Thus the manifold substantivalist is committed to a radical form of indeterminism. No specification of initial data on O can determine the subsequent evolution of g within O. For instance, suppose that \(\Sigma\) is a spatial slice “before” O (in a spacetime with temporal orientation). It follows that no amount of information about the metric (and sources) outside of O can be sufficient to determine the value of the metric at points within O. Then one is committed, it would seem, to the claim that ( M, g) and \((M,d^*g)\) represent different physical possibilities, corresponding to different assignments of metrical values to points within O but also that these two possibilities are empirically indistinguishable from one another. Now suppose that one accepts “manifold substantivalism”: that is, one believes that the manifold M represents spacetime, independently of any of the fields defined on M. We now have two spacetimes, ( M, g) and \((M,d^*g)\), with the same sources, which differ within O but which, by virtue of being isomorphic, do not differ in any of their observable properties, even in principle. (We assume that g and d have been chosen so that there exists some point in O at which \(d^*(g)\ne g\).) It follows from the foregoing that this spacetime, too, is a solution to Einstein’s equation, with source term T. Now consider the spacetime \((M,d^*g)\), where \(d^*\) is the pullback along d. To make the point especially striking and to follow Einstein’s original argument (though this is not essential), let us suppose ( M, g) is a solution to Einstein’s equation for some stress-energy tensor T that happens to vanish within O. Let \(d:M\rightarrow M\) be a diffeomorphism, which we assume to be the identity on M/ O, and to differ from the identify somewhere within O. Choose an open set O with compact closure (the “hole”), assumed to be a proper subset of M. The Earman–Norton version of the argument may be put as follows. Since then, hundreds of papers have appeared responding to the Earman–Norton argument, and much of the subsequent literature in the foundations of general relativity and elsewhere in philosophy of physics has been shaped by these debates. Footnote 4 Soon after, Norton, working now in collaboration with John Earman, argued that Einstein’s argument could be reconstructed in a way that was still relevant to contemporary debates in philosophy of physics, as a Leibniz-inspired reductio of a metaphysical conviction in the reality of spacetime points that they dubbed “manifold substantivalism”. Meanwhile, John Earman was working on the bearing of Leibniz’s thinking on a similar view, following some early thinking of Howard Stein. Working independently following this discovery, Stachel Footnote 3 and Norton argued that Einstein’s Entwurf theory and hole argument were not trivial blunders, but the result of deeper metaphysical convictions according to which, “point events of the spacetime manifold are incorrectly thought of as individuated independently of the field itself” Norton (, 256). Norton and John Statchel in the Einstein Archive at Princeton, miscatalogued as lecture notes from the University of Zurich. This simple perspective on the hole argument changed following a remarkable event in the history of science: the discovery of Einstein’s notebook of scratchpad calculations during those crucial years. In the Entwurf theory the preferred coordinate system arose in the description of a Newtonian limit, and in the hole argument it arose in the explicit expression of the metric. Over the subsequent decades, the hole argument would reappear in the work of various groups on quantum gravity, usually together with a standard story: Einstein’s 1913 blunder was a failure to realise that he had chosen a preferred coordinate system, and so he prematurely rejected general covariance. Footnote 1 But by the end of 1915, he had rejected the hole argument and the Entwurf theory, in part because he had found general relativity. In 1913, Einstein presented the hole argument in an effort to show that there could be no adequate “generally covariant” or diffeomorphism-invariant theory of gravity, instead advocating his erroneous Entwurf field equations. The history of the subject is perhaps well-known, but worth repeating. Few topics in the philosophy of physics have received more attention in the past forty years than Einstein’s hole argument.
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