Thanks a lot dear Aria Tsam for your interesting papers. I trust that gravitational dipole radiation is necessary and should be considered in GR. I also refer you to my recent paper at http://link.springer.com/article/10.1007%2Fs10509-009-0058-y

Dipole gravitational radiation in GR does not present because of conservation of the linear momentum of N-body system. It may be present in some alternative theories of gravity though. This problem has been well understood long ago.

In analogy with electromagnetism where we have positive and negative charges one could have simply generalized non-relativistic Newton's theory of gravitation by admitting negative masses, \pm M. Then, as in the electrostatics, one could define the gravitational dipole moment as the product of the absolute value |M| of M and the distance between two equal in absolute value masses. Following the path of contemporaries of Albert Einstein one could then write down the d'Alambert wave equation with the scalar source J=\rho for the scalar Newton's potential \Phi from which the static Poisson equation for Newton's gravity follows. In an exact analogy one could then develop the theory of (scalar, though) radiation. Not only the dipole radiation would be then present. However, in the theory based on the Galileo-Poincare-Einstein Relativity Principle and the Einsteinian Equivalence Principle the experimental fact that there are no negative inertial/gravitational masses, which Einstein created by 25th of November 1915, there is no `electric' type dipoles. On the other hand Einstein's General Relativity describes rotating objects. One of the quintessential aspects of Einstein's theory is that angular momentum produces gravitational fields and in the case of very strong fields the Kerr gravitational field illustrates the point that although there exists the `magnetic' type dipoles they do not radiate. Guess why?

Therefore the radiation starts with the time dependent quadrupole (there as usual are higher multipoles which radiate) and in the limit of weak fields the famous, and correct, Einstein's quadrupole formula gives the valid description of quadrupole radiation.

The question why there are no negative gravitational masses is even more interesting, but this note is already too long so I will not dwell on it. Instead I refer all interested parties to the recent gr-qc e-print by Jacob Bekenstein.

If you define the dipole moment of a system in the standard way, with respect to the system's center of mass as the origin,, and replace the charges q_i with masses m_i, then the mass dipole moment always vanishes, whether there is negative mass or not, by the definition of the center of mass.

One can of course postulate the existence of mass dipole (point) particles, where the dipole moment is instead taken with respect to the position of the particle.

I am not so sure if pure rigorous mathematicians (people such as Rich Schoen, S.T. Yau) were ever able to show the positivity of local mass. Does physicists just assume local positivity of mass? But Hawking seems to suggest otherwise. Could anyone clarify this for me. I can only understand people like S.T. Yau on these types of things. These mathematicians would be considered rigorous to the top scale. They will not make any assumptions except what is given a general relativity as given by Einstein and Riemannian manifold theory.

In my opinion, if rwo bodies orbiting each other as the Earth and the Moon, and one of them has electric charge it should radiate as accelerating charge. In this case there is disbalance of gravitational radiations of bodies and in addition to quadruple radiation some dipole tipe radiation must be.

Let us say that the mass-energy density be ρ(⃗r). Then monopole moment is ∫ρ(⃗r)d3r while the dipole moments are, ∫ρ(⃗r) ⃗r  d3r and ∫ρ(⃗r) ⃗r  x ⃗⃗v(⃗r) d3r. Monopole moment vanishes from the conservation of total mass energy. The second one is the CM frame mass energy, which is constant. Because there is no radiation seen in CM frame, in any other frame also radiation is absent. The third one is total angular momentum of the system, which is fixed. We can therefore conclude that there are no monopole type or dipole type gravitational radiation in a classical theory. 

Ref: https://www.astro.umd.edu/~miller/teaching/astr498/

lecture 24

Gravitational dipole moment in general relativity was mistakenly eliminated without due consideration. You have to see the linearized weak field limit of general relativity to understand why it was mistakenly thrown away. The center of mass of an object is a sacred one in Newtonian mechanics. It won't change unless an external force is applied to the object. Spin rotation won't do it as long as you stick to Newtonian mechanics.

