On scalar curvature invariants in three dimensional spacetimes

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


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).


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



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.


  1. 1.
    Coley, A.A., Hervik, S., Gibbons, G.W., Pope, C.N.: Class. Quant. Grav 25, 145017 (2008). [arXiv:0803.2438]ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Hervik, S., Pravda, V., Pravdova, A.: Class. Quant. Grav. 31, 215005 (2014). [arXiv:1311.0234]ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Coley, A., Hervik, S., Pelavas, N.: Class. Quant. Grav 23, 3053 (2006). [arXiv:gr-qc/0509113]ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Coley, A., Hervik, S., Pelavas, N.: Class. Quant. Grav 26, 125011 (2009). [arXiv:0904.4877]ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Coley, A.: Class. Quant. Grav 25, 033001 (2008). [arXiv:0710.1598]MathSciNetCrossRefGoogle Scholar
  6. 6.
    Coley, A., Fuster, A., Hervik, S.: Int. J. Mod. Phys. A 24, 1119 (2009). [arXiv:0707.0957]ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Coley, A.A.: Phys. Rev. Lett. 89, 281601 (2002).[arXiv:hep-th/0211062]
  8. 8.
    Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    MacCallum, M.A.H.: Spacetime invariants and their uses. In: Proceedings of International Conference on Relativistic Astrophysics, Lahore, February (2015, to appear). arXiv:1504.06857 [gr-qc]
  10. 10.
    Abdelqader, M., Lake, K.: Phys. Rev. D 91, 084017 (2015). [arXiv:1412.8757]
  11. 11.
    Page, D.N., Shoom, A.: Phys. Rev. Lett. 114, 141102 (2015). [arXiv:1501.03510]
  12. 12.
    Fulling, S.A., King, R.C., Wybourne, B.G., Cummings, C.J.: Class. Quant. Grav 9, 1151 (1992)ADSCrossRefGoogle Scholar
  13. 13.
    Coley, A., Hervik, S., Pelavas, N.: Class. Quant. Grav 26, 025013 (2009). [arXiv:0901.0791]ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Coley, A., Hervik, S., Pelavas, N.: Class. Quant. Grav 27, 102001 (2010). [arXiv:1003.2373]ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Zakhary, E., McIntosh, C.B.G.: Gen. Relativ. Gravit 29, 539 (1997)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Carminati, J., Lim, A.E.K.: J. Math. Phys. 47, 052504 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    MacCallum, M.A.H., Aman, J.E.: CQG 3(6), 1133–41 (1986)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Coley, A.A., MacDougall, A., McNutt, D.D.: Class. Quant. Grav. 31(23), 235010 (2014). [arXiv:1409.1185]ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Aliev, A.N., Nutku, Y.: Class. Quant. Grav. 12, 2913 (1995)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hall, G.S., Morgan, T., Perjes, Z.: Gen. Relativ. Gravit 19, 1137 (1987)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Cox, D.P.G.: Phys. Rev. D 68, 124008 (2003)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Barrow, J.D., Shaw, D.J., Tsagas, C.G.: Class. Quant. Grav. 23, 5291 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Sousa, F.C., Fonseca, J.B., Romero, C.: Class. Quant. Grav. 25, 035007 (2008)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Karlhede, A.: Gen. Relativ. Gravit 12, 693 (1980)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Milson, R., Wylleman, L.: Class. Quant. Grav. 30, 095004 (2013)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Thomas, T.Y.: Differential Invariants of Generalised Spaces. Cambridge University Press, Cambridge (1934)Google Scholar
  27. 27.
    Siklos, S.T.C., Fustero, X., Verdaguer, E.: Relativistic astrophysics and cosmology. In: Proceedings of the XIVth GIFT International Seminar on Ther-Phys, page 201. World Scientific, Singapore (1984)Google Scholar
  28. 28.
    Brooks, D., McNutt, D.D., Simard, J.P., Musoke, N.: [arXiv:1506.03415]

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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