What is Dipole Gravity and What It isn’t?

In a recent communication with a prominent astrophysicist, I noticed that there is a general misconception about the theory of dipole gravity. Somehow people seem to think dipole gravity is some kind of a modification of general relativity.

This is far from the truth. Dipole gravity is not a modification of general relativity. Although it may sound bizarre, the general relativistic gravitational field both the inside and out of a rotating spherical source has never been fully worked out. In Newtonian gravity, the gravitational field inside a spherical shell is zero. However, due to the equivalence principle, general relativity predicts that there will be an induced gravity field due the constant acceleration of the mass resulted by the rotational motion of the object.

Because of the enormous complexity of the integral calculation, the only known solution to this problem was available only at the close distance from the center of the sphere which has been worked out by Lense and Thirring in 1918. In their original paper, they found the general relativistic gravity force close to the center of the rotating spherical shell which was given by,


These forces have been known as Lense-Thirring force. The x and y component of the force shows the radially outgoing structure which has contributed to the notion that it is a manifestation of the centrifugal force in accordance with Mach’s principle.

However, the presence of the attractive harmonic z component of the force was enigmatic from the beginning. There are historical records showing that Thirring had correspondences with Einstein in several occasions regarding this problem. Obviously, it baffled Thirring as much as Einstein.

It must be emphasized that the above expression is valid only for small x, y, z which is very close to the center of the rotating spherical shell. In other areas beyond the center of the sphere, the integral calculation is simply impossible. This form of the force ceases to have any meaning as soon as the distance from the center increases beyond the closest proximity. And the known solution to the linearized theory of general relativity has stated that the rotating spherical mass does not have a meaningful dipole term, which is true only in the very far distances. So this problem has been left out as an open question in general relativity since its birth.

The conventionally known "gravitomagnetism", which is a modified version of Maxwell's equation, was one of the desperate attempts to understand the extension of the Lense-Thirring force and the acceleration induced gravity effect beyond the limit of the proximity to the center of the rotating source. Needless to say, there is no mention of the derivation of the Lense-Thirring force or any form close to it, from this formulation at the distance close to the center of the sphere.

There simply was no known general relativistic solution in the intermediate area away from the center to the relatively close distance from the surface of the rotating spherical source.

This means that the general relativistic gravity field induced by the rotational motion of a spherical object has never been fully understood to include the entire space beyond the close proximity to the center.

What the theory of dipole gravity has accomplished is that it calculated and showed the compact mathematical form describing all the details of the field inside and out of the rotating spherical source. This was achieved by dividing the sphere into two sectors of the hemispheres, and by calculating the fields individually and adding them together, which is possible because the potential function is a scalar quantity.

The field close to the center calculated from this method produced the Lense-Thirring force of the form,


Several points can be noticed.

1. Both forms have the same sign correlations, eg, the radial and the axial component of the forces have the opposite sign with respect to each other.

2. They have the equal functional form as second order differential equations.

3. There is a uniform difference of a constant factor 2/15 between the two expressions.

4. The form derived from dipole gravity has the missing velocity dependent component of the force.

In the theory of dipole gravity, it has been specifically pointed out that only the diagonal component of the metric tensor is considered. The velocity dependent force can be added later without the loss of generality.

The difference of the constant factor by 2/15 may be explained by considering the fact that the center of the sphere is close to the centers of the two separate hemispheres. The centers of the two hemispheres are singularity points where the dipole field becomes infinity(which is a mathematical artifact) and the field close to the center of the hemispheres within the range of R/2 will not be accurate(larger than actually it is), which explains the discrepancy. This problem can easily be fixed by introducing a form factor etha.


Now, the total potential without singularity can be written



However, this was not the end of the story. The further surprising irony was that the signs of the Lense-Thirring forces are all reversed as later found out. Once the Lense-Thirring force is identified as the residual force from the two oppositely superposed long ranged dipole fields within the rotating spherical shell, the continuity of the force lines all around the space becomes an important issue.

