Spacetime in General Relativity

The hole argument is an attempt to show that a manifold substantivalist conception of the spacetime implies that general relativity is indeterministic. This argument is now a standard piece of equipment in the philosopher of physics' toolbox. For this reason, a detailed analysis might seem like a bit of a waste of ink. However, although there are indeed many excellent presentations and analyses of this argument,130 there is, I think, still much more to be said, both about the argument itself, about the various responses to it that have been suggested in its aftermath, and about its connections to the issues presented in the previous chapters (especially the connection to gauge theory). Most importantly, however, the connections between this argument and the problems of time and change that crop up in canonical quantum gravity remain to be adequately explicated.131 I postpone this task until Chapter 7. The purpose of the present chapter is to get clear on exactly what allows the hole argument to get its grip, how the various responses work, and how both the argument and the responses relate to the issues of previous chapters.

My response matches up with what I had to say in those chapters: the in-determinism of the hole argument highlights the fact that general relativity is 'indifferent' to which points of spacetime the metric field is spread out over. This does not imply that spacetime points do not exist, nor that they do; nor does it rule out haecceitism, nor does it imply it. It does not mandate relationalism nor does it mandate substantivalism. In fact, the indeterminism of general relativity is not relevant to these conceptions of spacetime. I argue that a structuralist conception works best—though, again, general relativity does not mandate it, part of my reason for opting for structuralism is because many prima facie incompatible positions can be made compatible with general relativity (philosophically: with the appropriate tweaks to modality and identity; or technically: with the appropriate tweaks in one's conception of observables).

The plan of this chapter is as follows. In §4.11 begin by introducing some background material that is essential in what follows. This will involve a discussion of manifold substantivalism, and the relation between the models of general relativity and physically possible worlds. In §4.3 I then go on to introduce the hole argument in the language of gauge theory, and catalogue what the various inter

130 For 'six of the best' try Earman & Norton [1987] (the original presentation), Norton [1988], Butterfield [1989], Hoefer [1996], Maudlin [1988], or Stachel [1993]. Einstein's own rendition can be found in [Einstein, 1916].

131 Though Gordon Belot and John Earman [1999; 2001] have done much to remedy this unfortunate state of affairs. My debt to this pair of articles should be readily apparent.

pretive options for such theories look like in the context of general relativity. In Chapter 5 I then give a taxonomy of responses, split into three classes: (1) determinist; (2) modalist; and (3) relationalist. I show that each response can be seen more or less as on all fours with some method for dealing with gauge freedom. I find fault with each response and then in Chapter 6 I provide a response that is in line with the general account of symmetries that I have been suggesting in the previous chapters.

The target of the hole argument is a certain form of substantivalism known as "manifold substantivalism". The idea is to show that such a conception of spacetime leads to an indeterminism concerning how the dynamical fields are spread over the points. This indeterminism is taken to flow from a direct interpretation of the representation relation between the models of general relativity and physically possible worlds; this is seen by Earman and Norton as concomitant with manifold substantivalism. In the next section I examine Earman and Norton's reasons for believing that manifold substantivalism is the best form of substantivalism available, and then, in §4.2, show how this view connects with the understanding of the relation between models and worlds. I argue both that (1) manifold substantival-ism isn't the most defensible form of substantivalism; and (2) that substantivalism is not committed to a one-to-one interpretation of the representation relation.

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