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The Newtonian limit

  • Bernard F. Schutz
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 205)

Abstract

We discuss in detail the development of the Newtonian and post-Newtonian approximations to general relativity. By using an initial-value approach, we are able to show that the post-Newtonian hierarchy through gravitational-radiation-reaction order is an asymptotic approximation to general relativity, thereby verifying the validity of the quadrupole formula for radiation reaction. We also show with equal rigor that the radiation from nearly-Newtonian systems obeys the far-field quadrupole formula (Landau-Lifshitz formula).There are no divergent terms in these approximations at any order, although logarithmic terms in the expansion parameter do appear at high order. We discuss the relationships of observables to post-Newtonian quantities by the method of osculating Newtonian orbits. Finally we discuss the role exact solutions may play in shedding light on some of these questions.

Keywords

Asymptotic Approximation Gravitational Radiation Radiation Reaction Divergent Term Newtonian Limit 
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References

  1. 1.
    See for example, C. W. Misner, K. S. Thorne, & J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973) or B. F. Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 1984). We adopt the sign and notation conventions of Misner, et al, op cit. Google Scholar
  2. 2.
    J. Ehlers, A. Rosenblum, J. N. Goldberg, & P. Havas, Astrophys. J. 208, L77 (1976).Google Scholar
  3. 3.
    E. Cartan, Ann. É cole Norm. Sup. 40, 325 (1923) and 41, 1 (1924); see Misner, et al, op cit. (ref. 1) for an exposition.Google Scholar
  4. 4.
    S. Chandresekhar, Astrophys. J. 142, 1488 (1965); S. Chandrasekhar & Y. Nutku, Astrophys. J. 158, 55 (1969); S. Chandrasekhar & E. P. Esposito, Astrophys. J. 160, 153 (1970). Radiation reaction itself was also studied at this time by W. L. Burke, J. Math. Phys. 12, 401 (1971).Google Scholar
  5. 5.
    See Misner, et al, op cit.(ref.1).Google Scholar
  6. 6.
    L. D. Landau & E. M. Lifshitz, Classical Theory of Fields (Addison-Wesley, Reading, Mass., 1971).Google Scholar
  7. 7.
    See for example G. D. Kerlick, Gen. Rel. Grav. 12, 467 and 521 (1980).Google Scholar
  8. 8.
    M. Walker & C. M. Will, Astrophys. J. 242, L129 (1980).Google Scholar
  9. 9.
    B. F. Schutz, Phys. Rev. D 22, 249 (1980).Google Scholar
  10. 10.
    T. Futamase & B. F. Schutz, Phys. Rev. D 28, 2363 (1983).Google Scholar
  11. 11.
    T. Futamase, Phys. Rev. D 28, 2373 (1983).Google Scholar
  12. 12.
    T. Futamase & B. F. Schutz (in preparation).Google Scholar
  13. 13.
    J. L. Anderson & T. C. Decanio, Gen. Rel. Grav. 6, 197 (1975).Google Scholar
  14. 4.
    Note that this formula was incorrectly written down in ref. 10. This had no effect on subsequent equations.Google Scholar
  15. 15.
    B. Paczynski, B. & R. Sienkiewicz, Astrophys. J. 268, 825 (1983)Google Scholar
  16. 15a.
    S. Chandrasekhar, Phys. Rev. Lett. 24, 611 (1970); J. L. Friedman & B. F. Schutz, Astrophys. J. 222, 281 (1978).Google Scholar
  17. 16.
    E. F. L. Balbinski & B. F. Schutz, Mon. Not. R. astr. Soc. 200, 43P (1982)Google Scholar
  18. 16a.
    R. A. Saenz & S. L. Shapiro, Astrophys. J. 221, 286 (1978) and 229, 1107 (1979).Google Scholar
  19. 17.
    J. H. Taylor & J. M. Weisberg, Astrophys. J. 253, 908 (1982); V. Boriakoff, D. C. Ferguson, M. P. Haugan, Y. Terzian & S. Teukolsky, Astrophys. J. 261, L101 (1982).Google Scholar
  20. 18.
    B. F. Schutz in X. Fustero & E. Verdaguer, eds., Relativistic Astrophysics and Cosmology (World Scientific Publishers, 1984), p.35.Google Scholar
  21. 19.
    B. F. Schutz, Mon. Not. R. astr. Soc. 207, 37P (1984).Google Scholar
  22. 20.
    H. Stephani, General Relativity (Cambridge University Press, Cambridge, 1982).Google Scholar
  23. 21.
    R. A. Isaacson, Phys. Rev. 166, 1263 and 1272 (1968).Google Scholar
  24. 22.
    W. L. Burke, op cit.(ref.4); R. E. Kates, Phys. Rev. D 22, 1871 (1980); and J. L. Anderson, R. E. Kates, L. S. Kegeles & R. G. Madonna, Phys. Rev. D 25, 2038 (1982).Google Scholar
  25. 23.
    J. Winicour, J. Math. Phys. 24, 1193 (1983), and to be published.Google Scholar
  26. 24.
    Balbinski & Schutz, op cit. (ref. 16); and E. F. L. Balbinski, S. L. Detweiler, L. Lindblom, and B. F. Schutz, to be published.Google Scholar
  27. 25.
    T. Damour & N. Deruelle, C. R. Acad. Sci.Ser. B 293, II 5037 (1981)and 677 (1981); T. Damour in N. Deruelle and T. Piran, eds., Gravitational Radiation (North-Holland, Amsterdam, 1983), p.59.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Bernard F. Schutz
    • 1
  1. 1.Department of Applied Mathematics and AstronomyUniversity CollegeCardiffU.K.

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