ConfProGR · Conformal and Projective Geometry in General Relativity

The mathematical formalism of general relativity is grounded in Riemannian geometry, which endows the spacetime manifold with a Lorentzian metric whose curvature encodes the gravitational field. This framework has been remarkably successful in describing wide range of physical phenomena, from black holes to cosmology. Nevertheless, in certain regimes, such as asymptotic regions or near initial singularities, alternative structures arising from conformal or projective differential geometry provide insight inaccessible from the Riemannian perspective. The goal of the proposed project is to adapt a natural mathematical framework designed for conformal and projective differential geometry to study fundamental problems in general relativity that are beyond the scope of Riemannian methods.The specific research objectives are:(i) establish the relation between the conformal geometry and stress–energy tensor of asymptotically de Sitter spacetimes;(ii) derive quasi-local and global invariants of spacetimes with positive cosmological constant;(iii) develop a projective tractor calculus approach to asymptotically flat spacetimes.These goals will be pursued in collaboration with Piotr Chruściel (Center for Theoretical Physics, Polish Academy of Sciences) and Rod Gover (University of Auckland). Using the machinery of tractor calculus, the project develops a novel geometric approach to the asymptotic structure of spacetimes in general relativity. A key component will be the development of a computer algebra framework specifically designed for the study of conformal and projective geometry.