate the physical ment. This is a little too quick, as shown by Ashtekar and Tate state vectors produces what is called the Wheeler-DeWitt equation. ˆ ˆ ˆ ¯ Po ¯ This equation is a kind of degenerate Schrödinger equation in which (1994). The family of operators Xt¯ = Xo cos(ωt) + ω sin(ωt)– there is no time dependence, a feature that is thought to pose in- which are the quantum operator analogues of the position evolv- ˆ ˆ ˆ ing constants–and Pt¯ = ωXo sin(ωt¯) + Po cos(ωt¯)–which are the 22 John Earman Thoroughly Modern McTaggart corresponding family of momentum operator evolving constants– issue. A positive verdict in favor of evolving constants would add commute with the quantum super-Hamiltonian constraint. It turns support to the line I sketched in the preceding section. By the same out that requiring that this family of operators be self-adjoint fixes token a negative verdict would indicate that either the dynamics of the measure (x, t)dxdt for the inner product of solutions to (1) to GTR must be unfrozen if gravity is to be quantized or else that grav- be of the form (t)dxdt, where (t) is unconstrained. One is there- ity cannot be quantized by the canonical Hamiltonian procedure. fore free to choose (t) to get the desired normalization. Thus, as Turning now to statistical mechanics, it might seem even more Ashtekar and Tate conclude, “it is not necessary to isolate time in obvious than in quantum mechanics that unfreezing the dynamics is order to construct the Hilbert space of physical states” (1994, 6468). necessary since without identifying an unfrozen time variable and a Montesinos, Rovelli, and Thiemann (1999) have studied the Dirac corresponding non-zero Hamiltonian, ordinary statistical mechan- quantization of another toy model which more closely mimics the ics is not possible: Without time and change, how can thermal- constraint structure of GTR in that it has two non-commuting Hamil- ization occur? What replaces the familiar classical Gibbs equilib- tonian constraints. (Recall that GTR has a different Hamiltonian rium state exp( H), where H is the Hamiltonian and constraint for each point of space.) It was found that the require- is the inverse temperature? Nevertheless, Montesinos and Rovelli ment that the quantum operator analogues of a family of classical (2001) have argued that a Boltzmann-like approach is possible with- gauge invariant evolving constants be self-adjoint determines the out deparametrization. The idea is that even though the dynamics inner product for an apparently sensible Hilbert space of physical of a parametrized system is frozen, an appropriate analogue of ther- states. So again quantization does not imply that it is necessary to malization can be obtained by taking a large ensemble of identical find a time variable with respect to which the dynamics is unfrozen.34 parametrized systems and turning on a “small interaction” among Some skepticism has been expressed about whether the evolving them. One would expect that with the interaction turned on, only a constants scheme can lead to an adequate quantization of GTR via small number of global macroscopic quantities will be conserved. the Dirac canonical quantization scheme (see Hájicekˇ (1991)–and In place of the usual equilibrium statistical state one will have a Rovelli’s (1991b) reply–and also Kucharˇ (1992, 1993)). It is too state determinedR by the requirement that it maximize the entropy early to make predictions about the outcome of this highly technical S := k Γ˘ ln dz, where the integration is over the phase space ˘ 34 Γ = Γ1 x Γ2 x ... of the ensemble of identical parametrized sys- The 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 that tems, subject to the requirements that is normalized and that it re- quantizing using different families of evolving constants leads to different quan- turn the observed expectation values hOki for the conserved macro- 1 P tizations that are not equivalent even up to the usual operator ordering ambi- scopic quantitiesR Ok. then has the form Z exp( k khOki), guities. However, this should not obscure the conceptual point that the evolv- where Z := Γ˘ dz. The application of this scheme to the example ing constants idea shows how quantization is possible without unfreezing the dy- of Montesinos, Rovelli, and Thiemann (1999) has been worked out 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 the in Montesinos and Rovelli (2001). The parameters k that charac- prior isolation of time. terize the equilibrium state generally do not include temperature as
23 John Earman Thoroughly Modern McTaggart ordinarily conceived, with the upshot that from the point of view of structing a quantum theory of gravity. Its solution will most likely the proposed approach “temperature plays no fundamental role in require the contributions of both philosophers and physicists. While the statistical analysis. It may not be defined, or, if it is defined at dismissing or maneuvering around the problem remains an option, it all, temperature is just one of the intensive macroscopic parameters is not an option that is easy to implement. The alternative of taking characterizing the equilibrium configuration of the system” (Mon- the problem seriously and working out the implications for physics, tesinos and Rovelli (2001, 567)). Whether or not these radical ideas for the philosophy of science, and for the philosophy of mind is the will yield fruit when applied to statistical mechanics of black holes, course I recommend.35 It is a course fraught with many difficulties of matter interacting with gravity, etc. remains to be seen. But the and unpleasant surprises. But the potential payoff seems to me to exploration of the implications for statistical physics of modern Mc- make the effort worthwhile. Taggartism seem well worth pursuing.
11 Conclusion
There is indeed a real problem about change. But it is not the one posed by Grandpop McTaggart and debated endlessly in the philo- sophical literature. The real problem is not concerned with McTag- gart’s A-series, Broad’s Becoming, or any of the other metaphysical extravagances that dot the philosophical literature on time. The real problem about change is the result of substantive discoveries in mod- ern physics–in particular, Einstein’s GTR–which seem to imply that there is no temporal change in any genuine physical magnitude, i.e. there is no B-series or property change. This is not to say that in the picture of physical reality that emerges from these discoveries that there is no change tout court; for there is still a temporally ordered series of coincidence events (the D-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 property change, and physics provides no motivation for positing changing properties of the occupants of the series. 35An even more radical solution to the problem of time in classical GTR and Understanding the real problem of change requires understand- quantum gravity is given by Barbour’s (2000) version of presentism. See But- ing the foundations of classical GTR and the issues involved in con- terfield (2002) for a critical review. 24 John Earman Thoroughly Modern McTaggart
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