The existence of gravitational waves is one of the prediction, now empirically validated, of Einstein's General Relativity. The underlying mathematical model for this predication is a class of Einstein Lorentzian manifolds dubbed "Asymptotically flat space-times". These are particular case of conformally compact manifolds, the presence of gravitational waves being geometrically encoded in their asymptotics (i.e at the conformal boundary).
I will review some of the definitions and classical results associated to asymptotically flat-space times: with an emphasis on the intertwined aspects of both geometry and physics. I will then show how a generalisation of the tractor calculus from conformal geometry can be used to invariantly and intrinsically encode gravitational radiations as an "extra" layer of geometry at the conformal boundary: they amounts to a choice of Tractor connection (these are a particular case of Cartan connections). More precisely, a non-vanishing tractor curvature correspond to the presence of gravitational radiations while the moduli space of flat connections is physically associated to the so-called "degeneracy of gravity vacua".