Dear Eue J,
Is the theory covariant? Does it account for gravitational lensing? Can it explain the bullet cluster? These are all the sorts of questions people are probably asking you. I'm sure you have answers.
Thanks for your email with the questions. Let me try as best as I can. If you ask if Newtonian gravity is covariant, the answer will be no. Probably it's the same with dipole gravity. It's just a linearized weak field limit solution. But one may try to formulate a specific tensor metric to study the strong field regime of dipole gravity and then the question of covariance will become important. Just like Newtonian gravity becomes a black hole in the strong field regime of Schwarzschild metric, the result will be a predictable extension from the weak field limit behavior. One possible reason that it may be hard to find a tensor metric for dipole gravity will be that it is not a stable gravitational system. The system will not be local. In other words, in the strong relativistic field regime, the gravitational dipole moment will not be at the local spot where one expects it to be because of the extremely strong accelerating force it experiences from the rest of the universe. Theoretically, in such a case, how one can devise a tensor metric where the acceleration can be so strong that the locality of the system can not be well defined. If it is a moving system with a constant speed, it will be trivial, but how one can devise a tensor metric specifically for a constantly accelerating system while the constancy of the acceleration depends on the intrinsic(spin angular momentum and the longitudinal asymmetry) properties of the mechanical system under investigation. But I'm sure someone will come up with an answer. Basically, it's a metric where the source is a rotating hemispherical type (longitudinally asymmetric) rotor. Of course, nice thing about the linearized theory is that one can get a glimpse of the strong field regime without actually finding the metric sorely representing the gravitational dipole moment.
The jets from the rotating ultra compact stellar object is just as common as the winds and storms in the earth environment. And the matters ejected from the poles can take a long route to come back to the center by following the gravito magnetic force lines. Those matters in their transit toward the center of the rotating stellar object form dark matter halo in the surrounding space and the gravitational lensing becomes a natural consequence of it. It is surprising that the dark matters are actually found by studying the bullet clusters colliding with the stationary galaxy.
Regarding the bullet cluster, if a small galactic system has started with the initial condition in such a way that it has a longitudinally asymmetric configuration with non zero spin angular momentum, the cluster will run in one direction accelerating as predicted by dipole gravity. As the longitudinal asymmetricity flattens out as time goes by, due to many possible reasons, the constancy of the velocity will be set in and it becomes a bullet cluster.
The major development in the process of publishing the blog was the finding of the sign error in the Lens-Thirring force. After the correction, it flood gated all the subsequent understandings, ie, the consistent gravito magnetic force lines, the dark matter halo and the details of the intricate black hole jet engine mechanism.
Eue J Jeong
==Answer to the question regarding the covariance of dipole gravity prepared earlier==
It looks like this is a common question in the minds of astro physicists as I have encountered more than several times. When Einstein's field equation is linearized, the individual terms are not by themselves covariant. For example, Newtonian (monopole) gravity will not be covariant by itself. Neither is dipole gravity. While they are parts of the solution to Einstein's field equation, the exact validity of it will be diminished substantially as the system goes into the extreme relativistic regime.
However, one can assume with great confidence that the major property of either monopole or dipole gravity will not change drastically as the system develops into the strong field regime. Black hole's gravity potential is different from the linearized weak field monopole(Newtonian) gravity, only in the way its functional variation over the close distances. The fundamental radial character of the monopole gravity force will not change. For example, the monopole gravity will not change suddenly into the dipole gravity just because the system goes into the strong field regime.
So, one can see that dipole gravity is a totally new entity. Its weak field limit property will not change into something else(other than dipole gravity itself) even if the system goes to the extreme relativistic regime.
One can see that there can be two different ways to perform research in general relativity to discover a new physics. One is trying to find a totally new metric tensor that may reveal some type of new physics in general relativity, which is the way most of the gravitational physicists are focused into these days. The other method is to find an actual mechanical system that can be calculable in the weak field limit of general relativity, which is presented in the theory of dipole gravity as well as in the quadrupole gravitational radiation research, the path of which is limited and has not been sought by many physicists. While some type of metric tensors that have been found may not represent the actual universe, the linearized weak field solution found directly from the mechanical system will represent a part of the actual universe at least in the regime of the weakly gravitating source.
PS; If we look back at the development of general relativity, it is not hard to see that dipole gravity is the true crown jewel of general relativity. Because the first term from the linearized theory which is the monopole(Newtonian) gravity was a totally expected one that can not surprise anybody and the third term which is the radiative type of gravitational quadruple moment is two orders of (v/c) magnitude weaker than the monopole gravity, which makes it extremely difficult to detect its effect. And none of these two known terms of gravitation seems to explain the most prominent cosmological problems of today, namely, the jets and the dark matter problem.
The second term in the linearized theory which is the dipole gravity was a totally unexpected one since it doesn't exist in the context of the classical gravitational physics. And it was not obvious if it does exist and meaningful within general relativity either, so it was generally considered non existent and that the dipole term in general relativity was considered physically meaningless, although many hints were there suggesting that it could exist and be real. For example, Lens-Thirring force should have been taken more seriously because it carries the signature of dipole gravity. And also, since acceleration of mass creates gravitational field, according to the equivalence principle, the gravitational field from the rotating sphere should have been looked at more carefully.