Similarity, Topology, and Physical Significance in Relativity Theory

Abstract

Stephen Hawking, among others, has proposed that the topological stability of a property of spacetime is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single "right" topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Geroch and some propositions of my own show that the main candidates-and each possible choice, to some extent-faces the horns of a no-go result. I suggest that instead of trying to decide what the "right" topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology.

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“…22 Thus, the progression from a more to a less fundamental parametrization occurs not in the flow to smaller values of Λ, but in the projection of the RG flow of the 21 Within the philosophical literature, the subjects of reduction and emergence as they relate to renormalization and effective field theory have been examined at length in the work of Butterfield, Hartmann, Castellani, Bain, Crowther, Huggett, Williams, Franklin, and others (Butterfield, 2014), (Hartmann, 2001), (Castellani, 2002), (Bain, 2013), (Crowther, 2015), (Huggett and Weingard, 1995), (Williams, 2017), (Franklin, 2017). Recent discussion concerning the general methodology of reduction in physics can be found in Butterfield, Crowther, Fletcher, and Rosaler (Butterfield, 2011a), (Butterfield, 2011b), (Butterfield and Bouatta, 2012), (Butterfield, 2014), (Fletcher, 2015), (Crowther, 2015), (Crowther, 2017), (Rosaler, 2015), (Rosaler, 2017), (Rosaler, 2018), (Rosaler, 2013).…”

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“…The classic discussion of stability considerations in general relativity is Hawking (). For further discussion and references, see Fletcher (). For the importance of stability considerations in underwriting physical explanations in general, see Batterman ().…”

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“…We emphasize that the Weyl algebra is not the only possible choice-and more, mathematical physicists and philosophers have presented a number of arguments that it is the wrong choice (see, e.g., Fannes and Verbeure, 1974;Landsman, 1990a,b;Buchholz and Grundling, 2008;Grundling and Neeb, 2009;Feintzeig, 2018a). Moreover, if one were to make a different choice, one would arrive at a different state space, with a different weak* topology, and a 23 Here we echo a point made forcefully by Fletcher (2016), that different choices of topology encode different senses of approximation-and in any particular case, careful attention must be paid to whether a particular topology captures the salient sense. See also Feintzeig (2018b) for applications of Fletcher's ideas to quantum theories.…”