I've got a rather messy unsolved (AFAIK) problem I've been thinking about a bit, and I thought I'd share it with the group. The basic idea is fairly simply stated. Suppose we have an accelerating spaceship which has a constant proper acceleration g. The question is, what is the curve (pick any coordinates that make the problem simple) that has a constant velocity 'v' with respect to any observer who is "stationary" with respect to the accelerating ship. The notion of "stationary wrt to the ship" invites one to choose a coordinate system in which the metric is static (not a function of time) and in which the location of the spaceship is the origin. Then objects which are stationary in this coordinate system are also stationary with respect to the ship. Other more complex formulations of this notion are possible, but I think it's well defined and coordinate independent. (If anyone doesn't agree, we can argue about it more :-).) The velocity 'v' is measured by the local clocks and local ruler of said "stationary observer", when the observer and the moving object are at the same place at the same time. I'm particularly interested in the magnitude of the 4-acceleration of v, i.e. the "gravity" an observer on such a trajectory would experience.