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r-Fold Multivectors and Superenergy

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Clifford Algebras

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

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Abstract

A general structure combining Grassmann, Clifford and tensor products is presented. The r-fold multivectors provide the basis for the natural extension of Grassmann and Clifford algebras when several geometric entities are multilinearly related. Any tensor can be organized and understood as an r-fold multivector when its antisymmetries are taken into account. The r-fold Clifford algebra is contrasted with the multiparticle geometric algebra. The application of r-fold Clifford algebra to the study of superenergy tensors in physics is shown to provide their simplest definition. In addition, it constitutes a most efficient tool for obtaining and proving their essential properties, such as dominant positivity and conditions for their conservation.

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References

  1. C.J.L. Doran, Geometric Algebra and its Application to Mathematical Physics. Ph.D. thesis, University of Cambridge, 1994.

    Google Scholar 

  2. J.M. Pozo and J.M. Parra, Positivity and conservation of superenergy tensors. Class. Quantum Grav. 19, 967–983, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.M.M. Senovilla, Super-energy tensors. Class. Quantum Grav. 17, 2799–2842, 2000. Preprint gr-qc/9906087.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Bel, Les états de radiation et le problème de l’ énergie en relativité générale. Cahiers de Physique 16, 59–80, 1962. English translation: Gen. Ref. Grav. 32, 2047-78 (2000). And independently, I. Robinson unpublished 1959 (see also I. Robinson, On the Bel-Robinson tensor, Class. Quantum Grav. 14, A331-A333, 1997)

    MathSciNet  Google Scholar 

  5. M. Riesz, Clifford Numbers and Spinors. The Inst. for Fluid Dynamics and Applied Math., Lecture Series 38. Univ. of Maryland, University Park, 1958. (Reprinted as facsimile (Dordrecht: Kluwer 1993) ed. E.F. Bolinder and P. Lounesto).

    MATH  Google Scholar 

  6. W.K. Clifford, Applications of Grassmann’s extensive algebra. Am. J. Math. 1, 350–358, 1878.

    Article  MathSciNet  Google Scholar 

  7. F.J. Chinea, A Clifford algebra approach to general relativity. Gen. Rei. Grav. 21 (1), 21–44, 1989.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. J.M. Parra and J.M. Pozo, Einstein Equations with Spacetime Geometric Clifford Algebra, 194–197. In Bona et al. [30], 1998.

    Google Scholar 

  10. A. Lasenby, S. Gull and C. Doran, STA and the interpretation of quantum mechanics, In Clifford (Geometric) Algebras with Applications to Physics, Mathematics, and Engineering. W.E. Baylis (Editor), Birkhäuser, Boston, 1996, pp. 147–169. 1996.

    Google Scholar 

  11. S.S. Somaroo, A.N. Lasenby, and C.J.L. Doran, Geometric algebra and the causal approach to multiparticle quantum mechanics. J. Math. Phys. 40, 3327–3340, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.M.M. Senovilla, Remarks on superenergy tensors, 175–182. In Martín et al. [31], 1999. (Senovilla J.M.M. 1999 Preprint gr-qc/9901019).

    Google Scholar 

  13. G.T. Horowitz and B.G. Schmidt, Proc. R. Soc. A 381, 215, 1982.

    MathSciNet  Google Scholar 

  14. M.A.G. Bonilla and J.M.M. Senovilla, Some properties of the Bel and Bel-Robinson tensors. Gen. Rel. Grav. 29, 91–116, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Chevreton, Sur le tenseur de superénergie du champ électromagnétique. Nuovo Cimento 34, 901–913, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Bergqvist and M. Ludvigsen, Quasi-local momentum near a point. Class. Quantum Grav. 4, L29-L32, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  17. L.B. Szabados, On certain quasi-local spin-angular momentum expressions for small spheres. Class. Quantum Grav. 16, 2889–2904, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.D. Brown, S.R. Lau, and J.W. York, Canonical quasilocal energy and small spheres. Phys. Rev. D 59, 064028, 1999.

