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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?

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  • $\begingroup$ Note that in the electromagnetic analogue, a rotating charged shell would not induce a torque on a charged object inside. So I wouldn't expect a rotation at the linearized gravity level either (using gravitomagnetism as an analogy.) There might be non-linear effects, though. $\endgroup$
    – Michael Seifert
    May 31, 2022 at 19:50
  • $\begingroup$ @MichaelSeifert Is frame-dragging non-linear? $\endgroup$
    – MatterGauge
    May 31, 2022 at 21:14

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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).

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  • $\begingroup$ Great! "Einsteinischen Gravitation", corrected by W. Pauli! An historical jewel! How did you find that? $\endgroup$
    – MatterGauge
    May 31, 2022 at 23:30
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    $\begingroup$ In this particular case I found the reference in Pfister's paper and the second result when googling the title is the link above from the Österreichische Zentralbibliothek für Physik which incidentally is exactly the version Pfister mentions. Usually it is very hard to find old originals like this one digitally. So in this case we have to thank an Austrian digitalisation team for doing a good job and having a proper and open web portal. $\endgroup$
    – N0va
    Jun 1, 2022 at 1:10
  • $\begingroup$ If I'm not mistaken, the first term wouldn't apply a torque to a mass at rest ($\vec{v}=0$). I'm not immediately sure whether the second one would, though. $\endgroup$
    – Michael Seifert
    Jun 1, 2022 at 12:20
  • $\begingroup$ Going through the vector identities, the second term applies a specific torque of $3 (\vec{\omega}\cdot \vec{r}) (\vec{r} \times \vec{\omega})$. If you set up an integral of this quantity over a solid body, you can show that it must vanish for any body with a rotational symmetry axis parallel to $\vec{\omega}$. So at this level of approximation, a small spherical fluid drop at rest at the center of the sphere would not start rotating. $\endgroup$
    – Michael Seifert
    Jun 1, 2022 at 13:32
  • $\begingroup$ @MichaelSeifert So the drop needs "a handle" to pull on? $\endgroup$
    – MatterGauge
    Jun 2, 2022 at 18:18

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