Gravitational Effects of Gyros Near a Massive Object

[Barry Srase]

Given an object spinning on its own axis, in orbit around a planet with mass and that the object is traveling in the direction of its axis, does the axis continue to point at the same point in infinity as it rotates around the planet or does the axis follow the curvature of space around the planet and thus appear to a distant observer to rotate through 360 degrees with each rotation around the planet?

 

[Dale]

does the axis continue to point at the same point in infinity as it rotates around the planet or does the axis follow the curvature of space around the planet and thus appear to a distant observer to rotate through 360 degrees with each rotation around the planet?

Actually, neither. The gyro does not continue to point at the same point, but it precesses much less than 360 degrees.

This effect is known as "frame dragging" and is one of the predictions of general relativity. I believe that Gravity Probe B confirmed it to within 20%, which is not a fantastic error margin, but was the first measurement of its kind. Essentially, they did exactly the experiment you described.

 

[mfb]

For Gravity Probe B, the axis of rotation changed by about 6.6 arcseconds per year due to the gravity well and 0.04 arcseconds per year due to the rotation of earth. Tiny effects, but the satellite was good enough to measure them.

 

[Barry Srase]

Thanks for your replies, very helpful. I think I have some more learning to do!

 

[Matterwave]

Actually, neither. The gyro does not continue to point at the same point, but it precesses much less than 360 degrees.

This effect is known as "frame dragging" and is one of the predictions of general relativity. I believe that Gravity Probe B confirmed it to within 20%, which is not a fantastic error margin, but was the first measurement of its kind. Essentially, they did exactly the experiment you described.

Unless you are very near a Kerr black hole, I think the frame dragging effect will be very much suppressed in comparison to the Geodetic effect and Thomas precession. In addition, the OP did not specify that that central mass is rotating, which is a requirement for frame dragging to occur.

 

[Dale]

the OP did not specify that that central mass is rotating, which is a requirement for frame dragging to occur.

Oops, you are right. I somehow assumed the rotation of the massive body was part of the question although it was not there.

 

[WannabeNewton]

...and that the object is traveling in the direction of its axis...

There is only one radius ($r = 3M$) in Schwarzschild space-time (space-time outside a non-rotating spherical mass) at which a gyroscope can travel in the direction of its axis and still be in circular orbit around the central mass so perhaps you didn't mean to put such a constraint on the gyroscope's motion.

 

[WannabeNewton]

In addition, the OP did not specify that that central mass is rotating, which is a requirement for frame dragging to occur.

These are not all separate effects and frame dragging does not require a rotating mass in order to occur. Frame dragging includes a variety of effects one of which is the precession of a gyroscope stationary in a coordinate system; geodetic precession in one coordinate system, say for a circular orbit, can easily be converted into frame dragging in the corotating coordinates.

 

[Matterwave]

These are not all separate effects and frame dragging does not require a rotating mass in order to occur. Frame dragging includes a variety of effects one of which is the precession of a gyroscope stationary in a coordinate system; geodetic precession in one coordinate system, say for a circular orbit, can easily be converted into frame dragging in the corotating coordinates.

I was basing my comments off of Gravity probe B's results: "...analysis of the data from all four gyroscopes results in a geodetic drift rate of −6,601.8±18.3 milliarcsecond/year (mas/yr) and a frame-dragging drift rate of −37.2±7.2 mas/yr, to be compared with the general relativity predictions of −6,606.1 mas/yr and −39.2 mas/yr, respectively..." http://arxiv.org/pdf/1105.3456v1.pdf

 

[A.T.]

...frame dragging does not require a rotating mass in order to occur...

...geodetic precession in one coordinate system, say for a circular orbit, can easily be converted into frame dragging in the corotating coordinates.

Isn't the mass rotating in the co-rotating coordinates of the orbiting object?

 

[WannabeNewton]

Isn't the mass rotating in the co-rotating coordinates of the orbiting object?

It certainly orbits the gyroscope but the sense in which matterwave and DaleSpam were using the term "rotation" is that of a spin angular momentum of the central mass, which is coordinate independent.

 

[Dale]

Yes, that is what I was thinking of in my reply, although the OP did not specify spin angular momentum of the central mass, so my comments were a little off topic.

 

[A.T.]

It certainly orbits the gyroscope but the sense in which matterwave and DaleSpam were using the term "rotation" is that of a spin angular momentum of the central mass, which is coordinate independent.

I assume they (and the authors of the linked paper) use the term "frame dragging" in the sense of an effect of that coordinate independent rotation, not an effect introduced by transforming into rotating coordinates, like you did in post #8. Is that correct?

 

[WannabeNewton]

I assume they (and the authors of the linked paper) use the term "frame dragging" in the sense of an effect of that coordinate independent rotation, not an effect introduced by transforming into rotating coordinates, like you did in post #8. Is that correct?

You can't assume one or the other, "frame dragging" applies to both effects, there's no picking and choosing what the term means. The paper simply says there is a frame dragging effect due to the Earth's rotation, it never says that's the only kind of frame dragging effect possible so saying there is no frame dragging if there is no rotation of the central mass is not an accurate statement.

 

[A.T.]

The paper simply says there is a frame dragging effect due to the Earth's rotation,

The paper distinguishes between "geodetic drift" and "frame dragging". In post #8 you seemed to suggest that you cannot separate the two effects. So how did they do it in that paper?

...saying there is no frame dragging if there is no rotation of the central mass is not an accurate statement.

If the mass is not rotating (in a frame independent sense) is there "frame dragging" in the rest frame of that mass (In the sense the paper uses the term "frame dragging" and in the sense you use it, if there is a difference)?