arXiv:1304.8122v1 [gr-qc] 30 Apr 2013 New perturbative method for solving the gravitational N -body problem in general relativity Slava G. Turyshev 1 and Viktor T. Toth 2 1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA and 2 Ottawa, Ontario K1N 9H5, Canada (Dated: May 1, 2013) We present a new approach to describe the dynamics of an isolated, gravitationally bound as- tronomical N -body system in the weak field and slow-motion approximation of general relativity. Celestial bodies are described using an arbitrary energy-momentum tensor and assumed to possess any number of internal multipole moments. The solution of the gravitational field equations in any reference frame is presented as a sum of three terms: i) the inertial flat spacetime in that frame, ii) unperturbed solutions for each body in the system boosted to the coordinates of this frame, and iii) the gravitational interaction term. Such an ansatz allows us to reconstruct all features of the gravitational field and to develop a theory of relativistic reference frames. We use the harmonic gauge conditions to impose a significant constraint on the structure of the post-Galilean coordinate transformation functions that relate global coordinates in the inertial reference frame to the local coordinates of the non-inertial frame associated with a particular body. The remaining parts of these functions are constrained using dynamical conditions, which are obtained by constructing the relativistic proper reference frame associated with a particular body. In this frame, the effect of external forces acting on the body is balanced by the fictitious frame-reaction force that is needed to keep the body at rest with respect to the frame, conserving its relativistic linear momentum. We find that this is sufficient to determine explicitly all the terms of the coordinate transformation. The same method is then used to develop the inverse transformations. The resulting post-Galilean coordinate transformations have an approximate group structure that extends the Poincar´ e group of global transformations to the case of a gravitational N -body system. We present and discuss the structure of the metric tensors corresponding to the reference frames involved, the rules for trans- forming relativistic gravitational potentials, the coordinate transformations between frames and the resulting relativistic equations of motion. PACS numbers: 03.30.+p, 04.25.Nx, 04.80.-y, 06.30.Gv, 95.10.Eg, 95.10.Jk, 95.55.Pe I. INTRODUCTION Recent experiments have successfully tested Einstein’s general theory of relativity in a variety of ways to remarkable precision [1, 2]. Diverse experimental techniques were used to test relativistic gravity in the solar system, namely: spacecraft Doppler tracking, planetary ranging, lunar laser ranging, dedicated gravity experiments in space and many ground-based efforts [2–4]. Given this phenomenological success, general relativity became the standard theory of gravitation, especially where the needs of astronomy, astrophysics, cosmology and fundamental physics are concerned [2]. The theory is used for many practical purposes involving spacecraft navigation, geodesy and time transfer. It is used to determine the orbits of planets and spacecraft and to describe the propagation of electromagnetic waves in spacetime [2]. As we shall see, finding a solution to the Einstein’s equations in the case of an unperturbed one body problem is quite a simple task. However, it turns out that a generalization of the resulting post-Newtonian solution to a system of N extended arbitrary bodies is not straightforward. A neutral point test particle with no angular momentum follows a geodesic that is completely defined by the external gravitational field. However, the coupling of the intrinsic multipole moments of an extended body to the background gravitational field (present due to external gravitational sources), affects the equations of motion of such a body. Similarly, if a test particle is spinning, its equations of motion must account for the coupling of the body’s angular momentum to the external gravitational field. As a result, one must be able to describe the interaction of a body’s intrinsic multipole moments and angular momentum with the surrounding gravitational field. Multipole moments are well-defined in the local quasi-inertial reference frame generalizing what was defined for the unperturbed one-body problem. While transforming these quantities from one coordinate frame to another, one should account for the fact that the gravitational interaction is non-linear and, therefore, these moments interact with gravitational fields, affecting the body’s motion. When the Riemannian geometry of the general theory of relativity is concerned, it is well known that coordinate charts are merely labels. Usually, spacetime coordinates have no direct physical meaning and it is essential to