The standard view amongst physicists is that a gauge-theoretical understanding of diffeomorphism invariance implies that the localization of fields is relational (i.e. grounded by relations between fields rather than manifold points), and that this in turn implies spacetime relationalism, or at least anti-substantivalism. Rovelli sketches the supposed implication as follows:
[Diffeomorphism invariance] implies that spacetime localization is relational, for the following reason. If (^, Xn) is a solution of the equations of motion, then so is (0(^), 0(Xn)) [where 0 is a diffeomorphism]. But 0 might be the identity for all coordinate times t before a given to and differ from the identity for some t > to. the value of a field at a given point in M, or the position of a particle in M, change under the active diffeomorphism 0. If they were observable, determinism would be lost, because equal initial data could evolve in physically distinguishable ways respecting the equations of motion. Therefore classical determinism forces us to interpret the invariance under Diff^ as a gauge invariance: we must assume that diffeo-morphic configurations are physically indistinguishable. ([Rovelli, 2000], p. 3779)
This is then taken to imply relationalism since diffeomorphic configurations are only distinguished by their localization on the manifold. They are different in the sense that they ascribe different properties to the manifold points. However, if we demand that localization is defined only with respect to the fields and particles themselves, then there is nothing that distinguishes the two solutions physically ...It follows that localization on the manifold has no physical meaning ...In GR, general covariance is compatible with determinism only assuming that individual spacetime points have no physical meaning by themselves ...Reality is not made up of particles and fields on a spacetime: it is made up of particles and fields (including the gravitational fields), that can only be localized with respect to one another. No more fields on spacetime: just fields on fields. ([Rovelli, 2004], pp. 70-1)
Hence, the 'physical' aspects of a system are not given by specifying a single field configuration, but instead by the "equivalence class of field configurations
...related by diffeomorphisms" (ibid.), a geometry. The observables of such a system are then given by diffeomorphism invariant quantities. Such specifications of states and observables are clearly independent of any background spacetime: only gauge invariant quantities are to enter into such specification, and any reference to a background metric (via, for example, fixed coordinates or functions on M) yields non-gauge invariant quantities. Thus, diffeomorphisms change the localization of dynamical fields on M; this is represented in the Hamiltonian scenario by the action of the constraints. However, the localization is a gauge freedom, so any states or quantities involving localization to points will not be measurable. Rovelli, like so many others, sees a direct connection between taking the equivalence class of metrics and relationalism about spacetime since because the equivalence class appears to imply that the metric field is entangled with spacetime points. (But this entanglement can work for both relationalists and substantivalists alike, as we saw earlier.)
Rovelli is not alone in interpreting the hole argument as an argument for re-lationalism. As I said, it seems to be the majority view amongst physicists. For example, Lee Smolin writes that "the basic postulate that makes GR a relational theory is" that "[a] physical spacetime is defined to correspond, not to a single (M, gab, f), but to an equivalence class of manifolds, metrics, and fields under the action of Diff(M)" (, p. 206). The idea here is that removing the symmetries (by 'modding out' by the diffeomorphisms) is taken to correspond to relationalism. In other words, relationalism is being aligned with reduction: we saw in Chapter 2 that this is a non sequitur; we shall return to the matter again in Chapter 8.
What status are we to attribute to the manifold once remove dependency upon its coordinates? Rovelli suggests that it is an "auxiliary mathematical device for describing spatiotemporal relations between dynamical objects" ([Rovelli, 2000], p. 3780). It is not without usefulness. Of course, spacetime coordinates enter into many areas of physics, especially mechanics and field theories, i.e. as positions of objects (particles, string excitations, etc.) or as the argument of a local field operator. Many physicists believe that general relativity rules out just such absolute local quantities—I agree. This is, again, seen to follow from, or imply, the practice of taking an equivalence class of manifolds and metrics under diffeomorphisms as the correct description of a world of general relativity. Smolin claims that a consequence of this view is that there are no points in a physical spacetime ...[since] a point is not a diffeo-
morphism invariant entity, for diffeomorphisms move the points around.
