Reading guide for Thurs. 11/12: John Norton, “The Hole Argument,” Proceedings of the Biennial Meeting of the Philosophy of Science Association, 1988, vol. 2 (on JSTOR), §§1-3 (56-59)
Norton’s aim here is to present an argument against a certain sort of substantival view of spacetime in the context of Einstein’s general relativity. You can find another, less compact, presentation of this on line in §§1-7 of an article by Norton that is part of the Stanford Encylcopedia of Philosophy.
http://plato.stanford.edu/entries/spacetime-holearg/
In particular, that presentation has a series of illustrations that provide background for the main illustration in the article on JSTOR. The remaining sections of the encyclopedia article (i.e., §§8-10) provide further historical and philosophical context.
• You will find a fair bit of technical vocabulary in Norton’s JSTOR article. Virtually all of it is explained to the level required for his purposes there, so it is safe to rely on the meaning you can derive from the context. (I add some further comments on terminology below.) The most technical part of the article is in section 4 of the article (pp. 59-61), and it is OK to skip that. (If you do read it, note one typo: item (b) on p. 61 should read “physical situation II.”)
• Some terminology:
Differentiable manifold. This is a set of points (of spacetime, so Norton calls them “point-events”) that has enough structure to say whether a function is continuous (i.e., has no jumps in its values) and whether it has well-defined rates of change of various sorts (i.e., partial derivatives). It’s more important to note that “metrical” structure that is not included—that is, there is no notion of distance between points or measure of the size of angles.
Diffeomorphism. This is a function or mapping from one differentiable manifold to another that is reversible (i.e., has an inverse) and preserves the structure of the manifold. It need not preserve things like distances or angles, so you can think of it as stretching or compressing the manifold in various ways without tearing it.
World line. This is the path of an object in spacetime. For example, someone running in a circle would return to the starting point at a later time, so the world line traced would be a helix. Notice that the bold, roughly vertical, lines in Norton’s diagram (a couple with spiral galaxies attached) are world lines. (He speaks of the world lines of “smoothed out” matter of the galaxies because the account of matter built into general relativity is not thought to capture matter on the finest scale.)
Light cone. A light cone represents the paths of light converging on a point from the past and diverging from it to the future. In three dimensions it is not possible to represent in any very clear way a 4-dimensional cone, so the convention is to represent the cones of a 3-dimensional spacetime. Norton uses these to indicate metrical structure in his diagram because the path of light is distinguished in relativity theory by its metrical features—it is “straight” in the appropriate sense but also has length 0 given the way length is defined in spacetime.
Tensors g and T. Tensors are abstract mathematical objects (a generalization of vectors) that represent the features of the universe beyond those that are part of the manifold M. As Norton notes, g includes metrical features of spacetime while T captures the distribution of matter. Einstein’s theory is embodied in a single equation tying the two. The interaction between them that this equation states was expressed by the physicist J. A. Wheeler in a slogan: “Space acts on matter telling it how to move. In turn, matter reacts back on space, telling it how to curve” (Misner, Thorne, and Wheeler, Gravitation, W. H. Freeman, 1973, p. 5).
• The quotation from Wheeler suggests one reason substantivalism is still a live issue. Although Einstein was influenced by Mach—as is shown by the fact that “matter tells space how to curve”—space can seem to be an independent actor in his theory. This appears in two forms. First, although the metrical structure of space is influenced by matter, it is not dependent on matter for its existence; the theory allows for a case of a universe empty of matter, which would then have the sort of metrical structure described in Einstein’s early special theory of relativity. Secondly, the manifold M is the basis on which the tensor T is defined, so it can seem to constitute a sort of “container” for matter, even if a different sort of container than Newton’s absolute space (which had the full metrical structure described by ordinary Euclidean geometry).