Asymptotic flatness at null infinity in higher dimensional gravity

TL;DR

The paper addresses how to define a meaningful notion of an isolated system and associated conserved quantities at null infinity in higher-dimensional gravity, focusing on even spacetime dimensions $d>4$. It adopts a conformal-infinity framework with a background geometry to specify precise fall-off conditions, and then proves a linear stability theorem for the proposed asymptotic flatness. Using the Wald–Zoupas formalism, it derives an explicit Bondi-energy expression and a radiative flux at null infinity, extending the Bondi concept beyond four dimensions and ensuring the radiated energy flux is positive. The work clarifies crucial differences from the $d=4$ case and demonstrates that the construction does not extend to odd dimensions, while providing a natural higher-dimensional generalization of energy and momentum for radiating spacetimes.

Abstract

We give a geometrical definition of the asymptotic flatness at null infinity in spacetimes of even dimension $d$ greater than 4 within the framework of conformal infinity. Our definition is shown to be stable against perturbations to linear order. We also show that our definition is stringent enough to allow one to define the total energy of the system viewed from null infinity as the generator conjugate to an asymptotic time translation. We derive an expression for the generator conjugate within the Hamiltonian framework, and propose to take this notion of energy as the natural generalisation of the Bondi energy to higher dimensions. Our definitions of asymptotic flatness and the Bondi energy formula differ qualitatively from the corresponding definitions in $d=4$; although the asymptotic structure of null infinity in higher dimensions parallels that in 4-dimensions in some ways, the latter seems to be a rather special case on the whole compared to general $d>4$. Our definitions and constructions do not work in odd spacetime dimensions, essentially because the unphysical metric seems to have insufficient regularity properties at null infinity in that case.