Structure of the Gravitational Field at Spatial Infinity

A formalism is introduced for analyzing the structure of the gravitational field in the asymptotic limit at spatial infinity. Consider a three-dimensional surface S in the space-time such that the initial data on S is asymptotically flat in an appropriate sense. Using a conformal completion of S by a single point A ``at spatial infinity,'' the asymptotic behavior of fields on S can be described in terms of local behavior at A. In particular, the asymptotic behavior of the initial data on S defines four scalars which depend on directions at A. Since there is no natural choice of a surface S in a space-time, the dependence of these scalars on S is essential. The asymptotic symmetry group at spatial infinity, whose elements represent transformations from S to other asymptotically flat surfaces, is introduced. It is found that this group, which emerges initially as an infinite-dimensional generalization of the Poincaré group, can be reduced to the Lorentz group. A set of evolution equations is obtained: These equations describe the behavior of the four scalars under the action of the asymptotic symmetry group. The four scalars can thus be considered as fields on a three-dimensional manifold consisting of all points at spatial infinity. The notion of a conserved quantity at spatial infinity is defined, and, as an example, the expression for the energy-momentum at spatial infinity is obtained.