Asymptotic flatness at spatial infinity in higher dimensions

TL;DR

The paper extends the Ashtekar–Hansen framework for spatial infinity to $d\ge4$ dimensions under vacuum Einstein equations by defining a precise asymptotic flatness condition via conformal completion. It introduces electric and magnetic parts of the Weyl tensor on a timelike hypersurface near spatial infinity and analyzes their transformation under supertranslations, showing that additional higher-dimensional constraints are needed to realize the Poincaré symmetry. Conserved charges, namely the $d$-momentum and angular momentum, are constructed and shown to agree with the ADM formulae, ensuring consistency with established asymptotically flat spacetimes. The work lays groundwork for understanding higher-dimensional asymptotic structure, with future directions including links to null infinity, Bondi-type energies, and potential implications for black hole uniqueness 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.