A frame dragging experiment in an accelerated reference frame

Redhat

Hi,

I was thinking about an experiment that might demonstrate frame dragging via the equivalence principle.

The apparatus consists of a rotating massive cylindrical shell within a vacuum. Near the inside and the outside of the shell exist two gyroscopes with their axis both perpendicular to the axis and to the radius of the rotating shell.

GR predicts that each gyroscope should rotate slightly in opposite directions around an axis parallel to the rotating shell. However, the equivalence principle seems to imply that space-time is more curved on the inside of the shell than on the outside. So it would seem that the inner gyroscope would rotate at a greater angle than the outside one. The difference in rotation should be due to the effect of frame dragging within the accelerated reference frame.

Does this make sense?

Redhat

pmb_phy

GR does not imply that the spacetime inside the cylinder is zero. Your experiment only implies that the gravitational field inside the cylinder is non-zero.

Pete

pervect

Yes.

I find it convenient to approach the topic in terms of the "gravitomagnetic field".

http://en.wikipedia.org/wiki/Gravitoelectromagnetism

Much like the magnetic field in a solenoid, the gravitomagnetic field points along the axis of the rotating cylinder . However, the gravitomagnetic field points in different directions inside and outside the cylinder "up" vs "down".

This gravitomagnetic field provides a torque on a rotating gyroscope. There is a velocity x B_g force, much like there is a Lorentz force on a charge. These forces produce a net torque on the rotating disk.

Note that the gyroscopes are assumed to be stationary in the coordinate system in which the cylinder is rotating.

I'm sorry, I can't make any sense out of this last part at all. "More curved" is so loose, I'm not sure what you mean (greater Ricci scalar?). Why would "more curved" imply greater rotation? What reference frame here is "accelerating" anyway? We might even not be talking about quite the same problem in spite of your care in describing it :-(.

In the situation I'm describing, the inner and outer gyroscopes are stationary in a coordinate system in which the cylinder is rotating and the "fixed stars" are not rotating.

The inner gyroscpe's center of mass is following a geodesic path, but the outer gyroscope, because of the gravitational attraction of the cylinder, is not following a geodesic path.

Redhat

Sorry for being unclear. First let me describe the apparatus better. I kind of rushed through it before.

Consider a very massive cylinder rotating very fast within a stationary balanced cage containing freely rotating gyroscopes positioned very close to the inside and the outside of the moving cylinder. The gyroscopes’ spin axes are (initially) both parallel to a line on the plane perpendicular to the cylinder’s axis. The setup is in space so that the gyroscopes are essentially in free fall around the rotating cylinder.

When I say space-time at the inner surface of the cylinder is “more curved” than the outside, I am (probably using the term too loosely but) referring to the idea that a region of space-time with a higher gravitational field is more curved than a region with a lower gravitational field (assuming the geometries of both regions are equal).

The Equivalence Principle says that “all accelerated reference frames possess a gravitational field” (http://en.wikipedia.org/wiki/Gravita...time_dilation). Obviously a test mass weighs more at the inside of the rotating cylinder than on the outside. I would also propose that a photon emitted from the surface of the cylinder would be red-shifted on the inside of the cylinder and blue-shifted on the outside. Therefore one could say that the gravitational field on the inside of the cylinder is greater than on the outside – thus “space-time is more curved on the inside”.

BTW Thanks pervect! Your explanations were most helpful. The GEM equations are very interesting. It seems the one of interest in this case is the analog of Ampere’s law. It describes the curl of the gravitomagnetic field and how it changes sign (direction) from the inside to the outside of the cylinder and so explains why the gyroscopes would precess in opposite directions.

However if I might be so bold as to conjecture that that equation might be incomplete as there is no analog to the Equivalence Principle within electromagnetism. (As far as I know an accelerated reference frame cannot produce an electric field).

Just as the dE/dt term was a later addition to Ampere’s Law, perhaps in the GEM equation there should be a term accounting for the gravitational effects of an accelerated reference frame. In the case of my experiment, the EP effects would be asymmetric as one went from the inside to the outside of the rotating cylinder and “should” therefore produce an asymmetric precession of the gyroscopes. ??

This assumes that the gyroscopes in a non-accelerating reference frame can “feel” the effects of the accelerating reference frame of the cylinder. (But maybe this assumption is incorrect).

Redhat

pervect

This isn't really the right way to describe curvature. What is proportional to the curvature tensor in GR is not "gravitational force" as it is usually thought of, but "tidal gravitational force".

Curvature is related to gravity, but it's a mistake to think of them as being synonyms.

What you are calling the gravitational field is just the 4-accleleration of a particle that is following some particular curve. In this case, the curve is just a curve of constant coordinates.

This is also sometimes called the "connection", or the "Christoffel symbols".

This is, IMO anyay, what most people mean when they say "gravitational field". The term is rather vague, so some other people mean different things when they use exactly the same words.

Anyway, the sort of gravity you are talking about isn't directly realted to curvature in the sense of being a synonym.

An accelerated reference frame is just a change in viewpoint, a change in POV. Mathematically, the elemnt in GR which is closest to your idea of gravity are the Christoffel symbols. Because these symbols depend on the POV of the observer, though, they are not tensors. Tensors are reserved to those quantites that don't depened on the POV of any observer, they can be regarded philosphically as being independent of any observer. (They can also be regarded a bit like an object in an object-oriented programming language, in that they contain all the information needed to transform themselves to give their appearance according to any arbitrary observer).

Anyway, because they are not tensors, the Christoffel symbols aren't really a "field". There is basically no way that one is going to take things that are dependent on one's POV and make them not-dependent on one's POV. This is why the gravitational field isn't really a "field" in the usual sense, because to be a field in the usual sense, an object has to be representable by a tensor.

MeJennifer

So the true mechanics of our warped 4D space-time reality of which we can only see Lorentz transformed instances?