What Is an Observable in General Relativity

It is a curious fact about the hole argument that the indeterminism is not an observable feature; it is not an empirical kind of indeterminism that we could in any way detect. Even if it was true that general relativity was indeterministic in the sense that Earman and Norton say the manifold substantivalist is committed to, it is of such a strange kind that we could never tell one way or the other whether we lived in a world that ran according to such a scheme. For example, Hoefer writes:

Note that this indeterminism is just about what individual points will underlie what material processes (matter and metric fields). It does not entail a failure of determinism in terms of the nonindividualistically-expressed happenings in space and time, that is, nothing observable is made indeterminate by the hole argument. ([Hoefer, 1996], p. 9)

Why is this so? It clearly depends on what we take 'observable' to mean. Recall the two fields, g and $*g, that resulted from an active diffeomorphism.187 The fields are mathematically distinct: they are spread differently over the manifold, such that g(x) = 0*g(x). But this is the only way they are distinct, only by their localization relative to the manifold—i.e. with respect to their absolute location. The crucial question is: do they represent the same physical situation, the same possible world? As Norton explains, "It would be very odd if they did not. Both systems of fields agree completely in all invariants; they are just spread differently on the manifold. Since observables are given by invariants, they agree in everything observable" ([2003], p. 114). But what things are observable? According to Einstein, following Kretschmann's bashing, only spacetime coincidences of fields and particles. This rather narrows down the space of observables, perhaps far too much, but it avoids the indeterminism and the unmeasurability of the quantities defined with respect to the manifold's points (by simply doing away with such quanti

187 By active diffeomorphism here I mean a mapping of the manifold to itself that preserves the topological and differential structure of the manifold. We made use of the carrying along (by the diffeomorphism) of geometric object fields on the manifold, such that if our diffeomorphism sends the point x to the point y then field values at y (post-diffeomorphism) look the same as they did at x (pre-diffeomorphism). Or, as Rovelli puts it, "[a] field theory is formulated in manner invariant under passive diffs (or change of co-ordinates), if we can change the coordinates of the manifold, re-express all the geometric quantities (dynamical and non-dynamical) in the new co-ordinates, and the form of the equations of motion does not change. A theory is invariant under active diffs, when a smooth displacement of the dynamical fields (the dynamical fields alone) over the manifold, sends solutions of the equations of motion into solutions of the equations of motion" ([Rovelli, 2001], p. 122).

ties). But the modern answer is pretty close: not necessarily coincidences between particles and fields, but correlations between field-values (one of which will be the gravitational field)—the latter presumably include the former.

This, or something analogous, was the case with the Leibniz-shift argument, permutations of indistinguishable quantum particles, and the indeterminism of Maxwell's theory. All agree that the theories are indeed deterministic or unprob-lematic at the level of empirically observable ontology—putting aside quantum indeterminism of course—and that we can make well confirmed predictions within each theoretical framework. The same goes for general relativity too; it has yet to be disconfirmed in any of its predictions about the behaviour of empirically observable objects. The equations of motion of the theory of general relativity are sufficient to propagate all empirically observable components of that theory. The problem concerns the empirically unobservable (unmeasurable188) ontology (if, indeed, we take there to be such): which individuals play which roles in the structure? If we chose as our motto 'what we cannot see cannot hurt us', there would be no problem with any of the cases I have examined so far. But surely we have progressed beyond such straight-jacketed empiricism? Perhaps, but any interpretation that takes a stand with respect to the ontological status of these "individual" elements—such as spacetime points and absolute location with respect to them—on the basis of physics will have the spectre of the Quine-Duhem problem to deal with: there will be multiple incompatible interpretations compatible with the theory and the evidence.

