Spinning solutions in general relativity with infinite central density

Research Article


This paper presents general relativistic numerical simulations of uniformly rotating polytropes. Equations are developed using MSQI coordinates, but taking a logarithm of the radial coordinate. The result is relatively simple elliptical differential equations. Due to the logarithmic scale, we can resolve solutions with near-singular mass distributions near their center, while the solution domain extends many orders of magnitude larger than the radius of the distribution (to connect with flat space–time). Rotating solutions are found with very high central energy densities for a range of adiabatic exponents. Analytically, assuming the pressure is proportional to the energy density (which is true for polytropes in the limit of large energy density), we determine the small radius behavior of the metric potentials and energy density. This small radius behavior agrees well with the small radius behavior of large central density numerical results, lending confidence to our numerical approach. We compare results with rotating solutions available in the literature, which show good agreement. We study the stability of spherical solutions: instability sets in at the first maximum in mass versus central energy density; this is also consistent with results in the literature, and further lends confidence to the numerical approach.


Gravitation General relativity Numerical relativity 


  1. 1.
    Friedman, J.L., Ipser, J.R., Parker, L.: Astrophys. J 304, 115 (1986).  https://doi.org/10.1086/164149 ADSCrossRefGoogle Scholar
  2. 2.
    Stergioulas, N., Friedman, J.L.: Astrophys. J. 444, 306 (1995).  https://doi.org/10.1086/175605 ADSCrossRefGoogle Scholar
  3. 3.
    Komatsu, H., Eriguchi, Y., Hachisu, I.: Mon. Not. R. Astron. Soc. 237(2), 355 (1989).  https://doi.org/10.1093/mnras/237.2.355 ADSCrossRefGoogle Scholar
  4. 4.
    Komatsu, H., Eriguchi, Y., Hachisu, I.: Mon. Not. R. Astron. Soc. 239(1), 153 (1989).  https://doi.org/10.1093/mnras/239.1.153 ADSCrossRefGoogle Scholar
  5. 5.
    Cook, G.B., Shapiro, S.L., Teukolsky, S.A.: Astrophys. J. 398, 203 (1992).  https://doi.org/10.1086/171849 ADSCrossRefGoogle Scholar
  6. 6.
    Cook, G.B., Shapiro, S.L., Teukolsky, S.A.: Astrophys. J. 422, 227 (1994).  https://doi.org/10.1086/173721 ADSCrossRefGoogle Scholar
  7. 7.
    Cook, G.B., Shapiro, S.L., Teukolsky, S.A.: Astrophys. J. 424, 823 (1994).  https://doi.org/10.1086/173934 ADSCrossRefGoogle Scholar
  8. 8.
    Bonazzola, S., Gourgoulhon, E., Salgado, M., Marck, J.A.: Astron. Astrophys. 278, 421 (1993)ADSGoogle Scholar
  9. 9.
    Bonazzola, S., Gourgoulhon, E.: Class. Quantum Grav. 11, 1775 (1994)ADSCrossRefGoogle Scholar
  10. 10.
    Salgado, M., Bonazzola, S., Gourgoulhon, E., Haensel, P.: Astron. Astrophys. 291, 155 (1994)ADSGoogle Scholar
  11. 11.
    Stergioulas, N.: Living Rev. Relativ. 6(3) (2003).  https://doi.org/10.1007/lrr-2003-3
  12. 12.
    Gourgoulhon, E.: arXiv:1003.5015 [gr-qc] (2010)
  13. 13.
    Shibata, M.: Numerical Relativity. World Scientific, Hackensack (2016)MATHGoogle Scholar
  14. 14.
    Friedman, J.L., Stergioulas, N., Astron, B.: Soc. India 39, 21 (2011)Google Scholar
  15. 15.
    Friedman, J.L., Ipser, J.R., Sorkin, R.D.: Astrophys. J. 325, 722 (1988).  https://doi.org/10.1086/166043 ADSCrossRefGoogle Scholar
  16. 16.
    Friedman, J.L.: J. Astrophys. Astron. 17, 199 (1996).  https://doi.org/10.1007/BF02702304 ADSCrossRefGoogle Scholar
  17. 17.
    Baumgarte, T.W., Shapiro, S.L., Shibata, M.: Astrophys. J. Lett. 528(1), L29 (2000)ADSCrossRefGoogle Scholar
  18. 18.
    Takami, K., Rezzolla, L., Yoshida, S.: Mon. Not. R. Astron. Soc. 416, 1 (2011).  https://doi.org/10.1111/j.1745-3933.2011.01085.x ADSCrossRefGoogle Scholar
  19. 19.
    Chandrasekhar, S.: Phys. Rev. Lett. 12, 116 (1964)ADSCrossRefGoogle Scholar
  20. 20.
    Chandrasekhar, S.: Astrophys. J. 140, 417 (1964).  https://doi.org/10.1086/147938 ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Saijo, M., Shibata, M., Baumgarte, T.W., Shapiro, S.L.: Astrophys. J. 548, 919 (2001).  https://doi.org/10.1086/319016 ADSCrossRefGoogle Scholar
  22. 22.
    Baiotti, L., de Pietri, R., Manca, G.M., Rezzolla, L.: Phys. Rev. D 75(4), 044023 (2007).  https://doi.org/10.1103/PhysRevD.75.044023 ADSCrossRefGoogle Scholar
  23. 23.
    Manca, G.M., Baiotti, L., DePietri, R., Rezzolla, L.: Class. Quantum Grav. 24, S171 (2007).  https://doi.org/10.1088/0264-9381/24/12/S12 ADSCrossRefGoogle Scholar
  24. 24.
    Misner, C.W., Zapolsky, H.S.: Phys. Rev. Lett. 12, 635 (1964).  https://doi.org/10.1103/PhysRevLett.12.635 ADSCrossRefGoogle Scholar
  25. 25.
    Misner, C.W., Zapolsky, H.S.: Phys. Rev. Lett. 13, 122 (1964).  https://doi.org/10.1103/PhysRevLett.13.122 ADSCrossRefGoogle Scholar
  26. 26.
    Oppenheimer, J.R., Volkoff, G.M.: Phys. Rev. 55, 374 (1939).  https://doi.org/10.1103/PhysRev.55.374 ADSCrossRefGoogle Scholar
  27. 27.
    Tooper, R.F.: Astrophys. J. 140, 434 (1964).  https://doi.org/10.1086/147939 ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Gourgoulhon, E.: Astron. Astrophys. 252, 651 (1991)ADSMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Colorado School of MinesGoldenUSA

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