Communications in Mathematical Physics

, Volume 65, Issue 1, pp 45–76 | Cite as

On the proof of the positive mass conjecture in general relativity

  • Richard Schoen
  • Shing-Tung Yau


LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.


Neural Network General Relativity Complex System Nonlinear Dynamics Total Mass 
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  1. 1.
    Arnowitt, R., Deser, S., Misner, C.: Phys. Rev.118, 1100 (1960b)CrossRefGoogle Scholar
  2. 2.
    Geroch, R.: General Relativity. Proc. Symp. Pure Math.27, 401–414 (1975)Google Scholar
  3. 3.
    Brill, D., Deser, S.: Ann. Phys.50, 548 (1968)CrossRefGoogle Scholar
  4. 4.
    Choquet-Bruhat, Y., Fischer, A., Marsden, J.: Maximal hypersurfaces and positivity of mass. Preprint (1978)Google Scholar
  5. 5.
    Choquet-Bruhat, Y., Marsden, J.: Solution of the local mass problem in general relativity. Commun. math. Phys.51, 283–296 (1976)CrossRefGoogle Scholar
  6. 6.
    Jang, P. S.: J. Math. Phys.1, 141 (1976)CrossRefGoogle Scholar
  7. 7.
    Leibovitz, C., Israel, W.: Phys. Rev.1 D, 3226 (1970)Google Scholar
  8. 8.
    Misner, C.: Astrophysics and general relativity, Chretien, M., Deser, S., Goldstein, J. (ed.). New York: Gordon and Breach 1971Google Scholar
  9. 9.
    Geroch, R.: J. Math. Phys.13, 956 (1972)CrossRefGoogle Scholar
  10. 10.
    Chern, S.S.: Minimal submanifolds in a Riemannian manifold. University of Kansas (1968) (mimeographed lecture notes)Google Scholar
  11. 11.
    Finn, R.: On a class of conformal metrics, with application to differential geometry in the large. Comment. Math. Helv.40, 1–30 (1965)Google Scholar
  12. 12.
    Huber, A.: Vollständige konforme Metriken und isolierte Singularitäten subharmonischer Funktionen. Comment. Math. Helv.41, 105–136 (1966)Google Scholar
  13. 13.
    Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32, 13–72 (1957)Google Scholar
  14. 14.
    Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. Math.60, 237–247 (1954)Google Scholar
  15. 15.
    Alexander, H., Osserman, R.: Area bounds for various classes of surfaces. Am. J. Math.97 (1975)Google Scholar
  16. 16.
    Federer, H.: Geometric measure theory. Berlin, Heidelberg, New York: Springer 1969Google Scholar
  17. 17.
    Morrey, C. B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  18. 18.
    Hörmander, L.: Linear partial differential operators. Berlin, Heidelberg, New York: Springer 1969Google Scholar
  19. 19.
    O'Murchadha, N., York, J. W.: Gravitational Energy. Phys. Rev. D10, 2345–2357 (1974)CrossRefGoogle Scholar
  20. 20.
    Kazdan, J., Warner, F.: Prescribing curvatures. Proc. Symp. Pure Math.27, 309–319 (1975)Google Scholar
  21. 21.
    Schoen, R., Simon, L., Yau, S.-T.: Curvature estimates on minimal hypersurfaces. Acta Math.134, 275–288 (1975)Google Scholar

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© Springer-Verlag 1979

Authors and Affiliations

  • Richard Schoen
  • Shing-Tung Yau

There are no affiliations available

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