Gravitational Warping by Non-Spherical Celestial Bodies

[Gerinski]

Purely theoretically, if there were very massive celestial bodies which were not spherical, how would their gravitational warping be? Which shapes would the geodesical paths around them have? Would they still be elliptical orbits or would they have sudden changes of direction?
Say for example, a huge massive cube, or huge a pyramid, a rod or a giant horseshoe? If a freefalling object would approach those and fall into orbit, which kind of path would it follow?

 

[bcrowell]

In the examples you're describing, the main effect would be Newtonian, not relativistic. The orbits would of course not be elliptical. A less extreme example is that part of the perihelion precession of Mercury is due to the sun's oblateness.

Relativistically, the gravitational field is produced not just by the mass but by the stress-energy tensor, so it also depends on things like the pressure and shear stresses.