[The phrase (without the "not") is taken from a NASA release: 04-351, 21/10/2004. The disclaimer reads:
"This material is being kept online for historical purposes. Though accurate at the time of publication, it is no longer being updated."
The conclusion one reaches is that there must be a good reason for this!]
We read from the Wikipedial Gravitoelectromagnetism (GEM) article that it "refers to a set of formal analogies between the equations of electromagnetism and relativistic gravitation...Gravitomagnetism is a widely used term referring to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge."
Trying to convert Maxwell's circuit equations into a dynamic theory of gravity is an old matter. Where Maxwell failed, his protegee, Heaviside, went full course and identified vortex motion with the magnetic field. To Heaviside, any speed of propagation of gravitational waves was acceptable.
The sole basis for such an analogy is that in the general theory, the Ricci tensor linearizes to a wave equation in the weak field limit. In this limit, "an apparent field appear[s] in a frame of reference different from that of a freely moving inertial body," concludes the Wikipedia article. But it doesn't say whether this frame is inertial or not. Further, "[t]his apparent field may be described by two components that act respectively[?] like electric and magnetic fields, since these arise in the same way around a mass that a moving electric charge is the source of electric and magnetic fields." No, no, and, no!
Continuing, the article claims that the "main consequence of the gravitomagnetic field, or velocity-dependent acceleration, is that a moving object near a massive, non-axisymmetric, rotating object will experience acceleration not predicted by a purely Newtonian (gravitoelectric) gravity field. More subtle predictions, such as induced rotation of a falling object and precession of a spinning object are among the last predictions of general relativity to be tested."
Although the article was last edited 2 days ago, all experimental tests are not more recent than 2007 or so. This attests to the fact on how relevant these effects are and submit to experimental verification.
It is rather surprising (like including my book, Seeing Gravity, in the list of best books on black holes by the BookAuthority) that in the list of references to the Wikipedia article is a paper by Costa and Herdeio (2008) which associates the two fields with "tidal" tensors stemming from the Weyl tensor and is Hodge counterpart. There it is explicitly stated that the gravitational analogs of Maxwell equations do not contain inductive terms that would correspond to the Faraday effect, and the Maxwell displacemet current. No circuit equations No wave equation! Moreover, the term "tidal" is misleading and inappropiate since tidal force are describable in Euclidean space. That is to say, that once the forces are removed, the system relaxes to its pre-perturbed state. This is not the case when non-Euclidean geometries are called into play. Curvature remains even in the absence of impressed forces.
All this attests to the fact that physical similarities can be deceiving, and it produces the wrong physics. Let us now return to the beginning of the problem of determining affect that motion has on determining the mass of an object.
First, physicists at the end of the nineteenth century identified two masses: the electromagnetic mass and the electrostatic mass; the former equaling 4/3 of the latter. Second, there appeared to be a depedency of mass on rectilinear motion, eventhough the experiment consisted in deflecting electrons by passing them through a magnetic field (Kaufmann’s, as opposed to Thomson’s, experiments).
There one could distinguish the ‘transverse’, as opposed to the ‘longitudinal’, mass. The longitudinal mass has its force in the direction of the velocity, while in the transverse mass they are perpendicular to one another, like in the Lorentz force. At high speeds, the longitudinal mass is much greater than the transverse mass, and since it did not fit Kaufmann’s experimental results it was quickly swept under the carpet. The latter configuration corresponds to a uniformly rotating disc where the (centripetal) force is point inward to the center of the disc, and is, therefore, perpendicular to the velocity which is tangent to the disc. This corresponds to the magnetic component of the Lorentz force. The force is normal to the plane containing the velocity and the magnetic field.
Parenthetically, an even earlier prediction was made by JJ Thomson who, in 1887, determined an additional mass caused by the motion of a small sphere. He equated the energy of the magnetic field to that of the electric field and gave an expression for the square of the former, which he considered to be the kinetic energy. This was proportional to the square of the sphere’s charge so that when a charged sphere was set into motion, its mass increased. What he calculated was the “electromagnetic” mass, which was 4/3 that of the “electrostatic” mass.
