Skip to main content
SpringerLink
Log in

Spinor Formulations for Gravitational Energy-Momentum

  • Chapter
Clifford Algebras

Part of the book series: Progress in Mathematical Physics ((PMP,volume 34))

Abstract

We first describe a class of spinor-curvature identities (SCI) which have gravitational applications. Then we sketch the topic of gravitational energy-momentum, its connection with Hamiltonian boundary terms and the issues of positivity and (quasi)localization. Using certain SCIs, several spinor expressions for the Hamiltonian have been constructed. One SCI leads to the celebrated Witten positive energy proof and the Dougan—Mason quasilocalization. We found two other SCIs which give alternate positive energy proofs and quasilocalizations. In each case the spinor field has a different role. These neat expressions for gravitational energy-momentum have much appeal. However it seems that such spinor formulations just have no room for angular momentum, which leads us to doubt that spinor formulations can really correctly capture the elusive gravitational energy-momentum.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity in Gravitation: An Introduction to Current Research Ed. L. Witten, Wiley, New York, 1962, pp. 227–265.

    Google Scholar 

  2. R. Beig and N. Ó Murchadha, The Poincaré group is the symmetry group of canonical general relativity, Ann. Phys. 174 (1987), 463–498.

    Article  MATH  Google Scholar 

  3. J. D. Brown and J. W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993), 1407–1409.

    Article  MathSciNet  Google Scholar 

  4. C.-C. Chang, J. M. Nester and C.-M. Chen, Pseudotensors and quasilocal energy-momentum, Phys. Rev. Lett. 83 (1999), 1897–1901; arXiv: gr-qc/9809040.

    Article  MathSciNet  MATH  Google Scholar 

  5. C.-C. Chang, J. M. Nester and C.-M. Chen, Energy-momentum (quasi-) localization for gravitating systems, in Gravitation and Astrophysics Eds. Liao Liu, Jun Luo, X-Z. Li, J-P. Hsu, World Scientific, Singapore, 2000, pp. 163–173; arXiv: grqc/9912058.

    Google Scholar 

  6. C.-M. Chen and J. M. Nester, Quasilocal quantities for general relativity and other gravity theories, Class. Quantum Grav. 16 (1999), 1279–1304; arXiv: gr-qc/9809020.

    Article  MathSciNet  MATH  Google Scholar 

  7. C.-M. Chen and J. M. Nester, A symplectic Hamiltonian derivation of quasilocal energy-momentum for GR, Gravitation & Cosmology 6 (2000), 257–270; arXiv: gr-qc/000l088.

    MathSciNet  MATH  Google Scholar 

  8. C. M. Chen, J. M. Nester and R. S. Tung, Quasilocal energy-momentum for geometric gravity theories, Phys. Lett. A 203 (1995), 5–11.

    MathSciNet  Google Scholar 

  9. S. Deser, Positive classical gravitational energy from classical supergravity, Phys. Rev. D 27 (1983), 2805–2808.

    MathSciNet  Google Scholar 

  10. S. Deser and C. Teitelboim, Supergravity has positive energy, Phys. Rev. Lett. 39 (1977), 249–252.

    Article  Google Scholar 

  11. A. Dimakis and F. Müller-Hoissen, Clifform calculus with applications to classical field theories, Class. Quantum Grav. 8 (1991), 2093–2132.

    Article  MATH  Google Scholar 

  12. A. Dougan and L. Mason, Quasilocal mass constructions with positive energy, Phys. Rev. Lett. 67 (1991), 2119–2122.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Estabrook, Lagrangians for Ricci flat geometries, Class. Quantum Grav. 8 (1991), LI51–154.

    Article  MathSciNet  Google Scholar 

  14. M. T. Grisaru, Positivity of the energy in Einstein theory, Phys. Lett. 73B (1978), 207–208.

    Google Scholar 

  15. R. D. Hecht and J. M. Nester, A new evaluation of PGT mass and spin, Ph. vs. Lett. A 180 (1993), 324–331.

    Article  Google Scholar 

  16. R. D. Hecht and J. M. Nester, An evaluation of mass and spin at null infinity for the PGT and GR gravity theories, Phys. Lett. A 217 (1996), 81–89.

