General relativity does not admit enough observables

One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity.

Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is fundamentally a geometric theory and, as such, it oughts to be "generally covariant", i.e., invariant under change of coordinates by any element of the diffeomorphism group Diff(M) of the ambient manifold M. The Problem of Observables is a famous instance of the difficulties associated with general covariance, and one directly related to ineffectiveness of classical quantization recipes when it comes to GR. In a nutshell, the problem of observables asks whether GR admits a complete set of smooth observables. That is, whether the family of all diffeomorphism-invariant, real-valued, smooth maps on the space Ein(M) of all Einstein metrics on M is rich enough to separate all physical spacetimes. So far the only smooth observables known (when M=R^4) are the constant maps. In this talk, we will illustrate how descriptive set theory provides a convenient framework for framing and answering the problem of observables in the negative. I will then demonstrate an uncountable collection of vacuum solutions which cannot be separated by any "definable" observable. These results are inspired by old discussions with Marios Christodoulou and are based on recent work with George Sparling.