In Thomson’s theory there was contraction of the sphere into an ellipsoid in the direction of motion predicted by Lorentz’s theory. In fact, Lorentz openly admitted that Thomson held the priority on determining the increase in mass due to motion, but that his calculation was “somewhat different from that to which one is led in the modern theory of electrons.” The word that should be underlined is “modern” for it had nothing whatsoever to do with it, where it was modern or archaic. Thomson, the discover of the electron, stood by his results until the end of his life, and certainly was not a staunch supporter of “relativity”.
Now what happens if the sphere is uncharged? There then should be no magnetic field, and, therefore, no increase in mass proportional to its square. Certainly, there should be no induction whereby the rotation of one field is caused by the rate of change of the other. Right? However, the young Einstein considered that a “smidgeon” of charge could always be added to a neutral sphere without changing its physical properties so that everything that applies to a neutral sphere could be carried over to a charged sphere no matter how weak. And what about the contraction of a sphere into an ellipsoid due to its motion. Where did the size of an electron, no less its shape, disappear to?
So far, only rectilinear motion has been considered. Consider, now, a uniformly rotating disc at a constant angular velocity, or, equivalently, a constant magnetic field. This is the configuration to which the Lorentz force should apply. But, now, we don’t add a smidgeon of charge to the neutral mass, but, rather we write the Lorentz force with mass replacing charge. Then the magnetic term will, apart from a numerical factor of 2, have the form of a Coriolis force in a rotating frame of reference. And this is precisely what GEM (gravito-electrodynamics) has capitalized on. Notwithstanding appearances, “fictitious” forces are not the same as “inertial” forces, and EM forces belong to the latter.
Coriolis forces depend on the mass of the particle, not upon its charge. If you go to a coordinate system rotating at a uniform frequency about a given axis it will seem that there is no magnetic field because the frame is rotating with the particle which changes due to the magnetic field. But, it is precisely in such a rotating coordinate system that there is a Coriolis force is acting to deflect the particle. (In the northen hemisphere, where the angular velocity points out of the ground, the Coriolis force tens to deflect a projectile shot parallel to the earth’s surface to the right. In the southern hemisphere it reverses direction.) If we go to an inertial system, the Coriolis force disappears, and it would seem that the magnetic field acts to spin the particle around some axis parallel to the magnetic field itself at the given frequency.
The Lorentz force depends on charge, not mass, so that particles of different masses will have the same electromagnetic fields acting on them, but different Coriolis forces. The two live in different “worlds” (i.e. frames), but whereas no increase in mass, or dragging, can be associated with a Coriolis force, an increase in mass, or dragging can be associated with electromagnetic fields which operate in inertial frames. It is as if the charge particle in motion carries with it its own magnetic field (Thomson’s interpretation.) The relative E/B ratio depends on the relative motion of the source, whereas Coriolis forces do not depend on the motion of the sources. No “relativistic” effects can be associated with Coriolis forces. Hence, the statement that “frame-dragging can be viewed as the gravitational analog of EM induction” is devoid of meaning.
It is also inaccurte to say that to Newton, it was mass which creates the gravitational field, whereas to GEM it is a rotating mass that produces the field. Mass just doesn’t disappear when a change from a non-inertial to an inertial frame occurs. Hence, there can be no inductive effects in system with mas but no charge. If inductive effects are absent in gravitation, then there can be no circuit equations, like those of Maxwell, that would lead to a wave equation. And no wave equation there can be no gravitational waves.
Just because the Coriolis force has the same form as the magnetic component of the Lorentz force doesn’t make member of the same family. The difference lies in the velocities in those expressions as well as to the mechanism wherby mass is related to the EM fields not to non-inertial fields. Velocities in rotating frames of reference disappear in inertial frames.
As a final point, the uniform rotating disc was the primary thorn in Einstein’s general theory of relativity. According to where the observer is located on the disc, Einstein contended that he/she should be able to discern his/her position with respect to the center by the time kept by his/her watch. Since the disc does not belong to the Euclidean plane, it proves impossible for the observer to determine where he/she is because as he/she approaches the rim of the disc, his/her rulers and clocks shrink along with him/her. The uniformly rotating disc corresponds to the hyperbolic plane of constant curvature, where velocities just don’t add, and, thus, there is no rotational analog to rectilinear phenomena such as contraction and time dilatation. Moreover, it is a domain where Einstein’s principle of equivalence does not apply.