Tuesday, April 24, 2007

Sign Error in Lense-Thirring Force?

In the original paper published by Lense-Thirring, the sign of the radial component of the force in the direction perpendicular to the rotation axis was repulsive and harmonic which has been considered as the general relativistic confirmation of the centrifugal force and the Mach's principle. In his paper published in 1999, Jeong also used the result of Lense-Thirring to confirm that his calculation of the superposed dipole gravity force inside a rotating sphere matches with the previoulsy published results of Lense-Thirring.

However, after a careful investigation of the dipole gravity potential and comparing it with the observed cosmological data, it has been found out that there may be sign errors in both Lense-Thirring and Jeong's forces originally published in 1999.

In Lense-Thirring's case, obviously, it was their desire to prove that general relativity conforms to Mach's principle or vice versa. In Jeong's theory of dipole gravity case, it was his desire to make sure his result conforms to Lense-Thirring's in his presentation of the diagram.

It may be that both have been wrong in the signs of their forces.

The major reason for this suspicion is in the direction of the radial component of the dipole gravity (gravito magnetic) force in the equatorial plane. If the lines of the force resembles that of two opposing bar magnets within the sphere, the radial force line will be outgoing (repulsive) around the equatorial plane just like the way Lense-Thirring force behaves near the center of the sphere. And it should be that way if anyone wants to claim it is the evidence of the centrifugal force.

However, the problem with this is in the fact that the direction of the outgoing radial force in the equatorial plane is not consistent with the dark matter problem. Also at the poles, the direction of the dipole gravity force is incoming which makes it hard to explain the jets considering that the Newtonian gravity force line is also incoming. So, the jet phenomena can be explained more logical way if the Z directional potential is like below than its upside down form.

For example, in the above shape of the potential along the Z axis, where the domed side represents the repulsive force, any density overflow from the central region toward the poles will be automatically ejected along the rotating axis of the space. And the longer range Newtonian gravity will bring them back to form two way jet streams. The only problem with this reversion is that the original Lense-Thirring force has to have its sign reversed.

While the direction of the dipole gravity force can be made arbitrary, because the direction of the center of mass shift can be defined either way depending on the convention, it has to be consistent with the observed cosmological data.

To solve the dark matter problem, we need an additional long range radial gravity force in the equatorial plane that is not repulsive but attractive. If this is the case, Lense-Thirring force has its sign reversed.

The derivation of the radial force in the equatorial plane presented in the page has the correct sign, because it was derived directly from the original potential not read off from the diagram.

In the illustration above, the force lines at the center of the two superposed gravito magnets are incoming(attractive) and that of the both poles of the magnet are outgoing(repulsive). At the center of these two magnets facing the same type of poles(attractive) in the middle, the force lines along the equatorial plane will form array of horizontal incoming force lines. The extension of this force lines at the center between the two gravito magnets can not be outgoing, meaning that the following expression of Lense-Thirring force has the wrong sign(remember the centrifugal force is radially outgoing force ).

The fundamental physical reason for this is because the continuity of the force line either in magnetism or gravito magnetism doesn't allow the change of its direction on its path 180 degree.
Therefore, Lense-Thirring force has to be the attractive harmonic force toward the center in the equatorial plane in total contrast to their original claim and also the force along the rotation axis must behave like repulsive harmonic force as shown in the above potential diagram along the Z axis(notice the potential near the center(Z=0) of the diagram) instead of the attractive harmonic force shown in the above expression and in the previous saddle diagram.

This comes to the clear conclusion that Lense-Thirring force has nothing to do with the centrifugal force. It is rather a simple manifestation of the attractive radial dipole gravity force near the center of the rotating spherical shell.

Therefore, it is corrected at this point that the dipole gravity force in the rotating hemisphere is attractive on the flat side and repulsive on the domed side, instead of the other way around. It makes the jet phenomena explained in a much more simple and elegant way and so is the dark matter problem, although the way the dipole moves itself in space makes it not being very well stream lined in the normal common sense of the aerodynamics.

But then there is nothing common sense about cosmology anyways. The alternative method of detecting dipole gravity force experiment mentioned in the following page will prove on this point by showing the direction of its actual movement in space.

Now, it seems that the overall consistency is restored.

Let's blame it on Mach. It was all his fault. :)

Of course, I'm trying to be humorous here if anyone has noticed. It's Lense-Thirring's fault and Jeong's fault as well when he modified the dipole potential diagram to fit the result of Lens-Thirring's. But how about those who didn't check it out all the way through? Let's try to be happy at least we found the error and corrected it.

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