Since we are aware of the jets and the dark matter problems in cosmology, the conventionally known signs of the Lense-Thirring force become very problematic. The repulsive radial force is not consistent with the dark matter problem any more than the attractive axial component of the force with the jets. Dark matter problems will be solved easily if the radial component of the force were attractive and the jets would be explained easily if the axial component of the force were repulsive.

In fact, the original formulation from dipole gravity showed the reversed signs for the Lense-Thirring force. However, since there was no compelling reason to doubt the correctness of the signs of the 90 year old formula, before applying the force to the actual cosmological problems, the result was simply adjusted to conform to the known results. Even the jet problem seemed all right with the original signs of Lense-Thirring configuration.

It was only with the dark matter problems that the signs of the Lense-Thirring force looked awfully awkward. One can not have the accretion phenomenon with the repulsive radial force, let alone the fast rotating spiral form of the galaxy. And the jet phenomenon seemed more easily explainable with the corrected signs of the Lense-Thirring force.

The necessary presence of the dark matter halo which is an absolute requirement for the explanation of the flat rotation curves within dipole gravity and also in dark matter hypothesis was the final nail on the coffin of the original signs of the Lense-Thirring force. It simply can’t be the other way around. The matters ejected by the jets have to come back to the equatorial plane and eventually to the galactic center to be recycled. The force lines depicted by the corrected Lense-Thirring force matched perfectly with this picture. If the original signs of the Lense-Thirring force were correct, matters will be ejected radially from the equatorial center to the plane and come back to the both poles which is against all odds.

More specifically, if this is the case, since the direction of the dipole gravity force lines and the Newtonian gravity force lines are in the same direction at the both poles, that are attractive, there will be no jets visible, because the accompanying two forces lines do not allow the debris to collide among themselves.

The dark matters are basically the debris in space in transit following this dipole gravity force lines.

Einstein was inspired by Mach's view of the universe and of the origin of the centrifugal force when he formalized general relativity. Naturally he expected that his theory of gravity should reflect Mach's point of view. The Lense-Thirring force was at the right spot and at the right moment. In his mind and that of Lense-Thirring's, there was no doubt that the force they derived from the second order effect of gravity was the manifestation and proof of Mach's principle. But how many times in the history of physics, people are inspired by something and discover something else totally new. But in general, I think, there is a general tendency of a bias when people strongly expect something to come out of their research activities.

Now, it is clear that the total general relativistic gravity field including the Newtonian gravity can be written
for a rotating hemispherical source. In a multiply connected dipole configuration, the second term will be represented by sums of all the existing dipoles in the system. For an example, the dipole field from a rotating sphere has to be a sum of the two oppositely connected dipoles within the source.

Any cosmological problems involving a rotating source will need this formulation to accurately describe the mechanics of the system just like we use Newtonian gravity to describe the non-rotating(very slowly rotating) stellar configurations.

It is not surprising at all that one has to make all kinds of extra assumptions to account for the baffling problems in cosmology, when this second order dipole gravity term was not present, even to the degree that the whole Newtonian mechanics has to be modified, let alone the plasma and the magnetic field for the jet phenomenon.

The correct order of the approach to solve the problem would be to apply this dipole gravity for the jets and the dark matter problems first and if it still doesn't work then use any additional tools to account for the further minute details.

In this regards, I noticed that the GPB data have been processed using the theoretical results derived from the conventional gravitomagnetism which is a wrong theory of gravity. The dipole gravity has the entirely different topological property compared to the theory derived from the modified Maxwell's equation. A rotating spherical mass has four distinctive poles instead of the two according to the dipole solution of general relativity. So, if the GPB data don't fit the predictions, they have to suspect that it may not be because of the systematic experimental error but because of the incorrect theoretical assumptions.

I'm sure this is not the end of cosmology. I hope dipole gravity can inspire young minds and help them discover something much bigger than the surface it barely scratched. I also hope it becomes the beginning of the new era of the engineering of this new scientific concept for the future space adventure of the mankind.

What has inspired the whole concept of dipole gravity?