Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

Abstract

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS 3 algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions.The analysis is performed in terms of two-dimensional gauge fields for isl(2, R), being isomorphic to the Poincaré algebra in 3D. Although the algebra is not semisimple, the formulation can still be carried out à la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer k, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For k ≥ 1, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, so that they are in involution; while in the case of k = 0, they generate the BMS 3 algebra. In the special case of k = 1, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to the ones of a specific type of the Hirota-Satsuma coupled KdV systems. For k ≥ 1, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of k.Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is then recovered from the corresponding surface integrals in the canonical approach.

The anisotropic chiral boson

“…[10][11][12][13][25][26][27][28][29][30][31][32]. In the context of three-dimensional gravity, it has been shown that the anisotropic scale invariance can also be induced by suitable choices of boundary conditions [33][34][35][36][37][38]. In two spacetime dimensions, field theories with anisotropic scaling have also been discussed in e.g.…”

“…As expected, in the isotropic case (z = 1) the conformal transformation (3.29) becomes local. In fact, in this case the function in (3.30) reads 34) and hence (3.29) and (3.31) reduce to the transformation law and the generators of the conformal symmetry of the Floreanini-Jackiw action [9], given by…”

BTZ black hole with Korteweg–de Vries-type boundary conditions: Thermodynamics revisited

“…Other examples of this relationship between 2D integrable systems and gravity in 2+1, have been also made for the cases of "flat" and "soft hairy" boundary conditions in[25] and[26], respectively.…”

Revisiting the asymptotic dynamics of General Relativity on AdS3

“…As an ending remark, it is worth mentioning that different classes of boundary conditions relating three-dimensional gravity with integrable systems have been proposed in [30][31][32][33]. It would be interesting to explore whether a similar construction, as the one performed here, could be carried out in those cases.…”