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Equations of motion for isolated bodies with relativistic corrections including the radiation reaction force

Published online by Cambridge University Press:  04 August 2017

L. P. Grishchuk  and
S. M. Kopejkin
L. P. Grishchuk
Affiliation:
Sternberg State Astronomical Institute, University Prospect, 13, 119899 Moscow, USSR
S. M. Kopejkin
Affiliation:
Sternberg State Astronomical Institute, University Prospect, 13, 119899 Moscow, USSR

Abstract

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We have derived in an explicit form the equations of motion for two spherically-symmetric non rotating bodies in the slow motion approximation. The equations include relativistic corrections of order (v/c)2, (v/c)4 and (v/c)5 to the newtonian equations of motion. It is shown that the equations depend on the only parameter characterizing each body, namely on its relativistic mass, regardless of its internal structure and degree of compactness. This means that the equations can also be applied to bodies with a strong internal gravity, such as neutron stars and black holes. It is shown that in the (v/c)2 and (v/c)4 approximations the equations can be derived from a Lagrangian. The Lagrangian is given in an exact form. The integration of the equations of motion is performed by the method of osculating elements. The formulae for secular change of the semi-major axis and eccentricity coincide precisely with the standard ones whose derivation is based on a calculation of the energy flux in the outgoing gravitational waves.

Type
Dynamical Effects in General Relativity
Information
Symposium - International Astronomical Union , Volume 114: Relativity in Celestial Mechanics and Astrometry , 1986 , pp. 19 - 34
Copyright
Copyright © Reidel 1986 

