I think it is best to use a more precise formulation of the weak equivalence principle instead of focusing on possible misleading apparent paradoxes. The weak equivalence principle amounts to the statement that the gravitational behavior of a system is fully determined by the system's stress-energy-momentum tensor, and does not depend on properties not captured in that tensor, such as material composition, electric charge, color or whatever. Very obviously, the stress-energy-momentum tensor of the Earth-Moon system is different from the stress-energy-momentum tensor of the Earth-test body system, so it is unsurprising that their gravitational behavior differs, too. But if I were to replace the Moon with a body made from different materials but having the same stress-energy-momentum tensor (never mind how feasible it is to perform this in reality) the gravitational behavior of the system will not change.

The classic WEP experiments (e.g., astronaut dropping a feather and a rock on the Moon's surface, watching them fall at the same rate) work in part because the test bodies have negligible masses, so their contribution to the system's stress-energy-momentum tensor can be ignored. But in this particular case, there is also another reason: If you drop the feather and the rock at the same time, the tiny amount by which the rock attracts the Moon will move the Moon ever so slightly closer, but the rock and the feather would continue to approach the Moon at the same rate. The feather will hit the surface at the same time as the rock, but it will hit the surface an a tiny bit sooner than it would have if the rock were absent.

The classic WEP experiments (e.g., astronaut dropping a feather and a rock on the Moon's surface, watching them fall at the same rate) work in part because the test bodies have negligible masses, so their contribution to the system's stress-energy-momentum tensor can be ignored.

I agree

Same masses should not alter the stress tensor appreciably so that the WEP should hold. 

So according to GR and WEP there cannot be any alteration of a geodesic motion according to internal dynamics of bodies.

"So according to GR and WEP there cannot be any alteration of a geodesic motion according to internal dynamics of bodies" -- No, that is really not true, except in very limited cases (e.g., spherically symmetric body, and "internal dynamics" that does not change the spherical symmetry.) In the general case, what you call the "internal dynamics" would change the gravitational multipole moments of a body, hence its interaction with other bodies. As an example, lunar gravity causes the Earth to deform, changing its multipole moments that, in turn, alter the Moon's orbit. (Think of the multipole moments as a particularly convenient representation of the stress-energy-momentum tensor of an approximately spherical body that is non-relativistic, so its stress-energy-momentum tensor is dominated by its mass distribution, with negligible contributions from internal momenta, stresses and pressure.)

In the general case, what you call the "internal dynamics" would change the gravitational multipole moments of a body, hence its interaction with other bodies.

Ok. But the internal dynamics occurring in a test body (black box) should be negligible in the dynamics of the test body itself. Two 10 cubic meter cannisters should fall on earth at the same speed regardless of what is inside them?

As an example, lunar gravity causes the Earth to deform, changing its multipole moments that, in turn, alter the Moon's orbit.

tidal locking for example

Best to read:

Borchardt, Glenn, and Puetz, Stephen J., 2012, Neomechanical gravitation theory, in Volk, G., ed., Proceedings of the Natural Philosophy Alliance, 19th Conference of the NPA, 25-28 July: Albuquerque, NM, Natural Philosophy Alliance, Mt. Airy, MD, p. 53-58. [10.13140/RG.2.1.3991.0483]

Dear Stefano, you ask very good question.

It can be strictly shown (see my paper in the journal Gravitation and Cosmology, v.8, p.232, 2002 (also in arXiv: gr-qc/0010056) "Translational motion of rotating bodies and tests of the equivalence principle") that such internal structure as rotation of a body will violate the geodesic motion and the free fall of the body depends on the orientation of the rotation axis, though the value of the acceleration deviation is very small ~ dg/g = (v_rot / c )^2. Intriguingly, such test of the WEP has not been performed yet, but will be done soon in the binary systems with pulsars.

Dear Yurij,

thanks a lot. I knew from you something regarding rotating bodies which could violate the WEP.  In such case though the stress tensor, as Victor says, should vary such that some kinetic energy of the rotating body does not have the same gravitational property of non rotating mass.

I would like to check with you something which is quite important in order to unveil the free falling properties of bodies. According to EEP and WEP a free falling system behaves like an isolated system: no intrinsic dynamic should be able to make any difference. 

Two cases will be presented: the first is the  INERTIAL CASE 

INERTIAL CASE

1) two black boxes of equal mass are FIRST and SECOND

2) these boxes are  cilinders with height H with base A and base B

3) At the extreme A of each box there is some positrons and electrons separated by some electromagnetic confinement system. Most of the mass of the box is composed by such matter/antimatter so that most of the mass M is concentrated in A. The center of mass is close to A.

4) It is possible to perform the annihilation in A at once and generate a flux of photons which are totally absorbed in B without breaking or deforming permanently the box (this is a possibility over 10^50 but still possilble).

The center of mass of the SECOND box will move inside the box, both box will proceed at constant speed regardless of what occurs inside.

So far I did not say anything particularly strange or new. It is a Gedanken Experiment.

Dear Thierry,

I'm afraid it is not so simple. A massive object with a spring could not be suitable to perform such thing because it would feel gravitation unlike a flying photon.

"Universality of free fall" is based on Newton's weak equivalence principle (WEP) which states: "the property of a body called mass is proportional to the weight." In WEP Newton did not specify whether the mass is the rest mass or relativistic mass and no discussion of motion. In Einstein's notion, free fall indicates inertial mass as well as relativistic mass but again in consideration of Eotvos experiment only inertial mass is equated to gravitational mass.Besides, Eotvos experiment is a static experiment which does not involve moving masses.

But how does the assumption that the gravitational attraction exists between the relativistic masses reflect on the equivalence principle? It appears that this assumption does not violate any of the three equivalence principles, the weak (WEP), Einstein (EEP) and the strong (SEP).

