# Can rotation be distinguished from frame-dragging?

I am trapped inside the usual elevator and experiencing a comfortable gravitational field of about 1 g. Or perhaps it's an acceleration of 1 g. I cannot tell which.
However, I notice when I drop an object from the ceiling it does not quite drop vertically. Its path bends slightly - always by the same amount in the same direction.
I am thinking that perhaps the elevator is actually sitting inside a very large centrifuge and what I am seeing is the Coriolis effect. Or perhaps the elevator is actually stationary but suspended on a very long cable above the equator of a very large rapidly-rotating mass and I am seeing the effect of frame-dragging.
Or perhaps my imagination is simply too vivid and, in fact, the elevator is still sitting at the ground floor in my apartment building in Singapore while the Earth rotates.
What measurements can I make inside the elevator to decide which of these scenarios is correct?

• What makes you think that those phenomena have the same effect? – Wolphram jonny Nov 21 '20 at 8:23
• @ Wolphram jonny Good question. I think I can set all three scenarios up so that the 'field' in the elevator is about 1 g, either from a GM/r^2 effect or w^2r effect. Then, when I look at this reference en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precession I see two indistinguishable versions for the Coriolis effect. I'm not sure what other local properties of spacetime to check? – Roger Wood Nov 21 '20 at 18:51

In principle, you could measure how homogeneous are the acceleration and the Coriolis field inside the elevator. If experiments are precise enough to determine how these fields vary within the laboratory/elevator one could then distinguish between the alternatives presented.

For a reference that provides pedagogical discussion of both Coriolis effect in a rotating frame and frame dragging effect around rotating planet I would recommend the paper:

We take the approach and all of the equations below from that paper.

Let us assume that the elevator is nonrelativistic, and use the Newton–Cartan theory for a unified treatment of gravity and inertia. While NC is often considered a formalization of pure Newtonian gravity, its descriptive power is somewhat greater and it can also be used to describe the effects of frame-dragging by weakly relativistic rotating bodies. Also note, that NC theory is usually formulated in terms of Cartan connection, but one could also interpret this connection in a rigid frame in terms of acceleration and Coriolis fields.

In an inertial frame $$S$$ the gravitational field is characterized by the free fall acceleration field $$\bf g$$. Denoting the noninertial frame of the elevator as $$S'$$ we can write the equation of motion in $$S'$$ for a point mass: $$\begin{equation} m\ddot{{\bf r}}=m{\bf G}+m\dot{{\bf r}}\times{\bf H}+{\bf F}\ \end{equation}$$ where $${\bf F}=R^{-1}{\bf f}$$ is the nongravitational force expressed in $$S'$$, and we have defined the acceleration field $$\begin{equation} {\bf G}=R^{-1}({\bf g}-\ddot{{\bf x}}_{0})-{\bf \Omega}\times({\bf \Omega}\times{\bf r})-\dot{{\bf \Omega}}\times{\bf r}\ ,\label{G} \end{equation}$$ describing velocity independent part of acceleration due to combined gravity and inertia and the Coriolis field $$\begin{equation} {\bf H}=2{\bf \Omega}\ .\label{H} \end{equation}$$

Here $$R=R(t)$$ is the rotation matrix describing the change of orientation between axes in $$S$$ and $$S'$$ and $${\bf x_0}(t)$$ is the position of origin of $$S'$$ in $$S$$, and $$\bf \Omega$$ is the instantaneous angular velocity of rotation of $$S'$$.

An experimenter in the elevator could measure the value of acceleration field at a given point using e.g. a pendulum: its orientation gives the direction of $$\bf G$$ while its frequency provides its magnitude.

To measure the Coriolis field $$\bf H$$ one could use the torque from Coriolis force on a gyroscope. If the gyroscope angular momentum is $$\bf L$$ the it evolves according to the equation: $$\frac{d \mathbf{L}}{dt}= - \frac12 {\bf H }\times{\bf L},$$ that is, it precesses with angular velocity $$-\mathbf{H}/2$$.

In order to distinguish between possibilities outlined by OP (elevator in a centrifuge in space , elevator held stationary above rapidly rotating planet, elevator slowly rotating together with the surface of the planet) one needs to determine not only the values of $$\bf G$$ and $$\bf H$$ at some specific point in the elevator, but how these fields vary within the elevator. Let us write the expressions for different possibilites:

First possibility, rotating frame, no gravity, angular velocity oriented along $$z$$-axis: $$\begin{eqnarray} {\bf G}&=&\Omega^{2}r\,{\bf e}_{r}\\ {\bf H}&=&2\,\Omega\,{\bf e}_{z}\, . \end{eqnarray}$$

Second possibility, gravitational and frame-dragging fields near a rotating spherical planet as seen from a static reference frame is a combination of gravitostatic acceleration of a single mass and gravitomagnetic dipole Coriolis field: $$\begin{eqnarray} \mathbf{G}&=&-\frac{GM\mathbf{r}}{r^{3}}\\ {\bf H}&=&\frac{2G}{c^{2}}\left(\frac{{\bf J}}{r^{3}}-\frac{3({\bf J}\cdot{\bf r}){\bf r}}{r^{5}}\right)\label{eq:H-Earth}, \end{eqnarray}$$ where $$M$$ and $$\mathbf{J}$$ are mass and angular momentum of the planet and $$c$$ is the speed of light. The $$c^{-2}$$ coefficient in front of Coriolis field indicates that we are dealing with the first post-Newtonian (gravitomagnetic) approximation to the full GR expression. Note, that this Coriolis field is position-dependent, unlike the pure rotation.

Finally, the third possibility slowly rotating spherical planet as seen from the frame, co-rotating with the planet would be the combination of the first and second cases (while possibly, the slowness of rotation means that one could drop the frame dragging terms).

Since these expressions are analytic, finding enough derivatives of $$\bf G$$ and $$\bf H$$ at the position of the elevator with sufficient precision would in principle allow one to distinguish between possibilities and even calculate parameters such as planet's mass and its rotation.

Of course, this is more of a theoretical possibility, since in practice the fields of a planet vary only on the scales comparable to its size, while the size of the elevator (presumably only a few meters) is too small to register noticeable variations. On the other hand, if the centrifuge is of a reasonable size one could at least either confirm or exclude this possibility with ease.

• @ A.V.S. Super answer, thanks! If can summarize: <br> 1) if it's a centrifuge, the g field will drop linearly from the floor to the ceiling of the elevator and the Coriolis effect will be independent of position. <br> 2) If it's a nearby rotating mass, the g field will also drop but will not be exacly linear and will droop slightly half way up. The Coriolis effect will also diminish going from floor to ceiling. <br> As you say, given the size of the average elevator compared to the average planet, some of these measurements might be a tad tricky. – Roger Wood Nov 24 '20 at 6:33