Abstract
We consider some principal methodological problems that appear when the Einstein-Infeld-Hoffmann method is used to find approximate solutions of the general relativity equations and to obtain information about the motion of particles whose interaction force is much greater than the gravitational attraction force. Among these problems are normalizing approximate expressions by expanding exact solutions written in the same coordinate conditions used in the Einstein-Infeld-Hoffmann method, assigning the smallness orders depending on relations between the smallness parameters in play, and verifying cancellations of divergent terms arising in surface integrals. Solving these questions in accordance with the internal logic of the Einstein-Infeld-Hoffmann method results in new tools and techniques for applying the method. We demonstrate these tools and techniques in the example of the problem of the motion of two electrically charged pointlike particles in the (v/c)3-approximation.
Similar content being viewed by others
REFERENCES
A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math., 39, 65 (1938).
A. Einstein and L. Infeld, Can. J. Math., 1, 209 (1949).
J. L. Anderson, Phys. Rev. D, 56, 4675 (1977).
M. V. Gorbatenko and A. V. Pushkin, Voprosy At. Nauki i Tekhn. Ser. Teor. Prikl. Fiz., No. 2, 28 (1985).
M. V. Gorbatenko and T. M. Gorbatenko, Theor. Math. Phys., 140, 1028 (2004).
V. A. Brumberg, Relativistic Celestial Mechanics [in Russian], Nauka, Moscow (1972); English transl.: Essential Relativistic Celestial Mechanics, Adam Hilger, Bristol (1991).
A. A. Logunov and M. A. Mestvirishvili, The Relativistic Theory of Gravitation [Russian], Nauka, Moscow (1989); English transl., Mir, Moscow (1989).
L. D. Landau and E. M. Lifshitz, Classical Field Theory [in Russian] (Course of Theoretical Physics, Vol. 2, 7th ed.), Nauka, Moscow (1988); English transl. prev. ed., Pergamon, Oxford (1975).
R. P. Feynman, Phys. Rev., 74, 939 (1948).
P. A. M. Dirac, Proc. Roy. Soc. London A, 176, 148 (1938).
Author information
Authors and Affiliations
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 160–176, January, 2005.
Rights and permissions
About this article
Cite this article
Gorbatenko, M.V. Obtaining equations of motion for charged particles in the (v/c)3-approximation by the Einstein-Infeld-Hoffmann method. Theor Math Phys 142, 138–152 (2005). https://doi.org/10.1007/s11232-005-0014-0
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11232-005-0014-0