But try this in your thought experiment, consider the relativistic mass increase effect for a rotating hemisphere. The center of mass of a hemispherical shell in rest state is in the middle between the center of the full sphere and the top of the dome in the Z axis.  As soon as it starts rotating, the center of mass moves toward the flat side of the hemisphere however small it may be. This is not Newtonian phenomenon although it is completely physical according to special relativity. It violates and strikes the Newtonian principle of mechanics of motion on the face. This is the source of dipole gravity. It can explain, the jet phenomenon, dark matter problem, Saturn ring, GPB experiment., Lense-Thirring force etc. In other words, a rotating cone or hemisphere is a gravitational dipole magnet. In the mass filled universe, you can predict it will accelerate spontaneously toward the flat side of the hemisphere. On the other hand, a rotating sphere will be a doubly and oppositely stacked gravitational dipole magnets where the south and north poles become the source of the anti-gravity centers, which is the cause of the jets from the rotating black holes. I recently published this result and others in my book, "Physics of the New Millennium, Birth of the New Paradigm". There are many other physics discussions in the book.

The other stumbling block for the understanding of this mystery was because the rotating sphere does not have this relativistic shift of the center of mass due to cancelation. That doesn't mean the individual oppositely stacked half spheres don't have this center of mass shift effect. If you calculate the dipole effect from the oppositely stacked rotating sphere, you get the same result of the Lense-Thirring force purely r and the angle theta dependent force. A rotating sphere or a cone shaped object is basically a gravitational magnet. The faster they rotate, the stronger the effect of antigravity. This is the deepest secret of most of the dark matter related physical problems. It includes, jets from the rotating black holes, Saturn's ring, GPB experimental data, the distribution of the dark matters etc etc. This problem and other electromagnetic capacitor anomaly are explained in the book, "Physics of the New Millennium, Birth of the New Paradigm".

If you are a physicist and looking for answers to many of the cosmological problems we face at the moment, you must read this book. It was published in Oct 2015.  

Dear Arbab, Eue, and All

I found this discussion just now and felt I had something to add to the discussion. I'm looking at a a possible analytic expression for Newtonian Gravitation at https://www.researchgate.net/project/Analytic-expression-for-Newtons-Gravitational-Constant See attached file

At the same time I am reading a work by you Arbab at http://khartoumspace.uofk.edu/bitstream/handle/123456789/17614/On%20the%20gravitational%20radiation%20of%20gravitating%20objects.pdf?sequence=1

It seems the 'dipole gravitation' as it is called is a way in which gravitational systems achieve equilibrium at the solar system level of gravitation(different to the gravitation of quasars). In your introduction Arbab you talk about a gravitatiing pair of neutron stars spinning about each other and emitting quadrupole gravitational waves. But the dipole gravitational wave is a stream of photons at the solar system level that transits between say Earth and Sun, or between any two gravitating objects. This is a stream of photons that causes a stream of collisions to nudge the two massive objects around in their orbits. This is a dielectric and a magnetic moment physics and allows us to quantify the system in terms of the wavelengths and frequencies involved in the gravitational (two-body) system at the solar system level.

Dear All

In brief I think this is the birth of a galaxy that forms into a black hole and sustains a system of solar systems around it.

Dear All

I think we are looking at two separate cases of gravitational waves here, one is an equilibrium state ( sun and planet at the solar system level where EM wave-particles, atoms and molecules, and their gravitational binding energy, photons are balanced) while the other is a non-equilibrium state (two stars, pulsar or quasar possibly near the birth of a galaxy where eventually a range of nuclear wave-particles and their fields such as neutrinos, gluons, etc will be are balanced)

Dear All,

If there is a case that the center of mass of the object changes when it (spin) rotates, you can say safely that the system violated Newtonian law of motion. That means that the system won't respect momentum conservation law either.

This non Newtonian shift of the center of mass happens when a hemisphere, a cone or a funnel rotates due to the relativistic mass increase effect. The longitudinally uneven distribution of matter causes the center of mass to move toward the flat side of the hemisphere.

This, in return, causes the second term in the linearized theory of general relativity to have the meaning. Dipole gravity in general relativity is not radiative. It is dynamic yet static. It has long range force of gravity that changes sign from the angle theta 0 to 180 degree that is negative.

Negative sign in gravitational potential means repulsive force. So dipole gravity is like magnets. One side has the attractive pole ( flat side of hemisphere) and the other side repulsive.

The verification of this force is in the fact that it reproduces Lense-Thirring force close from the center of there sphere where two hemispheres are superposed in opposite configuration. The mystery of dipole gravity has been solved!!! Of course, this is the real cause of the dark matter problem and the flat rotational velocity curve enigma.

