Motions of self-gravitating bodies to the second post-Newtonian order of general relativity

We derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies using an approximation of general relativity known as the post-Newtonian approximation. The post-Newtonian (PN) approximation is an expansion of Einstein's equations in powers of epsilon∼ (v/c)2 ∼ Gm/rc 2, where v is the relative velocity, m = m1 + m2 is the total mass, r is the distance between the bodies, c is the speed of light, and G is the Newtonian gravitational constant. We work to order epsilon2, or second post-Newtonian (2PN) order beyond the Newtonian approximation. In particular we study the contributions of the internal structure of the bodies (such as self-gravitational binding energy) that would diverge if the size of the bodies were to shrink to zero holding their masses fixed. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the "bare" masses to include 1PN and 2PN self-gravity terms, we show that a class of potentially divergent terms cancels, leaving 2PN equations of motion that depend only on the renormalized masses (modulo tidal effects). This is further evidence of the Strong Equivalence Principle, an assertion of general relativity that the motion of bodies should be independent of their internal structure, and supports the use of post-Newtonian approximations to derive equations of motion for strong-field bodies such as neutron stars and black holes.