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Conformal Infinity – Development and Applications

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Beyond Einstein

Part of the book series: Einstein Studies ((EINSTEIN,volume 14))

Abstract

The notion of conformal infinity grew out of the desire to describe graviational waves in a coordinate independent way. In a sense it abstracts the notion of a far-field in electrodynamics in a geometric way. This contribution describes the development of this idea from the earliest attempts to geometrically capture the notion of gravitational radiation by Felix Pirani to the rigorous definition by Roger Penrose which is universally accepted today. The usefulness of the concept is demonstrated with some applications.

This article is dedicated to the memory of Jürgen Ehlers.

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Notes

  1. 1.

    Pirani uses Petrov’s classification of the Riemann tensor in terms of three types, which are not enough to characterise all possibilities. Petrov type I corresponds to the types {1111}, {22} and {–}, while Petrov type {II} contains type {211} and {4} and Petrov’s type II is type {31}.

  2. 2.

    The letter \({\mathcal {I}}\) is a ‘\({\mathcal {I}}\)’, from which the colloquial name ‘scri’ for null-infinity derives.

  3. 3.

    The name arises from the fact that when the part of Σ lying inside \(\tilde {\mathcal {M}}\) is considered as a hypersurface with induced geometry from the physical metric \(\tilde g_{ab}\) one finds that it is a manifold with asymptotically constant negative scalar curvature, just like the space-like hyperboloids in Minkowski space.

  4. 4.

    Strictly, speaking this refers only to non-stochastic sources.

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Acknowledgements

The author is very grateful to the organisers of the conference ‘Beyond Einstein’ for the opportunity to present the material of this contribution. Furthermore, it is a pleasure to thank Ted Newman, Roger Penrose and Engelbert Schücking for sharing their memories of the development of the subject.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9010, New Zealand

    Jörg Frauendiener

  2. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316, Oslo, Norway

    Jörg Frauendiener

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  1. Jörg Frauendiener
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Correspondence to Jörg Frauendiener .

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  1. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    David E. Rowe

  2. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    Tilman Sauer

  3. Centre François Viète, Université de Nantes, Nantes Cedex, France

    Scott A. Walter

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Frauendiener, J. (2018). Conformal Infinity – Development and Applications. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_16

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  • DOI: https://doi.org/10.1007/978-1-4939-7708-6_16

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