Monday, May 4, 2009

Spin, Angular Momentum And Worldlines

Spin -conceptualized as a ring of mass spinning about its center of mass-has basis in Relativistic descriptions as being the phenomena governing all physical motion. It appears that spin/rotation's relativistic effects are at the origin of force itself. We may visualize spin as simply an angular momentum vector in space-time. The angular momentum of spin of an electron can be visualized as analogous to the angular momentum of a spinning wheel.

A common framework could be expressed something like this: Consider a set up shown in the figure below.

On a ct-x plane, we plot the rotational motion of a point P on the rim of the spinning wheel as approximately the path a-b-c below.

The point's angular position shifts uniformly (since the wheel has a constant angular velocity), starting at time 0 at point 'a', moving about the rim, getting farther and farther away from its original position until its diametrically on the other side of the wheel (point 'b'), before moving back towards its original position at the end of the rotation period (point 'c'). Plotted on a ct-x plane, with the t on the vertical (y)axis and the position i.e., x on the horizontal axis, it would mean a straight line of a fixed slope specifically tied to the velocity of the point would start at 'a', reaching a mid-interval maximum before returning to the original position.

(The following derivation is performed by James Hartle in his book Gravity: An Introduction although he might disagree my representation of the trajectory abc as the spin of the wheel.)

We know that the distance element on the ct-x plane will be


If the angles are conveniently angle(a) = 45 degrees, angle(c) = 45 degrees, and sides ab=bc=L, we can evaluate the distance element, ds.

Since dS2=-(ct)2+dx2 =>ab=bc=-(L)2+L2 =0 On the other hand, ac = 2L

It is important to realize that ac is always the longest side while a path running ab+bc will always have a shorter length than ac, tending towards zero (if the angles a and c are greater than zero but less than 45degrees) or have zero length (if angles a and c are 45degrees i.e. at the speed of light) compared to ac.

dSa-b-c £ dSa-c

This geometric result obtained due to the nature of the distance element of spacetime has an important consequence for spin/rotation. Path ac represents the worldline of a non-rotating wheel. Path ab+bc represents the worldline of a point on the rim of a rotating wheel. Using the preceding result, we see that the rotating wheel always has a shorter path from a to c than a non-rotating wheel. In fact, what is actually guaranteed (by the construction of the spacetime distance element) is that ac will be the longer path under any circumstance ie. a nonrotating wheel has a longer spacetime distance between two events than a rotating one.

Energy absorbed by the wheel will take the shorter path (ab+bc) and arrive at event c on the ct-x plane out of phase with the energy absorbed by the supporting frame. Energy contained in the wheel (including its mass energy, ewheel = mwheel*c2) always arrives at point c with a phase difference w.r.t. the energy contained in the supporting frame (including its mass energy, eframe = mframe*c2).

This phase difference is the key process that is harnessed by the relativistic machine- it is exploited both for the angular momentum vector of the electrons that are spinning round the inductor coil's loops as well as for the angular momentum vector of the spinning wheels in the relativistic flying machine being discussed here in analogous ways to create resonance.

But before we talk about the machine, lets just ask the seemingly simple question:

What is an object?

At the risk of seeming to digress, we investigate what we mean when we refer to something as an ‘object’.

An Object is that aggregation, (the sequence of spacetime events corresponding to) whose history can be described in terms of a worldline running through its center of mass.

We may recall that in Einstein’s SR, if we picture two world lines starting at the same event in spacetime, but each following its own path for subsequent events, such a diagram may represent the decay of a particle into two others or the emission of one particle by another.

Now, consider the diagram below which represents the first half of the rotation of the wheel - that part of the rotation where the point on the rim of the wheel leaves its original location and traverses space, thereby increasing its distance from its original position, reaching a maximum when the point reaches a location that is diametrically opposite to its starting location. (Thereafter, the point only moves closer to its original location, with the cycle being completed when the particle completes one full rotation.) - Such a process maybe described as a decay into two objects.

Now we recall that in Einstein’s SR, two world lines that start out separately and then intersect will represent a collision/encounter. The diagram below represents the second half of the rotation of the wheel which represents exactly such a collision. During the collision process, force is exerted by one object on another.

The path abc represents the worldline of the spinning wheel. On the other hand, the diagram below represents the world line of the carriage that holds the spinning wheel. The spinning wheel is mounted on ballbearings which are embedded in the carriage.

Thus, for a carriage holding a spinning wheel, we would need TWO worldlines to describe its history accurately. Incidentally this is why Newton’s Laws cannot analyze this situation correctly. Newton’s Laws contain an implicit assumption that every object has exactly ONE worldline.

Thus we may assert that a carriage with a spinning wheel is in actuality 2 separate objects. We deduce that when we start spinning up the wheels, we trigger an energy trapping process which consists of repeated cycles of first a decay of one object into two objects and then a collision. Thus energy is transferred during the collision process from a formerly internal object to its formerly complimentary part. Energy can be pumped into the tank circuit during the decay process. Energy can be pumped from the tank, via an antenna, into the environment during the collision process.

During the Decay stage, the worldline of the entire set up splits into 2 wordlines
During the collision stage, the 2 worldlines of the set up merge into 1 line. In ElectroMagnetic resonance, this can be exploited to pump EM radiation by the tuned LC circuit or in the case of the Relativistic Machine presented in this paper, this results in the pumping of directed Kinetic Energy to the frame holding the machine.
This is the commonality between Electro Magnetism And Accelo Gravitic phenomena
and indicates rotation and its manipulation to be the fundamental process at work in both.

It is posited here that resonance is actually the harnessing of the decay-collision process to transfer energy from an internal frame (or potential energy contained within a tank) to the external frame (or into kinetic energy flowing to the inertial frame that contains such a ‘live’ energy transference process).

This implies that the arrangement of spinning wheels presented later on, can be used to constitute a mechanical (Gravitronic?) Capacitor-Inductor circuit.

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