Infinity in spacetime geometry: a projective formulation

par Dr Giampiero Esposito (Université de Naples, Italie)

jeudi 29 janv. 2026, 14:00 → 15:00 Europe/Paris

Salle 1180, bâtiment E2 (Salle des séminaires )

Points at infinity occur in several aspects of spacetime geometry, e.g., the action principle, definition of singular or singularity-free spacetimes, isolated gravitating systems, coordinate-free definition of asymptotic flatness, positive-mass theorems, fall-off conditions in the global approach to (quantum) field theory. In pure mathematics, the appropriate framework is provided by projective maps, that make it possible to "bring infinity down to a finite distance". In general relativity, besides the conformal embedding picture of Penrose, there is still room left for other conceptual paths. For this purpose, we first prove that a local diffeomorphism exists between a smooth 4-manifold M (not yet endowed with a metric) and real projective space RP4. If M is later replaced by a spherically symmetric spacetime (M,g), the projective counterparts of timelike, spacelike and null infinity are here obtained. Their nature as single points (timelike and spacelike infinity) or a three-manifold (null infinity) arises neatly from the projective map that we build. In the resulting spacetime, the associated coordinate transformation yields a non-diagonal but completely viable metric, as we show explicitly for Schwarzschild spacetime. The equations obtained suggest that the various kinds of infinity can be studied with the methods of algebraic geometry for algebraic surfaces and algebraic manifolds. Moreover, the differential forms studied in the projective differential geometry of surfaces in the twentieth century might find an original application to general relativity.