Dynamics of multiparticle systems on the symplectic extension of the Schwarzschild space-time manifold

N.N. Fimin1


Methods of extension of the Schwarzschild metrics to the 8D symplectic phase manifold are considered. The method used is based on Sasaki's metric of the tangent bundle over the pseudo-Riemannian 4D space-time manifold.


  1. V. P. Frolov and I. D. Novikov, Black Holes: Basic Concepts and New Developments (Birkhauser, New York - Boston - Heidelberg, 1997).
  2. J. Stewart, Advanced General Relativity (Cambridge University Press, Cambridge, 1996).
  3. L. P. Hugston and K. P. Tod, An Introduction to General Relativity (Cambridge University Press, Cambridge, 1994).
  4. M. Ludvigsen, General Relativity. A Geometric Approach (Cambridge University Press, Cambridge, 1999).
  5. R. K. Sachs and H. Wu, General Relativity for Mathematicians (Springer-Verlag, New York - Heidelberg -Berlin, 1977).
  6. N. Straumann, General Relativity and Relativistic Astrophysics (Springer-Verlag, Berlin - Heidelberg - New York - Tokyo, 1984).
  7. F. De Felicem and C. J. S. Clarke, Relativity on Curved Manifolds (Cambridge University Press, Cambridge, 1990).
  8. B. S. De Witt, Phys. Rep. 19, 295 (1975).
  9. N. D. Birrell and P. C. W. Davies, Quantum Field in Curved Space (Cambridge University Press, Cambridge, 1982).
  10. H. Stephani, Relativity: An Introduction to Special and General Relativity (Cambridge University Press, Cambridge, 2004).
  11. J. Karkowski, E. Malec, and Z. Swierczynski, Schwarzschild Black Holes and Propagation of Electromagnetic and Gravitational Waves, gr-qc/0204086.
  12. J. Tian, Ya. Gui, G. Guo, S. Zhang, and W. Wang, The Real Scalar Field in Schwarzschild - de Sitter Spacetime, gr-qc/0304009.
  13. K. D. Kokkotas and B. G. Schmidt, Quasi-Normal Modes of Stars and Black Holes, gr-qc/9909058.
  14. J. Ehlers and R. Geroch, Equation of Motion of Small Bodies in Relativity, gr-qc/0309074.
  15. S. Hayward, Phys. Rev. D 49, 6467 (1994).
  16. T. K. Das, Behavior of Matter Close to the Event Horizon, astro-ph/0312548.
  17. R. Narayan, R. Mahadevan, and E. Quataert, in: Proc. Reykjavic Symp. on Non-Linear Phenomena in Accretion Disks around Black Holes, eds. M. Abramowicz, G. Bjornsson, and J. Pringle (Reykjavik, 1997).
  18. N. E. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 67, 084033-1 (2003).
  19. A.N. Petrov and J. Katz, Proc. Roy. Soc. London 458, 319 (2002).
  20. G. Kalbermann, Diffraction of Wave Packet in Space and Time, quant-ph/0008077.
  21. G. 't Hooft, The Scattering Matrix Approach for the Quantum Black Hole, gr-qc/9607022.
  22. C. P. Burgess, Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory, http://relativity.livingreviews.org/lrr-2004-5 (2004).
  23. S. Liberati, Quantum Vacuum Effects in Gravitational Fields: Theory and Detectability, PhD Thesis (International School for Advanced Studies, Trieste, Italy, 2000).
  24. M. Heusler, Stationary Black Holes: Uniqueness and Beyond, ITP, University of Zurich, http://www-theorie.physik.unizh.ch/ heusler (1998).
  25. H. E. Brandt, Chaos, Solitons and Fractals 10, 267 (1999).
  26. H. E. Brandt, Found. of Physics Lett. 13, 307 (2000).
  27. H. E. Brandt, Found. of Physics Lett. 5, 221 (1992).
  28. E. R. Caianiello, Nuovo Cim. B 59, 350 (1980).
  29. E. R. Caianiello, Lett. Nuovo Cim. 32, 65 (1981).
  30. E.R. Caianiello, Rivista del Nuovo Cimento 15, 432 (1992).
  31. G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories (Reidel, 1985).
  32. G. S. Asanov and S. F. Ponomarenko, A Finsler Bundle over Space-Time, Associated Gauge Fields and Connections (Shtiintsa, Kishinev, 1989, in Russian).
  33. D. Bao, S. S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry (Springer, Berlin-Heidelberg, 1999).
