There are two classes of topologies most often placed on the space of Lorentz metrics on a fixed manifold. As I interpret a complaint of Geroch [Relativity (Plenum, 1970), p. 259 and Gen. Relativ. Gravitation 2, 61 (1971)], however, neither of these standard classes correctly capture a notion of global spacetime similarity. In particular, Geroch presents examples to illustrate that one, the compact-open topologies, in general seems to be too coarse, while another, the open (Whitney) topologies, in general seems to be too fine. After elaborating further the mathematical and physical reasons for these failures, I then construct a topology that succeeds in capturing a notion of global spacetime similarity and investigate some of its mathematical and physical properties.
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Thanks to Jim Weatherall and Chris Wüthrich for encouraging comments, and to audience members in Pittsburgh, Pasadena, Cambridge, and Nijmegen for illuminating discussion. Part of the research leading to the work on which this manuscript was based (Ref. 3, Chap. 4) was completed with the support of a National Science Foundation Graduate Research Fellowship.