${\rm BMS}_3$ invariant fluid dynamics at null infinity

Abstract

We revisit the boundary dynamics of asymptotically flat, three dimensional gravity. The boundary is governed by a momentum conservation equation and an energy conservation equation, which we interpret as fluid equations, following the membrane paradigm. We reformulate the boundary's equations of motion as Hamiltonian flow on the dual of an infinite-dimensional, semi-direct product Lie algebra equipped with a Lie-Poisson bracket. This gives the analogue for boundary fluid dynamics of the Marsden-Ratiu-Weinstein formulation of the compressible Euler equations on a manifold, $M$, as Hamiltonian flow on the dual of the Lie algebra of ${\rm Diff}(M)\ltimes C^\infty(M)$. The Lie group for boundary fluid dynamics turns out to be ${\rm Diff}(S^1) \ltimes_{\rm Ad} {\rm \mathfrak{vir}}$, with central charge $c=3/G$. This gives a new derivation of the centrally extended, three-dimensional Bondi-van der Burg-Metzner-Sachs (${\rm BMS}_3$) group. The relationship with fluid dynamics helps to streamline and physically motivate the derivation. For example, the central charge, $c=3/G$, is simply read off of a fluid equation in much the same way as one reads off a viscosity coefficient. The perspective presented here may useful for understanding the still mysterious four-dimensional BMS group.