Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity
Yannick Herfray
TL;DR
This paper develops an intrinsic, conformally invariant framework for the radiative sector of asymptotically flat spacetimes by constructing a tractor calculus on null infinity that handles degenerate conformal metrics. It introduces Poincaré operators, showing they encode asymptotic shear data and are in one-to-one correspondence with null-normal Cartan connections; the tractor curvature then captures gravitational radiation and the degeneracy of gravity vacua. By separating weak/universal structure from strong/radiative data, the approach unifies good-cut geometry, Möbius structures, and Cartan connections, yielding a robust toolkit for generating invariants of the radiative phase space. In particular, in four dimensions the asymptotic shear parametrizes an affine space of null-normal tractor connections, tying memory effects and soft modes to the geometry of scrI in a manifestly conformal, Cartan-theoretic language and suggesting extensions to 3D boundaries and nonzero cosmological constant settings.
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 generalisation 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 two 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 generalise 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 correspond 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, in particular, to proliferate invariants of the phase space.