Asymptotic Structure of Symmetry Reduced General Relativity

TL;DR

The paper develops a comprehensive framework for the asymptotic structure of symmetry-reduced general relativity in 3 dimensions, focusing on Einstein–Rosen-type spacetimes with a space-translation Killing field. By reducing 4D vacuum GR to 3D gravity coupled to a scalar, it analyzes null infinity, constructs a Bondi-like energy, and derives a flux formula while revealing a richer asymptotic symmetry group Diff(S^1) with super-translations. Key findings include the appearance of a nontrivial Bach tensor at I, a conical deficit encoding energy, and a precise mechanism to select translations via Bondi-type frames, with direct implications for the 4D theory and midi-superspace quantization. The work lays groundwork for both classical S-matrix analyses and prospective quantum gravity in reduced dimensions, highlighting fundamental differences from 4D, such as the dynamical leading-order metric and the role of matter fall-off.

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.