Asymptotic shear and the intrinsic conformal geometry of null-infinity

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.

Charge algebra in Al(A)dSn spacetimes

“…Similarly, in the context of asymptotically flat spacetimes, letting flux going through null infinity is a necessary condition to consider radiating solutions. In this case, the asymptotic shear C AB plays the role of source and the charges are non-integrable [23,24,31,112]. When asymptotically locally flat spacetimes are considered, the transverse boundary metric q AB also plays the role of source [30,49,50,58,124].…”

The Weyl BMS group and Einstein’s equations

“…In the language of [70,71], this conformal extension means that the universal structure on I is reduced from a null vector and degenerate metric (original BMS) to a (thermal) Carroll structure (BMSW) [28,31,72,73]; see also [74,75] for an intrinsic and conformally invariant geometrical description of null infinity.…”

Tractor geometry of asymptotically flat spacetimes

“…How are we, however, to interpret all this from the point of view of null-infinity, i.e from the point of view of the Carroll geometry? A hint of the solution is given by the recent work [20]. In this work, it was shown that a (d − 1)dimensional conformal Carroll geometry (n a , h ab , I ) (here bold notation is used to emphasised that the fields are weighted) is canonically associated to a (d+1)-dimensional vector bundle the "null-tractor bundle"…”

“…We now discuss precisely how this equivalence can arise. It was proven in [20] that any choice of trivialisation u : I → R for I → S d−1 canonically defines an isomorphism…”