Wednesday, November 2, 2011

Precession & Relativity

Experiment Update: The design of the new parts has been finalized down to the screws. P.H. has taken the lead to get this done. Vielen danke P.! Das ist wunderbar! Here's one small part of the new designs. I expect assembly in 2 weeks.

Motor Harness
 Prof. James Hartle makes clear in his book "Gravity: An Introduction to Einstein's Relativity" that gyroscopes experience precession in curved spacetime. The curvature maybe caused either due to the sheer presence of the mass (in which case the gyroscope is described as being in de Sitter or geodetic precession) or due to the rotation of mass. Since it would take a whole lot of mass to curve spacetime even slightly, we can safely discard de Sitter precession as having much to do with the behavior of the rel.machine prototype in experiment 4.60. The situation is less clear when we look at Lense-Thirring precession.

Just as in Lense-Thirring precession, in experiment 4.60 also, the spinning wheels' axes are shifting their Azimuth.

In just the way a lense-thirring precession effected spinning gyro orbits the spinning earth, each flywheel is orbiting the central axis of the machine in such a way that each flywheel interprets the other wheel as rotating about it. Could it be that this thereby causes each flywheel to experience Lense-Thirring style precession due to its orbiting of the other spinning flywheel.

The gravity probe B experiments have now shown that the cyrogenic precision gyroscopes in orbit around the Earth in a satellite experience about 0.037 degrees per year of precession due to this effect.

Is it possible that the spacetime curvature produced by a large slowly rotating body is comparable in some way to the spacetime curvature produced by a small fast rotating body? Could the earth serve as a good model for the former case and a spinning flywheel serve as a good model for the latter?

[Afteral, the rotation velocity of a point near the earth's surface (due to the rotation of the earth about its own axis) is 32792.16 cm/second. The velocity of a point on the flywheel's surface at around the experimental speed is around 4149 cm/second. That is to say the flywheels when they're spinning at the speeds at which experiment 4.60 was conducted, have 13% of the velocity and experience a centrifugal force about 9%, of that experienced by them as consequence of being situated on the earth's surface as opposed to free, empty space].

If so then, is it possible that the latter could cancel the former? Could it also be that the Lense-Thirring style precession also has a flip side to it, i.e, that as the gyro of the gravity probe B precesses (due to the rotation of the earth), the axes of the cryogenic gyros in the satellite also cause the combined earth-satellite frame to translate in space?

Only, since the earth is so vey massive this effect upon the earth from the gyros as they hurtle through space, is so tiny as to be not measurable by any instruments available. In the case of the rel.machine, given that the frame of the machine has about the same range of mass as the spinning wheels, is it possible then, that the frame experiences this translational momentum more strongly than the earth does due to the gravity probe B gyroscopes? Could it be that we could safely ignore such effects in the gravity probe B, but that we cannot ignore it in the case of the rel.machine? If so, could it be that we can exploit this precessive inertial motion of the flywheels to do useful work?

From this point of view, experiment 4.60 is but the tip of what can be achieved by the arrangement. The two flywheels are in Lense-Thirring style precession around one another.This is accompanied by a net angular momentum that would, if it were free to, take the flywheel system along a path radially away from the local gravity source.  Each flywheel is executing a spiral path upward. And that is why we see the flywheels coming in as they do alternately in experiment 4.60. It is the circular movement in the horizontal plane of a spiral trajectory. The vertical ascent part of the movement is experienced by the main frame frame and cage holding the flywheel because any effort on the part of the flywheels to rise will be transmitted to the main frame and cage due to their mechanical attachment along the z-axis. However, since the amount of lift induced in the arrangement is still relatively weak, we see the entire frame rising but unable to do more than move the center of mass to a slightly higher level but not taking off the ground. It seems we still need to amplify the lift considerably if we wish to see the frame lift clear off the ground.Right now, we have a very flat spiral.

de Sitter or Geodetic Precession

The following is quoted from James Hartle "Gravity: An Introduction to Einstein's General Relativity"

Begin Exceprt
First consider the behavior of a gyroscope in orbit around a nonrotating spherical body of mass M. For simplicity let's consider a circular orbit in the equitorial plane. An observer riding with the gyro will see its spin precess in the equitorial plane. In the observer's frame, where the gyro is at rest, the spin has only spatial components, its magnitude is constant, and the symmetry under reflections in the equitorial plane shows that it remains in the equitorial plane if it started in it. Thus limited, precession in the plane is all the gyro can do.
Suppose at the start of an orbit the observer orients the gyro in a direction in the equitorial plane (say in the direction of a distant star). General relativity predicts that on completion of an orbit, the gyro will generally point in a different direction making an angle delta phi (geodesic) with the starting one. That change in direction is called geodetic precession.

