I don't see what you mean by change of gravitational mass because of magnetic fields (at least not without utterly unobtainable magnetar like magnetic fields), but I can certainly see how even minute magnetisation effects in a strong magnetic field (by earth standards) could provide additional torques on a suspended test mass that would make Eotvos type experiments even more difficult than they are without a magnetic field. I would be interested to know how you think strong magnetic fields can validate the Eotvos experiment.

General relativity describes the interaction of matter and gravity as the coupling between the matter stress-energy tensor and the metric. Magnetic fields couple to the metric only through combinations that are symmetric tensors  of the electromagnetic field, since the electromagnetic field strength tensor is antisymmetric and can't couple directly to the metric. One such example is the stress-energy tensor, so it is, of course, of interest to calibrate the contribution of any other tensor combinations. 

The Eötvös experiment is non-trivial, precisely because it checks that the only matter properties that are sensitive to gravity, i.e. couple to the metric, are described by the matter stress-energy tensor and nothing else, in the Newtonian limit, when the statement becomes the equality of inertial and gravitational mass.

In Newtonian gravity the equality of gravitational mass (that enters in the expression for gravitational attraction) and inertial mass (that enters in the expression of Newton's second law) is a postulate; in general relativity the postulate is that matter couples to gravity only through the invariant combination of the product of the metric and the matter stress-energy tensor. In Newtonian gravity, however, it's not easy to parametrize deviations-in general relativity it's much easier: it suffices to introduce fields, beyond the metric, that can"t be eliminated from the Einstein-Hilbert action by field redefinitions. One example is scalar-tensor theories; another is supergravity theories.

It seems to me that similar problems were discussed in : Thorne K.S., Lightman A.P.,Lee D. The foundation of theory of gravitational theories // Phys.Rev.D 1973,vol.7,pp.3563-3571. The authors had argued that the Eotvos-Braginsky experiments force us to accept that gravity is a metric phenomenon; in particular that matter fields couple to gravity ONLY through the invariant combination of the product of the metric and the matter stress-energy tensor.

The Eötvös experiment isn't trivial, because there isn't any reason why the gravitational mass and the inertial mass of any object should be equal-it's the Eötvös experiment that  shows that this is so. Newtonian mechanics and gravity are consistent for any value of this ratio. 

General relativity assumes this equality, in that matter couples to gravity only through the energy-momentum tensor. There's a priori nothing that would prohibit that matter couple to gravity through other symmetric tensors, functions of the matter fields. That has consequences that can be checked. And it can be shown that the Newtonian limit of general relativity does lead to the equality of gravitational and inertial mass. So a different result of the Eötvös experiment would have consequences, also, for general relativity, in that it would indicate that additional fields, beyond the metric, determine spacetime geometry. Such fields have been studied: scalar fields, in scalar-tensor theories and scalar, spinor and vector fields, in supergravity. What is called gravimagnetism is, of course misleading, since the vector field that is involved is not the field that mediates electromagnetic forces. It's a vector field of the supergravity multiplet, that was called the ``graviphoton'', by Joël  Scherk, who introduced the name ``graviphoton'': http://inspirehep.net/record/142417 since as a spin-1 field, it would mediate repulsive interactions. 

This would not mean that general relativity is inconsistent-it would mean that it is incomplete. That's expected-what's not known is how it can be completed, already, as a classical theory.

GR isn't proven by the result of the Eötvös experiment, that was realized before GR was invented-it's consistent with the result of the E¨ötvös experiment. It isn't based on the bending of light and on the Newtonian limit-it predicts the bending of light and possesses a non-relativistic limit that's consistent with the Newtonian limit. Mass is a gauge invariant quantity in general relativity. Once more, in Newtonian mechanics when one does the Eötvös experiment one is measuring m_inertial/m_gravitational for different material compositions. There's no a priori theoretical constraint that implies that this ratio is equal to 1 for any given substance, nor that it is equal for different substances; it can't be calculated within Newtonian mechanics.  It's an experimental fact that this ratio has the same value  for different substances and is equal to 1-the two statements are independent, since the ratio is a dimensionless quantity.  General relativity is consistent with it being the same for different substances and for it being equal to 1. 

There were many attempts to describe Newtonian gravity and electrostatics, in particular,  in a common framework and they are, now, recognized as misguided-they didn't take relativity into account, for starters. The work of Kaluza and Klein tried to describe such a framework, but, also, wasn't successful, because the vector field isn't the same as that for electromagnetism. Now the approach of Kaluza and Klein is used in other contexts.

 The correct way to include additional fields in the description of spacetime was found by Brans and Dicke in the 1960s using a scalar field and by Scherk in the 1970s using supergravity. And the vector fields don't have anything to do with the photon. 

I think that in strong electric or a magnetic field a rate of (atomic) clocks should be less, than in weak field..

 “Change of the course of time in force field and imponderability”, V.P. Oleinik, Physics of Consciousness and Life,Cosmology and Astrophysics ,2, с.20-37, 2001.

