The Hole Argument The View From Gauge Theory

Recall that a model of general relativity is given by a triple M = (M,g^v, T^v)— where M, g^v, and T^v represent the spacetime manifold, the metric field, and the stress-energy field respectively. Such models are taken to represent the physically possible worlds of general relativity when they satisfy the field equations. The hole argument setup demands that (M,g^v) possesses a Cauchy surface, "so that the environment [is] as friendly as possible for determinism" (Earman [1989], p. 179).143

Although it is not often presented in such terms, the hole argument of Earman and Norton is based upon the initial-value formulation of general relativity. This allows one to consider general relativity as describing the evolution in (parameter) time of an initial data set comprising the canonical variables at an initial time. The initial value problem requires that spacetime (M,g^v) is globally hyperbolic. With this condition satisfied, we are free to foliate (M,g^v) by Cauchy surfaces St (parametrized by a global time function t). Viewed in this way, the Lorentzian spacetime metric g^v on (M,g^v) induces a (Riemannian) spatial 3-metric qab on each of the St, and also induces its determinant |q| and its inverse qab. One obtains the spacetime 4-geometry by selecting a point on St, specified by coordinates xi, and displacing it normally to the slice. The change in proper time dr of the point is given by the lapse function N, so that dt = Ndt. The spatial coordinates will generally be shifted in such a displacement, giving: xi(t + dt) = xi(t) - Ni dt, where Ni is the shift vector.

This formulation views spacetime as representing the history of a Riemannian metric on a hypersurface. The dynamical variable is the 3-metric on the hypersur-face.144 There are many details I have skipped over here, but we have the essential

143 A Cauchy surface is a hypersurface that intersects every non-extendible timelike curve once only. This restriction on the models allows one to pull back a global time function t: M ^ R, where the level surfaces of t are Cauchy surfaces, and such that t increases along the future direction of a timelike curve.

144 There are other formulations that begin by writing down general relativity in terms of a connection, in which case the connection on a hypersurface is the dynamical variable. See [Rickles, 2005] for the hole argument translated into these other formulations.

ones required for the hole argument (a full, technical account can be found in Wald [1984], Ch. 10). General relativity should, then, be able to predict the evolution in parameter-time of the spatial metric on a hypersurface. The hole argument says that, according to the manifold substantivalist's conception of spacetime, this is not possible: general relativity cannot determine which future point will underlie a certain field value. In general, local field quantities cannot be predicted.

The argument is based upon a conception of determinism whereby agreement about the initial data on some initial hypersurface should suffice to uniquely pick out a single set of data on some future hypersurface. This is violated in the following way. Consider the Cauchy data (S,g) on an initial (Cauchy) slice St (t = 0). To the future of St (t > 0) define a 'hole' H c (M — {St: t < 0}), such that within H the gravitational field is non-zero and any non-gravitational fields are set to zero— that is, H is as empty as we can get in the context of general relativity. Then define a hole diffeomorphism so that acts as the identity on the exterior of the hole (at x e (M — {St: t < 0})), and smoothly differs from the identity on the hole's boundary and in the interior of the hole (at x e H)—i.e. 4»n is a diffeomorphism with compact support. Thus, we see that q(x) = q(x) (Vx e (M — {St: t < 0})) but q(x) = q(x) (Vx e H).

The general covariance of Einstein's field equations gives us the following equivalences: (1) q' = q and (2) q'(4>H.x) = q(x). The first equivalence simply means that a metric (which solves the fields equations) and its drag-along are both solutions. The second equivalence is a local equivalence; it says that the metrics are equivalent when the dragged-along metric is evaluated at the dragged-along point. However, the metrics at the same point are not equivalent; i.e., q'(x) = q(x). General covariance therefore implies that there exists a pair of solutions that agree up to an initial data slice but diverge thereafter: different evolutions of the metric into H are compatible with the initial data plus the field equations.145

The question is whether or not q and q' are to be regarded as representing physically distinct solutions (or distinct physically possible worlds). Recall that the manifold substantivalist thinks of the points of M as having their identity and individuality settled independently of any fields (representing matter or energy sources) defined with respect to them. Therefore, the solutions are irrelevant to the identity of the points, and we can speak of the same point entering relations with both q and q'. The relations are going to be different between these cases since, in general, q'(x) and q(x) represent distinct assignments of geometrical properties to the same point x. It looks as though the manifold substantivalist is going to have to say that the diffeomorphic solutions do indeed represent distinct physical possibilities.

