Higher Dimensional Bondi Energy with a Globally Specified Background Structure

TL;DR

The paper addresses defining a higher-dimensional Bondi energy in even dimensions using a gauge with globally compact cross-sections of null infinity. It adopts a Gaussian null conformal gauge, introduces a regularized news tensor, and uses the Wald–Zoupas covariant phase space framework to construct a Bondi energy-momentum ${\cal H}_\xi$ with a flux formula. A central result is a theorem that ensures the divergence ${\tilde{\nabla}}_a P^a$ extends smoothly to ${\mathscr I}$ and yields a universal Bondi energy-momentum integrand $P^a$ for all even $d \ge 4$, together with a concrete energy-loss rate. The work clarifies asymptotic symmetry structure in higher dimensions (no generic supertranslations) and provides a foundation for comparing Bondi and ADM energies, with potential extensions to nonlinear regimes and braneworld scenarios.

Abstract

A higher (even spacetime) dimensional generalization of the Bondi energy has recently been proposed by gr-qc/0304054 within the framework of conformal infinity and Hamiltonian formalizm. The gauge condition employed in gr-qc/0304054 to derive the Bondi energy expression is, however, peculiar in the sense that cross-sections of null infinity specified by that gauge are anisotropic and in fact non-compact. For this reason, that gauge is difficult to use for explicit computations of the Bondi energy in general, asymptotically flat radiative spacetimes. Also it is not clear, under that gauge condition, whether apparent difference between the expressions of higher dimensional Bondi energy and the 4-dimensional one is due to the choice of gauges or qualitatively different nature of higher dimensional gravity from 4-dimensional gravity. In this paper, we consider instead, Gaussian null conformal gauge as one of more natural gauge conditions that admit a global specification of background structure with compact, spherical cross-sections of null infinity. Accordingly, we modify the previous definition of higher dimensional news tensor so that it becomes well-defined in the Gaussian null conformal gauge and derive, for vacuum solutions, the expression for the Bondi energy-momentum in the new gauge choice, which takes a universal form in arbitrary (even spacetime) dimensions greater than or equal to four.