Question

What are the equations of motion that model near light speed orbits of a massive body about incredibly massive bodies?

Answer 1

In Kip Thorne's recently published book, The Science of Interstellar, he describes, by means of an illustration, the complex nature of a spacecraft orbiting a massive black hole with velocities ~ 0.55c and similar to the orbits in this illustration  enter image description here  he compares these orbits to the more typical Newtonian elliptical orbits most people are familiar with.  Considering only two bodies (e.g. black hole and spacecraft), what are the the equations of motion that describe such complex orbits, and considering the parameters involved (mass, radius, speed for example), where is the dividing point where elliptical orbits begin to appear more chaotic?  An aside - the thing that interested me here is the similarity to trajectories often observed in just simple nonlinear differential equations (e.g. Lorentz equations). Here the ability to predict system states become nearly impossible as the calculations are extremely sensitive to the assumed initial conditions. So I'm wondering if predicting the ephemeris of a spacecraft orbiting black holes is subject to the same limitations rendering it an impossible ( or otherwise highly risky) pursuit.

Answer 2

The warping of space (e.g., from extreme gravity and/or gravitational waves) due to the rotation of the black hole causes perturbations to the orbit, thus disturbing the constant elliptical orbit under weaker gravity. It's been more than five years since my GR class but I am not sure if the Kerr metric is enough to cause these orbits.