What this implies is that a complete specification of the fields outside of the hole (given by the Cauchy data on a hypersurface) is not sufficient to uniquely determine the evolution of the fields within the hole. Hence, one cannot solve uniquely the Einstein field equations describing the fields within the hole: the evolution into the hole is underdetermined by the field equations. Instead, one has an

145 Equivalently: we can always find a pair of models (M,g) and (M,g') (such that g' = that agree up to some St (an initial temporal segment) but diverge thereafter. That g' = g implies that if either of these models is admissible then so is the other.

infinite class of solutions describing diffeomorphic metric fields, all compatible with the initial data. The statement that diffeomorphically related solutions represent one and the same physical state of affairs is called Leibniz Equivalence by Earman and Norton; they say that it is just this equivalence that the manifold sub-stantivalist must reject. It is this rejection that leads to the indeterminism.146

How does this manifest itself in a problematic way? Rovelli and Gaul give a nice example demonstrating the underdetermination of the metric field by the field equations:

Take for example two points P, Q g M and consider two metrics g^v(x) and g^v(x), which are both solutions of [Einstein's field equations]. Then the distance d between P and Q computed using the two metrics is different, i.e., dg(P, Q) = dg(P, Q). We have two distinct metrics on M which both solve Einstein's equations. ([Gaul and Rovelli, 2000], pp. 303-4)

Though we don't have a complete breakdown of determinism,147 we do have some problem of predictability (concerning which points 'sit under' which bits of the fields, or are assigned which field values), and some problem of indeterminism. The problem is that we cannot uniquely determine the evolution of any fields into the hole if we understand the equivalence of class of metrics (under diffeomor-phisms) as representing a class of distinct possibilities. This leads to the problem outlined by Rovelli and Gaul. Of course, the problem they mention fades away if we evaluate the dragged-along metric at the dragged-along (image) point. Then the computed distances would be identical. But this supposedly involves a relationalist move, for the metric field (and other fields) are taken to define the point's identities.148 However, as we've seen, the manifold substantivalist attributes an identity to the points over and above the fields, and must consider the evaluation of the metrics at the same point as a possible operation. This seems to be mandated by the active diffeomorphism too, which is precisely the invariance under transformations of the fields over the same points—i.e. one and the same point is assigned different properties.

I think that the best way to understand what is going on in the hole argument is along the lines of the underdetermination problem of gauge theories that resulted from gauge freedom (as explained in §3.3). The standard position of physicists as regards the status of general covariance (e.g., as one finds in textbooks on general relativity) is to interpret the general covariance of the field equations as expressing the gauge freedom of general relativity, which is then taken to be a gauge theory. One can find this viewpoint voiced explicitly in the following passage from Wald:

If a theory describes nature in terms of a spacetime manifold, M, and tensor fields, T(i), defined on the manifold, then if 0 : M ^ N is a diffeomorphism, the solutions (M, T(i)) and (N, 0*T(i)) have physically identical properties. Any physically meaningful statement about (M, T(i)) will hold with equal

146 Both the claim that the substantivalist must deny Leibniz equivalence, and that this denial leads to indeterminism can be rejected, as I show in the responses (Chapter 5).

147 It is not that we have no idea at all how the data will evolve: the initial-value problem is well posed up to diffeomorphism.

