Much of the known physics I quote in this entry is humbly borrowed from James B. Hartle's epic book Gravity: An Introduction to Einstein's Relativity. The speculations at the end on the other hand are solely mine. I clarify not so much as to declare zealous ownership but rather to spare the eminent professor of any misattribution of such speculations of mine, which prove not to be to his taste. I intend neither to mislead nor obfuscate. Yet progress for me is impossible unless I advance theory to suit observation and therefore I speculate. Sometimes I need a theory to foresee my future actions and therefore I speculate. At the root is the desire to explore a corner of rotational dynamics that still remains benighted in my brain.
I unreservedly recommend Hartle's book to those all amateur scientist-activists out there, even for those without any mathematical introduction to basic differential calculus provided that they have the intrepidity to grapple with the calculus aspects of some of the Jabberwockies they are sure to encounter in this doozie of a book. This is because often the real details, the real beauty of the concepts Hartle lays out are hidden in the differentials of first- and second- order differential equations involving angular velocity, angular momentum and angular acceleration.
Begin Excerpt from Gravity: An Introduction to Einstein's General Relativity by James B. Hartle
The geometry outside a non-rotating black hole or a star is given accurately by the spherically symmetric Schwarzschild geometry. However, no object in nature is exactly non rotating. The Sun for exmple, is rotating at the equator with a period of approximately 27 days. As a consequence of the resulting centripetal acceleration, the Sun is not exactly sperically symmetric but is slightly squashed along the rotation axis. But it is not very much out of round; an equitorial diameter is less than a part in a 100,000 longer than a diamter along the rotation axis. The small value of that difference is why the schwarzschild geometry is an excellent approximation to the curved spacetime geometry outside the Sun.
The curved spacetimes produced by rotating bodies have a richer and more complex structure than the Schwarzschild geometry, as discussion of rotating black holes (in the next chapter, of the book Gravity by James Hartle from which this passage is being quoted) illustrates. But there is one limiting case that is accessible. This is the case of slow rotation, when the body is rotating sufficiently slowly that only deviations from the spherically symmetric Schwarzschild metric that are first order in the angular velocity or angular momentum are of significance. Since centripetal accelerations are second-order in the angular momentum, the shape of the body is not rotationally distorted to first order. It remains spherical. Why then is there a change in the exterior geometry of spacetime? The answer is that general relativity predicts that curvature is produced, not only by the distribution of mass-energy, but also by its motion. When the curvature of spacetime is small and the velocities V of the sources are also small, these effects are typically of order V/c smaller than the GM/Rcsquare effects of the mas distribution itself. This is not unlike electromagnetism, where fields are produced not only by charge distributions but also by currents. Pursuing this analogy, these (V/c)(GM/[Rcsquare]) effects are sometimes called gravitomagnetic. (...) We explore one simple example of a gravitomagnetic effect- the dragging of inertial frames by a slowly rotating body. In this chapter the dragging is small; in the next chapter on rotating black hold it will be large.
Rotational Dragging of Intertial Frames
Consider my post of... where we noted that inertial frames of special relativity are not rotating with respect to the frame in which the distant matter in the universe is at rest. Were all the distant matter somehow to start rotating, the local inertial frames- those in which the plane of the Foucault pendulum would not precess- might be expected to rotate along with it. If only a small part of the matter in the universe is set into rotation, then the inertial frames might be dragged along slightly. General relativity predicts such rotational dragging of inertial frames.
Even the rotation of the Earth drags the inertial frames in its vicinity slightly. Dimensionally, at the surface of the Earth, the induced angular velocity of the inertial frames with respect to infinity, w, might be guessed to be related to the Earth's angular velocity omega earth by
w ~ (GMearth/[Rearth*csquare])*Omegaearth,
where Mearth and Rearth are the mass and radius of the Earth, respectively. Later in this chapter (not included in this brief excerpt- please refer to the original book by Hartle), we will confirm this estimate, which gives
w ~ .3"/yr
The inertial frames therefore rotate each year by an angle that is rougly that subtended by a football field on the Moon. Even so, at the time of writing, satellite experiments are underway to detect this small effect predicted by general relativity.
The gyroscope is a natural test body with which to observe the dragging of inertial frames because the spin of a gyro points in a fixed direction in an inertial frame. A discussion of gyroscopes in curved spacetime is therefore an appropriate place to begin the discussion of the dragging of inertial frames.
