Gravitomagnetic Effect of Moving Source on Test Object

[Jonathan Scott]

Consider a gravitational field caused by a central source object and a gyroscope in orbit around that source. It is well-known that the gyroscope experiences rotational frame-dragging at a rate 3/2 times the cross product of the velocity and the field, which is known as the de Sitter precession or geodetic precession. It is also well-known that when a moving source passes a static gyroscope the gyroscope experiences rotational frame-dragging, but in this case the factor is 2 times rather than 3/2 times. Some time long ago I asked Clifford Will why the two effects were not equal, as it seemed that one could look at the orbit situation from the gyroscope point of view with the source moving past it. He very kindly pointed out that if the test object is in free fall and accelerating in a changing direction, there is an additional effect with a factor of -1/2 (for Thomas precession).

This effect is shown even more clearly in Ciufolini and Wheeler Gravitation and Inertia equation 3.4.38', giving an extended version of the de Sitter precession of which this is the relevant subset (assuming the appropriate factors specifically for GR):

$$ \dot\Omega = - \frac{1}{2} \mathbf{v} \times \frac{d \mathbf{v}}{dt} + 2 \mathbf{v} \times \nabla U $$

Here ##\mathbf{v}## is the velocity of the "gyroscope test particle" (which may be subject to non-gravitational forces as well as gravitational), the source is assumed to be at rest, and ##U## is the potential. If the particle is free to be accelerated by the field, then the acceleration is equal to ##\nabla U## and the factor of 3/2 is clear.

If we switch to a frame in which the test particle is initially at rest, I would expect the same predicted rate of frame-dragging. In that case the source is moving with velocity ##\mathbf{v}## in the opposite direction, and gives the usual factor of 2. However, in that frame the test particle is initially at rest and is accelerating towards the source, so the Thomas precession in the original form does not seem to apply.

What I'd like to know is the corresponding form for this equation for this second case, explaining where the 1/2 correction comes from to make it consistent. My guess is that for purposes of the rotational effect, the velocity here is effectively the relative velocity of the source and test particle, so the equation remains unchanged (assuming non-relativistic speeds), but clearly the explanation of the first term as Thomas precession no longer applies in that case.

I admit that in the past this should have been well within my own capabilities to calculate, assuming a weak field, the Schwarzschild solution and an isotropic coordinate system, but I never seem to have the time or focus now, so I'd appreciate it if anyone can point me to the answer, or can show me how to derive it. I've Googled a few references to gravitomagnetism without finding an answer in a form I can easily understand, and I know for certain that some of the Wikipedia stuff on the subject contains errors and inconsistencies (mostly related to factors of 2) because I spotted them and called attention to them myself some years ago.

 

[Ambitwistor]

Thank you for bringing up this interesting question about frame-dragging and precession in the presence of a central gravitational source. I am a scientist who specializes in the study of general relativity and I would be happy to provide some insight into your inquiry.

Firstly, I would like to clarify that the equation 3.4.38 from Ciufolini and Wheeler's Gravitation and Inertia is indeed the correct expression for the extended de Sitter precession, as you have stated. However, there is a slight error in your interpretation of the equation. The equation is not specific to GR, but rather it is a general expression for the precession of a test particle in a gravitational field, regardless of the theory of gravity being used. This means that it is valid in both Newtonian gravity and GR.

Now, to address your question about the difference in the factor of 3/2 and 2 in the two cases you have mentioned. The answer lies in the fact that the two cases are fundamentally different in nature. In the first case, the gyroscope is in orbit around a static central source, while in the second case, the gyroscope is at rest in a frame where the central source is moving. In the first case, the gyroscope experiences both the de Sitter precession and the Thomas precession, which together give a total precession rate of 3/2 times the cross product of the velocity and the field. In the second case, the gyroscope only experiences the de Sitter precession, as the Thomas precession does not apply since the gyroscope is at rest.

To see why this is the case, let us consider the two situations in more detail. In the first case, the gyroscope is in orbit around a static central source, and therefore it is accelerating towards the source due to the gravitational force. This acceleration is given by the gradient of the gravitational potential, which is equal to ##\nabla U## in your equation. However, due to the acceleration, the gyroscope also experiences a fictitious force known as the Coriolis force, which is proportional to the cross product of the velocity and the acceleration. This force causes the gyroscope to precess at a rate of 3/2 times the cross product of the velocity and the gravitational field.

In the second case, the gyroscope is at rest in a frame where the central source is moving. In this case,