Sam Fletcher (MCMP, Minnesota): “On the Local Flatness of Spacetime”

Abstract: Many discussions of the foundations of general relativity put a special emphasis on describing every relativistic spacetime as “locally flat,” or as “locally Minkowskian”. Such claims are prima facie puzzling: after all, curvature is itself a local property, being described by a tensor field on spacetime. In general, relativistic spacetimes have non-vanishing curvature, so there is a straightforward sense in which they are not locally flat. Still, there is a natural intuition behind claims of “local flatness” arising from analogy with a sufficiently small region of a curved surface, like that of the Earth, which can to a good approximation be described as planar. But like many “principles” of general relativity, there does not seem to be much consensus regarding how to make this intuition more precise. Without attempting a comprehensive survey, we note three common articulations of what it could mean for spacetime to be “locally flat” or “locally Minkowskian,” arguing that each of them is unsatisfactory. We then explore a different, but precise and coordinate-independent sense in which relativistic spacetimes might be described as (approximately) locally flat.

This talk is based on a joint work with Jim Weatherall.