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We are in empty space and see a spherical drop of water. Around the drop we have placed a massive shell with uniform density. The drop is positioned at the center. Then we set the shell in rotational motion (by small rockets on the side). Will the drop start rotating (slowly)? Will frame drag cause a torque?

The Newtonian idea of gravity predicts a zero gravity field inside the sphere. General relativity predicts frame dragging. The mass-energy-momentum tensor includes momentum and that's what we see in this case.

So, will it rotate? Will the shell and the droplet be eventually rotating in tandem? Of course we must stop the acceleration before a black hole develops...

Can we say the rotating sphere induces torsion?

MatterGauge
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Apparently Thirring computed this in 1918: Phys. Z. 19, 33 (1918) (in German) the central (corrected result) for the acceleration of a test particle inside a slowly rotating mass shell (of mass $M$, radius $R$, and angular momentum $\vec{\omega}$) is given by $$ \vec{a}=−2d_1(\vec{\omega} \times\vec{v} )−d_2[\vec{\omega} \times(\vec{\omega} \times\vec{r})+2(\vec{\omega} \cdot\vec{r})\vec{\omega} ], $$ with the constants $d_1 = 4MG/3Rc^2$ and $d_2 = 4MG/15Rc^2$ for the Coriolis- and centrifugal contributions respectively, according to H. Pfister (2005) On the history of the so-called Lense-Thirring effect. This expression is valid only close to the center of the sphere: $|\vec{r}|\ll R$.

A macroscopic fluid drop in the center of the mass shell should start/be differntially rotating. So yes the shell would apply torque to the droplet taking into effects of general relativity (namely the dragging of interal frames/Lense–Thirring effect).

N0va
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