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 L(ξ) satisfies the Poisson algebra {L(ξ); L(η){P.B. = L([ξ, η]L), 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.
Highlights
FormalismIn (2+2) formalism of Einstein’s theory [2], the 4-dimensional spacetime E4 is regarded as a fiber bundle that consists of a 2-dimensional base space M1+1 of the Lorentzian signature and a 2-dimensional spacelike fiber N2 at each point on M1+1
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]
When we identify v with a physical time in this formalism, the induced metric on the hypersurface Σ of v = constant is given by ds2|Σ = −2hdu2 + eσρab(dya + Aa+du)(dyb + Ab+du)
Summary
In (2+2) formalism of Einstein’s theory [2], the 4-dimensional spacetime E4 is regarded as a fiber bundle that consists of a 2-dimensional base space M1+1 of the Lorentzian signature and a 2-dimensional spacelike fiber N2 at each point on M1+1. The horizontal lifts ∂ˆ± of the tangent vector fields ∂± are defined by. Where [A±, T ]Lab···cd··· denotes the Lie derivative of Tab···cd··· with respect to A± := Aa±∂a. It would be helpful to define the following in- and out-going null vector fields for understanding geometrical meanings of quasilocal angular momentum; n. When we identify v with a physical time in this formalism, the induced metric on the hypersurface Σ of v = constant is given by ds2|Σ = −2hdu2 + eσρab(dya + Aa+du)(dyb + Ab+du). Where L(u, v; ξ) is quasilocal angular momentum of gravitational fields defined by d2y(ξaπa) + L0. It was proved that the sufficient and necessary condition that quasilocal angular momentum L(ξ) of a two-surface along a given vector field ξa is insensitive to the ways of labeling the surface is. The definition of L(ξ) makes sense only if ξa is a divergence-free vector field on N2
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