Asymptotic flatness at spatial infinity in higher dimensions

TL;DR

This paper extends the concept of asymptotic flatness at spatial infinity to higher dimensions ($d extgreater= 4$), analyzing symmetries and conserved quantities, and identifies additional conditions needed for $d>4$.

Contribution

It provides a conformal completion-based definition of asymptotic flatness in higher dimensions and reveals new conditions necessary for symmetry and conserved quantities in $d>4$.

Findings

Asymptotic symmetry in higher dimensions parallels four dimensions with specific conditions.

Additional conditions are required in $d>4$ for Poincaré symmetry and conserved angular momentum.

The magnetic part of the Weyl tensor must vanish faster in higher dimensions.

Abstract

A definition of asymptotic flatness at spatial infinity in $d$ dimensions ($d\geq 4$) is given using the conformal completion approach. Then we discuss asymptotic symmetry and conserved quantities. As in four dimensions, in $d$ dimensions we should impose a condition at spatial infinity that the "magnetic" part of the $d$-dimensional Weyl tensor vanishes at faster rate than the "electric" part does, in order to realize the Poincare symmetry as asymptotic symmetry and construct the conserved angular momentum. However, we found that an additional condition should be imposed in $d>4$ dimensions.