Post-Newtonian effects in N -body dynamics: conserved quantities in hierarchical triple systems

Conventional approaches to incorporating general relativistic effects into the dynamics of N-body systems containing central black holes, or of hierarchical triple systems with a relativistic inner binary, may not be adequate when the goal is to study the evolution of the system over a timescale related to relativistic secular effects, such as the precession of the pericenter. For such problems, it may be necessary to include post-Newtonian (PN) 'cross terms' in the equations of motion in order to capture relativistic effects consistently over the long timescales. Cross terms are PN terms that explicitly couple the two-body relativistic perturbations with the Newtonian perturbations due to other bodies in the system. In this paper, we show that the total energy of a hierarchical triple system is manifestly conserved to Newtonian order over the relativistic pericenter precession timescale of the inner binary if and only if PN cross-term effects in the equations of motion are taken carefully into account.