[Paper Review] Asymptotic flatness at null infinity in higher dimensional gravity

Stefan Hollands, Akihiro Ishibashi|ArXiv.org|Nov 19, 2003

Black Holes and Theoretical Physics3 references18 citations

TL;DR

This paper proposes a geometric definition of asymptotic flatness at null infinity for even-dimensional spacetimes with d > 4, using conformal infinity methods. It derives a Hamiltonian generator for asymptotic time translations, defining a manifestly positive Bondi energy in higher dimensions that generalizes the 4D concept and ensures energy flux is always positive, with the formula differing qualitatively from the d=4 case due to distinct fall-off behaviors of gravitational perturbations.

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.

Motivation & Objective

• To define asymptotic flatness at null infinity in higher-dimensional spacetimes (d > 4, even) using conformal infinity methods.
• To ensure the definition is stable under linear perturbations.
• To construct a Hamiltonian generator conjugate to asymptotic time translations, enabling a definition of total energy (Bondi energy) in higher dimensions.
• To generalize the 4D Bondi energy concept to d > 4, showing qualitative differences in fall-off behavior and structure.
• To establish a manifestly positive energy flux for gravitational radiation in higher dimensions.

Proposed method

• Uses the conformal compactification of spacetime, introducing an unphysical metric g_ab = Ω² g̃_ab to bring null infinity to a finite boundary.
• Imposes fall-off conditions on the unphysical metric and its derivatives, particularly n_a n^a = O(Ω^{(d+2)/2}) and ∇_a n_b = O(Ω^{(d-2)/2}), to define asymptotic flatness.
• Applies the Wald-Zoupas Hamiltonian framework to derive the generator H_ξ conjugate to asymptotic time translations ξ^a.
• Derives the Bondi energy formula as an integral over a cross section of null infinity, involving the Ricci tensor, Weyl tensor, and news tensor N_ab.
• Fixes the conformal gauge by imposing conditions on the background metric, ensuring consistency and uniqueness of the energy expression.
• Expresses the energy in terms of the Weyl tensor’s Coulomb part and radiative contributions via the news tensor, with explicit dependence on Ω^{-(d-4)}.

Experimental results

Research questions

• RQ1 How can asymptotic flatness at null infinity be consistently defined in even-dimensional spacetimes with d > 4?
• RQ2 What is the correct generalization of the Bondi energy to higher dimensions, and how does it differ from the 4D case?
• RQ3 Is the proposed energy definition stable under linear perturbations in d > 4?
• RQ4 Can a manifestly positive energy flux for gravitational radiation be derived in higher dimensions using Hamiltonian methods?
• RQ5 Why do the definitions and formulas in d > 4 differ qualitatively from those in d = 4, particularly in the fall-off behavior of gravitational fields?

Key findings

• The proposed definition of asymptotic flatness is stable under linear perturbations in even dimensions d > 4.
• The Bondi energy in d > 4 is given by a Hamiltonian generator that explicitly separates Coulomb and radiative contributions via the Weyl tensor and news tensor.
• The energy flux through null infinity is always positive for future-directed time translations, ensuring physical consistency.
• In the d-dimensional Schwarzschild spacetime, the Bondi energy reduces to c(d−2)A_{d−2}/(16πG), matching the ADM mass, confirming consistency.
• The formula differs qualitatively from the 4D case, where the Ricci term is replaced by a combination involving S_ab and ρ_ab, indicating a distinct asymptotic structure in higher dimensions.
• The method fails in odd spacetime dimensions due to insufficient regularity of the unphysical metric at null infinity.