Let us consider a mass M of radius R_o (Fig. 1).
   At a distance h from the center of this mass M, we place a spherical shell of mass m₁ and radius R.
   Moreover, at the center of the spherical shell we place another point mass m₂, (m₁ ≠ m₂).
   Now t= 0 (Phase I), we allow these three masses m₁, m₂, M to move freely under the influence of the force of universal attraction.
   Let us also assume that after a time dt (Phase ΙΙ), υ₁, υ₂ and V are respectively the velocities of masses m₁, m₂ and Μ, relative to an inertial observer Ο.
   Note: Masses m₁, m_2, and M are considered to be homogeneous and absolutely solid bodies.
   In addition, velocities υ₁, υ₂ and V are considered to be positive numbers (that is, only the meters of their magnitudes are taken into account), while mass m₂ is considered to be found always within the spherical shell m_1.

   The basic question that is being raised is the following:
   At what velocities do masses m₁ και m₂ fall in the gravitational field of Mass M once simultaneously dropped in free fall from a height h? 
   The answer to this question is given below:

PROOF

  CASE Ι: When masses m₁ and m₂ are equal (m₁ = m₂).
   In this case, and at time t = 0, in accordance with Newton’s first law on universal attraction, mass M attracts mass m₁ by a force F.
   Also, mass Μ attracts mass m₂ by a force F΄.
   Therefore,

   However, given that masses m₁ and m₂ are equal, then forces F και F΄ whereby mass M attracts masses m₁ and m₂ will also be equal, that is:

F = F΄         (4)

Therefore, relations (1), (2), (3) και (4), yield:

   From relation (5), since masses m₁ and m₂ are equal (m₁ = m₂), it results that:

                                 υ₂ = υ₁                           (6)

   Thus, based on what precedes, we are led to the following conclusion:
   Two equal masses m₁ and m₂, when simultaneously dropped in free fall from a height h in the gravitational field of a mass M, fall always at the same velocities υ₁ and υ₂, (υ₁ = υ₂).

   CASE ΙΙ: When masses m₁ and m₂ are unequal, (m₁ ≠ m₂).
   As demonstrated in Case Ι, when masses m₁ and m₂ are equal (m₁=m₂), they always fall at the same velocity (υ₁ = υ₂), in the gravitational field of a mass M.
   The question that is being raised is the following:
   If the two masses m₁ and m₂ are different (m₁ ≠ m₂), then which of the two falls at greater velocity in the gravitational field of a mass M, the smaller mass or the larger one?
   The answer to this question is given below:
   Let us assume that from the two masses m₁ and m₂, m₁ < m₂.
   a) In this case and at time t = 0, in accordance with Newton’s first law on universal attraction, mass M attracts mass m₁ by a force F and mass m₁ also attracts mass Μ by an equal force F (based on Newton’s third law on action - reaction).
   Therefore, in this case, the following relations will apply for both masses m₁ and Μ:

                                       F dt = m₁ dυ              (7)
                   and F dt = M dV₁΄            (8)
       where, dυ = (υ₁ – 0)  and dV₁΄ = (V₁ – 0)      (9)

From relations (7), (8) and (9) it results that:

                             m₁ υ₁ = ΜV₁         (10)

 b) Similarly, mass M attracts mass m₂ by a force F΄ and mass m₂ also attracts mass Μ by the same force F΄.
   Thus, in this case, the following relations will apply for both masses m₂ and Μ:

     F΄ dt = m₁ dυ΄             (11)
and  F΄ dt = M dV₂΄        (12)
where, dυ΄ = (υ₂ – 0)  and dV₂΄ = (V₂ – 0)      (13)

Relations (11), (12) and (13) yield:

 m₂ υ₂ = ΜV₂               (14)

   Because, however, force F΄ is greater than force F, (F΄ > F), since attraction force F΄ between mass m₂ and mass M is greater than attraction force F between mass m₁ and mass M, given that m₁ < m₂, then from relations (8) and (12) it results that:

                           F dt = M dV₁΄            (15)
                   and F΄ dt = M dV₂΄            (16)

   and given that F΄ > F as stated above, relations (15) and (16), yield:

                       M dV₂΄ > M dV₁΄          (16.1)

  And based on relations (9) and (13), from relation (16.1) it results that:

                         V₂ > V₁                           (17)

   Therefore, the larger mass m₂ imparts to mass Μ greater velocity V₂, while the smaller mass m₁ imparts to mass Μ smaller velocity V₁.
   So, from relations (10) and (14) we obtain:

and because from relation (17) V₂ > V₁, then relations (18) and (19) yield:

   Relation (21) will from now onwards be called the “basic relation” in our problem.

