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The Weyl BMS group and Einstein's equations

Laurent Freidel, Roberto Oliveri, Daniele Pranzetti, Simone Speziale

TL;DR

The work extends the asymptotic symmetry group of flat spacetime to the Weyl BMS (BMSW) group, encompassing supertranslations, sphere diffeomorphisms, and Weyl rescalings. Using a Noetherian split and a generalized Barnich–Troessaert bracket, it shows that renormalized charges furnish a centerless representation of BMSW on every cross-section of null infinity and that flux-balance laws reproduce the full set of asymptotic Einstein equations. A holographic renormalization at null infinity yields finite charges and fluxes and reveals new conjugate pairs in the asymptotic phase space, including a sphere-scale factor and renormalized energy. The Vacua of asymptotically flat spacetimes are labeled by BMSW data, and the formalism unifies existing BMS extensions while clarifying the role of internal Lorentz charges and the relation to BT charges. These results enhance the link between asymptotic symmetries, gravitational dynamics, and holographic perspectives on quantum gravity at null infinity.

Abstract

We propose an extension of the BMS group, which we refer to as Weyl BMS or BMSW for short, that includes, besides super-translations, local Weyl rescalings and arbitrary diffeomorphisms of the 2d sphere metric. After generalizing the Barnich-Troessaert bracket, we show that the Noether charges of the BMSW group provide a centerless representation of the BMSW Lie algebra at every cross section of null infinity. This result is tantamount to proving that the flux-balance laws for the Noether charges imply the validity of the asymptotic Einstein's equations at null infinity. The extension requires a holographic renormalization procedure, which we construct without any dependence on background fields. The renormalized phase space of null infinity reveals new pairs of conjugate variables. Finally, we show that BMSW group elements label the gravitational vacua.

The Weyl BMS group and Einstein's equations

TL;DR

The work extends the asymptotic symmetry group of flat spacetime to the Weyl BMS (BMSW) group, encompassing supertranslations, sphere diffeomorphisms, and Weyl rescalings. Using a Noetherian split and a generalized Barnich–Troessaert bracket, it shows that renormalized charges furnish a centerless representation of BMSW on every cross-section of null infinity and that flux-balance laws reproduce the full set of asymptotic Einstein equations. A holographic renormalization at null infinity yields finite charges and fluxes and reveals new conjugate pairs in the asymptotic phase space, including a sphere-scale factor and renormalized energy. The Vacua of asymptotically flat spacetimes are labeled by BMSW data, and the formalism unifies existing BMS extensions while clarifying the role of internal Lorentz charges and the relation to BT charges. These results enhance the link between asymptotic symmetries, gravitational dynamics, and holographic perspectives on quantum gravity at null infinity.

Abstract

We propose an extension of the BMS group, which we refer to as Weyl BMS or BMSW for short, that includes, besides super-translations, local Weyl rescalings and arbitrary diffeomorphisms of the 2d sphere metric. After generalizing the Barnich-Troessaert bracket, we show that the Noether charges of the BMSW group provide a centerless representation of the BMSW Lie algebra at every cross section of null infinity. This result is tantamount to proving that the flux-balance laws for the Noether charges imply the validity of the asymptotic Einstein's equations at null infinity. The extension requires a holographic renormalization procedure, which we construct without any dependence on background fields. The renormalized phase space of null infinity reveals new pairs of conjugate variables. Finally, we show that BMSW group elements label the gravitational vacua.

Paper Structure

This paper contains 44 sections, 240 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Notation & nomenclature:
  3. Summary of the results
  4. Flux-balance relation
  5. Future null infinity
  6. Phase space renormalization
  7. BMSW group and Einstein's equations
  8. Geometric interpretation of the expansion coefficients
  9. The BMSW group
  10. Residual diffeomorphisms
  11. Boundary symmetry Lie algebra
  12. Extended BMS and Generalized BMS as Lie sub-algebroids
  13. Generalized bms
  14. Extended bms
  15. Action on the asymptotic phase space
  16. ...and 29 more sections

Figures (1)

  • Figure 1: The figure shows an idealized $r=const$ hypersurface, the null vector $l$, which can be used to study its radial evolution, and the time-like vector $t$ tangent to the hypersurface.