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Tractor geometry of asymptotically flat spacetimes

Yannick Herfray

TL;DR

<3-5 sentence high-level summary>The paper develops a tractor-based description of AF spacetimes that unifies interior spacetime geometry with the boundary conformal Carroll structure at null infinity. It shows that the null-tractor bundle on $\mathscr{I}$ is canonically derived from the interior spacetime tractor bundle, and that compatible normal connections on this bundle are in one-to-one correspondence with the leading-order germ of the interior, encoding asymptotic shear and, in 4D, gravitational radiation through the tractor curvature. In dimensions $d=3,4$ the curvature data map to mass, angular momentum, and NP coefficients, while for $d\ge5$ the curvature captures zero modes with radiation encoded in subleading data. The framework uses BMS coordinates and the Thomas operator to provide intrinsic, coordinate-invariant statements about null infinity and its dynamics, suggesting a boundary Chern-Simons-type interpretation for radiative degrees of freedom.

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.

Tractor geometry of asymptotically flat spacetimes

TL;DR

<3-5 sentence high-level summary>The paper develops a tractor-based description of AF spacetimes that unifies interior spacetime geometry with the boundary conformal Carroll structure at null infinity. It shows that the null-tractor bundle on is canonically derived from the interior spacetime tractor bundle, and that compatible normal connections on this bundle are in one-to-one correspondence with the leading-order germ of the interior, encoding asymptotic shear and, in 4D, gravitational radiation through the tractor curvature. In dimensions the curvature data map to mass, angular momentum, and NP coefficients, while for the curvature captures zero modes with radiation encoded in subleading data. The framework uses BMS coordinates and the Thomas operator to provide intrinsic, coordinate-invariant statements about null infinity and its dynamics, suggesting a boundary Chern-Simons-type interpretation for radiative degrees of freedom.

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.

Paper Structure

This paper contains 64 sections, 13 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Asymptotically flat spacetimes: the conformal approach
  3. Asymptotically flat spacetimes and conformal geometry
  4. Conformal geometry
  5. Tractors
  6. The tractor bundle
  7. Example : Energy-Momentum tractor
  8. Example : Infinity tractors and Thomas operator
  9. The normal tractor connection
  10. Example : Infinity tractors and Energy-momentum tractors
  11. Almost Einstein manifolds
  12. Einstein Equations
  13. Almost Einstein manifolds
  14. Differentiability
  15. Zeroth order structure at null infinity, the null-tractor bundle
  16. ...and 49 more sections

Key Result

Proposition 2.1

Let $\left(M , \bm{g}_{\mu\nu}\right)$ be a conformal manifold. There exists a representative $\tilde{g}_{\mu\nu} := \Omega^{-2} g_{\mu\nu}$ which satisfies Einstein vacuum equations if and only if there exists a covariantly constant section of the tractor bundle such that $I^I \bm{X}_I \in \Gamma\left[ L \right]$ is nowhere vanishing. Then $\bm{\Omega} = I^I \bm{X}_I$, $I^I$ is an infinity tract

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 14 more