## Manifold Substantivalism

It is the manifold that Earman and Norton claim is the best candidate for what represents spacetime for a substantivalist. Hence, it is the M component of the spacetime models that we discussed in §1.1 that the substantivalist should be committed to. This commitment to the manifold amounts to a realism about the points of spacetime along with their topological and differential properties. Note that this part of the model classes as a background structure in that it is fixed across the physically admissible models of the theory (though general relativity can be formulated on different manifolds)—this matches the definition I gave in §1.1).

Why do they choose this structure as being the correct representation of spacetime? They claim that such a view is naturally extractable from the local formulation of spacetime theories:

We take all the geometric structure, such as the metric and derivative operator, as fields determined by partial differential equations. Thus we look upon the bare manifold—the 'container' of these fields—as spacetime. ([Earman and Norton, 1987], pp. 518-9)

This is most evident, say Earman and Norton, in the context of general relativity. They back this up by pointing out that the metric in general relativity "now incorporates the gravitational field"; "carries energy and momentum132", and is

132 Compare this to the following remark of Feynman's about the status of the electromagnetic field: "The fact that the electro-magnetic field can possess momentum and energy makes that field very real..." [1962], Vol. 1, Ch. 10, 9). Hence, the carrying of energy and momentum are taken to signify a robust form of reality of the carrier.

such that "a gravitational wave propagating though space [could have] its energy ...collected and converted into other types of energy, such as heat or light, or even massive particles".133 They say: "If we do not classify such energy bearing structures as the wave as contained within spacetime, then we do not see how we can consistently divide between container and contained" ([Earman and Norton, 1987], p. 519). So presumably, Earman and Norton see the most defensible form of spacetime substantivalism as involving an entity without the usual spatiotemporal properties: the manifold merely has dimension and a notion of betweenness for points. The distance relations, and spatial and temporal relations come from a material field in spacetime!

Robert Rynasiewicz [1996] essentially says that we can't consistently (unambiguously) divide between container and contained in this way in the context of general relativity as we can in the context of pre-GR (non-field) theories. For this reason he claims that the substantivalist/relationalist debate is "outmoded". I remarked in §2.1.1 that a crucial feature of the relationalist/substantivalist debate was the availability of a clear-cut distinction between 'matter' and 'space(time)'. This was required in order that the fundaments of the debate, concerning the relative ontological priority of matter and space(time), have a clear expression. For space, for a substantivalist, is defined to be that which functions as an independent container for everything else; whereas for a relationalist space is defined by matter: the distinction between container and contained is collapsed. Hence Rynasiewicz's rejection of the debate in the context of general relativity.134

Clearly, the substantivalism on offer in the context of general relativity is a very different beast to the ones considered earlier. Substantivalism about Newtonian, neo-Newtonian, and Minkowskian spacetimes incorporated the metric structure; they could do this because it was a background structure, dynamically decoupled from matter-energy. But, as I just mentioned, the only available background structure in general relativity is the manifold. In their discussion, Earman and Norton do in fact consider the metric as representing spacetime, only to reject it on the grounds that:

classifying the metric as part of the container spacetime leads to trivialisa-tion of the substantivalist view in unified field theories of the type developed by Einstein, in which all matter is represented by a generalised metric tensor. For there would no longer be anything contained in spacetime, so that the substantivalist view would in essence just assert the independent existence of the entire universe. ([Earman and Norton, 1987], p. 519)

In any case, according to Rynasiewicz, the metric field in general relativity does not definitively admit an interpretation as contents or container; neither substantival structure nor relative structure. Hoefer [1998] disagrees, and argues that at least one side of the debate, substantivalism, can be given a clear definition and,

133 This belief is amplified by Kuchar, who writes: "[t]he ripples of gravitational radiation can travel around, interfere, attract each other, and amplify. They can hold themselves together in a gravitational geon. Part of the gravitational radiation can leak out, part of it may collapse and form a black hole" ([1993], p. 4).

134 Rynasiewicz also extends his thesis to the classical theory of electromagnetism. I do not discuss this aspect of his argument here; for a nice discussion see Hoefer ([1998], pp. 454-7).

given that relationalism is just the denial of substantivalism, the full debate can be formulated: "for better or worse the debate still goes on". Hoefer disregards the importance of the 'container/contained' distinction, and instead focuses the dispute on the ontological status accorded to the metric field. I agree with Hoe-fer that the debate can be formulated as he suggests, but the debate also admits a formulation in terms of the manifold, as Earman and Norton suggest. Which is correct? I have to agree with Hoefer, for the reason that the metric carries all the significant aspects concerning spacetime. The manifold only has topological and differential structure, and, as I intimated above, these are hardly sufficient to ground spatiotemporal properties and relations. It simply isn't equipped with enough structure to represent spacetime (cf. [Maudlin, 1988], p. 87).

