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Do you mean accelerate the sphere you are in? If I remember right you will remain in the same location as the sphere, until you collide into the side. Since the net forces on you inside are zero, you should remain still.
Someone else read J. Richard Gott's Time Travel in Einstein's Universe. He proposes if you take the mass of about 6 Jupiter's, collapse it into a spherical shell 10m diameter of the hollow and 12m overall diameter, you will bend space enough that your time will slow down significantly.
Just some fun effects of gravity.
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No, I mean accellerate inside the sphere.
Relatavistic effects should make the sphere no longer spherical, to a small degree. Would that change the math at all?
If the sphere weighed ALOT (like, ALOT), these effects might be detectable even at small amounts of accelleration or velocity. You may be able to avoid black-hole problems by making the sphere larger. . .
Random: might the effects behave alot like inertia?