The theory given below conforms with Eotvos experiment when two gravitating bodies are in equilibrium and at rest in the same coordinate system. For example, if an object is thrown radially upwards in the coordinate system of the earth, it will attain a maximum height and then freely fall back to the earth. Momentarily when the object is at the maximum height, both the object and the earth are perfectly at rest in the same coordinate system. At this moment, the relativistic mass of the object is the same as its rest mass. Therefore at this moment of equilibrium, the gravitational mass is equal to inertial mass. Rest of the time it is the relativistic mass which is equal to the gravitational mass. This is the dynamic version of WEP introduced in the theory given below which states that the gravitational mass of a body is equal to its relativistic mass. Whether one is working with the coordinate time of the central potential or with the proper time of the orbiting body, one of the two masses would certainly be the relativistic mass. So this factor of relativistic mass needs to be considered.

I think the earth moon system does fall towards each other as per the WEP but the free fall forces are counterbalanced by the oscillatory centrefugal and centrepetal forces due to the kinetic energy of the orbital motion. This is what causes the elliptical orbit.

When we talk about gravitational force depending on the composition and natural frequency of the body we may feel inclined to think that it would violate the WEP of Newton and Einstein. If we look at the proposal from the point of view of the dynamic WEP which states that the gravitational mass is equal to the relativistic mass, we find that there is no violation because the gravitational frequency shift of massive body would cause its relativistic mass to alter and not its rest mass. This alteration of the relativistic mass is reflected in its motion, i.e. its velocity. If one is determined to test the static WEP with respect to the gravitational frequency shift proposal, one should have two very massive objects of different composition but exactly the same inertial (rest) mass subjected to a very large gravitational gradient over a distance of few AU or few kpc and then there is a chance that one would discover both objects experiencing different gravitational force.

Article Periodic relativity: Basic framework of the theory

Dear Vikram,

Eotvos experiment is a static experiment which does not involve moving masses.

Yesssss

The theory given below conforms with Eotvos experiment when two gravitating bodies are in equilibrium and at rest in the same coordinate system. For example, if an object is thrown radially upwards in the coordinate system of the earth, it will attain a maximum height and then freely fall back to the earth. Momentarily when the object is at the maximum height, both the object and the earth are perfectly at rest in the same coordinate system.

I agree

The issues is double:

a) regarding the relavitistic momentum, or relativistic energy which is not considered in the Schwartzschild solution in radial free falling.

b) regarding the local violation of the wep due to internal dynamics bodies.

Dear Stefano, you got it right. I don't know what Thierry is trying to say. He should elaborate a little more.

I agree with Thierry: only relativistic momentum and energy exist. The relativistic mass is something which does not exist..

I also agree with both of you. Relativistic mass is simply the measure of kinetic energy of the particle added to its rest mass. But in my case all the derivatives of the position are zero at the instance when the particle is at rest. Secondly relativistic mass has nothing to do with the longitudinal and transversal gravity fields. Relativistic mass has to do with the fact that gravity not only acts on the masses but also on the energy. It can be kinetic energy or it can be energy of electric charge. This is why gravitational mass is equal to relativistic mass in my theory.

Vikram,

I don't know if a material body which goes close to the speed of light will have the same gravitational mass as its inertial mass...

What would it be its inertial mass?? 

I have  my doubt that the gravitational interaction works the same for very high kinetic energies.. there is a lot to talk about the retarded potentials..

Material body moving at speed close to light cannot have the same gravitational mass as its inertial mass. But it will have a same gravitational mass as its relativistic mass. I have demonstrated this for light in the attached article which has a zero rest mass. In this, gravitational redshift has been calculated using the relativistic mass in equation 4.4 and same result as GRT is obtained in eqn. 4.7 without using Einstein's field equations. And bending of light is calculated using relativistic mass in eqn. 4.16 and same result as GRT is obtained in eqn. 4.17 without using Einstein's field equations. Light has only kinetic energy and gravity attracts this kinetic energy through relativistic mass. So in my theory gravitational interaction works the same for very high kinetic energy because of the use of relativistic mass. But it will not work the same in GRT because it uses the inertial mass. Therefore to fix this problem perhaps you may need to look at retarded potentials. But these are all patchwork with many approximations. Retarded potentials are usually used in linearized general relativity and gravitational waves.

Article Periodic relativity: Basic framework of the theory

Vikram,

"But it will not work the same in GRT because it uses the inertial mass."

here I agree..

but I have also say that the SCHW. solution in GR for circular orbits is very accurate I don't know if it can be outperformed. In such case though there is not an energy exchange between the two bodies like in the radial free falling case.

"Therefore to fix this problem perhaps you may need to look at retarded potentials."

the relativistic energy /momentum  has certainly a diverging weight in term of  inertia, it is very clear in accelerators. It is not so clear what happens to the gravitational interaction though.

Stefano:

SCHW. solution in GR for circular orbits is very accurate. This is true for weak fields. At stronger gravitational field I have noticed some deviation from the limiting radius of black hole in GR. See eqns. 4.17 to 4.23 in my article. However this requires experimental verification.

Dear Vikram,

you may be right that for strong fields even in circular orbits SCHW solution may be not be accurate, but the GR is in these cases. In such cases infact the geodesic motion occurs without any changing in relative mass/energy of the bodies, the metric does not change and everything should work fine. But this is only an ideal situation which never occurs, this is why GR is just an approximation of actual gravitation and in some cases can have very low accuracy.

This is to be discussed.

Article The incompatibility of Einstein EP with the local Gravitatio...

A discussion on this matter is made in this paper I recently wrote,

please have a look.

Article A revisitation of the Schiff's derivation of the time dilati...