   COMMENT: At this point, it should be stressed that by adding sides in relations (10) and (14), we obtain:

   where V is the resultant velocity which is simultaneously imparted to mass M by both masses m₁ and m₂.
   Specifically:

                             V = V₁ + V₂                       (23)

   As it can be observed, relation (22) expresses (in absolute numbers) “the principle of conservation of momentum” in the system of the three masses m₁, m₂ and Μ.
   Consequently, since in our problem m₁ < m₂, that is, m₁ / m₂ < 1, then relation (21) assumes only the following three forms:

   Let us now consider the above three cases (a),(b) and (c) :

   1.  In the first place, relation (24) is rejected because:
   According to the basic relation (21) in our problem:

   Now let us say:

   where, k = positive number (k < 1), that corresponds to two certain masses m₁ and m₂, (m₁ < m₂).
   (e.g. if m₁ = 5 kg and m₂ = 100 kg, then m₁ / m₂ = k = 5 / 100 = 0,05).
   Thus, from relation (26), we obtain:

As it can be observed, inequality (28), is verified for all values of fraction υ₂ / υ₁ greater than k, Fig. 1 (a).

   Therefore, since fraction υ₂ / υ₁ is verified for all values which are greater than k, then it will apparently be verified, too, for value υ₂ / υ₁ = k+ε, where ε positive number ε ® 0. Evidently, is k+ε<1.
   All the above apply for the basic relation (21) in our problem.

   2. However, in relation (24) fraction υ₂ / υ₁ is verified only for values greater than 1, Fig. 1(a).
   Therefore, relation (24) is never verified for value of fraction υ₂ / υ₁ =k+ε , (as it should occur according to the basic relation (21) in our problem).
   Thus, for this reason relation (24) is rejected.
   Similarly, for the same reason, relation (25.1) must also be rejected.
   So from the three relations (24), (25)  and (25.1) , relation (25) will necessarily apply, that is:

   Relation (30) reveals that fraction υ₂ / υ₁ of velocities υ₂ and υ₁ of the two unequal masses m₁ < m₂, during their free fall in the gravitational field of mass M, is always between fraction k of masses m₁ / m₂ = k and number 1.
   Relation (29) expresses the fundamental law of free fall of two unequal masses m₁ and m₂, (m₁ < m₂) in the gravitational field of mass M.
   According to the fundamental law of free fall of bodies given by relation (29), we observe that relative to an inertial observer O, the smaller mass m₁ falls at a greater velocity υ₁ and that the larger mass m₂ falls at a smaller velocity υ₂ in the gravitational field of mass M.
   NOTE: From three relations (24),(25) and (25.1) only relation (25), (namely the accepted relation) holds (simultaneously) for the three values m₁/m₂<1, υ₂/υ₁<1 and m₁/m₂>0, fig. (1.1)

fig. (1.1)

Relation (24) never holds (simultaneously) for the three values m₁/m₂<1, υ₂/υ₁>1 and m₁/m₂>0, fig. (1.2)

fig. (1.2)

Similarly, relation (25.1) never holds (simultaneously) for the three values m₁/m₂<1, υ₂/υ₁=1 and m₁/m₂>0, fig. (1.3)

fig. (1.3)

   Yet, in the case where the two masses  m₁ and m₂ are equal (m₁ = m₂), (and therefore, velocities υ₁ και υ₂ will also be the same (υ₂ = υ₁), as proven in Case I), the fundamental law of free fall is formulated as follows:

   Finally, as it can be noted, the fundamental law of free fall, relation (29), mentioned above, clashes with the all-too-familiar Galileo’s law of free fall, according to which “all bodies, irrespective of their mass, fall at the same velocity in the gravitational field of a mass M”.
   Consequently, on the basis of proof provided above, the law of free fall as formulated by Galileo is utterly erroneous.

   Example: If, for instance, we simultaneously let one tank (armored fighting vehicle) and a cork (e.g. from a bottle) fall freely from a height h above the surface of the Earth (Moon, Asteroid, etc), then the cork will fall at a greater velocity than the tank, and these two bodies will never fall at the same velocity as Galileo erroneously asserts based on his Leaning Tower of Pisa Experiment.

