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On a Five-Dimensional Version of the Goldberg-Sachs Theorem

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Relativity and Gravitation

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 157))

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Abstract

The recently developed generalization of the Goldberg-Sachs theorem to five-dimensional Einstein spacetimes is summarized. This generalization involves two steps. First it has been proven that in arbitrary dimension an Eistein spacetime admitting a multiple WAND admits also a multiple geodetic WAND. Second, in five dimensions, the \(3 \times 3\) optical matrix of such geodetic multiple WAND can be cast to one of three canonical forms, each determined by two free parameters.

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Notes

  1. 1.

    An Einstein spacetime is a solution of the vacuum Einstein equation, possibly with a cosmological constant, i.e. with the Ricci tensor \(R_{ab}=(R/d)g_{ab}\) in \(d\) dimensions.

  2. 2.

    Shear is defined as traceless symmetric part of the optical matrix.

  3. 3.

    Sufficient conditions on \(\rho _{ij}\) for \(\varvec{\ell }\) to be a multiple WAND are not in full generality known, but it has been shown [5, 6] that \(\rho _{ij}=0\) (Kundt class) and \(\rho _{ij} \propto \delta _{ij}\) (Robinson-Trautman class) are examples of such sufficient conditions.

  4. 4.

    \(\bar{g}_{\mu \nu }\) is a metric of constant curvature and KS vector \({\varvec{k}}\) is null with respect to \(\bar{g}_{\mu \nu }\) and thus also with respect to \(g_{\mu \nu }\).

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Acknowledgments

The authors acknowledge support from research plan RVO: 67985840 and research grant no P203/10/0749.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Marcello Ortaggio, Vojtěch Pravda & Alena Pravdová

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Harvey S. Reall

Authors
  1. Marcello Ortaggio
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  2. Vojtěch Pravda
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  3. Alena Pravdová
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  4. Harvey S. Reall
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Correspondence to Marcello Ortaggio .

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Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Jiří Bičák

  2. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Tomáš Ledvinka

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Ortaggio, M., Pravda, V., Pravdová, A., Reall, H.S. (2014). On a Five-Dimensional Version of the Goldberg-Sachs Theorem. In: Bičák, J., Ledvinka, T. (eds) Relativity and Gravitation. Springer Proceedings in Physics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-06761-2_23

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