Symmetries and conservation laws of Navier-Stokes equations

206

314

The asymptotic group of symmetries at null infinity of flat spacetimes in three and four dimensions is the infinite dimensional Bondi-Metzner-Sachs (BMS) group. This has recently been shown to be isomorphic to non-relativistic conformal algebras in one lower dimension, the Galilean Conformal Algebra (GCA) in 2d and a closely related non-relativistic algebra in 3d [1]. We provide a better understanding of this surprising connection by providing a spacetime interpretation in terms of a novel contraction. The 2d GCA, obtained from a linear combination of two copies of the Virasoro algebra, is generically non-unitary. The unitary subsector previously constructed had trivial correlation functions. We consider a representation obtained from a different linear combination of the Virasoros, which is relevant to the relation with the BMS algebra in three dimensions. This is realised by a new space-time contraction of the parent algebra. We show that this representation has a unitary sub-sector with interesting correlation functions. We discuss implications for the BMS/GCA correspondence and show that the flat space limit actually induces precisely this contraction on the boundary conformal field theory. We also discuss aspects of asymptotic symmetries and the consequences of this contraction in higher dimensions. arXiv:1203.5795v2 [hep-th]

Section: An Aside: Non-relativistic Hydrodynamicsmentioning

confidence: 77%

BMS/GCA redux: towards flatspace holography from non-relativistic symmetries

Bagchi

Fareghbal

2012

206

314

“…It was found that the GCA could be given an infinite dimensional lift for all space-time dimensions. This also turned out to be related to the symmetries of non-relativistic hydrodynamic equations [5]. In two dimensions, the infinite GCA was shown to arise very naturally from the contraction of two copies of the infinite Virasoro symmetry [6].…”

Section: Introductionmentioning

confidence: 88%

Tensionless strings and Galilean Conformal Algebra

Bagchi

2013

116

125

Abstract:We find an intriguing link between the symmetries of the tensionless limit of closed string theory and the 2-dimensional Galilean Conformal Algebra (2d GCA). 2d GCA has been discussed in the context of the non-relativistic limit of AdS/CFT and more recently in flat-space holography as the proposed symmetry algebra of the field theory dual to 3d Minkowski spacetimes. It is best understood as a contraction of two copies of the Virasoro algebra. In this note, we link this to the tensionless limit of bosonic closed string theory. We show how it emerges naturally as a contraction of the residual gauge symmetries of the tensile string in the conformal gauge. We also discuss a possible "dual" interpretation in terms of a point-particle like limit.

“…Beyond diffeomorphisms, the search in gravity has in general been somewhat limited [15,36], however in the presence of a spacetime isometry, the symmetry group becomes remarkably large [35], particularly for vacuum spacetimes. For symmetries of the Navier-Stokes equations see [31], and with regards to the conformal group [12,32]. In the light of the fluid/gravity correspondence, one may ask whether the symmetries of these systems are linked.…”

Section: Duality In the Context Of Holographymentioning

confidence: 99%

“…In the case of the vacuum-to-vacuum spacetime Ehlers group, the Ernst scalar transforms according to the Möbius map (31). If conjugation of the Ernst scalar is to belong to this map, we must have…”

Section: Generalized Versus Spacetime Ehlersmentioning

confidence: 99%

The Navier–Stokes equation and solution generating symmetries from holography

Berkeley

Berman

2013

The fluid-gravity correspondence provides us with explicit spacetime metrics that are holographically dual to (non-)relativistic nonlinear hydrodynamics. The vacuum Einstein equations, in the presence of a Killing vector, possess solution-generating symmetries known as spacetime Ehlers transformations. These form a subgroup of the larger generalized Ehlers group acting on spacetimes with arbitrary matter content. We apply this generalized Ehlers group, in the presence of Killing isometries, to vacuum metrics with hydrodynamic duals to develop a formalism for solution-generating transformations of Navier-Stokes fluids. Using this we provide examples of a linear energy scaling from RG flow under vanishing vorticity, and a set of Z 2 symmetries for fixed viscosity.