General Relativity pp 527-545 | Cite as

# The Positive Mass Theorem

## 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## References

## Textbooks on General Physics and Astrophysics

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

## Mathematical Tools: Modern Treatments of Differential Geometry for Physicists

- 40.T. Frankel,
*The Geometry of Physics*(Cambridge University Press, Cambridge, 1997) zbMATHGoogle Scholar

## Mathematical Tools: Selection of Mathematical Books

- 41.S. Kobayashi, K. Nomizu,
*Foundations of Differential Geometry*, vol. I (Interscience, New York, 1963) zbMATHGoogle Scholar - 42.S. Kobayashi, K. Nomizu,
*Foundations of Differential Geometry*, vol. II (Interscience, New York, 1969) zbMATHGoogle Scholar

## Research Articles, Reviews and Specialized Texts: Chapter 3

- 110.R. Schoen, S.T. Yau, Commun. Math. Phys.
**65**, 45 (1976) MathSciNetADSCrossRefGoogle Scholar - 111.R. Schoen, S.T. Yau, Phys. Rev. Lett.
**43**, 1457 (1979) MathSciNetADSCrossRefGoogle Scholar - 112.R. Schoen, S.T. Yau, Commun. Math. Phys.
**79**, 231 (1981) MathSciNetADSzbMATHCrossRefGoogle Scholar - 113.R. Schoen, S.T. Yau, Commun. Math. Phys.
**79**, 47 (1981) MathSciNetADSCrossRefGoogle Scholar

## Research Articles, Reviews and Specialized Texts: Chapter 4

- 136.S. Chandrasekhar,
*The Mathematical Theory of Black Holes*(Oxford University Press, London, 1983) zbMATHGoogle Scholar

## Research Articles, Reviews and Specialized Texts: Chapter 9

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