   COMMENT: In Fig. 1, another solution to the problem, that will reveal Galileo’s, as well as Einstein‘s great error (as regards the Equivalence Principle of the General Theory of Relativity), is to calculate quantitatively (numerically) relative to the inertial observer O velocities  υ₁, υ₂, V as functions of time t, that is:

when masses m₁, m₂, M, (m₁ ≠ m₂), as well as height h are known.
Apparently, velocities υ₁ and υ₂ that will result from relations (32) will be υ₁ ≠ υ₂.
NOTE: Any physicist, mathematician, university who/which wishes to formulate and publish the above three functions (32) is welcome to do so.

 2^nd PROOF FOR THE PROBLEM
(THE REVERSE PROOF)

In Fig. 1 after a time dt > 0 (phase ΙΙ, with mass m₂ being found at all times within the spherical shell ),m₁ m₂ , if we apply

1. The “principle of conservation of energy”, and
2. The “principle of conservation of momentum”,

to the system of the three masses m₁, m₂, M, we obtain:

where, G = theuniversalgravitationalconstant. 

Also, relations (R.1) can be written as follows:

where:

and

Β = MV                                 (R.4)

Also, Α > 0 and Β > 0..

Note: In relations (R.1), υ₁, υ₂, V, h, h₁, h₂ are considered to be positive numbers.
Also at time dt > 0, if the intervals covered by masses are m₁, m₂, M are dS₁,dS₂ ,dS respectively, then apparently the following relations are valid:

By solving the system of equations (R.2), relative to υ₁ and υ₂, we obtain:

where:

In relations (R.5) number Κ is apparently,

 Therefore, we will have two cases:
Ι. Κ = 0 or
ΙΙ. Κ > 0.

CASE Ι: When Κ = 0.

In this case, because Κ = 0, relations (R.5) yield:

Therefore, from relations (R.8) its results that:

Thus, when Κ = 0 then υ₁=υ₂.
Now we will prove that if υ₁=υ₂ , then m₁=m₂.

Proof

a. At first, we will prove that υ₁=υ₂ m₁υ₁=m₂υ₂ .
Hypothesis: Let us assume that if υ₁=υ₂ then relation m₁υ₁=m₂υ₂ .is valid.

According to Fig. 1 of our problem, the following relations are valid:

Furthermore, from relation  m₁υ₁=m₂υ₂ of the hypothesis we obtain:

On the basis of relation (R.11), relations (R.10) yield:

where F is the force of universal attraction between masses m₁ and Μ at time t = 0, namely:

and F’ is the force of universal attraction between masses m₁ and Μ at time t = 0, namely:

On the basis of relations (R.12.1) and (R.12.2), relation (R.12) yields:

However, (according to proof 1 of our problem), when masses m₁ and m₂ are equal (m₁=m₂), then the following will be valid:

Yet, the fact that relation (R.12.4), to which we have concluded, is in accordance/consistent with relation  υ₁=υ₂ of our hypothesis, signifies that this hypothesis is valid.
Therefore, it has been proven that υ₁=υ₂ m₁υ₁=m₂υ₂

b. We will then prove that m₁υ₁=m₂υ₂   m₁=m₂
In the above accepted relation m₁υ₁=m₂υ₂ by considering that υ₁=υ₂ (according to our hypothesis), we obtain:

Therefore, it has been proven that   m₁υ₁=m₂υ₂   m₁=m₂
On the basis of everything analyzed above, we come to the following conclusion:

Conclusion Ι

In our problem (Fig. 1), when Κ = 0 (i.e. υ₁=υ₂), then m₁υ₁=m₂υ₂ and m₁=m₂.
In other words, equal velocities correspond to equal masses.
More analytically:

CASE ΙΙ: Όταν είναι Κ > 0.
Because υ₁, υ₂, m₁ ,m₂ , Β, Κ, > 0 in relations (R.5), in the second of relations (R.5), number , is at all times a positive number, namely >0. Apparently, number , is a appositive number also, that is,   >0
Moreover, in relations (R.5), υ₁> υ₂,  namely υ₁- υ₂>0 , because:

Following calculations, relation (R.13) yields:

Given that the second side of relation (R.14) is a positive number, then:

Therefore, from relation (R.15) it results that if in relations (R.5) Κ > 0, then υ₁> υ₂.
Now we will prove that if υ₁> υ₂, then m₁< m₂.

Proof

a. At first, we will prove that υ₁υ₂ m₁υ₁m₂υ₂ .

