General relativity clocks?

Question

A hole is drilled through uniformely dense Earth, from North to South pole.
Four clocks were syncronized at North pole.

One clock A was dropped into the hole and reappeared 84 minutes later.
Clock B was launched into low polar orbit, and landed at North pole 84 minutes later.
Clock C was fired upward with carefully chosen inital velocity, and landed simultaneously with A and B 84 minutes later.
Clock D remained at rest at North pole.

Which clocks are ahead and which are behind?
(Note that A,B and C remained in free fall and followed geodesic lines)

Answer 1

I confess, I don't know the answer, but I have some ideas.

While all geodesic lines are paths which minimize the Lorentz interval. Not all are created equal. See http://www.physicsforums.com/showthread.php?t=103154 , and in particular, the last comment. We know that at least one geodesic will create the least absolute time dilation, but which if any of our test particles follows that particular geodesic? And how do the others fit in with respect to a polar observer?

For observer D, Clock A would be gravitationally redshifted by earth's gravity well for the whole transit (and the path would probably take less than 84 minutes due to the concentration of mass at the core). And Clock C, fired upwards would be gravitationally blueshift for its whole journey.

As for Clock B in ground level orbit, here is a site which says that the special relatitivity dialation based on speed of orbit exceeds the general relativity effect of speeding up time based on altitude gain, at least up to a radial distance of 10,000 m from the earth's center (apprx 4000m orbit). http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Note, both time dilation based on speed and on gravity are both effects of general relativity, but the article breaks out the speed as special relativity] Satellites below a 10k radial orbit have a frequency shift down and above 10k, shift up, So B's clock would be slow.

So I am opining A (hole) slowest, B (orbit) slow, D (on ground) normal, C (upward) fast.

In other words, although A, B, and C are all geodesics, only C offers the shortest path.

Answer 2

A,B, and C will appear to be behind (slow) when compared to clock D.

Answer 3

A ahead, D normal, B behind, C behind (more so that B i think).