Skip to main content
SpringerLink
Log in

Generalized theorem on positive definiteness of energy in the general relativity theory

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We introduce the notion of the energy of gravitational and material fields with respect to an arbitrary time-like vector field and a space-like hypersurface as an integral over a nonholonomic hypersurface. In the case of an asymptotically Minkowskian space, by developing the approach of Witten and Nester, we obtain an expression for the energy functional in terms of the spinors associated with a differential-geometric distribution. By applying the Sen-Witten generalized equation, we prove the nonnegativity of this functional.

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. Yu. S. Vladimirov,Reference Systems in the Theory of Gravity [in Russian], Atomizdat, Moscow (1982).

    Google Scholar 

  2. O. S. Ivanitskaya,Lorentzian Basis and Gravitational Effects in the Einstein Theory of Gravity [in Russian], Nauka i Tekhnika, Minsk (1979).

    Google Scholar 

  3. N. V. Mitskevich, A. P. Efremov, and A. I. Nesterov,Dynamics of Fields in the General Relativity Theory, [in Russian], Énergoatomizdat, Moscow (1985).

    Google Scholar 

  4. V. O. Pelykh, “Spinors associated with the distributionV 4 3 and generalizations of the Sen-Witten equation,”Mat. Metody Fiz.-Mekh. Polya,39, No. 2, 133–139 (1996).

    MathSciNet  Google Scholar 

  5. R. F. Polishchuk, “Nonholonomic generalization of the Stokes theorem,”Vestn. Mosk. Gos. Univ., No. 2, 229 (1973).

    Google Scholar 

  6. Y. Choquet-Bruhat, “Mathematical problems of the general relativity theory,”Usp. Mat. Nauk.,40, Issue 6, 3–39 (1985).

    MATH  MathSciNet  Google Scholar 

  7. R. Arnowitt, S. Deser, and C. Misner, “The dynamics of general relativity,” in: L. Witten (editor),Gravitation, An Introduction to Current Research, Wiley, New York (1962), pp. 227–265.

    Google Scholar 

  8. A. Ashtekar and G. T. Horowitz, “Phase space of general relativity revisited: A canonical choice of time and simplification of the Hamiltonian,”J. Math. Phys.,25, No. 5, 1473–1480 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Brill and S. Deser, “Variational methods and positive energy in general relativity,”Ann. Phys.,50, No. 3, 548–570 (1968).

    Google Scholar 

  10. D. Brill, S. Deser, and L. D. Faddeev, “Sign of gravitational energy,”Phys. Lett.,26A, No. 11, 538–539 (1969).

    Google Scholar 

  11. Y. Choquet-Bruhat, “Construction des solutions radiatives approachées des equations d'Einstein,”Comm. Math. Phys.,12, No. 1, 16–35 (1969).

    Article  MathSciNet  Google Scholar 

  12. Y. Choquet-Bruhat, “Positive-energy theorems,” in: B. S. DeWitt and R. Stora (editors),Relativité, Groupes et Topologie, Elsevier, Amsterdam (1984), pp. 740–785.

    Google Scholar 

  13. Y. Choquet-Bruhat and D. Christodoulou, “Elliptic systems in ℋ s,δ spaces on manifolds which are Euclidean at infinity,”Acta Math.,146, No. 2, 124–150 (1981).

    MathSciNet  Google Scholar 

  14. A. Dimakis and F. Muller-Hoissen, “Spinors fields and the positive-energy theorem,”Class. Quantum Grav.,7, No. 3, 283–295 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  15. S. W. Hawking and G. F. R. Ellis,The Large-Scale Structure of Space-Time, Cambridge University Press, Cambridge (1973).

    MATH  Google Scholar 

  16. J. Nester, “A new general energy expression with a simple positivity proof,”Phys. Lett.,83A, No. 6, 241–244 (1981).

    MathSciNet  Google Scholar 

  17. J. Nester, “A positive gravitational energy proof,”Phys. Lett.,139A, No. 3, 4, 112–114 (1989).

    MathSciNet  Google Scholar 

  18. J. Nester, “Gauge condition for orthonormal three-frames,”J. Math. Phys.,30, No. 3, 624–626 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Penrose and W. Rindler,Spinors and Space-Time: Vol. 1, Cambridge University Press, Cambridge (1984).

    Google Scholar 

  20. O. Reula, “Existence theorem for solutions of Witten's equation and nonnegativity of total mass,”J. Math. Phys.,22, No. 8, 810–814 (1981).

    Google Scholar 

  21. R. Schoen and S.-T. Yau, “On the proof of the positive mass conjecture in general relativity,”Comm. Math. Phys.,65, No. 1, 47–76 (1979).

    Article  MathSciNet  Google Scholar 

  22. E. Witten, “A new proof of the positive energy theorem,”Commun. Math. Phys.,80, No. 3, 381–402 (1981).

    Article  MathSciNet  Google Scholar 

Download references

Authors
  1. V. O. Pelykh
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences. L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 26–34, April–June, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pelykh, V.O. Generalized theorem on positive definiteness of energy in the general relativity theory. J Math Sci 99, 1548–1556 (2000). https://doi.org/10.1007/BF02674178

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02674178

Keywords

Search

Navigation