Timelike infinity and asymptotic symmetry

By extending Ashtekar and Romano's definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike inf inity for an isolated system with a source is proposed. The treatment provi des unit spacelike three-hyperboloid timelike infinity and avoids the intro duction of the troublesome differentiability conditions which were necessar y in the previous works on asymptotically flat space-times at timelike infi nity. Asymptotic flatness is characterized by the falloff rate of the energ y-momentum tensor at timelike infinity, which makes it easier to understand physically what space-times are investigated. The notion of the order of t he asymptotic flatness is naturally introduced from the rate. The definitio n gives a systematized picture of hierarchy in the asymptotic structure, wh ich was not clear in the previous works. It is found that if the energy-mom entum tensor falls off at a rate faster than similar to t(-2), the space-ti me is asymptotically flat and asymptotically stationary in the sense that t he Lie derivative of the metric with respect to partial derivative(t) falls off at the rate similar to t(-2). It also admits an asymptotic symmetry gr oup similar to the Poincare group. If the energy-momentum tensor falls off at a rate faster than similar to t(-3), the four-momentum of a space-time m ay be defined. On the other hand, angular momentum is defined only for spac e-times in which the energy-momentum tensor falls off at a rate faster than similar to t(-4). (C) 1998 American Institute of Physics. [S0022-2488(98)0 1812-X].