  34. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (New York, Marcel Dekker Inc., 1973).
  35. V. G. Bagrov, G. S. Bisnovatyi-Kogan, V. A. Bordovitsyn et al., Relativistic Particles' Radiation Theory (Fizmatlit, Moscow, 2002, in Russian).
  36. A. Kawaguchi, Transactions of the A. M. S. 44, 158 (1938).
  37. S. Sasaki, Tohoku Math. J. 10, 338 (1958).
  38. R. C. Tolman, Proc. Nat. Acad. Sci. US 20, 169 (1934).
  39. J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
  40. C. Hellaby and K. Lake, Astroph. J. 290, 381 (1985).
  41. C. J. S. Clarke and N. O'Donnell, Rend. Sem. Mat. Univ. Pol. Torino 50, 39 (1992).
  42. C. Hellaby, Some Properties of Singularities in the Tolman Model, PhD Thesis (Queen's University of Ontario, Kingston, Canada, 1985).
  43. L. D. Landau and E. M. Lifshitz, Classical Theory of Fields (Fizmatlit, Moscow, 2001, in Russian).
  44. S. Lang, Introduction to the Theory of Differentiable Manifolds (IL, Moscow, 1967). [Original edition???]
  45. Ya. B. Zeldovich and L.P. Grishchuk, Mon. Not. Roy. Astron. Soc. 207, 23 (1984).
  46. J. L. Synge, Relativity: the Special Theory (North-Holland Publishing Company, Amsterdam, 1956).
  47. J. L. Synge, Relativity: the General Theory (North-Holland Publishing Company, Amsterdam, 1960).
  48. S. R. de Groot, W. A. van Leeuwen, and Gh. G. van Weert, Relativistic Kinetic theory: Principles and Applications (North-Holland Publishing Company, Amsterdam - New York - Oxford, 1980).
  49. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  50. D. V. Gal'tsov, Phys. Rev. D 66, 025016 (2002).
  51. D. V. Gal'tsov and P. A. Spirin, Grav. Cosmol. 12, 1 (2006).
  52. D. V. Gal'tsov and P. A. Spirin, Grav. Cosmol. 13 241 (2007).
  53. E. Poisson, The Motion of Point Particles in the Curved Spacetime, Living Rev. Relativity 7, http://www.livingreviews.org/lrr-2004-6 (2004).
  54. Y. Mino, M. Sasaki, and T. Tanaki, Phys. Rev. D 55, 3457 (1997).
  55. T. C. Quinn and R. M. Wald, Phys. Rev. D 56, 3381 (1997).
  56. W. Hikida, H. Nakano, and M. Sasaki, Self-Force Regularization in the Schwarzschild Spacetime, gr-qc/0411150.
  57. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman and Company, San Francisco, 1973).
  58. R. Miron, D. Hrimiuc, H. Shimada, and S. Sabau, The Geometry of Hamilton and Lagrange Spaces (Kluwer Academic Publishers, New York - Boston - Dordrecht - London - Moscow, 2002).
  59. R. Miron and M. Anastaiei, The Geometry of Lagrange Spaces: Theory and Applications (Kluwer Academic Publishers, New York - Boston - London, 1994).
  60. A. Sandovici, Rend. Sem. Mat. Univ. Pol. Torino 63, 255 (2005).
  61. S. Capozziello, A. Feoli, G. Lambiase, G. Papini, and G. Scarpetta, Massive Scalar Particles in a Modified Schwarzschild Geometry, gr-qc/0003087.
  62. M. Crasmareanu, Novi Sad J. Math. 33, 11 (2003).
  63. D. Husemoller, Fibre Bundles (MvGraw - Hill Book Company, New York - St. Louis - San Francisco, 1966).
  64. J. D. Bekenstein, The Relation between Physical and Gravitational Geometry, preprint UCSB-TH-92-41 (November 1992).
  65. D. McDuff and D. Salamon, Introduction to Symplectic Topology (Clarendon Press, Oxford, 1998).
  66. I. Bucataru, Metric Nonlinear Connections, math.DG/0412109.
  67. M. Crampin, J. London Math. Soc. 2, 178 (1971).
  68. J. Grifone, Ann. Inst. H. Poincare 22, 287 (1972); 22, 291 (1972).
  69. Yu. N. Orlov, Fundamentals of Quantization of Degenerate Dynamic Systems (MFTI, Moscow, 2004).
  70. G. C. McVittie, General Relativity and Cosmology (Chapman and Hall Ltd., London, 1956).
  71. I. Bucataru, Some Basic Properties of Lagrangians, math.DG/0507560.
For more information about this paper please visit Springer's Home Page of this paper.

Back to The Contents Page