On solar system dynamics in general relativity
Abstract
Recent work in the literature has advocated using the EarthMoonplanetoid Lagrangian points as observables, in order to test general relativity and effective field theories of gravity in the solar system. However, since the threebody problem of classical celestial mechanics is just an approximation of a much more complicated setting, where all celestial bodies in the solar system are subject to their mutual gravitational interactions, while solar radiation pressure and other sources of nongravitational perturbations also affect the dynamics, it is conceptually desirable to improve the current understanding of solar system dynamics in general relativity, as a first step towards a more accurate theoretical study of orbital motion in the weakgravity regime. For this purpose, starting from the Einstein equations in the de DonderLanczos gauge, this paper arrives first at the LeviCivita Lagrangian for the geodesic motion of planets, showing in detail under which conditions the effects of internal structure and finite extension get canceled in general relativity to first postNewtonian order. The resulting nonlinear ordinary differential equations for the motion of planets and satellites are solved for the Earth’s orbit about the Sun, written down in detail for the SunEarthMoon system, and investigated for the case of planar motion of a body immersed in the gravitational field produced by the other bodies (e.g. planets with their satellites). At this stage, we prove an exact property, according to which the fourthorder time derivative of the original system leads to a linear system of ordinary differential equations. This opens an interesting perspective on forthcoming research on planetary motions in general relativity within the solar system, although the resulting equations remain a challenge for numerical and qualitative studies. Last, the evaluation of quantum corrections to location of collinear and noncollinear Lagrangian points for the planar restricted threebody problem is revisited, and a new set of theoretical values of such corrections for the EarthMoonplanetoid system is displayed and discussed. On the side of classical values, the general relativity corrections to Newtonian values for collinear and noncollinear Lagrangian points of the SunEarthplanetoid system are also obtained. A direction for future research will be the analysis of planetary motions within the relativistic celestial mechanics set up by Blanchet, Damour, Soffel and Xu.
 Publication:

International Journal of Geometric Methods in Modern Physics
 Pub Date:
 2017
 DOI:
 10.1142/S0219887817501171
 arXiv:
 arXiv:1607.06298
 Bibcode:
 2017IJGMM..1450117B
 Keywords:

 Planetary motions;
 Lagrangian points;
 04.60.Ds;
 95.10.Ce;
 Canonical quantization;
 Celestial mechanics;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 26 pages, 3 figures