There are hence no observables of the form of the value of some field at a given point of a manifold, x. ([Smolin, 2000], p. 5)
The latter point, that there are no local (i.e. localized to a particular spacetime point) observables in general relativity, is, as I have said, perfectly true of course. I think this is the real 'lesson' of the hole argument. Unfortunately, Smolin, like Rovelli, explicitly draws relationalist conclusions from the fact that the observables of general relativity are relational. Firstly, it does not follow that there are no points: that the observables are indifferent to matters of spacetime point role does not imply there are no spacetime points. Secondly, the fact that points of the manifold are problematic does not mean that there is no other notion of a point on the table. As Robert Dicke remarks, describing the view of J.L. Synge:
general relativity describes an absolute space ...certain things are measurable about this space in an absolute way. There exist curvature invariants that characterize this space, and one can, in principle, measure these invariants. Bergmann has pointed out that the mapping of these invariants throughout space is, in a sense, labeling of the points of this space with invariant labels (independent of coordinate system). These are concepts of an absolute space, and we have here a return to the old notions of an absolute space. ([Dicke, 1964], pp. 124-5)
Here, as we mentioned in the previous chapter, the idea is to get a set of coordinate conditions that allow one to define a set of intrinsic coordinates. One constructs the complete set of scalars from the metric and its first and second derivatives, which for the matter-free case leaves four non-zero scalars that take different values at different points of the manifold. Hence, one achieves a complete labeling of the manifold in an intrinsic gauge invariant way—this follows from the fact that we are dealing with scalars which do not change their values under diffeomorphisms. These points can then be used to localize quantities which become gauge invariant as a result of the gauge invariance of the scalars. For Synge, the only difference between this space and Newton's is that the geometric properties of the Einsteinian space are "influenced by the matter contained therein"—that is, the latter is background independent. Of course, since we are dealing with invariants of the metric here, it is open to the relationalist to call this a 'material' field. So continues the interminable interpretive tug-of-war! The point is, it is always open for the sub-stantivalist to construct an alternative interpretation within the framework that is supposed to be fit only for relationalists.
Connecting relational localization explicitly to the observables, Rovelli writes that since the only physically meaningful definition of location within GR is relational ...GR describes the world as a set of interacting fields including g^v (x), and possibly other objects, and motion can be defined only by positions and displacements of these dynamical objects relative to each other. [...] All this is coded in the active diffeomorphism invariance ...of GR. Because active diff invariance is gauge, the physical content of GR is expressed only by those quantities, derived from the basic dynamical variables, which are fully independent from the points of the manifold. [.] [Diff invariance] gets rid of the manifold. ([Rovelli, 2001], p. 108)
This is a fairly common view too, as I intimated above, but it is also a non sequitur: substantivalism is perfectly compatible with the view that observables of general relativity are relational, with a gauge invariant conception of the observables of the theory, the sophisticated substantivalists have demonstrated this—I gave two strategies above. There is no necessity to "get rid of the manifold" to help oneself to gauge invariance: Saunders' PII and his preservation of PSR could be wielded by the determined substantivalist here. Rovelli assumes that the view that the observables are relative beasts describing the relative location and evolution of dynamical objects implies relationalism about spacetime, but all this shows is that the observables are indifferent to matters concerning spacetime points. The gauge variant is by no means restricted to physicists. Weinstein too makes the same mistake: "formulating observables in [general relativity] as diffeomorphism-invariant objects eliminates any reference to the underlying spacetime manifold (arguably making the theory non-absolutist)" ([Weinstein, 2001], p. 68)—I assume that by "non-absolutist" Weinstein means "non-substantivalist". Once again, we have an interpretative underdetermination: both substantivalists and relationalists can lay claim to this feature of gauge invariance.197
However, aside from the fact that Rovelli wrongly aligns his position to rela-tionalism, I think that there is much to recommend it. However, I prefer a conception based on neither substantivalism nor relationalism, nor on a distinction between matter and spacetime. The best way to overcome the problems raised by the symmetry arguments considered so far, and still to come, is to 'go structural' and eschew the 'subject-based' foundations upon which the traditional interpretations are predicated: the relational gauge invariant (or, what I shall refer to from now on as the correlation) view of observables is formally correct but the interpretation is wrong (based as it is on a position, relationalism, that is well under-determined by this view of the observables). Thus, a kosher observable will be a correlation between field values (e.g. the value of the curvature of the gravitational field where and when the electromagnetic field strength takes on the value F198). These contain no dependence on the manifold, and determine locations and times via relations between quantities. The relational aspect suggests that the correlation can be broken down into components that have some ontological weight independently of their role in the correlation.199 This is what a structuralist position denies. I elaborate this view further in the next chapter where I use it to defend Rovelli's evolving constants of motion and partial observables strategies for dealing with the problem of time from objections that focus upon the relational interpretation. I then defend the position in more detail, and compare it with other views, in the final chapters.
197 Of course, the gauge invariance here takes on a special form since it is mixed up with background independence. Strictly speaking, it is the way these two interact that lies underneath the various claims that relationalism is uniquely selected. Because there are no background fields we get the diffeomorphism symmetry from the manifold; hence, the states and observables of the theory must be constructed entirely from the dynamical degrees of freedom.
198 There are obvious symmetry problems here: if there are parts of the field with the same value then the approach fails. This is a problem with the Bergmann-Komar approach too: it rules out symmetries. However, one can form the observable with more than a pair of fields; one can bring in other degrees of freedom too.
199 Indeed, we see that this forms a class of objection to these positions in the context of the problem of time (cf. §7.4.2).
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