A Note on the Newman-Unti Group and the BMS Charge Algebra in Terms of Newman-Penrose Coefficients

Glenn Barnich,Pierre-Henry Lambert

doi:10.1155/2012/197385

Glenn Barnich, Pierre-Henry Lambert

Open Access

https://doi.org/10.1155/2012/197385

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Abstract

The symmetry algebra of asymptotically flat spacetimes at null infinity in four dimensions in the sense of Newman and Unti is revisited. As in the Bondi-Metzner-Sachs gauge, it is shown to be isomorphic to the direct sum of the abelian algebra of infinitesimal conformal rescalings with𝔟𝔪𝔰4. The latter algebra is the semidirect sum of infinitesimal supertranslations with the conformal Killing vectors of the Riemann sphere. Infinitesimal local conformal transformations can then consistently be included. We work out the local conformal properties of the relevant Newman-Penrose coefficients, construct the surface charges, and derive their algebra.

Highlights

The definitions of asymptotically flat four dimensional spacetimes at null infinity by Bondi et al 1, 2 BMS and Newman-Unti NU 3 in 1962 merely differ by the choice of the radial coordinate
NU allow the leading part of the metric induced on Scri to undergo a conformal rescaling
When this generalization is considered in the BMS setting, it turns out that the symmetry algebra is the direct sum of the BMS algebra bms[4 4] with the abelian algebra of infinitesimal conformal rescalings 5, 6

Summary

Introduction

The definitions of asymptotically flat four dimensional spacetimes at null infinity by Bondi et al 1, 2 BMS and Newman-Unti NU 3 in 1962 merely differ by the choice of the radial coordinate. The standard, globally well-defined symmetry algebra bmsg4lob consists in restricting to the globally well-defined conformal Killing vectors of the sphere which correspond to infinitesimal Lorentz transformation, while the supertranslation generators are expanded into spherical harmonics This local versus global versions of the symmetry algebra are not related to the BMS gauge choice, but will occur in alternative characterizations of the asymptotic symmetry algebra where the conformal Killing vectors of the sphere play a role. Examples of this are the geometrical approach of Geroch 8 based on Penrose’s definition of null infinity 9 and as we will explicitly discuss in this paper, the asymptotic symmetries in the NU framework. The algebra of these charges is derived and shown to involve field dependent central charges in the case of bms[4] which vanish for bmsg4lob

NU Metric Ansatz for Asymptotically Flat Spacetimes

Asymptotic Symmetries in the NU Approach

Explicit Relation between the NU and the BMS Gauges

Surface Charge Algebra