A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity

Abstract

A new definition of asymptotic flatness in both null and spacelike directions is introduced. Notions relevant to the null regime are borrowed directly from Penrose’s definition of weak asymptotic simplicity. In the spatial regime, however, a new approach is adopted. The key feature of this approach is that it uses only those notions which refer to space–time as a whole, thereby avoiding the use of a initial value formulation, and, consequently, of a splitting of space–time into space and time. It is shown that the resulting description of asymptotic flatness not only encompasses the essential physical ideas behind the more familiar approaches based on the initial value formulation, but also succeeds in avoiding the global problems that usually arise. A certain 4-manifold—called Spi (spatial infinity) —is constructed using well-behaved, asymptotically geodesic, spacelike curves in the physical space–time. The structure of Spi is discussed in detail; in many ways, Spi turns out to be the spatial analog of ℐ. The group of asymptotic symmetries at spatial infinity is examined. In its structure, this group turns out to be very similar to the BMS group. It is further shown that for the class of asymptotically flat space–times satisfying an additional condition on the (asymptotic behavior of the ’’magnetic’’ part of the) Weyl tensor, a Poincaré (sub-) group can be selected from the group of asymptotic symmetries in a canonical way. (This additional condition is rather weak: In essence, it requires only that the angular momentum contribution to the asymptotic curvature be of a higher order than the energy–momentum contribution.) Thus, for this (apparently large) class of space–times, the symmetry group at spatial infinity is just the Poincaré group. Scalar, electromagnetic and gravitational fields are then considered, and their limiting behavior at spatial infinity is examined. In each case, the asymptotic field satisfies a simple, linear differential equation. Finally, conserved quantities are constructed using these asymptotic fields. Total charge and 4-momentum are defined for arbitrary asymptotically flat space–times. These definitions agree with those in the literature, but have a further advantage of being both intrinsic and free of ambiguities which usually arise from global problems. A definition of angular momentum is then proposed for the class of space–times satisfying the additional condition on the (asymptotic behavior of the) Weyl tensor. This definition is intimately intertwined with the fact that, for this class of space–times, the group of asymptotic symmetries at spatial infinity is just the Poincaré group; in particular, the definition is free of super-translation ambiguities. It is shown that this angular momentum has the correct transformation properties. In the next paper, the formalism developed here will be seen to provide a platform for discussing in detail the relationship between the structure of the gravitational field at null infinity and that at spatial infinity.