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A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients

Glenn Barnich, Pierre-Henry Lambert

TL;DR

The paper analyzes asymptotic symmetries at null infinity in the Newman–Unti framework, proving the NU algebra is the direct sum of abelian conformal rescalings and the BMS4 algebra, with the latter being a semi-direct product of supertranslations and sphere conformal Killing vectors. It develops the Newman–Penrose coefficient transformation laws under these symmetries, and builds the corresponding surface charges using a modified Lie bracket to account for metric dependence. A detailed gauge relation with the BMS framework is provided, including a dictionary mapping NU data to BMS data and clarifying how conformal rescalings enter. The resulting charge algebra features a field-dependent central extension, receding to the standard (central-charge-free) BMS4 on global sections, and yields the Bondi mass-loss formula within this NU setting.

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 with bms4. The latter algebra is the semi-direct 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.

A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients

TL;DR

The paper analyzes asymptotic symmetries at null infinity in the Newman–Unti framework, proving the NU algebra is the direct sum of abelian conformal rescalings and the BMS4 algebra, with the latter being a semi-direct product of supertranslations and sphere conformal Killing vectors. It develops the Newman–Penrose coefficient transformation laws under these symmetries, and builds the corresponding surface charges using a modified Lie bracket to account for metric dependence. A detailed gauge relation with the BMS framework is provided, including a dictionary mapping NU data to BMS data and clarifying how conformal rescalings enter. The resulting charge algebra features a field-dependent central extension, receding to the standard (central-charge-free) BMS4 on global sections, and yields the Bondi mass-loss formula within this NU setting.

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 with bms4. The latter algebra is the semi-direct 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.

Paper Structure

This paper contains 6 sections, 1 theorem, 59 equations, 2 tables.

Table of Contents

  1. Introduction
  2. NU metric ansatz for asymptotically flat spacetimes
  3. Asymptotic symmetries in the NU approach
  4. Explicit relation between the NU and the BMS gauges
  5. Transformation laws of the NU coefficients characterizing asymptotic solutions
  6. Surface charge algebra

Key Result

Theorem 3.1

The spacetime vectors $\xi[Y,\widetilde{T},{\widetilde{\omega}};g]$ realize the extended $\mathfrak{bms}_4$ algebra in the modified Lie bracket, in the bulk of an asymptotically flat spacetime in the sense of Newman and Unti.

Theorems & Definitions (1)

  • Theorem 3.1