References

Alexander, M. E.: 1973, Astroph. Space Sci., 23, 459.Google Scholar
Anderson, J. L. and Decanio, T. C.: 1975, Gen. Rel. Grav. 6, 197.CrossRefGoogle Scholar
Barker, B. M. and O'Connell, R. F.: 1980, Can. J. Phys. 58, 1659.Google Scholar
Bekenstein, J. D.: 1973, Ap. J. 183, 657.Google Scholar
Bel, L., Damour, T., Deruelle, N., Ibanez, J. and Martin, J.: 1981, Gen. Rel. Grav. 13, 963.CrossRefGoogle Scholar
Breuer, R. A. and Rudolph, E.: 1981, Gen. Rel. Grav. 13, 777.CrossRefGoogle Scholar
Breuer, R. A. and Rudolph, E.: 1981, Gen. Rel. Grav. 14, 181.CrossRefGoogle Scholar
Brumberg, V. A.: 1972, ‘Relativistic Celestial Mechanics’ Nauka, Moscow.Google Scholar
Burke, W. L.: 1971, J. Math. Phys. 12, 401.CrossRefGoogle Scholar
Caporali, A.: 1980, Max-Planck Institut Prepint, MPI-PAE/Astro 219.Google Scholar
Carmeli, M.: 1965, Nuov. Cim. 37, 842.Google Scholar
Chandrasekhar, S.: 1965, Ap. J. 142, 1488.CrossRefGoogle Scholar
Chandrasekhar, S. and Esposito, F. P.: 1970, Ap. J. 160, 153.Google Scholar
Chandrasekhar, S. and Nutku, Y.: 1969, Ap. J. 158, 55.Google Scholar
Damour, T.: 1983, In ‘Gravitational Radiation’, Deruelle, N. and Piran, T. eds., North-Holland, Amsterdam, 59.Google Scholar
D'Eath, P. D.: 1975a, Phys. Rev. D11, 1387.Google Scholar
D'Eath, P. D.: 1975b, Phys. Rev. D12, 2183.Google Scholar
Demianski, M. and Grishchuk, L. P.: 1974, Gen. Rel. Grav. 5, 673.CrossRefGoogle Scholar
Dixon, W. G.: 1979, In ‘Isolated Gravitating Systems in General Relativity’, Ehlers, J. ed., North-Holland, Amsterdam, 156.Google Scholar
Droste, J.: 1916, Proc. Roy. Acad. Sci. Amsterdam, 19, 447.Google Scholar
Duboshin, G. N.: 1975, ‘Celestial Mechanics: Fundamental Problems and Methods’, Nauka, Moscow.Google Scholar
Ehlers, J.: 1980, Ann. N.Y. Acad. Sci., 336, 279.CrossRefGoogle Scholar
Ehlers, J., Rosenblum, A., Goldberg, J.N. and Havas, P.: 1976, Ap. J. Lett., 208, 77.CrossRefGoogle Scholar
Ehlers, J. and Rudolph, E.: 1977, Gen. Rel. Grav., 8, 197.Google Scholar
Einstein, A., Infeld, L. and Hoffman, B.: 1938, Ann. Math., 39, 65.CrossRefGoogle Scholar
Fock, V.A.: 1959, ‘Theory of Space, Time and Gravitation’, Pergamon, London.Google Scholar
Fichtenholz, I. G., 1950, Zh. Eksp. Teor. Fiz., 20, 233.Google Scholar
Futamase, T.: 1983, Phys. Rev., D28, 2373.Google Scholar
Futamase, T. and Schutz, B.F.: 1983, Phys. Rev., D28, 2363.Google Scholar
Gel'Fand, I. M. and Shilov, G. E.: 1959, ‘The theory of distributions’, Fizmatgiz, Moscow.Google Scholar
Grishchuk, L.P. and Kopejkin, S. M.: 1983, Pis'ma Astron. Zh., 9, 436 (Sov. Astron. Lett., 9, 230).Google Scholar
Grishchuk, L.P., Petrov, A.N. and Popova, A.D.: 1984, Comm. Math. Phys. 94, 379.CrossRefGoogle Scholar
Hulse, R. A. and Taylor, J. H.: 1975, Ap. J. Lett., 195, 51.Google Scholar
Infeld, L.: 1954, Acta Phys. Polon., 13, 187.Google Scholar
Infeld, L. and Michalska-Trautman, R.: 1969, Ann. Phys. (N.Y.), 55, 561.CrossRefGoogle Scholar
Kates, R. E.: 1980, Phys. Rev. D22, 1853.Google Scholar
Kerlick, G. D.: 1980, Gen. Rel. Grav., 12, 467 and 521.CrossRefGoogle Scholar
Kislik, M. D., Kolyuka, Yu. F., Kotel'nikov, V. A., Petrov, G. M. and Tikhonov, V. F.: 1980, Dokl. AN SSSR, 255, 545.Google Scholar
Kopejkin, S. M.: 1985, Astron. Zh., (to appear).Google Scholar
Landau, L. D. and Lifshitz, E. M., 1975, ‘The Classical Theory of Fields’, Pergamon, Oxford.Google Scholar
Linet, B.: 1981, C.R. Acad. Sci. Paris, 292, ser. II, 1425.Google Scholar
McCrea, J. D.: 1981, Gen. Rel. Grav., 13, 397.CrossRefGoogle Scholar
Misner, C. W., Thorne, K. S. and Wheeler, J. A.: 1973, ‘Gravitation’, Freeman, San Francisco.Google Scholar
Ohta, T., Okamura, H., Kimura, T. and Hiida, K.: 1974, Prog. Theor. Phys., 51, 1220.CrossRefGoogle Scholar
Papapetrou, A.: 1951, Proc. Phys. Soc. Lond., A64, 57 and 302.CrossRefGoogle Scholar
Papapetrou, A. and Linet, B.: 1981, Gen. Rel. Grav., 13, 335.CrossRefGoogle Scholar
Petrova, N. M.: 1949, Zh. Eksp. Teor. Fiz., 19, 989.Google Scholar
Petrova, N. M. and Sandina, I. V.: 1974, Dokl. AN SSSR, 217, 319.Google Scholar
Peters, P. C.: 1964, Phys. Rev., 136, 1124.Google Scholar
Peters, P. C. and Mathews, J.: 1963, Phys. Rev., 131, 435.Google Scholar
Schäfer, G.: 1982, Prog. Theor. Phys., 68, 2191.Google Scholar
Schäfer, G.: 1984, Phys. Lett., 100A, 128.Google Scholar
Schattner, R.: 1979, Gen. Rel. Grav., 10, 377 and 395.Google Scholar
Shapiro, I. I.: 1979, in ‘Astrofisica e cosmologia gravitazione quanti e relatività’, Guinti Barbèra, Firenze.Google Scholar
Spyrou, N.: 1977, Gen. Rel. Grav., 8, 463.Google Scholar
Spyrou, N.: 1978, Gen. Rel. Grav. 9, 519.Google Scholar
Thorne, K. S. and Hartle, J. B., 1984, A Caltech preprint, GRP, 015.Google Scholar
Weisberg, J. M. and Taylor, J. H.: 1984, Phys. Rev. Lett., 52, 1348.Google Scholar
Will, C. M.: 1981, ‘Theory and experiment in gravitational physics’, Cambridge University Press, Cambridge.Google Scholar
Zhang, X. H.: 1984, A Caltech preprint, GRP, 030.Google Scholar