jartsa said:
the orbiting observer takes a curved path through the space-time, while a clock on Earth takes a straight path through the space-time.
No. In curved spacetime, not all straight paths have maximal length, and straight paths are not always longer than curved paths.
The clock on Earth's path is curved, because it has nonzero proper acceleration. The orbiting clock's path might be straight or it might be curved, depending on the orbital speed; for the right orbital speed, the one that allows a free-fall orbit, the orbiting clock's path will be straight. The OP specified an orbital speed close to the speed of light, though, so in that case the orbiting clock's path will be curved, since it will take a very large
inward proper acceleration to keep it in orbit. But even a clock in free-fall orbit, for low enough orbital altitudes, will have a shorter path through spacetime than a clock on Earth, even though a free-fall orbiting clock's path through spacetime is straight.
Even in curved spacetime, there will always be
some straight path that is of maximal length, but it might take some effort to find it. In the case under discussion, consider this straight path: a clock is moving upward, radially, and passes the orbiting clock at the same instant the orbiting clock is directdly overhead of the Earth clock. (We are idealizing the Earth as non-rotating for this thought experiment; the Earth's rotation adds further complications.) The radially moving clock has just the right velocity so that it rises upward, decelerates, comes to a stop, starts falling back downward, and passes the orbiting clock again at the same instant the orbiting clock has completed exactly one orbit and is again directly overhead of the Earth clock. The radially moving clock will then have the longest path through spacetime between the two meetings.