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BMS invariance and the membrane paradigm

Robert F. Penna

TL;DR

This work demonstrates a precise equivalence between the infinite set of BMS conservation laws for asymptotically flat spacetimes and the infinite family of membrane charges in the black hole membrane paradigm. By employing the Damour-Navier-Stokes equations and the membrane stress-energy on null surfaces, it derives horizon and null-infinity charges and shows they reproduce the standard BMS charges, while also generalizing to arbitrary subregions of arbitrary null surfaces and to nonstationary spacetimes. It clarifies the identification of the superrotation subgroup as Diff(S^2) within the membrane framework and discusses antipodal matching between null boundaries, with implications for black hole information and radiation. The paper also extends the analysis to electrodynamics, illustrating charge conservation at every angle as a manifestation of large gauge symmetries in the membrane picture.

Abstract

The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetime. It is infinite dimensional and entails an infinite number of conservation laws. According to the black hole membrane paradigm, null infinity (in asymptotically flat spacetime) and black hole event horizons behave like fluid membranes. The fluid dynamics of the membrane is governed by an infinite set of symmetries and conservation laws. Our main result is to point out that the infinite set of symmetries and conserved charges of the BMS group and the membrane paradigm are the same. This relationship has several consequences. First, it sheds light on the physical interpretation of BMS conservation laws. Second, it generalizes the BMS conservation laws to arbitrary subregions of arbitrary null surfaces. Third, it clarifies the identification of the superrotation subgroup of the BMS group. We briefly comment on the black hole information problem.

BMS invariance and the membrane paradigm

TL;DR

This work demonstrates a precise equivalence between the infinite set of BMS conservation laws for asymptotically flat spacetimes and the infinite family of membrane charges in the black hole membrane paradigm. By employing the Damour-Navier-Stokes equations and the membrane stress-energy on null surfaces, it derives horizon and null-infinity charges and shows they reproduce the standard BMS charges, while also generalizing to arbitrary subregions of arbitrary null surfaces and to nonstationary spacetimes. It clarifies the identification of the superrotation subgroup as Diff(S^2) within the membrane framework and discusses antipodal matching between null boundaries, with implications for black hole information and radiation. The paper also extends the analysis to electrodynamics, illustrating charge conservation at every angle as a manifestation of large gauge symmetries in the membrane picture.

Abstract

The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetime. It is infinite dimensional and entails an infinite number of conservation laws. According to the black hole membrane paradigm, null infinity (in asymptotically flat spacetime) and black hole event horizons behave like fluid membranes. The fluid dynamics of the membrane is governed by an infinite set of symmetries and conservation laws. Our main result is to point out that the infinite set of symmetries and conserved charges of the BMS group and the membrane paradigm are the same. This relationship has several consequences. First, it sheds light on the physical interpretation of BMS conservation laws. Second, it generalizes the BMS conservation laws to arbitrary subregions of arbitrary null surfaces. Third, it clarifies the identification of the superrotation subgroup of the BMS group. We briefly comment on the black hole information problem.

Paper Structure

This paper contains 15 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The membrane paradigm
  3. Stationary spacetimes
  4. Event horizons
  5. Null infinity
  6. Superrotations
  7. General null surfaces
  8. Nonstationary spacetimes
  9. Null infinity
  10. Physical interpretation
  11. Black hole information problem
  12. Antipodal matching
  13. Charge conservation at every angle
  14. Lienard-Wiechert fields
  15. Symmetries

Figures (1)

  • Figure 1: Penrose diagram for the exterior of an asymptotically flat, eternal black hole. This is the causal diamond for observers who remain forever in the black hole exterior. Future and past black hole horizons (${\cal H}^+$ and ${\cal H}^-$) and future and past infinity (${\cal I}^+$ and ${\cal I}^-$) are indicated. The stretched horizon (dotted red) and stretched infinity (dotted blue) are timelike. Heavy colored arrows indicate the unit normals of the membranes. Heavy black arrows indicate $k_\alpha$, the normals of the null surfaces.