Quasilocal angular momentum of gravitational fields in (2+2) formalism

Abstract

Abstract. Recently the Poisson algebra of a quasilocal angular momentum of gravitational fields L(ξ) in (2+2) formalism of Einstein's theory was studied in detail [1]. In this paper, we will briefly review the definition of L(ξ) and its remarkable properties. Especially, it will be discussed that, up to a constant normalizing factor, and this algebra reduces to the standard SO(3) algebra at null infinity. It will be also argued that our angular momentum is a quasilocal generalization of A. Rizzi's geometric definition. FormalismIn (2+2) formalism of Einstein's theory [2], the 4-dimensional spacetime E 4 is regarded as a fiber bundle that consists of a 2-dimensional base space M 1+1 of the Lorentzian signature and a 2-dimensional spacelike fiber N 2 at each point on M 1+1 . Then, the Einstein's theory can be interpreted as a gauge theory defined on M 1+1 with diff(N 2 ) as a gauge symmetry, the diffeomorphism group of N 2 [3]. Let us introduce a coordinate system {u, v, y a : a = 2, 3} on E 4 , where {u, v} and {y a } are coordinates on M 1+1 and N 2 , respectively. The most general line element is given by [4],All the metric components are functions of u, v, and y a , since we don't assume any spacetime isometry. Throughout this paper, the subscripts + and − denote u and v, respectively, and N 2 is assumed to be compact. The horizontal lifts∂ ± of the tangent vector fields ∂ ± are defined bŷwhere A a ± are the gauge connections of diff(N 2 ). The diff(N 2 )-covariant derivative of a diff(N 2 )-tensor density T ab··· cd··· with weight w is defined bywhere [A ± , T ] Lab··· cd··· denotes the Lie derivative of T ab··· cd··· with respect to A ± := A a ± ∂ a . It would be helpful to define the following in-and out-going null vector fields for understanding geometrical meanings of quasilocal angular momentum;