Tractor geometry of asymptotically flat spacetimes

TL;DR

This paper explores the geometric structure of null infinity in asymptotically flat spacetimes, revealing how tractor bundles encode physical quantities like mass, angular momentum, and gravitational radiation across different dimensions.

Contribution

It establishes a canonical derivation of null-tractor bundles from interior spacetime geometry and links normal connections to physical asymptotic data in all dimensions.

Findings

In d=3, tractor connection encodes mass and angular momentum.

In d>=4, it encodes asymptotic shear.

In d=4, tractor curvature indicates gravitational radiation.

Abstract

In a recent work it was shown that conformal Carroll geometries are canonically equipped with a null-tractor bundle generalizing the tractor bundle of conformal geometry. We here show that in the case of the conformal boundary of an asymptotically flat spacetime of any dimension d>=3, this null-tractor bundle over null infinity can be canonically derived from the interior spacetime geometry. As was previously discussed, compatible normal connections on the null-tractor bundle are not unique: We prove that they are in fact in one-to-one correspondence with the germ of the asymptotically flat spacetimes to leading order. In dimension d=3 the tractor connection invariantly encodes a choice of mass and angular momentum aspect, in dimension d>=4 a choice of asymptotic shear. In dimension d=4 the presence of tractor curvature correspond to gravitational radiation. Even thought these results are by construction geometrical and coordinate invariant, we give explicit expressions in BMS coordinates for concreteness.