Investigations of isotropy and homogeneity of spacetime in first-order logic

Abstract

We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in certain cases, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in certain cases. An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics – together with all the continuity and smoothness apparatus that would entail – the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatised by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed.