Seminars and short courses

The set of (smooth) metrics that can be placed on a Riemannian manifold defines an infinite-dimensional "superspace" that, remarkably, can itself be imbued with the structure of a (Fréchet) manifold. The subspace pertaining to (spatially-sliced) Einstein metrics was explored in detail by Wheeler and collaborators back in the late 50s, as it provides a means to describe a collection of spacetimes purely in terms of geometry through the famous words "mass without mass; charge without charge". At least in some restricted contexts, a natural basis relates to multipole moments which provide a tool to decompose a spacetime into a set of numbers. I will describe the construction of such superspaces, how to define inner products and (weak) Riemannian metrics there, and how they may be useful to provide astrophysical intuition. For instance, geodesics can be computed on Met(M) which allows one to define a single number that tells you how "distant" two spacetimes (e.g., two Kerr black holes) are from one another.