148 This is certainly how Rovelli sees it, and Stachel too; but Hoefer and others see this as compatible with substantivalism. I discuss their arguments in Chapter 5.

validity for (N, 0*T®) .. .Thus, the diffeomorphisms comprise the gauge freedom of any theory formulated in terms of tensor fields on a spacetime manifold. In particular, the diffeomorphisms comprise the gauge freedom of general relativity. ([Wald, 1984], p. 438)

Thus, even though we 'move' the tensor fields with the action of the diffeomorphisms (via the drag-along), the tensor fields 'retain' their structure, they are essentially the same. Though they may be formally very different, diffeomorphic models pick out exactly the same physically measurable properties—though the problem of explicating these remains. This is, of course, very much like the indifference situations I presented in the previous chapters, especially the kinematic shift argument from the Leibniz-Clarke correspondence: formal distinctness coupled with physical indistinguishability.

Conceived in this way, the indeterminism issuing from the hole argument is simply a natural consequence of the underdetermination resulting from the gauge freedom of the theory; as we have seen, this is something to be found in any gauge theory. Thus, according to this 'gauge theoretic' account, the hole argument's conclusion is simply yet another example of the underdetermination that results from direct interpretations of the representation relation. Clearly, the argument hits manifold substantivalism because that view is seen as underwriting a direct interpretation of the models of general relativity.

Bearing these points in mind, let me now present general relativity as a gauge theory, and show how the hole argument arises as a natural consequence of a direct interpretation. The following account builds on the material that I presented in Chapter 3 and the previous subsection.

First, fix a compact three manifold £. Consider globally hyperbolic (vacuum) solutions to the field equations for general relativity, (M,g), such that the Cauchy surfaces of the solutions are diffeomorphic to £. Then consider an embedding of £ in (M,g), given by a diffeomorphism $ : £ ^ M—hence, $(£) is a Cauchy surface of (M,g); call it S. Then g induces a geometry on S characterized by a pair of tensors qab and Kab: q is a Riemannian metric on S (aka the first fundamental form); K is the extrinsic curvature of S describing how it is embedded in (M,g) (aka the second fundamental form). The map $ is used to pull-back q and K from S to £. £ can be embedded in (M,g), provided q and K satisfy the following (constraint) equations:

These are called the Gauss and Codazzi constraints respectively—q enters only in the definition of the scalar curvature R and the covariant derivative V. The pair (q, K) is taken to represent a dynamical state of the gravitational field just in case it satisfies these two constraints at each point x e £.

Given this setup, it is natural to choose the space of Riemannian metrics on £ as the configuration space: Q = Riem(£). Hence, the extended (full) phase space is just T* Q. A symplectic structure is induced by taking the canonically conjugate momentum to q to be pab =df 1/|qj(Kab — Kcqab) (where |q| is the determinant of q).

However, the extended phase space will contain points that are dynamically 'inaccessible', corresponding (at best) to models that are not solutions to the field equations. The 'true' (physical) phase space is then the constraint surface C D T*Q on which the following (first-class) constraints hold:

These are now called the scalar (Hamiltonian) and vector (diffeomorphism) constraints respectively—there are infinitely many, since they must hold for all x e £. Restricting the symplectic form to C and setting the Hamiltonian to zero (since it is just a sum of the constraints) yields the gauge theoretic formulation of general relativity characterized by a presymplectic geometry. This presentation contains all the information we need to extract the hole argument (and most of the details we need for the problem of frozen dynamics considered in Chapter 7).

The hole argument is generated as follows. Firstly, we note that the constraints generate Hamiltonian vector fields on the constraint surface C, such that the vectors are null with respect to the presymplectic form on C. Now, consider two points (p, q) and (p', q') lying in the same gauge orbit [(p, q)] on C, and such that they can be joined by an integral curve of the vector constraint. Then there is a diffeomor-phism 0 : £ ^ £ such that 0*p = p' and = q', implying that (p, q) and (p', q') agree on the geometrical structure of £. They can be seen as disagreeing only with respect to which points of £ play which roles; i.e., as to the geometrical properties assigned to the points x e £. Hence, the vector constraint generates gauge transformations that act by permuting the points of a spatial slice, rearranging their geometrical properties. This results in inflation, and if there is some commitment (for whatever reason) to the inflated possibility set then haecceitistic differences will be implicated in ones ontology.