Hypothesis: Let us assume that if υ₁υ₂ , then relation m₁υ₁=m₂υ₂ will be valid.
According to Fig. 1, the following relations will be valid:

Also, from relation  m₁υ₁₌m₂υ₂  of the hypothesis we obtain:

On the basis of relation (R.15.2), relations (R.15.1) yield:

where F is the force of universal attraction between masses m₁ and Μ at time t = 0, namely:

and F’ is the force of universal attraction between masses  m₂ and Μ at time t = 0, namely:

Based on relations (R.15.4) and (R.15.5) relation (R.15.3) yields:

However, (according to proof 1 of our problem), when masses m₁ = m₂ are equal, the following will be valid:

Nevertheless, the fact that relation (R.15.7), to which we have concluded, is not consistent with relation υ₁υ₂ in our hypothesis signifies that the hypothesis we have formulated is not valid.
Consequently, when υ₁υ₂, then relation m₁υ₁=m₂υ₂ is not valid. Thus, relation m₁υ₁m₂υ₂will necessarily be valid.
Therefore, it has been proven that , υ₁υ₂ m₁υ₁m₂υ_2.

b. We will then prove that , m₁υ₁<m₂υ₂ m₁<m₂.
In our problem, when K > 0, then υ₁>υ₂, will apply, i.e. υ₁υ₂ (according to relation (R.15) proven earlier). This signifies that
υ₁>υ₂ m₁υ₁m₂υ₂ , based on the conclusion
υ₁υ₂ m₁υ₁m₂υ₂ , referred to above.
Therefore, in the second of relations (R.2), i.e. in relation

(where B = MV), for the unequal positive numbers  m₁υ₁ and m₂υ₂, the following two cases will apply:

Let us now examine the two cases (1) and (2) individually.

b.1 In case (1):

If relation (R.15.11) is valid, then the following relation will be necessarily valid, as well: 

where ε is a positive number for which , .
Yet, according to our problem  υ₁>υ₂, namely:

which means that since relation (R.15.13) is valid, the following relation will also be valid:

From relations (R.15.12) and (R.15.14) we obtain:

However, in relation (R.15.15), number 1 – 2ε is, 1 – 2ε < 1.
Therefore, from relation (R.15.15) it results that:

Consequently, it has been proven that  m₁υ₁<m₂υ₂ m₁<m₂.

b.2 In case (2) we have:

So if relation (R.15.17) is valid, then the following relation will also be valid:

However, according to our problem υ_1>υ₂, namely:

This means that since relation (R.15.19) is valid, then the following relation will be valid, as well:

Relations (R.15.18) and (R.15.20) yield:

However, (as proven earlier – proof 1), when masses m₁ and m₂ are equal, (m₁ = m₂), the corresponding velocities υ₁ and υ₂ are equal, too (υ₁=υ₂); likewise for the corresponding momentums m₁υ₁ and m₂υ₂ are equal, too, namely:

Yet, relation (R.15.22), clashes with relation (R.15.17) of case (b.2) in which

Therefore, we come to the conclusion that case (b.2) is not valid.
Thus, of the two cases (b.1) and (b.2), only case (b.1) is valid, namely, ,m₁υ₁<m₂υ₂ m₁<m₂ .
After everything analyzed above, the following conclusion is drawn:

Conclusion ΙΙ

In our problem (Fig.1), when Κ > 0 (i.e. υ₁ > υ₂ ), then m₁υ₁<m₂υ₂ and m₁<m₂ .
In other words, unequal velocities correspond to unequal masses and the greater velocity corresponds to the smaller mass.
More analytically:

VERIFICATION

Α. Verifying Conclusion Ι: When Κ = 0 (i.e. υ₁=υ₂), then m₁=m₂.

Proof

The second of relations (R.2) yields:

And based on what has been proven earlier, when υ₁=υ₂, then:

Relations (R.16) and (R.16.1) yield

Therefore, in our case (Κ = 0), since on the basis of relation (R.9) υ₁=υ₂, relations  (R.16.2) and (R.16.3) will yield

Consequently, according to relation (R.16.4), Conclusion Ι is verified.

Β. Verifying Conclusion ΙΙ: When Κ > 0 (i.e. υ₁>υ₂), then m₁<m₂.

Proof

We earlier proved that when υ₁>υ₂, then:

Relations (R.5) yield:

Furthermore, relations (R.16.5) and (R.16.6) yield:

In relation (R.16.7), given that Κ > 0 and Β > 0, in order for relation (R.16.7) to be valid, we must have:

Therefore, according to relation (R.16.8), Conclusion ΙΙ is verified.

VARIOUS CONCLUSIONS

We earlier proved that when υ₁>υ₂, then:

Relation (R.17) successively yields:

However, given that the kinetic energy Ε₁ of mass m₁ is:

and the kinetic energy Ε₂ of mass m₂ is:

relations  (R.17.1), (R.17.2) and (R.17.3) yield:

In other words, the kinetic energy Ε₁ of the smaller mass _{m1 is} less than kinetic energy Ε₂ of the greater mass m₂, (m₁<m₂). 