    Article  Google Scholar 

  17. G. T. Horowitz and A. Strominger, On Witten’s expression for gravitational energy, Phys. Rev. D 27 (1983), 2793–2804.

    MathSciNet  Google Scholar 

  18. J. Isenberg and J. M. Nester, Canonical gravity, in: General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein Vol I., Ed. A. Held, Plenum, New York, 1980, pp. 23–97.

    Google Scholar 

  19. J. Kijowski and W. M. Tulczyjew, A Symplectic Frameworkfor Field Theories Lecture Notes in Physics, Vol. 107, Springer, Berlin, 1979.

    Book  Google Scholar 

  20. F. F. Meng, Quasilocal Center-of-Mass Moment in GR, MSc. Thesis, National Central University, 2002 (unpublished).

    Google Scholar 

  21. E. W. Mielke, Geometrodynamics of Gauge Fields—On the Geometry of Yang-Mills and Gravitational Gauge Theories Akademie, Berlin, 1987.

    MATH  Google Scholar 

  22. C. W. Misner, K. Thorne and J. A. Wheeler, Gravitation Freeman, San Fransisco, 1973.

    Google Scholar 

  23. J. M. Nester, A new gravitational energy expression with a simple positivity proof, Phys. Lett. A 83 (1981), 241–242.

    Article  MathSciNet  Google Scholar 

  24. J. M. Nester, The gravitational Hamiltonian, in Asymptotic Behavior of Mass and spacetime Geometry Lecture Notes in Physics, Vol. 202, Ed. F. Flaherty, Springer, Berlin, 1984, pp 155–163.

    Chapter  Google Scholar 

  25. J. M. Nester and R. S. Tung, Another positivity proof and gravitational energy localizations, Phys. Rev. D 49 (1994), 3958–3962.

    Article  MathSciNet  Google Scholar 

  26. J. M. Nester, R. S. Tung and V. Zhytnikov, Some spinor curvature identities, Class. Quant. Grav. 11 (1994), 983–987.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. M. Nester and R. S. Tung, A quadratic spinor Lagrangian for general relativity, Gen. Rel. Grav. 27 (1995), 115–119.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys. 88 (1974), 286–319.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math Phys. 65 (1979), 45–76.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Szabados, Two-dimensional Sen connections in general relativity, Class. Quantunl Grav. 11 (1994), 1833–1847.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. S. Tung and J. M. Nester, The quadratic spinor Lagrangian is equivalent to the teleparallel theory, Phys. Rev. D 60 (1999), 021501.

    Article  MathSciNet  Google Scholar 

  32. K. H. Vu, Quasilocal Energy-Momentum and Angular Momentum for Teleparallel Gravity, MSc. Thesis, National Central University, 2000 (unpublished).

    Google Scholar 

  33. E. Witten, A simple proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381–402.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, National Taiwan University, Taipei, 106, Taiwan, China

    Chiang-Mei Chen

  2. Department of Physics, National Central University, Chung-Li, 320, Taiwan, China

    James M. Nester & Roh-Suan Tung

Authors
  1. Chiang-Mei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James M. Nester
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roh-Suan Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Tennessee Technological University, 5054, TN 38505, Cookeville, USA

    Rafał Abłamowicz

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Birkhäuser Boston

About this chapter

Cite this chapter

Chen, CM., Nester, J.M., Tung, RS. (2004). Spinor Formulations for Gravitational Energy-Momentum. In: Abłamowicz, R. (eds) Clifford Algebras. Progress in Mathematical Physics, vol 34. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2044-2_27

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2044-2_27

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3525-1

  • Online ISBN: 978-1-4612-2044-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation