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General Relativity and Gravitation

, Volume 34, Issue 6, pp 837–852 | Cite as

Geometry and Dynamics of Relativistic Particles with Rigidity

  • Manuel Barros
Article

Abstract

The simplest models describing spinning particles with rigidity, both massive and massless, are reconsidered. The moduli spaces of solutions are completely exhibited in backgrounds with constant curvature. While spinning massive particles can evolve fully along helices in any three-dimensional background, spinning massless particles need anti De Sitter background to be consistent.

The main machinery used to determine those moduli in AdS3 is provided by a pair of natural Hopf mappings. Therefore, Hopf tubes, B-scrolls and specially the Hopf tube constructed on a horocycle in the hyperbolic plane, play a principal role in this program.

Spinning massless and massive particle moduli spaces of solutions anti De Sitter background Hopf mappings 

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REFERENCES

  1. 1.
    Arroyo, J., Barros, M., and Garay, O. J. (2000). Some examples of critical points for the total mean curvature functional. Proc. Edinburgh Math. Soc. 43, 587.Google Scholar
  2. 2.
    Arroyo, J., Barros, M., and Garay, O. J. (2001). Holography and total charge. J. Geomet. Phys. Google Scholar
  3. 3.
    Barros, M., (1997). General helices and a theorem of Lancret. Proc. A. M. S. 125, 1503.Google Scholar
  4. 4.
    Barros, M., (1998). Free elasticae andWillmore tori in warped product spaces. Glasgow Math. J. 40, 265.Google Scholar
  5. 5.
    Barros, M., (2000). Willmore-Chen branes and Hopf T-duality. Class. Quantum Grav. 17, 1979.Google Scholar
  6. 6.
    Barros, M., Ferrández, A., Lucas, P. and Merońo, M. A. (1995). Hopf cylinders, B-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti De Sitter space. C. R. Acad. Sci. Paris 321, 505.Google Scholar
  7. 7.
    Barros, M., Cabrerizo, J. L. and Fernández, M., (2000). Elasticity and conformal tension via the Kaluza-Klein mechanism. J. Geomet. Phys. 34, 111.Google Scholar
  8. 8.
    Besse, A. L. (1987). Einstein manifolds. Springer Verlag, Berlin Heidelberg.Google Scholar
  9. 9.
    Cheng, S.-Y. (1976). Eigenfunctions and nodal sets. Comment. Math. Helv. 51, 43.Google Scholar
  10. 10.
    Erbacher, J., (1971). Reduction of the codimension of an isometric immersion. J. Differential Geometry 5, 333.Google Scholar
  11. 11.
    Fenchel, W., (1929). Über die Krümmung und Windung geschlossenen Raumkurven. Math. Ann. 101, 238.Google Scholar
  12. 12.
    Graves, L. (1979). Codimension one isometric immersions between Lorentz spaces. Trans. Amer. Math. Soc. 252, 367.Google Scholar
  13. 13.
    Greub, W., Halperin, S., and Vanstone, R. (1972,1973,1976). Connections, Curvature and cohomology. 3 Vols. Academic Press, New-York.Google Scholar
  14. 14.
    Husain, V., and Jaimungal, S., (1999). Topological holography. Physical ReviewD 60, 061501-1/5.Google Scholar
  15. 15.
    Kazdan, J. L. and Warner, F. W. (1975). Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. of Math. 101, 317.Google Scholar
  16. 16.
    Maldacena, J. (1998). Adv. Theor. Math. Phys. 2, 231.Google Scholar
  17. 17.
    Nersessian, A., Massless particles and the geometry of curves. Classical pictures. QFTHEP'99, Moscow 1999. hep-th/9911020.Google Scholar
  18. 18.
    Nersessian, A. (2000). D-dimensional massless particle with extended gauge invariance. Czech. J. Phys. 50, 1309.Google Scholar
  19. 19.
    Nersessian, A., and Ramos, E. (1999). A geometrical particle model for anyons. Mod. Phys. Lett. A 14, 2033.Google Scholar
  20. 20.
    Nesterenko, V. V., Feoli, A., and Scarpetta, G., (1995). Dynamics of relativistic particle with Lagrangian dependent on acceleration. J. Math. Phys. 36, 5552.Google Scholar
  21. 21.
    Nesterenko, V. V., Feoli, A., and Scarpetta, G., (1996). Complete integrability for Lagrangian dependent on acceleration in a space-time of constant curvature. Class. Quant. Grav. 13, 1201.Google Scholar
  22. 22.
    Ody, M. S. and Ryder, L.H. (1995). Time-independent solutions to the two-dimensional non-linear O.3/ sigma model and surfaces of constant mean curvature. Int. J. Mod. Phys. A 10, 337.Google Scholar
  23. 23.
    O'Neill, B., (1983). Semi-Riemannian Geometry. Academic Press, New-York, London.Google Scholar
  24. 24.
    Plyushchay, M. S. (1989). Massless point particle with rigidity. Mod. Phys. Lett. A 4, 837.Google Scholar
  25. 25.
    Plyushchay, M. S. (1990). Massless particle with rigidity as a model for the description of bososns and fermions. Phys. Lett. B 243, 383.Google Scholar
  26. 26.
    Plyushchay, M. S. Commemt on the relativistic particle with curvature and torsion of world trajectory. hep-th/9810101.Google Scholar
  27. 27.
    Prokuskin, S. F. and Vasiliev, M. A. (1999). Currents of arbitrary spin in AdS3. Phys. Lett. B 464, 53.Google Scholar
  28. 28.
    Ramos, E. and Roca, J. (1995). W-symmetry and the rigid particle. Nuclear Physics B 436, 529.Google Scholar
  29. 29.
    Smale, S. (1961). Generalized Poincare conjecture in dimension greater than four. Ann. of Math. 74, 391.Google Scholar
  30. 30.
    Witten, E. (1998). Adv. Theor. Math. Phys. 2, 253.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Manuel Barros
    • 1
  1. 1.Departamento de Geometría y TopologíaUniversidad de Granada 18071GranadaSpain

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