The Positive Mass Theorem

  • Norbert Straumann
Part of the Graduate Texts in Physics book series (GTP)

Abstract

The important positive mass theorem roughly says that—in contrast to Newtonian gravity theory—it is impossible to construct an object out of ordinary matter, i.e., matter with positive local energy density, whose total energy (including gravitational contributions) is negative. In this chapter we will give essentially E. Witten’s proof of the positive energy theorem, which makes crucial use of spinor fields. To be complete, we develop the necessary tools on spinors in GR in an appendix to this chapter. It is very remarkable that spinors have turned out to be so useful in simplifying the proof of an entirely classical property of GR. We add some remarks on the Penrose inequality, which can be regarded as a sharpening of the positive energy theorem for black holes.

Keywords

Black Hole Apparent Horizon Spinor Field Spacelike Hypersurface Positive Mass Theorem 
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References

Textbooks on General Physics and Astrophysics

  1. 34.
    N. Straumann, Relativistische Quantentheorie (Springer, Berlin, 2004) Google Scholar

Mathematical Tools: Modern Treatments of Differential Geometry for Physicists

  1. 40.
    T. Frankel, The Geometry of Physics (Cambridge University Press, Cambridge, 1997) MATHGoogle Scholar

Mathematical Tools: Selection of Mathematical Books

  1. 41.
    S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I (Interscience, New York, 1963) MATHGoogle Scholar
  2. 42.
    S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. II (Interscience, New York, 1969) MATHGoogle Scholar

Research Articles, Reviews and Specialized Texts: Chapter  3

  1. 110.
    R. Schoen, S.T. Yau, Commun. Math. Phys. 65, 45 (1976) MathSciNetADSCrossRefGoogle Scholar
  2. 111.
    R. Schoen, S.T. Yau, Phys. Rev. Lett. 43, 1457 (1979) MathSciNetADSCrossRefGoogle Scholar
  3. 112.
    R. Schoen, S.T. Yau, Commun. Math. Phys. 79, 231 (1981) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 113.
    R. Schoen, S.T. Yau, Commun. Math. Phys. 79, 47 (1981) MathSciNetADSCrossRefGoogle Scholar

Research Articles, Reviews and Specialized Texts: Chapter  4

  1. 136.
    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, London, 1983) MATHGoogle Scholar

Research Articles, Reviews and Specialized Texts: Chapter 9

  1. 286.
    E. Witten, Commun. Math. Phys. 80, 381 (1981) MathSciNetADSCrossRefGoogle Scholar
  2. 287.
    J. Stewart, Advanced General Relativity (Cambridge University Press, Cambridge, 2003) Google Scholar
  3. 288.
    J. Nester, Phys. Lett. A 83, 241 (1981) MathSciNetADSCrossRefGoogle Scholar
  4. 289.
    R. Penrose, W. Rindler, Spinors and Space-Time (Cambridge University Press, Cambridge, 1984) MATHCrossRefGoogle Scholar
  5. 290.
    R.P. Geroch, J. Math. Phys. 9, 1739 (1968) MathSciNetADSMATHCrossRefGoogle Scholar
  6. 291.
    O. Reula, J. Math. Phys. 23, 810 (1982) MathSciNetADSMATHCrossRefGoogle Scholar
  7. 292.
    T. Parker, C.H. Taubes, Commun. Math. Phys. 84, 223 (1982) MathSciNetADSMATHCrossRefGoogle Scholar
  8. 293.
    G.W. Gibbons, S.W. Hawking, G.T. Horowitz, M.J. Perry, Commun. Math. Phys. 88, 295 (1983) MathSciNetADSCrossRefGoogle Scholar
  9. 294.
    R. Penrose, Ann. N.Y. Acad. Sci. 224, 125 (1973) ADSCrossRefGoogle Scholar
  10. 295.
    G. Huisken, T. Ilmanen, J. Differ. Geom. 59, 353 (2001) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Norbert Straumann
    • 1
  1. 1.Mathematisch-Naturwiss. Fakultät, Institut für Theoretische PhysikUniversität ZürichZürichSwitzerland

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