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On quadratic first integrals of the geodesic equations for type {22} spacetimes

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Abstract

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.

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Authors and Affiliations

  1. Department of Mathematics, Birkbeck College, London, England

    Martin Walker & Roger Penrose

  2. Department of Physics, University of Texas at Austin, 78712, Austin, Texas, USA

    Martin Walker & Roger Penrose

Authors
  1. Martin Walker
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  2. Roger Penrose
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Walker, M., Penrose, R. On quadratic first integrals of the geodesic equations for type {22} spacetimes. Commun.Math. Phys. 18, 265–274 (1970). https://doi.org/10.1007/BF01649445

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  • DOI: https://doi.org/10.1007/BF01649445

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