The N-body problem in general relativity

The form of the metric tensor in the first and two and one-half post-Newtonian approximations (PNA) is derived, and the method for its evaluation in the second PNA is presented for a system of N bodies, which are spherical, homogeneous, and rotate uniformly around axes through their centers. The energy, linear-momentum, and angular-momentum integrals up to the first PNA are also derived for the above system. The expressions found consist of 'pointlike,' 'structure,' and 'rotation' terms. The 'pointlike' terms coincide with those given by previous authors. The 'structure' and 'rotation' terms depend on the radius, internal energy, and angular velocity of the spheres. In a particular numerical application the first PNA correction in the energy integral is given. This increases with increasing mass, decreasing radius, and decreasing distance of the bodies, and becomes comparable to the Newtonian value of the energy for small distances and for radii approaching the Schwarzschild radius.