This dissertation explores in particular a few ways in which similarity bears on general relativity, our best physical theory of space and time. The first way concerns the role of the concept of “physicality” in theorizing and using spacetime models, which manifests in at least two different ways. One is whether particular models are (or have properties that are) “physically unreasonable“—they are deemed to be pathological or otherwise artifactual, and so must excluded from the theory. Another is whether particular properties of those models are “physically significant”—warranted to be inferred about the properties of a physical system they represent. This distinction, which I draw in more detail in chapter 2, separates these two senses of physicality, the former modal-metaphysical and the latter inferential-epistemic. It is this latter that more intimately involves the notion of similarity, particular notions of which can be encoded through the use of topology. I argue then that there is no canonical topology on the models of general relativity. Rather, the choice of topology must be nontrivially dependent on the context of investigation.

In chapter 3, I bring attention to another way in which similarity, as encoded in topology, can be put to substantive use, namely in better understanding the nature of the reductive relationship between general relativity and its predecessor, the Newtonian theory of space, time, and gravitation. In particular, I show how this relationship can be made both perfectly general—applying to any situation the theories describe—and explanatory of the Newtonian theory's success.

Finally, in chapter 4, I explicate what I think are the deeper structural issues with certain examples due to Robert Geroch that are intended to show a certain topology commonly used in the physics literature in fact has several counterintuitive features. I show that there is in fact no topology that meets a strong version of Geroch's demands, but if those demands are weakened slightly, such a topology may then be constructed, one that corresponds to a natural interpretation for spacetime similarity in terms of certain classes of global observables.