Asymptotic Flatness and Bondi Energy in Higher Dimensional Gravity

TL;DR

This work extends the notion of asymptotic flatness and Bondi energy to even spacetime dimensions greater than four by employing conformal infinity and the Wald–Zoupas formalism. It establishes linear stability of the proposed asymptotic structure, constructs a generator for asymptotic time translations, and derives a local Bondi energy formula whose radiative flux is manifestly positive and reduces to the ADM mass for Schwarzschild-like solutions. A key finding is the absence of a direct higher-dimensional analog of the 4D supertranslations, and the framework does not straightforwardly extend to odd dimensions due to regularity issues at null infinity. The results pave the way for incorporating matter fields and higher-derivative gravity in a similar asymptotic analysis, while underscoring qualitative differences from the familiar 4D case.

Abstract

We give a general geometric definition of asymptotic flatness at null infinity in $d$-dimensional general relativity ($d$ even) within the framework of conformal infinity. Our definition is arrived at via an analysis of linear perturbations near null infinity and shown to be stable under such perturbations. The detailed fall off properties of the perturbations, as well as the gauge conditions that need to be imposed to make the perturbations regular at infinity, are qualitatively different in higher dimensions; in particular, the decay rate of a radiating solution at null infinity differs from that of a static solution in higher dimensions. The definition of asymptotic flatness in higher dimensions consequently also differs qualitatively from that in $d=4$. We then derive an expression for the generator conjugate to an asymptotic time translation symmetry for asymptotically flat spacetimes in $d$-dimensional general relativity ($d$ even) within the Hamiltonian framework, making use especially of a formalism developed by Wald and Zoupas. This generator is given by an integral over a cross section at null infinity of a certain local expression and is taken to be the definition of the Bondi energy in $d$ dimensions. Our definition yields a manifestly positive flux of radiated energy. Our definitions and constructions fail in odd spacetime dimensions, essentially because the regularity properties of the metric at null infinity seem to be insufficient in that case. We also find that there is no direct analog of the well-known infinite set of angle dependent translational symmetries in more than 4 dimensions.