Pedagogy on Dipole (Anti)Gravity

For those who are not familiar with the origin of dipole gravity, it would be informative to quote from a textbook on gravitation. The following is a clip from the book by A.K. Raychaudhuri, S.Banerji and A. Banerjee titled "General relativity, Astrophysics, and Cosmology". There is no special reason for quoting from this book, because it is the same in "Gravitation" by Kip Thorne et al. as well.

Linear expansion in mathematics is a convenient way to sort out a major complex function into a series of simplified terms in the order of its magnitude or around the region of particular interest. The first major term in the weakly gravitating linearized theory of general relativity which is the strongest is Newtonian monopole and naturally the second one is the gravitational dipole.



The above is quoted from the chapter on "External field of a weakly gravitating source" on page 43.

"The dipole term vanishes, if the origin O is also the center of mass of the source" is what it says and agreed upon. The integral is basically the definition of the dipole gravitational moment that is the length element times the density integrated over the whole volume of the object. It must be noted that this is a very broad general statement regardless of the geometrical shape of the source in consideration.

However, when the rotating hemisphere is introduced as a source, it becomes obvious that the meaning of the statement "if the origin O is also the center of mass of the source" becomes very ambiguous. Although the standard choice of the geometrical shape of the source has been a sphere since it is generally the easiest and the simplest shape to integrate the Newtonian potential over the volume element, this doesn't necessarily have to be the case for the dipole gravitational moment.

Now the question is "which center of mass?" since the rotating hemisphere now has the duality in its center of mass. The one at rest or the one in rotation? If one had set up the origin of the coordinate system at the center of mass of the hemisphere when it was at rest, the new center of mass in rotation would have been shifted due to the relativistic mass increase effect. One may say the origin of the coordinate system has to be the one that has already been shifted. But the problem is not that simple, because no one asks how fast the object is going to rotate before setting up the coordinate system. And furthermore, this question did not exist for the rotating sphere. Why?

This shift is energy dependent and it's not a fluke of accident caused by the misaligned coordinate system. Here we postulate that this relativistic center of mass shift is the real cause of the gravitational dipole moment.

However, it must be noted that even without this postulate, if Lens-Thirring had their calculation using the rotating hemisphere, they would have gotten the same answer. They could have come up with all the details of the dipole gravity field around the rotating hemisphere. The reason for their result which covers the region only close to the center of the sphere was because of the wrong choice of the (gravitational) source(spherical shell). If one chooses the rotating sphere as the source, the integral calculation for the dipole gravity field becomes very difficult other than the region close to the center of the sphere.

It seems like the unique way a finite length element can be related to the rotational motion of an object is by the relativistic center of mass shift from the rotation of a longitudinal axially asymmetric object.

In any case, the rest is to see if the result of this "postulate" conforms with the known results in physics, for example, Lens-Thirring force etc and furthermore, to see if the results can explain cosmological phenomena hitherto unexplained, ie, jet phenomena, dark matter problems and others.

The importance of this force is that it is the strongest long range gravity force right next to Newtonian gravity. And any other higher order gravity effect will be weaker at least by the factor of v/c.

Some time ago, there was a debate regarding the theory with Dr. Choptuik at UT Austin. It must have been almost 9 years ago. He said "dipole gravity is not general relativity". He may be right because Einstein did not say anything about it. In fact, if you have special relativity, and apply it in the same manner to any advanced theory of gravity(for example, Brans-Dicke theory of gravitation), the second order linearized term will show the same result ie, dipole gravity. So, in a way, dipole gravity does not help differentiating general relativity from other similar theories.

However, is it really fair to say dipole gravity is not general relativity? I don't think you can say DPG is not general relativity. But then also, you can't say it is the trade mark of general relativity either. After all, Newtonian gravity can be derived out of both general relativity and Brans-Dicke theory solely because they are designed that way. If your theory doesn't produce Newtonian gravity in its linearized form, your theory of gravity would be simply wrong.

If you say "dipole gravity is not general relativity", isn't it the same as saying "Newtonian gravity is not general relativity" as if Newtonian gravity has nothing to do with general relativity, which is not true. Both Newtonian gravity and dipole gravity are parts of the broader theory like general relativity and/or Brans-Dicke theory. However, neither of them(NG and DPG) may provide the crucial test to differentiate between the two( general relativity and Brans-Dicke theory).

Dipole gravity is a part of general relativity and is also a part of Brans-Dicke theory as well. It is a theory that can not be avoided once special relativity is proven to be correct. Apparently, the rotating hemisphere violates Newton's second law of motion. Because its center of mass changes without the external force in the direction of the shifted center of mass.

We can not take both principles to be absolutely correct when they are contradicting each other in such a glaring fashion. http://dipoleantigravity.blogspot.com/2007/05/what-is-at-stake.html