End Excerpt
Speculation: As Hartle points out, rotation produces its own curvature of spacetime. This curvature causes frame dragging. Now, theoreticians have been comfortably ignoring third order deviations, stopping with second order deviations in angular acceleration as the last distinct phenomena to study. Certainly second order deviations in angular accleration have been proven capable of producing not first order shape distortions, but rather second order shape distortions (i.e. the Sun remains more or less round, but with some deviation, or 'squashing' at the poles and some bulging at the equator.
Can it, perhaps be so that third order deviations in the prototype are capable of producing neither first order shape distortions, nor second order shape distortion, but simply third order position distortion, i.e. inertial motion? For I propose that what the prototype is doing in experiment#3 is attempting to maintain an inertial frame in the face of the disturbance imposed on the spinning wheels by the constant torque.
At present I am confronted with the dilemma of how to explain the results of the experiments I posted in my last entry. How do we explain that reversing the spin of one of the wheels cause such a radical change in the behavior of the prototype? Let me recap the results of the experiments where we impose a constant 2A torque for 30 seconds - observe the video closely, several times and you will notice the following facts about the video of the experiment:
a) When the wheels weren't spinning the wheels' z-axes remained radial w.r.t the center of the main rotation we are imposing upon the cage and the cage speeded up, theretically indefinitely but practically, the experment was suspended after about 20 seconds as the the speed was reaching unsafe levels
b) When the wheels are reinforcing each other's spins in the initial condition, there is no more indefinite acceleration. The steady state result is a tangential positioning of the wheels with the spin axis of the wheels aligned tangentially (as in along the vectorial direction of the instantaneous velocity of any object undergoing circular motion).
c) When the wheels are not reinforcing each other's spins i.e., they are cancelling each other out, there is also no indefinite accleration, but this time the wheels perform a dance, with each wheel taking its turn to perform a self-rotation i.e. each wheel's motor platform can be seen coming inward into the center of the rotation and swinging out. The two wheels take turns and never come together into the center. They are scrupulous in this behavior which I have documented and verified several times.
Now why would they do that? Why would they execute these self-rotations, when according to Newtonian dynamics the easiest thing for them to do is simply behave as if there is no net momentum in the system, which there isn't, when you analyze the initial condition? Lets go one step further and also ask, why the second case i.e. the experiment which had the wheels' spins reinforcing each other, didn't do this? Afteral, with rotational friction force almost zero, there is no difference in the physical restrictions imposed on the configuration for these experiments.
It seems to be that if we understand the prototype as not a simple object, i.e. an object with a single center of mass. We need to introduce the brand new concept of a 'compound object' we can explain this dilemma easily. I have proposed earlier in a different post that the prototype consists not of one center of mass, but rather multiple centers of mass.
A compound object has primary and secondary centers of mass. The prototype contains one primary center of mass and two secondary centers of mass. During the third experiment, we have two secondary centers of mass (the centers of each of the rotating wheels) with net spin, and one primary center of mass, the center of mass of the overal prototype without a net spin - remember that in the initial condition, the two wheel spins are oriented opposite to one another, thereby giving the overal prototype a net zero spin.
Under such conditions, we can say that third order deviations of angular surge manifest neither as a shape change in the primary center of mass, but nor as as a shape change the secondary centers of mass, but rather as a position change ie. uniform motion of the spinning wheels i.e self-rotation. This self-rotation is identical for the two wheels, (as they are identically impressed upon by the constant torque and both wheels have identical moments of inertia as well as spin velocity) and therefore appear to take turns in their self-rotation, with each motor seeming to come into the center and swing out alternately.
Viewing the phenomena in this way allows us to see that while theoreticians have analyzed rotation and its effects on spacetime geometry accurately to first and second order, they have hitherto neglected the third order effects. It now seems that experiments are proving to us that even third order effects are significant, especially in the case of gyroscopes with reasonably fast rotors and small masses (compared to a star or a planet).
Why then does the second experiment NOT show this effect? Because, in the case of the second experiment, the net spin of the primary center of mass is not zero - witness that the entire object has a net spin (given by the sum total of the angular momenta of the two wheels in the initial condition.) This being the case, it is the primary center of mass that is to be analyzed as the one with a net angular momentum and the
two secondary centers of mass that are seen as being without spin, therefore the effects are seen as a position change of the primary center of mass, i.e. the constant gyroscopic reaction of the main frame of the machine, causing it to lift up about a constantly changing point situation on the circumference of the base of the prototype. The spinning wheels therefore simply 'fall' into a tangential position and maintain it as itis the lowest energy position.
Experimental Update: Upon carefully considering the results of my previous experiments, I have decided that there is a need to make one more modification to the the prototype. All major parts necessary have been procured. I expect to begin testing the modified prototype within one week. </div>
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