Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited

It is argued that the symmetry algebra of asymptotically flat spacetimes at null infinity in 4 dimensions should be taken as the semi-direct sum of supertranslations with infinitesimal local conformal transformations and not, as usually done, with the Lorentz algebra. As a consequence, two dimensional conformal field theory techniques will play as fundamental a role in this context of direct physical interest as they do in three dimensional anti-de Sitter gravity. Research Director of the Fund for Scientific Research-FNRS. b Research Fellow of the Fund for Scientific Research-FNRS. 2 BARNICH, TROESSAERT In the study of gravitational waves in the early sixties [1, 2], a lot of efforts have been devoted to specifying both local coordinate and global boundary conditions at null infinity that characterize asymptotically flat 4 dimensional spacetimes. The group of non singular transformations leaving these conditions invariant is the well-known Bondi-MetznerSachs group. It consists of the semi-direct product of the group of globally defined conformal transformations of the unit 2-sphere, which is isomorphic to the orthochronous homogeneous Lorentz group, times the abelian normal subgroup of so-called supertranslations. What seems to have been overlooked so far is the fact that, when one focuses on infinitesimal transformations and does not require the associated finite transformations to be globally well-defined, the symmetry algebra is the semi-direct sum of the infinitesimal local conformal transformations of the 2-sphere with the abelian ideal of supertranslations, and now both factors are infinite-dimensional. This is already obvious from the details of the derivation of the asymptotic symmetry algebra by Sachs in 1962 [3]. Let x = u, x = r, x = θ, x = φ andA,B, · · · = 2, 3. Following [3] up to notation, the metric gμν of an asymptotically flat spacetime can be written in the form ds = e V r du − 2edudr + gAB(dx A − Udu)(dx − Udu) (1) where β, V, U, gAB(det gAB) are 6 functions of the coordinates, with det gAB = rb for a function b(u, θ, φ). Sachs fixes b = sin θ, but the geometrical analysis by Penrose [4] suggests to keep it arbitrary throughout the analysis. In order to streamline the derivation below, it turns out convenient to use the parametrization |b| = 1 4 e , which implies in particular that g∂αgAB = ∂α ln ( r 4 4 e ). The fall-off conditions for gAB are gABdx dx = r2γ̄ABdx dx +O(r), (2) where the 2-dimensional metric γ̄AB is conformal to the metric of the unit 2-sphere, γ̄AB = e 2φ 0γAB and 0γABdxdx = dθ + sin θdφ. In terms of the standard complex coordinates ζ = e cot θ 2 , the metric on the sphere is conformally flat, dθ + sin θdφ = Pdζdζ̄, P (ζ, ζ̄) = 1 2 (1+ζζ̄). We thus have γ̄ABdxdx = e dζdζ̄ with φ̃ = φ−lnP . In the following we denote by D̄A the covariant derivative with respect to γ̄AB and by ∆̄ the associated Laplacian. In the general case, the remaining fall-off conditions are β = O(r), U = O(r), V/r = −2r∂uφ̃+ ∆̄φ̃+O(r ). (3) The transformations that leave the form of the metric (1) invariant up to a conformal rescaling of gAB, i.e., up to a shift of φ̃ by ω̃(u, x), are generated by spacetime vectors BMS ALGEBRA REVISITED 3 satisfying Lξgrr = 0, LξgrA = 0, g LξgAB = 4ω̃, Lξgur = O(r ), LξguA = O(1), LξgAB = O(r), Lξguu = −2r∂uω̃ − 2ω̃∆̄φ̃+ ∆̄ω̃ +O(r ). (4) The general solution to these equation is    ξ = f, ξ = Y A + I, I = −f,B ∫ ∞ r dr(eg), ξ = − 2 r(D̄Aξ A − f,BU B + 2f∂uφ̃− 2ω̃), (5) with ∂rf = 0 = ∂rY . In addition,