Skip to main content
SpringerLink
Log in
Book cover

3+1 Formalism in General Relativity pp 157–183Cite as

Asymptotic Flatness and Global Quantities

  • Chapter
  • First Online:

  • 3614 Accesses

Part of the book series: Lecture Notes in Physics ((LNP,volume 846))

Abstract

After providing a definition of asymptotic flatness, we introduce the global quantities that one may associate to the spacetime or to each slice of the 3 + 1 foliation: the ADM mass, the ADM linear momentum, the total angular momentum, the Komar mass and the Komar angular momentum. For each of these quantities, we derive expressions in terms of the 3+1 objects and provide some concrete examples.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    See Sect. 4.2.1.

  2. 2.

    In index notation, \(-\overrightarrow{\user2{T}}(\user2{\it v})\) is the vector \({-T^\alpha}_{\mu} v^\mu.\)

  3. 3.

    Actually the first condition proposed by York, Eq. (90) of Ref. [3], is not exactly (8.46) but can be shown to be equivalent to it; see also Sec. V of Ref. [16].

  4. 4.

    The Killing equation follows immediately from Eq. (8.53) with the Lie derivative expressed via Eq. (2.92), along with \(\varvec\nabla\user2{g}=0.{}\)

References

  1. Jaramillo, J.L., Gourgoulhon, E.: Mass and angular momentum in general relativity. In: Blanchet, L., Spallicci, A., Whiting, B. (eds.) Mass and motion in general relativity. Fundamental Theories of Physics, vol. 162, p. 87. Springer, Dordrecht (2011)

    Google Scholar 

  2. Szabados, L.B.: Quasi-local energy-momentum and angular momentum in general relativity. Living Rev. Relativity 12, 4. http://www.livingreviews.org/lrr-2009-4 (2009)

  3. York, J.W.: Kinematics and dynamics of general relativity. In: Smarr, L.L. (ed.) Sources of Gravitational Radiation, p. 83. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  4. York, J.W.: Energy and momentum of the gravitational field. In: Tipler, F.J. (ed.) Essays in General Relativity, a Festschrift for Abraham Taub, p. 39. Academic Press, New York (1980)

    Google Scholar 

  5. Straumann, N.: General Relavity, with Applications to Astrophysics. Springer, Berlin (2004)

    Google Scholar 

  6. Ashtekar, A.: Asymptotic structure of the gravitational field at spatial infinity. In: Held, A. (ed.) General Relativity and Gravitation, One Hundred Years After the Birth of Albert Einstein, vol. 2, p. 37. Plenum Press, New York (1980)

    Google Scholar 

  7. Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)

    MATH  Google Scholar 

  8. Henneaux, M.: Hamiltonian Formalism of General Relativity. Lectures at Institut Henri Poincaré, Paris, http://www.luth.obspm.fr/IHP06/ (2006)

  9. Ashtekar, A., Hansen, R.O.: A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity. J. Math. Phys. 19, 1542 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  10. Regge, T., Teitelboim, C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. (N.Y.) 88, 286 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Poisson, E.: A Relativist’s Toolkit, The Mathematics of Black-Hole Mechanics. Cambridge University Press, Cambridge, http://www.physics.uoguelph.ca/poisson/toolkit/ (2004)

  12. Martel, K., Poisson, E.: Regular coordinate systems for Schwarzschild and other spherical spacetimes. Am. J. Phys. 69, 476 (2001)

    Article  ADS  Google Scholar 

  13. Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79, 231 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  15. Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, p. 227. Wiley, New York, available at http://arxiv.org/abs/gr-qc/0405109 (1962)

  16. Smarr L., York J.W.: Radiation gauge in general relativity. Phys. Rev. D 17, 1945 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  17. Chruściel P.T.: On angular momentum at spatial infinity. Class. Quantum Grav. 4, L205 (1987)

    Article  MATH  Google Scholar 

  18. Komar A.: Covariant conservation laws in general relativity. Phys. Rev. 113, 934 (1959)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Ansorg, M., Petroff, D.: Negative Komar mass of single objects in regular, asymptotically flat spacetimes. Class. Quantum Grav. 23, L81 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Beig, R.: Arnowitt-Deser-Misner energy and \(g_{00}.\) Phys. Lett. 69A, 153 (1978)

    Article  MathSciNet  Google Scholar 

  21. Ashtekar, A., Magnon-Ashteka, A.: On conserved quantities in general relativity. J. Math. Phys. 20, 793 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  22. Katz, J.: A note on Komar’s anomalous factor. Class. Quantum Grav. 2, 423 (1985)

    Article  ADS  Google Scholar 

  23. Gourgoulhon, E., Jaramillo, J.L.: A 3+1 perspective on null hypersurfaces and isolated horizons. Phys. Rep. 423, 159 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  24. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lab. Univers et Théories (LUTH) UMR 8102 du CNRS, Observatoire de Paris, Université Paris Diderot, place Jules Janssen 5, 92190, Meudon, France

    Éric Gourgoulhon

Authors
  1. Éric Gourgoulhon
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gourgoulhon, É. (2012). Asymptotic Flatness and Global Quantities. In: 3+1 Formalism in General Relativity. Lecture Notes in Physics, vol 846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24525-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24525-1_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24524-4

  • Online ISBN: 978-3-642-24525-1

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation