Gravitational radiation reaction and balance equations to post-Newtonian order

Luc Blanchet

doi:10.1103/PhysRevD.55.714

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Gravitational radiation reaction and balance equations to post-Newtonian order

Luc Blanchet

Département d’Astrophysique Relativiste et de Cosmologie, Centre National de la Recherche Scientifique (UPR 176), Observatoire de Paris, 92195 Meudon Cedex, France

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Phys. Rev. D 55, 714 – Published 15 January, 1997

DOI: https://doi.org/10.1103/PhysRevD.55.714

Abstract

Gravitational radiation reaction forces and balance equations are investigated to 3/2 post-Newtonian (1.5PN) order beyond the quadrupole approximation, corresponding to the 4PN order in the equations of motion of an isolated system. By matching a post-Newtonian solution for the gravitational field inside the system to a post-Minkowskian solution (obtained in a previous work) for the gravitational field exterior to the system, we determine the 1PN relativistic corrections to the “Newtonian” radiation reaction potential of Burke and Thorne. The 1PN reaction potential involves both scalar and vectorial components, with the scalar component depending on the mass-type quadrupole and octupole moments of the system, and the vectorial component depending in particular on the current-type quadrupole moment. In the case of binary systems, the 1PN radiation reaction potential has been shown elsewhere to yield consistent results for the 3.5PN approximation in the binary’s equations of motion. Adding up the effects of tails, the radiation reaction is then written to 1.5PN order. In this paper, we establish the validity to 1.5PN order, for general systems, of the balance equations relating the losses of energy, linear momentum, and angular momentum in the system to the corresponding fluxes in the radiation field far from the system.

References (64)

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L. Blanchet and T. Damour, Phys. Lett.104A, 82 (1984).
J. L. Anderson and T. C. DeCanio, Gen. Relativ. Gravit.6, 197 (1975).
A. Papapetrou and B. Linet, Gen. Relativ. Gravit.13, 335 (1981).
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L. Blanchet and T. Damour, Ann. Inst. Henri Poincaré Phys. Théor.50, 377 (1989).
T. Damour and B. R. Iyer, Ann. Inst. Henri Poincaré Phys. Théor.54, 115 (1991).
L. Blanchet, Phys. Rev. D51, 2559 (1995).
L. Blanchet and T. Damour, Phys. Rev. D37, 1410 (1988).
L. Blanchet, Phys. Rev. D47, 4392 (1993) [referred in the text to as paper I].
B. R. Iyer and C. M. Will, Phys. Rev. Lett.70, 113 (1993).
B. R. Iyer and C. M. Will, Phys. Rev. D52, 6882 (1995).
W. B. Bonnor and M. A. Rotenberg, Proc. R. Soc. LondonA265, 109 (1961).
A. Papapetrou, Ann. Inst. Henri Poincaré, A14, 79 (1971).
J. D. Bekenstein, Astrophys. J.183, 657 (1973).
W. H. Press, Phys. Rev. D15, 965 (1977).
A. Peres, Phys. Rev.128, 2471 (1962).
Our conventions and notation are the following: signature $- + + +$ ; greek indices =0,1,2,3; latin indices =1,2,3; $g = \det$ ${(g}_{μ ν})$ ; $η_{α β} = η^{α β}$ = flat metric = diag(-1,1,1,1); $r = | x | {= (x}_{1}^{2} {+ x}_{2}^{2} {+ x}_{3}^{2})^{1 / 2}$ ; $n^{i} {= n}_{i} {= x}^{i} / r$ ; $\partial_{i} = \partial / \partial x^{i}$ ; $x^{L} {= x}_{L} {= x}_{i_{1}} x_{i_{2}} \dots x_{i_{l}}$ and $\partial_{L} = \partial_{i_{1}} \partial_{i_{2}} \dots \partial_{i_{l}}$ , where ${L = i}_{1} i_{2} \dots i_{l}$ is a multi-index with $l$ indices; $x_{L - 1} {= x}_{i_{1}} \dots x_{i_{l - 1}}$ , $x_{aL - 1} {= x}_{a} x_{i_{1}} \dots x_{i_{l - 1}}$ , etc…; $x ̂_{L}$ and $\partial ̂_{L}$ are the (symmetric) and trace-free parts of $x_{L}$ and $\partial_{L},$ for instance $x ̂_{ij} {= x}_{i} x_{j}$ $-$ $\frac{1}{3}$ $δ_{ij} r^{2}$ ; $x ̂_{L}$ is denoted also $x_{〈 L 〉};$ $T_{(ij)} =$ $\frac{1}{2}$ ${(T}_{ij} {+ T}_{ji})$ ; the superscript $(n)$ denotes $n$ time derivatives.
G. Schäfer, Lett. Nuovo Cimento36, 105 (1983).
V. A. Fock, Theory of Space, Time and Gravitation (Pergamon, London, 1959).
In Appendix B of Ref. [49] the expression of the 1PN-conserved energy $E^{1 P N}$ as given by the equations (B3) and (B7) is in error. The correct expression is (4.18) in the present paper. The agreement between the equations (B3) and (B7) was proved in Ref. [49] only when both equations take their correct form.
R. Epstein and R. V. Wagoner, Astrophys. J.197, 717 (1975).
K. S. Thorne, in Gravitational Radiation [19], p. 3.

[c1] R. A. Hulse and J. H. Taylor, Astrophys. J., Lett. Ed.195, L51 (1975).

[c2] J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London)277, 437 (1979).

[c3] T. Damour, Phys. Rev. Lett.51, 1019 (1983).

[c4] J. H. Taylor, A. Wolszczan, T. Damour, and J. M. Weisberg, Nature (London)355, 132 (1992).

[c5] J. H. Taylor, Class Quantum Grav.10, S167 (1993).

[c6] C. Cutler, T. A. Apostolatos, L. Bildsten, L. S. Finn, E. E. Flanagan, D. Kennefick, D. M. Markovic, A. Ori, E. Poisson, G. J. Sussman, and K. S. Thorne, Phys. Rev. Lett.70, 2984 (1993).

[c7] L. S. Finn and D. F. Chernoff, Phys. Rev. D47, 2198 (1993).

[c8] C. Cutler and E. Flanagan, Phys. Rev. D49, 2658 (1994).

[c9] E. Poisson, Phys. Rev. D47, 1497 (1993).

[c10] C. Cutler, L. S. Finn, E. Poisson, and G. J. Sussmann, Phys. Rev. D47, 1511 (1993).

[c11] H. Tagoshi and T. Nakamura, Phys. Rev. D49, 4016 (1994).

[c12] M. Sasaki, Prog. Theor. Phys.92, 17 (1994).

[c13] H. Tagoshi and M. Sasaki, Prog. Theor. Phys.92, 745 (1994).

[c14] E. Poisson, Phys. Rev. D52, 5719 (1995).

[c15] L. Blanchet, T. Damour, and B. R. Iyer, Phys. Rev. D51, 5360 (1995).

[c16] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett.74, 3515 (1995).

[c17] C. M. Will and A. G. Wiseman, Phys. Rev. D54, 4813 (1996).

[c18] L. Blanchet, Phys. Rev. D54, 1417 (1996).

[c19] T. Damour, in Gravitational Radiation, edited by N. Deruelle and T. Piran (North-Holland, Amsterdam, 1983), p. 59.

[c20] W. L. Burke, Ph.D. thesis, California Institute of Technology, 1969 (unpublished).

[c21] K. S. Thorne, Astrophys. J.158, 997 (1969).

[c22] W. L. Burke, J. Math. Phys. (N.Y.)12, 401 (1971).

[c23] S. Chandrasekhar, Astrophys. J.158, 45 (1969).

[c24] S. Chandrasekhar and Y. Nutku, Astrophys. J.158, 55 (1969).

[c25] S. Chandrasekhar and F. P. Esposito, Astrophys. J.160, 153 (1970).

[c26] J. Ehlers, A. Rosenblum, J. N. Goldberg, and P. Havas, Astrophys. J.208, L77 (1976).

[c27] M. Walker and C. M. Will, Astrophys. J.242, L129 (1980).

[c28] L. Blanchet and T. Damour, Phys. Lett.104A, 82 (1984).

[c29] J. L. Anderson and T. C. DeCanio, Gen. Relativ. Gravit.6, 197 (1975).

[c30] A. Papapetrou and B. Linet, Gen. Relativ. Gravit.13, 335 (1981).

[c31] J. Ehlers, Ann. (N.Y.) Acad. Sci.336, 279 (1980).

[c32] G. D. Kerlick, Gen. Relativ. Gravit.12, 467 (1980).

[c33] G. D. Kerlick, Gen. Relativ. Gravit.12, 521 (1980).

[c34] R. Breuer and E. Rudolph, Gen. Relativ. Gravit.13, 777 (1981).

[c35] R. Breuer and E. Rudolph, Gen. Relativ. Gravit.14, 181 (1982).

[c36] G. Schäfer, Ann. Phys. (N.Y.)161, 81 (1985).

[c37] T. Damour and N. Deruelle, Phys. Lett.87A, 81 (1981).

[c38] T. Damour and N. Deruelle, C. R. Seances Acad. Sci. II, Mech.-Phys. Chim. Sci. Terre Sci. Universe293, 537 (1981).

[c39] T. Damour, C. R. Seances Acad. Sci. II, Mech.-Phys. Chim. Sci. Terre Sci. Universe294, 1355 (1982).

[c40] L. Blanchet and T. Damour, Philos. Trans. R. Soc. London, Ser. A320, 379 (1986).

[c41] L. Blanchet, Proc. R. Soc. London, INSA409, 383 (1987).

[c42] L. Blanchet and T. Damour, Phys. Rev. D46, 4304 (1992).

[c43] W. B. Bonnor, Philos. Trans. R. Soc. London, Ser. A251, 233 (1959).

[c44] W. B. Bonnor and M. A. Rotenberg, Proc. R. Soc. London, Ser. A289, 247 (1966).

[c45] A. J. Hunter and M. A. Rotenberg, J. Phys. A2, 34 (1969).

[c46] K. S. Thorne, Rev. Mod. Phys.52, 299 (1980).

[c47] L. Blanchet and T. Damour, Ann. Inst. Henri Poincaré Phys. Théor.50, 377 (1989).

[c48] T. Damour and B. R. Iyer, Ann. Inst. Henri Poincaré Phys. Théor.54, 115 (1991).

[c49] L. Blanchet, Phys. Rev. D51, 2559 (1995).

[c50] L. Blanchet and T. Damour, Phys. Rev. D37, 1410 (1988).

[c51] L. Blanchet, Phys. Rev. D47, 4392 (1993) [referred in the text to as paper I].

[c52] B. R. Iyer and C. M. Will, Phys. Rev. Lett.70, 113 (1993).

[c53] B. R. Iyer and C. M. Will, Phys. Rev. D52, 6882 (1995).

[c54] W. B. Bonnor and M. A. Rotenberg, Proc. R. Soc. LondonA265, 109 (1961).

[c55] A. Papapetrou, Ann. Inst. Henri Poincaré, A14, 79 (1971).

[c56] J. D. Bekenstein, Astrophys. J.183, 657 (1973).

[c57] W. H. Press, Phys. Rev. D15, 965 (1977).

[c58] A. Peres, Phys. Rev.128, 2471 (1962).

[c59] Our conventions and notation are the following: signature $- + + +$ ; greek indices =0,1,2,3; latin indices =1,2,3; $g = \det$ ${(g}_{μ ν})$ ; $η_{α β} = η^{α β}$ = flat metric = diag(-1,1,1,1); $r = | x | {= (x}_{1}^{2} {+ x}_{2}^{2} {+ x}_{3}^{2})^{1 / 2}$ ; $n^{i} {= n}_{i} {= x}^{i} / r$ ; $\partial_{i} = \partial / \partial x^{i}$ ; $x^{L} {= x}_{L} {= x}_{i_{1}} x_{i_{2}} \dots x_{i_{l}}$ and $\partial_{L} = \partial_{i_{1}} \partial_{i_{2}} \dots \partial_{i_{l}}$ , where ${L = i}_{1} i_{2} \dots i_{l}$ is a multi-index with $l$ indices; $x_{L - 1} {= x}_{i_{1}} \dots x_{i_{l - 1}}$ , $x_{aL - 1} {= x}_{a} x_{i_{1}} \dots x_{i_{l - 1}}$ , etc…; $x ̂_{L}$ and $\partial ̂_{L}$ are the (symmetric) and trace-free parts of $x_{L}$ and $\partial_{L},$ for instance $x ̂_{ij} {= x}_{i} x_{j}$ $-$ $\frac{1}{3}$ $δ_{ij} r^{2}$ ; $x ̂_{L}$ is denoted also $x_{〈 L 〉};$ $T_{(ij)} =$ $\frac{1}{2}$ ${(T}_{ij} {+ T}_{ji})$ ; the superscript $(n)$ denotes $n$ time derivatives.

[c60] G. Schäfer, Lett. Nuovo Cimento36, 105 (1983).

[c61] V. A. Fock, Theory of Space, Time and Gravitation (Pergamon, London, 1959).

[c62] In Appendix B of Ref. [49] the expression of the 1PN-conserved energy $E^{1 P N}$ as given by the equations (B3) and (B7) is in error. The correct expression is (4.18) in the present paper. The agreement between the equations (B3) and (B7) was proved in Ref. [49] only when both equations take their correct form.

[c63] R. Epstein and R. V. Wagoner, Astrophys. J.197, 717 (1975).

[c64] K. S. Thorne, in Gravitational Radiation [19], p. 3.

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Gravitational radiation reaction and balance equations to post-Newtonian order

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