The present status of Einsteinian relativistic celestial mechanics.
The present status of Einsteinian relativistic celestial mechanics is reviewed. Starting from a conceptual description of the problem at the Newtonian level, we compare the basic concepts of relativistic celestial mechanics to those of Newtonian one. Some problems to be solved in the years to come are formulated. In the following we want to consider the gravitational N-body problem, i.e. we deal with the dynamics of N distinct bodies of arbitrary shape and composition under the innuence of their mutual gravitational interaction. Other, non-gravitational forces will not be considered here. For didactical reasons let us rst summarize the Newtonian treatment of this problem. Newtonian celestial mechanics is coined by the existence of a class of preferred Cartesian inertial coordinates, x = (ct; x i). Such inertial coordinates exist globally in Newton's framework, i.e. they cover the entire space-time manifold. Moreover, these preferred coordinates in Newton's theory have direct physical meaning: they are directly related with the observables. The basic variables of Newtonian celestial mechanics are the local matter variables (mass-density), v (the velocity eld of matter) and t ij (the stress-tensor, containing pressure, elastic stresses, etc.) and the gravitational eld variable U (Newtonian potential). These variables obey the local equations of motion @ @t + r(v) = 0; (1) @v i @t + @ @x j h v i v j + t ij i = @ @x i U : (2) The rst equation describes the conservation of mass (continuity equation) and the second one is the force density equation (the Euler equation). The eld equation (Poisson equation), U = ?4GG; (3) completes our set of diierential equations. If one adds some equations of state of matter (e.g., pressure as function of density for cold matter or stress-strain relations) in principle these equations would be suucient to formulate Newtonian celestial mechanics in one Cartesian inertial 1