Institute for Advanced Study/Princeton University Joint Astrophysics Colloquium

The increasing role of general relativity in the dynamics of stellar systems with central massive black holes, in the generation of extreme mass-ratio inspirals, and in the evolution of hierarchical triple systems inspires a close examination of how post-Newtonian effects are incorporated into N-body dynamics. Most approaches incorporate relativity by adding to the Newtonian N-body equations the standard two-body post-Newtonian terms for a given star around the black hole or for the close binary in a triple system. We argue that, for calculating the evolution of such systems over timescales comparable to the relativistic pericenter advance timescale, it is essential to include ``cross terms'' in the equations of motion. These are post-Newtonian terms that represent a coupling between the potential of the central black hole or the inner binary system and the potential due to the other star(s) in the system. Over pericenter precession timescales, the effects of such terms can actually be ``boosted'' to amplitudes of Newtonian order. We display the post-Newtonian N-body equations of motion including a central black hole in a truncated form that includes all the relevant cross terms, in a format ready to use for numerical implementation. For hierarchical triple systems, we show explicitly the effects of cross terms on the orbit-averaged equations for the orbit elements of the inner binary for the special case where the third body is on a circular orbit, and present the results of numerical integrations of those equations.