Supertranslations at Timelike Infinity

TL;DR

This paper defines asymptotic flatness at timelike infinity in four dimensions, analyzes supertranslations' effects on fields, and shows charges can be computed on surfaces around sources, with no contributions from timelike infinity.

Contribution

It introduces a new framework for asymptotic flatness at timelike infinity and derives explicit expressions for supertranslation and Lorentz charges.

Findings

The Lee-Wald symplectic form has no contributions from timelike infinity.

Supertranslation and Lorentz charges are explicitly computed.

Charges can be evaluated on any surface surrounding sources at timelike infinity.

Abstract

We propose a definition of asymptotic flatness at timelike infinity in four spacetime dimensions. We present a detailed study of the asymptotic equations of motion and the action of supertranslations on asymptotic fields. We show that the Lee-Wald symplectic form $\Omega (g, \delta_1 g, \delta_2 g)$ does not get contributions from future timelike infinity with our boundary conditions. As a result, the "future charges" can be computed on any two-dimensional surface surrounding the sources at timelike infinity. We present expressions for supertranslation and Lorentz charges.