The Two Extended Bodies Problem in General Relativity within the post-Newtonian Approximation

The dynamics of extended bodies is a fundamental problem in any gravitational theory. In the case of General Relativity, this problem is under study since the theory was published. Several methods have been developed and different approaches are avalaible in the literature to interpret the relativistic contributions in the motion under gravity influence. The main goal in this thesis is to study a general method to face the equation of motion for extended bodies in General Relativity. We started with a proposal in the Newtonian theory, which consists in a multipolar expansion for the gravitational potentials as a function of the mass density moments and other physical variables as the stress tensor. The methodology give us the equation of motion for an isolated and self-gravitating system of extended bodies in Newtonian mechanics. A geometrical approach to get the equation of motion is also used for the Newtonian problems, it allows us to extend the methodology to General Relativity. In General Relativity, some new concepts are necessary: world tube, world line, generalized ideas of momentum, angular momentum, torque and center of mass are introduced in a general context. General expressions for the equation of motion in the case of extended bodies are written without any restriction. In order to gain some physical understanding, we compute the Papapetrous's equation of motion for a test extended body in a static and isotropic metric. Finally, we study a system of two extended bodies in the post-Newtonian approximation. We define the mass multipole moments and momentum from the gravitational potentials (metric functions) to the first post-Newtonian order. We follow the Landau-Liftshitz formalism to find out the equation of motion for the moments and applying the standard coordinate transformation for this theory, we write the traslational equation of motion