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Near-horizon BMS symmetries as fluid symmetries

Robert F. Penna

Abstract

The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity. Recently, Donnay et al. have derived an analogous symmetry group acting on black hole event horizons. For a certain choice of boundary conditions, it is a semidirect product of ${\rm Diff}(S^2)$, the smooth diffeomorphisms of the two-sphere, acting on $C^\infty(S^2)$, the smooth functions on the two-sphere. We observe that the same group appears in fluid dynamics as symmetries of the compressible Euler equations. We relate these two realizations of ${\rm Diff}(S^2)\ltimes C^\infty(S^2)$ using the black hole membrane paradigm. We show that the Lie-Poisson brackets of membrane paradigm fluid charges reproduce the near-horizon BMS algebra. The perspective presented here may be useful for understanding the BMS algebra at null infinity.

Near-horizon BMS symmetries as fluid symmetries

Abstract

The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity. Recently, Donnay et al. have derived an analogous symmetry group acting on black hole event horizons. For a certain choice of boundary conditions, it is a semidirect product of , the smooth diffeomorphisms of the two-sphere, acting on , the smooth functions on the two-sphere. We observe that the same group appears in fluid dynamics as symmetries of the compressible Euler equations. We relate these two realizations of using the black hole membrane paradigm. We show that the Lie-Poisson brackets of membrane paradigm fluid charges reproduce the near-horizon BMS algebra. The perspective presented here may be useful for understanding the BMS algebra at null infinity.

Paper Structure

This paper contains 13 sections, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plunging star
  3. Metric
  4. Momentum density
  5. Near-horizon BMS symmetries as fluid symmetries
  6. Non-expanding black hole
  7. Discussion
  8. Infinite dimensional symmetries of the Euler equations
  9. The vortex
  10. General formulation of fluid dynamics
  11. Right-invariance of fluid dynamics
  12. Noether's theorem
  13. Charge algebra

Figures (3)

  • Figure 1: We have zoomed in on the north pole. Initially circular rings are squeezed along $\theta$ and stretched along $\phi$ by the passage of the star.
  • Figure 2: Slicing transformations, $t\rightarrow t+f(x^A)$, do not commute. After a slicing transformation, the normal of the new spatial slice is tilted. The commutator of two infinitesimal slicing transformations is a spatial diffeomorphism.
  • Figure 3: The Lagrangian, Eulerian, and convected momenta are related by a triangle of maps.