A gyroscope in orbit about a spherically symmetric, nonrotating body with an orbital velocity small compared to the speed of light. In this spacetime diagram, time points upward and space in horizontal. The scale of time has been made about a factor of five smaller than the scale of space to get the diagram to fit on the page. The tube is the world sheet of the surface of the body about which the world line of the gyro twists. The spin s is perpendicular to the four-velocity of the gyro u, although that relationship is not so evident with the reduced scale of time. The spin remains fixed in a local inertial frame falling with the gyro but precesses with respect to infinity because of the curvature of spacetime produced by the body. This is called geodetic precession.
End Excerpt

Now, since geodetic precession is a product of mass and we do not have significant enough mass to cause relativistic spacetime curvature, we may safely discard geodetic precession as having a major part to play in the theoretical construct we seek to explain experiment 4.60 and to also design a better prototype that might exploit the behavior we have seen to perhaps do useful work.

Lense-Thirring Precession refers to the precession experienced by the spin axis of a gyroscope in orbit (i.e. physically bound to the object whose spin is causing the test gyroscope to spin, but still having a certain degree of freedom atleast to alter its azimuth say, wrt a distant star) around a ROTATING massive object (the Earth for example). I have capitalized the word 'rotating' to emphasize that it is not the mass of the earth that directly produces Lense-Thirring precession of the spin axis of the orbiting gyro (that effect is discussed above under the topic of geodetic precession), but rather it is the rotaton of the massive object about which the test gyroscope happens to be in orbit.

General Relativity predicts that this rotation of the mass of the earth also causes a curvature of spacetime around it. That curvature is responsible for the precession of the spin axis of the test gyro in addition to the geodetic precession that the earth's mass will produce in the spin axis of the test gyroscope. (The total curvature being a sum of the two.) The following is quoted from James Hartle "Gravity: An Introduction to Einstein's General Relativity"

Begin Excerpt
Gyroscopes in the Spacetime of a Slowly Rotating Body To illustrate how the effects of rotation on the geometry of spacetime can be studied with gyroscopes, we consider the thought experiment shown schematically in figure 14.3. A laboratory carrying a gyroscope falls freely down the rotation axis of the slowly rotating Earth. Initially the spin axis of the gyro is oriented perpendicular to the rotation axis pointing in an azimuthal direction phi.
Were the Earth not rotating, the guro's spin axis would remain fixed as it falls- always point along the same azimuthal angle phi. This can be verified by solving the gyroscope equation (14.6) but it follows mor eimmediately from the symmetry of the Shwarzschild metric under phi -> -phi. The gyso could not precess without breaking this symmetry. The geodesit precession is. therefore, zero for this orbit. But the rotation of the Earth breaks this symmetry and the gyrscope precesses with time, as we now calculate. The precession of the gyro on its downward plunge is determined by the gyroscope equation (14.6) in the metric (14.22) because it is following a geodesic. We expect the rate of precessionto be small for the Earth.
End Excerpt
The following are important relavant illustrations regarding Lense-Thirring Precession.

Note: The Gravity Probe B experiment was launched shortly after this edition of the book was published. lists the results of the experiments. Final results of the GP-B experiment were announced at NASA HQ in Washington DC on 4 May 2011. The experimental results are in agreement with Einstein's theoretical predictions of the geodetic effect (0.28% margin of error) and the frame-dragging effect (19% margin of error).

Thus while frame dragging has been confirmed, its lower value (~37.2) as against its prediction (~.39.2) leaves open the question of what happened to the remaining energy? Could it just be that that energy was infact transformed into translational energy of the earth-gyro frame about their solar orbit?

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