Observables are generally understood to be those quantities described by physical theories that are measured in physical interactions between systems; but they go beyond the empirically observable, for is no necessity that we can observe them. They encode information about the state of a system, and their values should be able to be predicted by the theory. Clearly, some of the candidates for observable ontology cannot be predicted, for one has at best a range of possible values connected by symmetry; one cannot determine the unique value from within this range. What I have been pushing for is an indifference or insensitivity of the laws of physics (specifically concerning observables) to certain kinds of unobservable ontology (qualitatively indistinguishable individuals). Specifically, those involving elements (objects) connected by gauge-type symmetries, such that if those elements were to appear permuted in different scenarios then the scenarios would be indistinguishable. In other words, the observables should not register haecceitistic differences—but this does not imply that there are none, simply that the physics is, or ought to be, a qualitative enterprise. This is borne out by gauge invariance principles, where this condition is built-in: in gauge theories, the observables give the same value on such elements, so that for some observable O and elements x and y related by a gauge transformation, i.e. x ~ y, we have O(x) = O(y). General relativity is a gauge theory, with the gauge freedom given by the dif-

188 Recall that the unmeasurability stems from the fact that the local values of certain fields are underdetermined by the theory and the world's qualitative properties and relations; since there will be many assignments of values to the fields that are compatible with the equations of the theory and the empirical structure (e.g. the electromagnetic field in the case of Maxwell's theory). Thus, one will not know whether one has measured A^ or A^ + df in the electromagnetic case, so one cannot view measurement as possible in this case: what is the result of the measurement? No measurement (observable) could ever distinguish between these cases.

feomorphisms of spacetime; a diffeomorphism's action on the fields amounts to a gauge transformation. However, in order to make proper sense of this proposal, we need to know what the observables are. I have dealt with the case of elec-tromagnetism already, and showed that there were a number of candidates; if we wanted to escape the indeterminism we had to make them gauge invariant (invariant under U(1)-transformations of the fields). In this section I discuss the much more complicated problem of observables in general relativity.189 In this case the observables will have to be diffeomorphism-invariant if the theory is to avoid the indeterminism. Since the diffeomorphisms comprise the gauge part of the theory, such observables will be thereby classed as gauge invariant as well. Any fields related by diffeomorphism will thus be understood as the same as far as the observables of the theory are concerned (i.e. as far as the physics goes). Since the observables will be the same in such cases, and since just such cases comprise the problem cases of the hole argument (namely metrics painted differently, yet diffeomorphically, on to the points of M), the hole argument is avoided. Let us discuss this resolution further, before considering its philosophical consequences.

One of the first things we notice about when we consider general relativity as a gauge theory is that the metric variables with respect to which the hole argument is defined do not class as observables, and so any local (i.e. defined with respect to manifold points190) quantity constructed from the metric cannot be observable either.191 Why not? Because the observables must be indifferent to permutations of the points, for such permutations (invertible ones) comprise the gauge freedom in the theory. The diffeomorphisms underlie the general covariance of general relativity, and this tells us that the models that are thus related will be indistinguishable as far as quantities not connected to the pointwise location of the fields are concerned. Therefore, the hole argument fails to get a grip: general relativity isn't meant to predict values for these quantities, nor for any quantities localized to points of the (bare) manifold regardless of how well we can specify them in connected regions of spacetime. The kinds of quantity that do class as observables in general relativity are relational (or highly non-local), much as Einstein argued, though not quite so narrowly defined. (I agree with this classification, but below I tame the view of many physicists that this implies relationalism about spacetime.) Let us say a little more about these observables, and give some examples that connect with measurements.

189 I only discuss the problem briefly here, focusing specifically on how the problem relates to the hole argument. In the next chapter I discuss the relations between the observables, the hole argument and the problem of time. This section can be seen as a bridge connecting the hole argument with the problems of time since those problems result from the application of the resolution of the hole argument presented in this section. In the final chapter I home in on ontological issues.

190 There is a proof (for the case of closed vacuum solutions of general relativity) that there can be no local observables at all [Torre, 1993]—'local' here means that the observable is constructed as a spatial integral of local functions of the initial data and their derivatives.

191 Though, you will recall, the metric does contain invariant components. However, these are best used to define local events and coordinates, rather than functioning as self-contained observables. One localizes observables with respect to these invariants.

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