Sorry, I point out a mistake where 'm' should be replaced by 'md' in last sentence as ".....assumed that the value of m (should be md) is very less or say almost zero."

Hi Bishal

Let me try to understand your questioning to try to help to avance in the discussion: when you talk about the equivalence you certainly know that we have to make a distintion between the weak and strong equivalence principles, rigth?

The distinction is already existed, this question is addressed to the both kind. And here, it is asked from the sense of WEP. In one sentence, the free fall concept is actually not self-sustained that it has ignored some terms which questions on its foundation; one is the usual tidal effect based on gravity variance in terms of acceleration but this question actually seeks comparison of falling accelerations between two falling masses and the usual way of notation of free fall.

Dear Bishal

I like the following formulation of WEP (Di Casola, Liberati, Sonego):

Weak equivalence principle (WEP): Test particles with negligible self-gravity behave, in a gravitational field, independently of their properties.

Gravitational weak equivalence principle (GWEP): Test particles behave, in a gravitational field and in vacuum, independently of their properties.

Accordingly, GWEP includes the WEP in the limit the ratio between the gravitational energy of a body (Gm_g^2/r) and its rest energy (mc^2) -> zero. m_g is the gravitational mass and m the rest mass of the particle.

The formulation of the GWEP explicit, differently from WEP, the independence of the behavior of two test particles not only under the action of a gravitational field but also on vacuum (an empty background).

The assumption: In a gravitational field all bodies fall with the same acceleration regardless of their mass and composition, which corresponds to the Universality of Free Fall (UFF) or the Weak Equivalence Principle (WEP), have been tested for different materials in the field of the Earth and the Sun. The tests have found no violation on the level of 10^-13 for the quantity ∆a/a, where ∆a denotes the difference in acceleration between two test bodies falling in a gravitational field; a represents the mean acceleration associated to the two test particles. Accordingly, given any experimental apparatus, the larger the driving acceleration, the more sensitive the UFF/WEP test will be.

Of course, the WEP is a fundamental postulate, so it seen to me very important to push its limits by higher orders of magnitude through new experiments and new theoretical formulations.

With respect to this point, Nobili and Anselmi express that “An experiment capable of testing UFF to extremely high precision can potentially break this deadlock. The situation is reminiscent of that at the end of the 19th century, when Michelson and Morley tested by very precise light interferometry the propagation of the newly discovered electromagnetic waves through the hypothetical ether. Their null experimental result showed beyond question that although its existence was generally assumed, there was in fact no ether; which led to the special theory of relativity. While Michelson and Morley knew which precision, their interferometer had to achieve in order to detect the relative velocity between the Earth and the ether, we do not know which precision a test of UFF-WEP should reach to detect a violation, if any.”

The cancellation of gravity in a freely falling frame holds locally for each frame. However, the gravitational field of a body may be non-uniform. This may give rise to tidal forces between test bodies if their centers of mass do not coincide. So, tests on this aspect are welcome. However, should remember that most tests are limited by the non-uniformity of the gravity field of physical bodies in the universe.

According to Hall, “An alternative statement of WEP is that the trajectory of a freely falling body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, ...”

Anyway, tests have shown that the WEP is accurate to within one part in 10 trillion, or an uncertainty of 10-13 of the acceleration of gravity. Test of accuracy of WEP to an uncertainty of 10-17 are underway. Probably, at this level we are reaching another limit of validity of GR. A new physics may be needed.

The one thing that is not mentioned in my post is the time difference of falling such masses which by usual way is shown almost zero performing the experiments at very low height difference (from Galileo to now) but we have not tested it with very large height difference. Talking to WEP and its violation I would suggest to read my paper Preprint A new course of relative acceleration within Classical regime

Moreover, the any kind of experiment we performed to test WEP is quite not acceptable as inertial mass is linked to gravity by the application of centripetal force which itself is not a true inertial force rather a kinetic force that takes central reference. And the reason to rise of relational mechanics.

Hi

I saw. It's about Newtonian physics. Nothing to do with GR.

Of course, the paper is almost about the Newtonian physics. What I was intended to mean is, see the impact of my paper to the definition of inertial mass in terms of inertial force execution or say straightly, there is no true existence of inertial force that a body experiences in complete ideal space. The freefall would be not of inertial rather as a consequence of gravitational effect supporting the Mach's idea. As Einstein himself said that his theory did not included the Mach's principle as he was intended to include Mach's principle as his basic foundation of GR, and many physicist now working to include this principle in GR through relational mechanics.

I suggest you read " 3.1 Ful llment on mass-discrepancy|alternative to MOND" section specially how it violet the WEP taking consideration of usual way to link centripetal force for inertial type of force.