It is obvious that Earman and Norton's manifold substantivalist will be forced into considering the different points on a gauge orbit as representing distinct states of affairs, since the points of the spatial slice have different geometrical properties and have their identities fixed independently of these properties; for example, according to (p, q) it is the point x that has the largest scalar curvature value, whereas according to (p', q') it is the point x'. If this is the case, then it is indeed true that general relativity is indeterministic for it can, at best, determine the geometrical structure of £, it cannot determine how this structure is distributed over the points. This conclusion follows from the premise that (p, q) and (p', q') represent distinct physically possible worlds; i.e. from a direct interpretation of the formalism. It is, then, no different, in kind, from the Leibniz-shift, permutation symmetry, and electromagnetism scenarios. All four are simply a manifestation symmetry, and, in the case of general relativity and electromagnetism, of gauge freedom.

Diagnosing the indeterminism of the hole argument in terms of gauge freedom leads to a number of possible resolutions of the hole argument. These options are fairly standard moves used when dealing with gauge freedom, and not surprisingly are essentially the same as those given in §3.3. However, it should be remembered that the manifold substantivalist is at liberty to uphold her views in the face of the indeterminism: it is not observable after all! (Belot [1996] has even argued that the surplus structure that results from one-one interpretations can be a positive feature when it comes to dealing with the problem of time.) In this context the methods look like this:

• One might implement Leibniz equivalence directly by 'quotienting out' the diffeomorphism symmetry and moving to the reduced space Q0 = Riem(£)/ Diff(£). Points of the reduced space, called "superspace", are then equivalence classes of diffeomorphic metrics on £ .If the construction procedure given above is then repeated with Q0 replacing Q then we eradicate the need for the vector constraint: diffeomorphically related metrics are identified at the first step. This gives us a partially reduced phase space, with the scalar constraint remaining. One then reapplies the same procedure with the scalar constraint, identifying points related by gauge transformations that it generates. In this way a standard Hamiltonian system with no gauge freedom is recovered. This route faces both technical and conceptual problems when it comes to quantizing the theory (discussed in Chapter 7). There is also the interpretive question of what conception of spacetime such a move would underwrite. It is generally supposed that such a move is only available to the relationalist; however, there have been claims that a suitably modified substantivalism can also follow this option. I discuss this briefly in Chapter 6 where I argue that both parties can help themselves to this method and, indeed, that this underdetermination leads into a structuralist conception of spacetime.

• One might 'gauge fix' the theory, as we have seen, essentially amounting to choosing a particular model from an equivalence class of gauge equivalent models. There are ways of doing this in general relativity, but they involve choosing a fixed foliation of spacetime. Once more, we face the question of what conception of spacetime is underwritten by this approach. In the next section I show that certain 'modalist' substantivalist responses to the hole argument achieve something analogical to gauge fixing. Once again, relationalists too can adopt this method.

• We also have the option of a gauge invariant interpretation, according to which the observables of the theory are precisely those functions that commute with all of the constraints. According to this method, each gauge orbit is taken to represent a single physically possible state of affairs. The representation relation between the points of phase space and physically possible worlds is many-one. Clearly, the reduced phase space method would classify as gauge invariant since the observables on such a space would correspond to gauge invariant quantities on the full phase space. Such a move is generally aligned with relationalist conceptions of spacetime; however, again, it has been argued that a modified substantivalism might also be compatible with this move. I will argue that this is indeed the case.

Thus, we have a counterpart for each of our methods for dealing with gauge freedom in the context of the hole argument of general relativity. Many of the responses can be seen as implementing one or another of these methods, although their connection to gauge theory is never made explicit. But the methods are neutral with respect to the ontology of spacetime. This motivates my broadly structuralist account based upon the general account of symmetries sketched in the previous chapters. I make some headway on this position in §6.2.

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