Whoosh - linear frame dragging papers?

[pervect]

I'm wondering if there are any papers that consider the effect of a rapidly moving mass on a spinning gyroscope.

The basic idea is you have some gyroscope, pointing at a distant "guide star", a "whoosh" as a massive relativistic object moves by, and then you look to see if your gyroscope is still pointing at said guide star.

I intuitively expect the gyroscope to be deflected through some angle, at least for some orientations of the gyroscopes, as measured by the distant guide star. But I don't have any detailed calculation of this, and it's tricky enough that I'd like to see a paper addressing it.

I imagine one could also consider the massive object to be stationary, and start fermi-walker transporting the spin axis along the orbit...but I'd still rather read about it than try to attempt it from scratch...

 

[bcrowell]

I imagine one could also consider the massive object to be stationary, and start fermi-walker transporting the spin axis along the orbit...but I'd still rather read about it than try to attempt it from scratch...

In that frame, the gyroscope is an ultrarelativistic particle, so doesn't it basically act just like a light ray being deflected in a Schwarzschild metric? I would think that in the case where the gyro's initial spin was parallel to its direction of motion, the resulting twist in the direction of its spin would be exactly equal to the well known expression for the deflection angle of a ray of light as it passes by the sun. I guess if you wanted the answer in the gyro's rest frame, you'd have to transform the angle into that frame. I don't know if there's any similarly easy way to get the result in the case where the spin wasn't initially longitudinal.

Does the usual idea of splitting the effect into geodetic and frame-dragging really apply in this situation?

 

[Jonathan Scott]

I'm wondering if there are any papers that consider the effect of a rapidly moving mass on a spinning gyroscope.

The basic idea is you have some gyroscope, pointing at a distant "guide star", a "whoosh" as a massive relativistic object moves by, and then you look to see if your gyroscope is still pointing at said guide star.

I intuitively expect the gyroscope to be deflected through some angle, at least for some orientations of the gyroscopes, as measured by the distant guide star. But I don't have any detailed calculation of this, and it's tricky enough that I'd like to see a paper addressing it.

I imagine one could also consider the massive object to be stationary, and start fermi-walker transporting the spin axis along the orbit...but I'd still rather read about it than try to attempt it from scratch...

This is just another form of the Lense-Thirring effect, which counts as rotational frame-dragging. I expect Ciufolini and Wheeler "Gravitation and Inertia" contains the basic information needed to calculate this, but it's the simplest case of gravitomagnetism, analogous to the magnetic field of a moving charge. As the gravitomagnetic field is equivalent to an angular velocity, I expect the overall effect is the time integral of that angular velocity for the whole encounter. However, I wouldn't care to calculate as in gravitomagnetism I usually get signs wrong and gain or lose factors of 2.

The term "linear frame-dragging" usually refers to the theoretical frame-dragging effect of linear acceleration of nearby objects, requiring a tiny additional force to keep the test object at rest relative to the distant stars. This effect is predicted by GR but in practice this is orders of magnitude too small to measure because it is swamped by the usual gravitational field. It is however very important in Machian theories of inertia.

 

[Jonathan Scott]

In my old notes, I've found something presumably based on a textbook (I don't know which one) which says that an object with mass $m$ and speed $v$ moving tangentially at distance $r$ induces a gravitomagnetic field $4Gmv/{r^2 c^2}$ which rotates the frame of a test object with an angular velocity of half of that, $2Gmv/{r^2 c^2}$.

At the time (early 1997) I noticed a curiosity that if you consider a gyroscope in a circular orbit about the sun, then the "Fokker - De Sitter" precession rate is $(3/2) Gmv/{r^2 c^2}$, made up of the Thomas precession plus the effect of space curvature, but if instead you consider the situation from the gyroscope's point of view as a frame-dragging effect due to the sun going round the gyroscope then the rotation rate is $2Gmv/{r^2 c^2}$, which is 4/3 as much, and I wondered why these were not equal.

I got so puzzled by it that eventually I sent an e-mail to Clifford Will, who very kindly explained that in the case described from the gyroscope's point of view, there is an additional precession effect of -1/2 affecting the apparent direction of the fixed stars, so that the precession relative to the stars as seen in the gyro's rest frame is 3/2 overall, matching the other result.