 Abstract: It is shown that the force in relativistic mechanics is not only the cause of acceleration of particle relative to an inertial frame of reference, but also the cause of change of the course of time along the particle's trajectory. Therein lies the physical content of the dynamical principle underlying special relativity (relativistic mechanics). The general formula for the relative course of time between the points lying on the trajectory of motion of particle under the action of a force field in an inertial reference frame is derived. The applications of the theory developed to homogeneous fields - to the field of gravity and electromagnetic field and to the gravitational field produced by a point mass particle are considered. Physical properties of the state of imponderability of particle in an external force field are investigated.

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Я полагаю, что в сильном электрическом или магнитном поле темп хода (атомных) часов должен быть меньше, чем в слабом.

«Изменение хода времени в силовом поле и невесомость» В.П. Олейник, Физика сознания и жизни, космология и астрофизика, 2, с.20-37, 2001.

 Аннотация. Показано, что в релятивистской механике сила является не только причиной ускорения частицы относительно инерциальной системы отсчета, но и причиной изменения хода времени вдоль траектории движения частицы. В этом состоит физическое содержание динамического принципа, лежащего в основе специальной теории относительности (релятивистской механики). Выведена общая формула для относительного хода времени между точками, лежащими на траектории движения частицы под действием силового поля в инерциальной системе отсчета. Рассмотрено приложение теории к однородным полям - полю силы тяжести и электромагнитному полю и к гравитационному полю, порождаемому точечной массой. Исследованы физические свойства состояния невесомости частицы во внешнем силовом поле. Отмечается, что изменение хода времени в силовом поле никак не связано с изменением метрики пространства-времени и является прямым следствием принципа причинности релятивистской механики.

@Thiery de Mees. Gravito magnetism is just a linearised form of general relativity valid in the weak field limit and . (see e.g. http://arxiv.org/pdf/gr-qc/0311030.pdf)

@Stam Nicholis. The inertial vs. gravitational constant of proportionality is not one but the square root of the gravitational constant (perhaps with an extra factor 4pi thrown in by convention).  Of course this is a matter of conventional choice of unit (i.e. not working in some form of Planck units by defining G away as a number not unlike c), but in the proposed new definition of the kg (which takes the definition of a second as defined by the fine structure transition of Cs rather than Planck time) it is a constant that has to be determined experimentally.

@Thiery de Mees.  I am afraid your annex does not convince me. More to the point,  nobody is fooling you, nobody is telling you a lie and was it really necessary to  SHOUT?  Please keep this a civilised forum.

Gravimagnetism relies on the correspondence between the Lorentz force equation and the equation of motion of a particle in a rotating frame: indeed the Coriolis term for mechanics does correspond to the contribution of the magnetic field in electromagnetism, and it's a standard exercise to show that it's possible to determine the Coriolis force contribution by imposing a known magnetic field. However there exists an additional term in the force law for a rotating frame, that's proportional to Ωx(Ωxr), that does not have a magnetic analog so the correspondence doesn't work.  There isn't an analog of, Newtonian, frame dragging in magnetism. Additionally, in the Lorentz equation and the Coriolis force equation, B and Ω respectively are external fields and radiation effects are not taken into account. 

The equations of motion for B (and E, that in the equation for the Coriolis force correspond to the acceleration term) of course, are the Maxwell equations and, of course, the Coriolis force is obtained in the Newtonian limit of general relativity, as well as from any generalizations, that include additional fields, beyond the metric, to describe spacetime. And any apparent similarities between the expressions of  the forces, in the non-relativistic limit, are overshadowed by the consequences of relativistic invariance, that implies that excitations of odd spin mediate repulsive forces and excitations of even spin mediate attractive forces. 

There are several experiments ongoing that are testing or will test the equivalence principle, using cold atoms, confined in electromagnetic traps.

A review of tests of the equivalence principle is available here: http://relativity.livingreviews.org/open?pubNo=lrr-2006-3&page=articlesu1.html

Once more, the issue isn't that general relativity represents the final word on the description of gravitational phenomena-it is to understand that it is consistent with the  Newtonian description in the appropriate limit and to try to understand what happens, when the approximation it represents isn't applicable. Gravimagnetism represents an imperfect analogy for, not a consistent description of,  Newtonian gravity and the analogy is used all the time in that context, for experiments, since it's possible to control magnetic fields very precisely, so it's nothing extraordinary as such.  It's not called gravimagnetism, of course-for the fact that forces, acting on the same body, can be added does not mean that they couple to the same properties of the body in question. 

That magnetic fields, especially non-uniform fields, can exert forces on macroscopic bodies in ways that does depend on their  composition is, of course, well known and is what's behind a toy called the Levitron. Frogs have been measured to levitate in a strong magnetic field. Therefore introducing any such magnetic field, inevitably, will introduce systematic effects, that must, of course, be calculated, as background to any possible violations of the equivalence principle, that doesn't involve magnetism.

Use of the strong magnetic field changes nothing because this converts one  well posed problem into another one.  This is not an engineering.This is science. Theorems are being proven given such and such conditions.Change the conditions, and the theorem no longer survives. It is being replaced by another theorem. So what?  Another one, whatever its proof could be, is not adding or subtracting a thing from the Eotvos experimental outcome.

