Incompleteness Theorems for Observables in General Relativity

Explore the fundamental challenges in quantum gravity through this mathematical physics lecture that examines the Problem of Observables in General Relativity using advanced techniques from Descriptive Set Theory. Discover why General Relativity's general covariance—its invariance under arbitrary coordinate transformations—creates profound technical and epistemological difficulties when attempting to formulate a quantum theory of gravity. Learn about groundbreaking research demonstrating that no non-trivial diffeomorphism-invariant quantities exist across all spacetimes, and understand the mathematical proof showing that even within vacuum solutions, no complete observables can be Borel definable. Gain insight into how this fundamental limitation parallels classical impossibility results, with the Problem of Observables representing to mathematical analysis what the ancient Delian problem represents to geometric construction with straightedge and compass. The presentation draws from collaborative research employing sophisticated methods from Descriptive Set Theory to establish these incompleteness theorems, providing a rigorous mathematical framework for understanding why certain observables in general relativity remain fundamentally elusive.