Timelike Infinity and Asymptotic Symmetry

TL;DR

This paper extends the concept of spacelike infinity to timelike infinity, proposing a new definition of asymptotic flatness that clarifies the hierarchy of asymptotic structures and their physical implications.

Contribution

It introduces a novel, less restrictive definition of asymptotic flatness at timelike infinity, clarifying the hierarchy and physical interpretation of asymptotic structures in isolated systems.

Findings

Spacetimes with energy-momentum tensor falling faster than t^{-2} are asymptotically flat and stationary.

Asymptotic symmetry group similar to Poincaré group is admitted.

Four-momentum and angular momentum are definable under specific fall-off conditions.

Abstract

By extending Ashtekar and Romano's definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike 3-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat spacetimes at timelike infinity. Asymptotic flatness is characterized by the fall-off rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what spacetimes are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than $\sim t^{-2}$, the spacetime is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to $\ppp_t$ falls off at the rate $\sim t^{-2}$. It also admits an asymptotic symmetry group similar to the Poincar\'e group. If the energy-momentum tensor falls off at a rate faster than $\sim t^{-3}$, the four-momentum of a spacetime may be defined. On the other hand, angular momentum is defined only for spacetimes in which the energy-momentum tensor falls off at a rate faster than $\sim t^{-4}$.