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On scalar curvature invariants in three dimensional spacetimes

  • N. K. Musoke
  • D. D. McNutt
  • A. A. Coley
  • D. A. Brooks
Research Article

Abstract

We wish to construct a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some order of differentiation in three dimensional (3D) Lorentzian spacetimes. In order to do this we utilize the Cartan–Karlhede equivalence algorithm since, in general, all Cartan invariants are related to scalar polynomial curvature invariants. As an example we apply the algorithm to the class of 3D Szekeres cosmological spacetimes with comoving dust and cosmological constant \(\Lambda \). In this case, we find that there are at most twelve algebraically independent Cartan invariants, including \(\Lambda \). We present these Cartan invariants, and we relate them to twelve independent scalar polynomial curvature invariants (two, four and six, respectively, zeroth, first, and second order scalar polynomial curvature invariants).

Keywords

Scalar curvature invariants Three dimensions Equivalence  Cartan–Karlhede Scalar polynomial curvature invariants Algebraic independence 

Notes

Acknowledgments

We would like to thank Malcolm MacCallum and Robert Milson for helpful comments. This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada
  2. 2.Perimeter Institute For Theoretical PhysicsWaterlooCanada

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