Are Maxwell Gravitation and Newton-Cartan Theory Theoretically Equivalent?

Abstract

A recent flurry of work has addressed the question whether Maxwell gravitation and Newton-Cartan theory are theoretically equivalent. This paper defends the view that there are plausible interpretations of Newton-Cartan theory on which the answer to the above question is “yes”. Along the way, I seek to clarify what is at issue in this debate. In particular, I argue that whether Maxwell gravitation and Newton-Cartan theory are equivalent has nothing to do with counterfactuals about unactualised matter, contra the appearance of previous discussions in the literature. Nor does it have anything to do with spacetime and dynamical symmetries, pace recent claims by Jacobs (2023). Instead, it depends on some rather subtle questions concerning how facts about the geodesics of a connection acquire physical significance, and the distinction between dynamical and kinematic possibility.