Asymptotic Structure of Symmetry Reduced General Relativity

TL;DR

This paper explores the asymptotic structure of symmetry-reduced general relativity with a focus on gravitational waves with a space-translation Killing field, revealing novel features of null infinity in three dimensions.

Contribution

It demonstrates how 4D Einstein vacuum equations reduce to 3D with matter sources, enabling analysis of asymptotic properties and symmetries in a simplified setting.

Findings

Regular null infinity completion in 3D despite 4D asymptotic flatness failure

Introduction of 3D Bondi energy-momentum and flux formula

Surprising features of null infinity unique to 3D description

Abstract

Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula. The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions.