Asymptotic shear and the intrinsic conformal geometry of null-infinity
Yannick Herfray
Abstract
In this article, we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presence of gravitational waves can be understood to be a generalization of the tractor calculus of conformal manifolds adapted to the case of degenerate conformal metrics. It follows that the whole formalism is, by construction, manifestly conformally invariant. We first show that a choice of asymptotic shear amounts to a choice of linear differential operator of order 2 on the bundle of scales of null-infinity. We refer to these operators as Poincaré operators. We then show that Poincaré operators are in one-to-one correspondence with a particular class of tractor connections, which we call “null-normal” (they generalize the normal tractor connection of conformal geometry). The tractor curvature encodes the presence of gravitational waves, and the non-uniqueness of flat null-normal tractor connections corresponds to the “degeneracy of gravity vacua” that has been extensively discussed in the literature. This work thus brings back the investigation of the radiative phase space of gravity to the study of (Cartan) connections and associated bundles. This should allow us, in particular, to proliferate invariants of the phase space.