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Asymptotic flatness and Bondi energy in higher dimensional gravity

J. Math. Phys. 46, 022503 (2005); doi:10.1063/1.1829152

Published 27 January 2005

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Stefan Hollands
Enrico Fermi Institute, Department of Physics, University of Chicago, Chicago, Ilinois 60637

Akihiro Ishibashi
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
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 falloff 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 space–times 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 space–time 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 four dimensions. ©2005 American Institute of Physics
History: Received 15 April 2003; accepted 14 July 2004; published 27 January 2005
Permalink: http://link.aip.org/link/?JMAPAQ/46/022503/1
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KEYWORDS and PACS

Keywords
PACS
  • 04.60.-m
    Quantum gravity
  • 95.30.Sf
    Relativity and gravitation in astrophysics
  • 04.20.Gz
    Spacetime topology, causal structure, spinor structure in general relativity
  • 04.62.+v
    Quantum field theory in curved spacetime
  • YEAR: 2005

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (22)
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  22. Note that throughout this paper, our convention for the * operation of a p-form is (*A)a1[centered ellipsis]ad–p = epsilona1[centered ellipsis]ad–pb1[centered ellipsis]bpAb1[centered ellipsis]bp.

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