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I am not a physicist.

let's say we have a space with an object in it, where all other gravitational bodies are so far away that their affect on the shape of the space is negligible.

let's say the object is moving, and rotating such that we can say (for simplicity's sake) that it is moving to the right, and rotating clockwise (so we are considering rotation around the z axis, and movement in the x axis). therefore the +y side of the object is moving faster than the -y side of the object relative to the underlying space.

it seems to me that this object should experience frame-dragging between the body and the underlying space and so curve its path such that it appears to curve very very very slightly upwards in y.

is this correct?

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No, it's not correct. Let's assume the object is axially-symmetric for simplicity. If the object is light (so we can assume spacetime really is flat), then we can change to an inertial frame where its movement is, momentarily, purely rotation. Now it is obvious by symmetry that it is not going to start moving on any direction, because no direction is now preferred: in other words its movement in that frame is always purely rotation. The argument for a non-axially-symmetric object is more hairy (read: beyond me at this time of night) but similar. – tfb 2 days ago

The object doesn't move relative to fixed space--there is no fixed space per special relativity--so forget that. The question is "why is there frame dragging?": because there is normal Newtonian-like gravitational attraction, and you need the frame-dragging field to make the whole thing relativistically invariant--that is, consistent for all inertial observers. I mean this in the same sense that the electric field needs a magnetic field to build a fully Lorentz invariant theory.

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