Precisely, the ``mathematical form'' is not the same-there are terms present in the expression for the Coriolis force that are absent in the Lorentz force. And all this is known and is the material of standard exercises in undergraduate physics courses. So there's nothing new mathematically about ``gyromagnetism'', or ``gravitomagnetism'' nor is there any new physical content to that mathematical approximation. The fact that the Coriolis force can be canceled by an appropriate magnetic field doesn't imply anything particular about the Coriolis force itself and nothing that can lead to any insight about the Eötvös experiment. If one does arrange the magnetic field to cancel the Coriolis contribution, there remain two terms: the gravitational term, proportional to (m_g/m_i)g and the dragging term Ωx(Ωxr). Arranging an electric field can then help in isolating one and/or the other. This is all standard experimental procedure. Assuming, of course the fields are static-if they're not, as the experiments on levitating macroscopic objects show, effects appear, that are understood, although amusing to apprehend. The theoretical  basis is, however, known and understood: http://www.levitron.com/physics.html

There's more to non-inertial motion than the Coriolis force, however and the material can be found in textbooks on mechanics and fluid mechanics on vortex motion. That it's possible to express the same approximations in many different ways doesn't mean that they're distinct. So the fact that it's possible to write expressions in the Newtonian limit, in many different ways, doesn't mean that they are anything beyond  a different parametrization of this limit. Making exaggerated claims, however doesn't help. 

The Eötvös experiment can be understood as illustrating the equivalence principle and how that happens is, now, known and can be used to calibrate new experiments.  The way to look for deviations isn't by including explicit effects, like a magnetic field, that could *mask* any deviations, however, for other reasons, that don't have anything to do with the equivalence principle. 

Of course: for the ``partial'' resemblance leads to predictions that certain effects are present or certain effects should be absent-that's what ``resemblances'' are good for, to facilitate the search for new effects.  And these can be tested. It's no issue that non-inertial motion is very interesting-just that it isn't described by  the analogs of Maxwell's equations. Electric charges don't have any equivalence principle-the response of matter to an electric and magnetic field is through the product of current and density and of  the individual value of the electric charge, with the corresponding potentials, not, just on the current and density.  In the expression for the Lorentz force appear both mass and charge, not charge alone, for instance. So were this analogy useful, it would lead to quite strong violations of the equivalence principle. Gravity has this special property, that matter's gravitational properties are described  through the corresponding current, the  stress-energy tensor, with a unique coupling constant, Newton's constant, whatever the matter distribution. The Newtonian limit then can recover all known effects. That's how it's possible to understand the Eötvös experiment, in particular.

Already the fact that the current for gravitational effects is a tensor implies that the gravitational field isn't a vector, like in electromagnetism, it's a, symmetric, tensor. Only its scalar part plays a role in the Newtonian limit, hence Poisson's equation for describing gravitational dynamics. Everything else is an application to fluid mechanics. 

Indeed, what's happened is that it can be shown that magnetic systems can be described by equations that are identical to the equations of motion of particles on curved manifolds, whose dynamics is described by the equations deduced from general relativity: The magnetization vector corresponds to the position vector of the fictitious particle. 

Dear Zi-Hua! What do you mean saying "strong" magnetic field? Is the field with H about 10^16 Gauss strong, or intermediate, or weak? (The Earth magnetic field is about 0.6 Gauss in average).

Please, be more accurate when you are saying "big/small", "strong/weak", "long/short" etc: give a scale!

It doesn't matter what the strength of the magnetic field is, in principle. The only effect a magnetic field can have, that's relevant for this discussion, in practice,  is in realizing levitation effects-cf. http://www.physics.ucla.edu/marty/diamag/diajap00.pdf  due to the known magnetic properties of matter. These do depend on the composition of the object. However, since the idea is to investigate, precisely, these effects, it doesn't appear to be useful to introduce a magnetic field on purpose-it can only mask any deviations from the equivalence principle,  since any effects the composition of the masses might have will be, precisely, masked by the effects due to the presence of the magnetic field. It can only serve for calibration purposes of the measurement.

There isn't any second (or first) field of gravity: gravity, in general relativity, is described by a symmetric tensor of rank 2, and, in the Newtonian limit, by a scalar, the scalar potential. The evolution of stars depends on their equation of state, that describes the energy-momentum tensor of matter, that composes the star. Cf. http://www.jb.man.ac.uk/~smao/starHtml/equationState.pdf

It is there that, for instance, electromagnetic interactions of the constituents of matter are taken into account. And, of course, they affect gravitational properties.  However many different equations of state can define the same energy-momentum tensor. So the equivalence principle can be described as the statement that all equations of state that give rise to the same energy-momentum tensor, affect spacetime the same way. 

@Thiery de Mees. The evolution of the sun towards a red giant has nothing to do with gravitomagnetism, but is the result of changes in burn rate and burning process in  the sun  due to changes in changing chemical composition and mass distribution after billlions of years of fusion burns through a significant part of the hydrogen.