Advertisement

New conserved currents for vacuum space-times in dimension four with a Killing vector

  • Alfonso García-Parrado Gómez-Lobo
Research Article
  • 194 Downloads

Abstract

A new family of conserved currents for vacuum space-times with a Killing vector is presented. The currents are constructed from the superenergy tensor of the Mars-Simon tensor and using the positivity properties of the former we find that the conserved charges associated to the currents have natural positivity properties in certain cases. Given the role played by the Mars-Simon tensor in local and semi-local characterisations of the Kerr solution, the currents presented in this work are useful to construct non-negative scalar quantities characterising Kerr initial data (known in the literature as non-Kerrness) which in addition are conserved charges.

Keywords

Conserved current Conserved charge Mars-Simon tensor Kerr solution 

Mathematics Subject Classification

83C15 83C05 58A25 

Notes

Acknowledgments

We thank Prof. J. M. M. Senovilla for reading the manuscript and comments. Supported by the project FIS2014-57956-P of Spanish “Ministerio de Economía y Competitividad” and PTDC/MAT-ANA/1275/2014 of Portuguese “Fundação para a Ciência e a Tecnologia”.

References

  1. 1.
    Ashtekar, A.: Lectures on Non-perturbative Canonical Gravity. Advanced Series in Astrophysics and Cosmology. World Scientific, Singapore (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Bäckdahl, T., Kroon, J.A.V.: Geometric invariant measuring the deviation from Kerr data. Phys. Rev. Lett. 104, 231102, 4 (2010)Google Scholar
  3. 3.
    Bäckdahl, T., Kroon, J.A.V.: On the construction of a geometric invariant measuring the deviation from Kerr data. Ann. Henri Poincaré 11, 1225–1271 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bäckdahl, T., Kroon, J.A.V.: The ‘non-Kerrness’ of domains of outer communication of black holes and exteriors of stars. Proc. R. Soc. A 467, 1701–1718 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bäckdahl, T., Kroon, J.A.V.: Constructing “non-Kerrness” on compact domains. J. Math. Phys. 53, 042503, 13 (2012)Google Scholar
  6. 6.
    Bergqvist, G., Eriksson, I., Senovilla, J.M.M.: New electromagnetic conservation laws. Class. Quantum Gravity 20, 2663–2668 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bergqvist, G.: Positivity of general superenergy tensors. Commun. Math. Phys. 207, 467–479 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eriksson, I.: Conserved matter superenergy currents for hypersurface orthogonal Killing vectors. Class. Quantum Gravity 23, 2279–2290 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eriksson, I.: Conserved matter superenergy currents for orthogonally transitive abelian \(G_2\) isometry groups. Class. Quantum Gravity 24, 4955–4968 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    García-Parrado, A., Kroon, J.A.Valiente: Kerr initial data. Class. Quantum Gravity 25, 205018, 20 (2008)Google Scholar
  11. 11.
    García-Parrado, A., Senovilla, J.M.M.: A set of invariant quality factors measuring the deviation from the Kerr metric. Gen. Relativ. Gravit. 45, 1095–1127 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Heusler, M.: Black Hole Uniqueness Theorems, Cambridge Lecture Notes in Physics, vol. 6. Cambridge University Press, Cambridge (1996)CrossRefMATHGoogle Scholar
  13. 13.
    Ionescu, A.D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lazkoz, R., Senovilla, J.M.M., Vera, R.: Conserved superenergy currents. Class. Quantum Gravity 20, 4135–4152 (2003)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mars, M.: A spacetime characterization of the Kerr metric. Class. Quantum Gravity 16, 2507–2523 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mars, M.: Uniqueness properties of the Kerr metric. Class. Quantum Gravity 17, 3353–3373 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mars, M.: Spacetime Ehlers group: transformation law for the Weyl tensor. Class. Quantum Gravity 18, 719–738 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mars, M., Senovilla, J.M.M.: Spacetime characterizations of \(\Lambda \)-vacuum metrics with a null Killing 2-form, http://arxiv.org/abs/1604.07274v1
  19. 19.
    Martín-García, J.M.: xAct: efficient tensor computer algebra, http://www.xact.es
  20. 20.
    Penrose, R., Rindler, W.: Spinors and Space-Time, vol. 1. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  21. 21.
    Senovilla, J.M.M.: Super-energy tensors. Class. Quantum Grav. 17, 2799–2841 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Física TeóricaUniversidad del País VascoBilbaoSpain
  2. 2.Centro de MatemáticaUniversidade do MinhoBragaPortugal

Personalised recommendations