Equations of motion for compact binary systems in general relativity: Do they depend on the bodies' internal structure at the third post-Newtonian order?

We present and discuss the possibility, derived from work carried out 20 years ago, that the equations of motion for compact binary neutron stars at the third post-Newtonian (3PN) order in general relativity might actually depend on the internal structure of the bodies. These effects involve integrals over the density and internal gravitational potentials of the bodies that are independent of the mass and radius of the bodies, but dependent on their equations of state. These effects could alter the coefficients in the 3PN equations derived using ``point mass'' methods by as much as 100 percent. They were found in independent calculations done at Washington University using the Direct Integration of the Relaxed Einstein Equations (DIRE) approach, and at the Institut d'Astrophysique de Paris using the Multipolar post-Minkowskian (MPPM) approach. Neither calculation was completed because of the enormous complexity of the algebraic computations and the limitations of software of the day (Maple, Mathematica), and because of an assumption that the effects would somehow cancel or be removable by some transformation. This assumption was rooted in the Strong Equivalence Principle (SEP), which would suppress such effects up to the stage where tidal interactions become important, effectively 5PN order for compact bodies. SEP was well supported at lower PN orders and in special examples. We argue that this assumption needs to be verified by calculations. If the results show that these terms exactly cancel or can be absorbed into renormalized masses or shifted positions of each body, this would provide support for the Strong Equivalence Principle. But if they do not cancel and are not incorporated into gravitational waveforms, they could impact efforts using next-generation gravitational-wave interferometers to extract information about the equation of state for neutron star matter.