Many, however, have thought that the dynamical nature of the metric field in general relativity, coupled with its role as determiner of chronogeometrical properties and relations pushes strongly towards a relationalist conception. Hence, Rynasiewicz's point can simply be reinstated at this level: is the metric to be conceived of as spacetime or matter? The reasons for choosing the latter option are to do with the features mentioned by Earman and Norton above, that the metric is dynamical, has energy, and so on.135 This relationalist understanding of the metric field certainly seems to be the prevalent one amongst physicists (cf. [Rovelli, 1997] and [Smolin, 2006], for example), but the view is that the metric field also represents spacetime. However, I agree here with Howard Stein on the falsity of the view that the metric field's having these properties leads automatically to relationalism if we view it as representing spacetime. Let me quote him at length:

It is often claimed that the general theory of relativity has demonstrated the correctness of Leibniz's view. This is a drastic oversimplification. It is no more true in the general theory than it in Newtonian dynamics that the geometry of space-time is determined by relations among bodies. If the general theory does in a sense conform better to Leibniz's views than classical mechanics does, this is not because it relegates "space" to the ideal status ascribed to it by Leibniz, but rather because the space—or rather the space-time structure—that Newton requires to be real, appears in the general theory with attributes that might allow Leibniz to accept it as real. The general theory does not deny the existence of something that corresponds to Newton's "immobile being"; but it denies the rigid immobility of this "being," and represents it as interacting with the other constituents of physical reality. ([Stein, 1967], p. 271)

Hence, Newton was committed to the existence of absolute substantival space and time; but the absoluteness was not a necessary part of substantivalism about spacetime: they can be safely separated. As long as one believes that whatever is

135 I should point out that the claim that substantive energy is carried by gravitational waves has been questioned by Hoefer ([1996], p. 13). His argument involves the fact that this energy must be represented by a pseudo-tensor, and is not unique (essentially because there is no canonical choice of time). To a large extent, the issues of the hole argument and, even more so, the problem of time are relevant to this problem (and it is relevant to them): how one deals with the energy content gravitational waves will depend on ones treatment of the problem of time (on which, see Chapter 7).

accorded the status of spacetime in a theory is treated as a robust entity that exists independently of matter, then this counts as a substantivalist interpretation regardless of whether the thing that is spacetime is absolute or dynamical. Clearly though, Stein is talking about something other than manifold substantivalism, for his version includes the metric field in the representation of spacetime. Hence, Newton was not a manifold substantivalist, and the old debate was founded on the fact that the (fixed) metric was an essential part of space and time. The importance, and distinctiveness of the metric field filters through into general relativity too, for there is no such thing as a part of space without a gravitational field—i.e. the metric field is nowhere vanishing.

The fact that one and the same object is being utilised to defend opposing position strongly suggests, however, that Rynasiewicz's objection is not without support. As I show in the chapter dealing with the responses to the hole argument, we find that explicit defenses of relationalism and substantivalism in fact look almost identical! This has lead some to seek a tertium quid between these opposing positions. For example, Mauro Dorato [2000] has outlined a position he calls "structural spacetime realism", according to which the relational nature of spacetime is accepted, but that this structure exists independently of physical objects and events. However, the position makes the same moves as the so-called "sophisticated substantivalists" who wish to endorse Leibniz equivalence. Moreover, it strikes me that Dorato has simply given another name to Stachel's relationalist position—see §5.3.2. We can view it as relationalist or substantivalist depending on how we view the metric field, and we are thus no further on. But I do agree with Dorato that the terms 'matter' and 'spacetime' are problematic as far as the metric field is concerned; it doesn't sit comfortably under either banner. Better, I say, is a Steinian structuralism according to which we simply be realist about the structure that is exemplified.136 If one wants to point out the fact that the metric field is an entity that can exist independently of other fields, and so is substantival, then one will always face Rynasiewicz's problem: is it 'material-like' or is it 'space-like'? One can always find a way to defend a relationalist position or a substantivalist position from such a spot. Better to wipe the spot away. This is, of course, the un-derdetermination of metaphysics by physics again. Previously, I intimated that we should be structuralists about gauge theories and classical spacetime because the interpretation of the symmetries always allowed us to set up reasonable opposing views. The same point applies here too.

Let us put these complications aside until the final chapter, where I fully defend my position. For now let us grit our teeth and agree with Earman and Norton that the manifold is the best object to represent spacetime for the substantivalist, that it functions as a container for the other fields (including the metric field), and that a manifold substantivalist will be committed to the view that points of this manifold represent physical spacetime points. We still need to say something about the relation between the models of general relativity and worlds.

136 This is more or less in keeping with the general account of structures admitting gauge-type symmetries that I have been pushing for the past two chapters. I consider Dorato's proposal, and defend a different version of structuralism in Chapter 8.

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