    Article  MathSciNet  Google Scholar 

  19. J.M.M. Senovilla, (super)n-energy for arbitrary fields and its interchange: conserved quantities. Mod. Phys. Lett. 15, 159–166, 2000. Preprint gr-qc/9905057.

    Article  MathSciNet  Google Scholar 

  20. D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space. Princeton Univ. Press, Princeton, 1993.

    MATH  Google Scholar 

  21. G. Bergqvist and J.M.M. Senovilla, Null cone preserving maps, causal tensors and algebraic Rainich theory. Class. Quantum Grav. 18, 5299–5325, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Garcia-Parrado and J.M.M. Senovilla, Causal relationship: a new tool for the causal characterization of Lorentzian manifolds. Class. Quant. Grav. 20, 625–664, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  23. G.W. Gibbons, Quantum gravity/strings/m-theory as we approach the 3rd millennium. In N Dadhich and J Narlikar, editors, Gravitation and Relativity: At the turn of the Millenium (Proc. of the GR-15 Conference), pp. 259–280, Pune, India, 1998. IUCAA. gr-qc/9803065.

    Google Scholar 

  24. S. Deser, The immortal Bel-Robinson tensor, 35–43. In Martín et al. [31], 1999.

    Google Scholar 

  25. G. Bergqvist, Positivity of general superenergy tensors. Commun. Math. Phys. 207 467–479, 1999.

    Article  MathSciNet  Google Scholar 

  26. J.M. Pozo and J.M. Parra, Clifford algebra approach to superenergy tensors, 283–287. In Ibáñez (32], 2000. Preprint gr-qc/9911041).

    Google Scholar 

  27. M. Riesz, Sur certain notions fondamentales en theorié quantique relativiste. In C. R. 10e Congrès Math. Scandinaves (Copenhagen), pp. 123–148, 1946. Marcel Riesz Collected Papers ed. L. Gårding and L. Hörmander (Berlin: Springer-Verlag 1988), 545–570.

    Google Scholar 

  28. D. Hestenes, Space-Time Algebra. Gordon & Breach, New York, 1966.

    MATH  Google Scholar 

  29. P. Teyssandier, Superenergy tensors for a massive scalar field, 319–324. In Ibáñez [32], 2000. (see also Teyssandier P 1999 Preprint gr-qc/9905080).

    Google Scholar 

  30. C. Bona, J. Carot, L. Mas, and J. Stela, editors, Analytical and Numerical Approaches to Relativity: Sources of Gravitational Radiation, Proc. of the Spanish Relativity Meeting, ERE97 (Palma de Mal/orca). Universitat de les Illes Balears, 1998.

    Google Scholar 

  31. J. Martín, E. Ruiz, F. Atrio, and A. Molina, editors, Relativity and Gravitation in General, Proc. of the Spanish Relativity Meeting in Honour of the 65 th Birthday of L. Bel, ERE98 (Salamanca). World Scientific, Singapore, 1999.

    Google Scholar 

  32. J. Ibáñez, editor, Recent Developments in Gravitation, Proc. of the Spanish Relativity Meeting, ERE99 (Bilbao). Universidad del Pais Vasco, 2000.

    Google Scholar 

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Authors and Affiliations

  1. Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, E-08028, Barcelona, Spain

    José María Pozo & Josep Manel Parra

Authors
  1. José María Pozo
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  2. Josep Manel Parra
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Editor information

Editors and Affiliations

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

    Rafał Abłamowicz

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© 2004 Birkhäuser Boston

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Pozo, J.M., Parra, J.M. (2004). r-Fold Multivectors and Superenergy. In: Abłamowicz, R. (eds) Clifford Algebras. Progress in Mathematical Physics, vol 34. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2044-2_33

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  • DOI: https://doi.org/10.1007/978-1-4612-2044-2_33

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3525-1

  • Online ISBN: 978-1-4612-2044-2

  • eBook Packages: Springer Book Archive

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