# Is spacetime flat inside a rotating hollow sphere in general relativity?

Newton himself proved the Shell theorem, stating that inside a hollow sphere there is no gravitational force on a point mass. This theorem relies on the fact that Newtonian gravity falls off like $$1/r^2$$ . So it is not trivial that this theorem also holds in general relativity there it is known as Birkhoffs theorem. This theorem predicts that inside a hollow sphere the gravitational field is zero meaning the spacetime is flat (see Is spacetime flat inside a spherical shell?).

However I recently had a hunch that this might not be true inside a rotating hollow sphere where we could expect something like the Lense-Thirring effect.

So is spacetime flat also inside a rotating hollow sphere according to general relativity? Or will it be dragged around by the rotating mass of the hollow sphere?

is spacetime flat also inside a rotating hollow sphere according to general relativity? Or will it be dragged around by the rotating mass of the hollow sphere?

These are not necessarily mutually exclusive options. Remember, that uniform gravitational field in Newtonian theory ($$\mathbf{g}=\mathrm{const}$$) is equivalent to non-inertial linearly accelerating frame. But in general relativity such situation corresponds to a flat (at least locally) spacetime. Only when the Newtonian gravitational field spatially varies ($$\nabla \mathbf{g}\ne 0$$) can we say that relativistic spacetime is curved at that point.

Similarly, rotating non-inertial frame in Newtonian theory could be described by a uniform Coriolis field. And in general relativity such uniform Coriolis field (a.k.a. gravitomagnetic field) corresponds to a flat spacetime while non-uniform gravitomagnetic/Coriolis fields indicate spacetime curvature.

So, if we consider the gravitational field of a thin rotating spherical shell in general relativity then, in the weak field approximation, spacetime inside the shell is flat, but characterized by nonzero constant value of gravitational potential (which means that time goes slower for an observer inside than a static observer far away from the shell) and a constant Coriolis (gravitomagnetic) field. This last means that an observer inside with zero angular momentum (usually abbreviated as ZAMO in literature) would be rotating with nonzero angular velocity relative to asymptotic observers outside the shell. Such effect serves as a good illustration of relativity of rotation in discussions of Mach's principle.

If we try to move beyond the weak field approximation we would run into some ambiguities just from trying to define what is “rotating spherical shell” means in general relativity. Note, that such ambiguities are already present in special relativity (see e.g. Ehrenfest paradox), so a choice must be made, do we try to have the geometry of rotating shell as close to spherical as possible or do we try to maintain the flat space inside the shell? For the first option the spacetime inside the shell would gain some curvature in addition to purely flat space frame-dragging, but that curvature would be a higher order effect (third, in powers of angular velocity). For the second option the spacetime inside the shell would be perfectly flat (yet rotating relative to asymptotic observers) but the shell itself would deviate from spherical geometry and would possess differential rotation (points on the shell with different latitudes would rotate with different angular velocities). For technical details see this paper and links therein.

• +1 but not clear to me if, in your opinion, there is Lense-Thirring effect inside the rigidly rotating shell, as asked by the OP. (Note: it seems to me that there is no problem having such a shell rotating rigidly, see e.g Sec. 3.4.3 in arxiv.org/abs/1003.5015). Commented Feb 16, 2023 at 10:36
• @Quillo: if, in your opinion, there is Lense-Thirring effect inside This largely depends on how one defines LT effect. Gyroscope inside the shell would be precessing if viewed by an asymptotic observer, however this precession could not be observed by an experimenter inside without access to outside references. there is no problem having such a shell rotating rigidly I did not say this is a “problem”. But rigidly relativistically rotating shell would have curved interior. Commented Feb 16, 2023 at 12:26
• I agree, let me reformulate: the spacetime of a rotating shell is "circular", meaning that there are 2 killing fields (a time-like and an azimuthal one wrapping around the rotation axis). Now, the metric can be decomposed in 1+1+2, where 1+1 are timelike+azimuthal directions defined by the Killing fields. Inside the spehre, is the restriction of the metric to the 1+1 slices diagonal or necessarily non-diagonal in such decomposition? This does not depend on gyroscopes. Commented Feb 16, 2023 at 13:00
• is the restriction of the metric to the 1+1 slices diagonal or necessarily non-diagonal I am unsure what would that accomplish? If we decompose w.r.t. global KVF then non-diagonal, of course, but this is precisely what I mean by access to outside references: observer inside must be told which of its local KVF are global. Without that information they would find 10 local KVF and would be free to choose whatever. Commented Feb 16, 2023 at 14:43
• That accomplishes that there is frame dragging inside the rotating shell, no matter which KVF you choose (the fact that there is frame dragging outside is known since the works of Hartle if I remember correctly). Commented Feb 16, 2023 at 16:38