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My question is really about the gravitomagnetic frame-dragging and the Lense Thirring effect. My question is not whether the frame dragging effect exists but rather it's manifestation in affecting precession. From Wikipedia:

Josef Lense and Hans Thirring ... predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess.

If I understand it right we can extrapolate the effect to frame dragging in general. So let's say we have a large rotating ball floating on the surface of a swimming pool, dragging the water around it and there is a smaller ball rotating nearby. To me the scene looks perfectly symmetrical and I can not see where the precession is coming from.

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The flow generated by the large rotating ball will have a non-vanishing vorticity more or less because the farther out you go the slower the water will be dragged around by said ball. The vorticity will cause the smaller ball to start precessing.

This is (loosely!) analogous to what happens with frame dragging due to gravitomagnetic fields in stationary, axisymmetric space-times. The rotating central mass generates a gravitomagnetic vector potential and this potential has a non-vanishing curl; more specifically, the potential comes from the Killing vector field that generates time translations and this vector field has a non-vanishing vorticity.

You can again loosely think of this vorticity as coming from the fact that relative to the central mass, or relative to the distant stars, the space-time is locally "dragged" into orbital rotation around the mass at a decreasing rate as we move away from it. As a result a gyroscope at rest relative to the distant stars will start precessing relative to them.

If you want to gain more intuition for this I would recommend checking out chapter 3 of "Black Holes: The Membrane Paradigm" by Thorne et al.

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