Skip to main content
SpringerLink
Log in

Formulations of the 3+1 evolution equations in curvilinear coordinates

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

Following Brown (Phys Rev D79:104029, 2009), in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both the Nagy–Ortiz–Reula and the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulations, we specialize to the case of spherical symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified BSSN formulation at work in spherical symmetry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnowitt, R., Deser, S., Misner, C.W.: In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 227–265. New York, Wiley (1962)

  2. York, J.: In: Smarr, L. (ed.) Sources of Gravitational Radiation. Cambridge University Press, Cambridge (1979)

  3. Alcubierre M.: Introduction to 3+1 Numerical Relativity. Oxford University Press, New York (2008)

    Book  MATH  Google Scholar 

  4. Shibata M., Nakamura T.: Phys. Rev. D52, 5428 (1995)

    MathSciNet  ADS  Google Scholar 

  5. Baumgarte T.W., Shapiro S.L.: Phys. Rev. D59, 024007 (1998)

    MathSciNet  ADS  Google Scholar 

  6. Campanelli M., Lousto C.O., Marronetti P., Zlochower Y.: Phys. Rev. Lett. 96, 111101 (2006)

    Article  ADS  Google Scholar 

  7. Campanelli M., Lousto C.O., Zlochower Y.: Phys. Rev. D73, 061501(R) (2006)

    MathSciNet  ADS  Google Scholar 

  8. Baker J.G., Centrella J., Choi D.I., Koppitz M., van Meter J.: Phys. Rev. Lett. 96, 111102 (2006)

    Article  ADS  Google Scholar 

  9. Pretorius F.: Phys. Rev. Lett. 95, 121101 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  10. Brown J.D.: Phys. Rev. D79, 104029 (2009)

    ADS  Google Scholar 

  11. Nagy G., Ortiz O.E., Reula O.A.: Phys. Rev. D70, 044012 (2004)

    MathSciNet  ADS  Google Scholar 

  12. Bona C., Massó J.: Phys. Rev. D40, 1022 (1989)

    ADS  Google Scholar 

  13. Bona C., Massó J.: Phys. Rev. Lett. 68, 1097 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Bona C., Massó J.: Int. J. Modern Phys. C Phys. Comput. 4, 88 (1993)

    Google Scholar 

  15. Bona C., Massó J., Seidel E., Stela J.: Phys. Rev. Lett. 75, 600 (1995)

    Article  ADS  Google Scholar 

  16. Bona C., Massó J., Seidel E., Stela J.: Phys. Rev. D56, 3405 (1997)

    ADS  Google Scholar 

  17. Alcubierre M., Brügmann B., Diener P., Koppitz M., Pollney D., Seidel E., Takahashi R.: Phys. Rev. D67, 084023 (2003)

    ADS  Google Scholar 

  18. Baker J.G., Centrella J., Choi D.I., Koppitz M., van Meter J.: Phys. Rev. D73, 104002 (2006)

    ADS  Google Scholar 

  19. Pretorius F.: Class. Quant. Grav. 22, 425 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Alcubierre M., González J.A.: Comp. Phys. Comm. 167, 76 (2005)

    Article  ADS  MATH  Google Scholar 

  21. Ruiz M., Alcubierre M., Nuñez D.: Gen. Relativ. Gravit. 40, 159 (2008)

    Article  ADS  MATH  Google Scholar 

  22. Arbona A., Bona C.: Comput. Phys. Commun. 118, 229 (1999)

    Article  ADS  MATH  Google Scholar 

  23. Hannam M., Husa S., Pollney D., Brugmann B., O’Murchadha N.: Phys. Rev. Lett. 99, 241102 (2007)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, Mexico D.F., 04510, Mexico

    Miguel Alcubierre & Martha D. Mendez

Authors
  1. Miguel Alcubierre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martha D. Mendez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel Alcubierre.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alcubierre, M., Mendez, M.D. Formulations of the 3+1 evolution equations in curvilinear coordinates. Gen Relativ Gravit 43, 2769–2806 (2011). https://doi.org/10.1007/s10714-011-1202-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-011-1202-x

Keywords

Search

Navigation