Eue Jeong

Contrary to conventional thinking, certain systems of masses can and do produce dipole gravitational radiation.

Every closed system of masses (including their radiation) conserves momentum and therefore has constant mass dipole moment. But that does not mean that dipole gravitational radiation from the system vanishes.

For example, consider a small mass speeding past a much larger mass with a kinetic energy much greater than the gravitational potential energy between them, so the small mass is not captured into a bound orbit. This unbound quadrupole does produce dipole gravitational radiation in the far (radiation) zone.

An exact weak-field calculation of the gravitational fields of both masses in this example shows that the acceleration fields of the masses do not completely interfere in the radiation zone. The reason is that the power spectral density and phases of the gravitational fields of each of the masses are shifted in the radiation zone owing to the differing speeds and changes in position of the masses.

For unbound quadrupoles, such as in this example, the first calculation of dipole gravitational radiation was done in:

Article Dipole gravity waves from unbound quadrupoles

The calculation later appeared in the book, Einstein's Inertial Field:

Book Einstein's Inertial Field: How it accounts for the Pioneer a...

For bound quadrupoles, like orbiting masses, no one has done a similar calculation. (Since the potential energy is greater than the kinetic energy, the weak-field approximation does not work.) But experimental evidence, such as from binary pulsars, seems to suggest that bound quadrupoles produce no dipole gravitational radiation.

Hi Franklin, Eue and All

I think we are saying something similar. Have look at the paper Article Self-field Theory — A Possible Gravitational Structure for Galaxies

If the galaxy is controlled by a quadrupole I see this as perhaps a differential of the strong nuclear forces within the nucleus (integrated over their volumes); notice I am not using GR but a new mathematics that is intimately related to quantum theory but is a deterministic version that does not use probabilities but gets the same answers as QT.

Dipole gravitational radiation in GR does not present because of conservation of the linear momentum of N-body system .

It doesn't. The second order term in the linearized theory of general relativity vanishes only because they (early researchers) couldn't figure out cases when the physical center of mass changes when rotating. Of course the reason is because they only considered typical spherical case as in Newtonian mechanics.

Rotating sphere doesn't change its center of mass when rotates but rotating hemisphere does change its center of mass because of the special relativistic mass increase effect. This effect puts the entire paradigm of gravitational physics upside down. You have now a case that the effective center of mass of an object changes by simple spinning rotation. Energy momentum conservation principle in gravitational physics breaks down spectacularly in this case.

This effect has been studied and published in the journal Physical Scripta in the year 1999.

https://archive.org/details/PhysicaScripta

The only problem with the paper was that the direction of Lense-Thirring force was reversed. Einstein claimed early that the radial component of Lense-Thirring force was due to centrifugal force. This mistake hampered the understanding of dipole gravity force in general relativity for a long time. Lense-Thirring's radial component of force was supposed to be directed inward. Once the direction of Lense-Thirring force is corrected, it starts solving many problems in gravitational physics.

According to the theory of dipole gravity, in essence, rotating hemisphere is a gravitational magnet. The dipole gravity force at the dome of the rotating hemisphere is repulsive and the same force at the flat side of the hemisphere is attractive. Once the two gravity dipole of the same size are attached face to face in a rotating sphere and rotate fast, it becomes a source of black hole jets at the two poles, and the center plane of the rotation become the horizontal plane of the rotating galaxies. It's not too difficult to imagine this same effect will cause Saturn's ring and the strange hexagonal marks at its poles.

The GPB( Gravity Probe B) experiment has already reported Lissajous curve 8 from the gyro mounted in the satellite that took polar orbits, which is a solid proof of dipole gravity which is caused by the spinning effect of the Earth.

The dipole gravity force lines around the rotating galactic center produces matter distribution that is conducive to the distribution of dark matters that cause the flat rotational velocity distribution curve.

This fulfills the biggest original yet divine purpose of general relativity that is the revelation of dipole gravity that is the strongest gravity force next to Newtonian.

It is, of course, a paradigm shifting revelation in fundamental physical sciences.

Truth may be hidden, but can not be silenced.

Dipole gravity is not radiative. It's static. Dynamic shift of the center of mass creates dynamic dipole gravity force which is the key secret of reactionless propulsion space travel.

This paper was published in the refereed journal in 1999. People need to wake up.