NOTABLE REMARK

According to proof 1 of our problem (Fig. 1), it has been demonstrated that:
a. If the two masses are equal, then their velocities will be equal too; and
b. If the two masses are unequal, then their velocities will also be unequal, and the smaller mass will move at a greater speed.
Reversely, according to proof 2 of our problem (Fig. 1), it has been demonstrated that:
a. If the velocities of the two masses are equal, then the masses themselves will be equal, too; and
b. If the velocities of the two masses are unequal, then the masses will be unequal, too, and the greater velocity will correspond to the smaller mass.
Proof 2 will be called “reverse proof” of our problem.
Apparently, proof 1 and 2 are equivalent to each other, that is, they express the exact same outcome.
As it can be observed, the result of proof 2 is exactly the same as the one of proof 1.

RECAPITULATION

In our problem (Fig. 1), as proven earlier, the following apply:

Α. On the basis of proof 1.

CASE Ι: When masses m₁ and m₂, are equal m₁= m₂, then:
1. υ₁=υ₂
2. J₁=J₂  (J₁=m₁υ_{1 ,} J₂=m₂υ₂ )
3. Ε₁=Ε₂ 
where, υ₁,J₁ ,Ε₁ are respectively the velocity, momentum and kinetic energy of mass m₁ and υ₂,J₂ ,Ε₂ are respectively the velocity, momentum and kinetic energy of mass m₂.

CASE ΙΙ: When masses m₁ and m₂ are unequal, m₁< m₂, then:
1. υ₁>υ₂
2. J₁<J₂ 
3. Ε₁<Ε₂

Reversely:

Β. On the basis of proof 2.

CASE Ι: When velocities υ₁ and υ₂, are equal  υ₁=υ₂ (i.e. Κ = 0), then:
1. m₁=m₂
2. J₁=J₂ 
3. Ε₁=Ε₂

CASE ΙΙ: When velocities   υ₁ and υ₂ , are unequal υ₁ >υ₂ (i.e. Κ>0), then:
1. m₁<m₂
2. J₁<J₂ 
3. Ε₁<Ε₂

CONCLUSION

On the basis of proof 2 –and proof 1, as well– (Fig. 1) referred to above, it is clearly demonstrated that the gravitational fields (e.g. the gravitational fields of mass Μ) do not have the property of imparting to all bodies (irrespective of their mass) the same acceleration (e.g. to masses  m₁<m₂), as proven earlier.
Unfortunately, Galileo (in conducting the Tower of Pisa Experiment) and Einstein (in formulating the “equivalence principle”) maintain exactly the opposite. Consequently, it is proven beyond any doubt that Galileo and Einstein were totally wrong!

NOTE

1. Only the fields of inertial forces have the property of imparting the same acceleration to all bodies (irrespective of the latter’s mass); gravitational fields never do so, as it has been proven above (See Proof 1 & 2) 
This is where Einstein greatly erred in formulating the “equivalence principle” of the General Theory of Relativity.
2. As theoretically proven above, (Proof 1 & 2), the “basis” of the “equivalence principle” is anything but solid.
This signifies that there is no need for us to carry out any experiment at all (such as the Gravity Probe b Experiment, etc) in order to verify whether the Theory of Relativity is accurate or not.
The foregoing Proofs 1 & 2 alone demonstrate clearly and beyond any doubt that the equivalence principle (the “weak equivalence principle” and the “strong equivalence principle”) is wrong and therefore the Theory of Relativity is an erroneous theory of Physics. 

EXAMPLE

According to Conclusion ΙΙ formulated above, if e.g. we simultaneously let the Moon and a pin drop freely from a height h above the surface of the Earth, then the pin will strike the surface of the Earth sooner than the Moon!!!
(Note: For reasons of simplicity, the force of universal attraction between the Moon and the pin is considered to equal zero, while the masses of the Moon and the pin are considered to be point masses).

COMMENT

Honestly, I cannot help wondering why all the “great” physics professors fail to perceive these “simple things” explained above, and insist to this day on the accuracy of the “equivalence principle” of the General Theory of Relativity?!
Forgive me for saying so, but it seems that the blind loyalty of these “great” professors to Einstein and their adoration of him hinder them from realizing that Einstein erred!! Unfortunately, the “equivalence principle” as demonstrated above, is totally wrong. 
This is so simple that it can be clear to any physics freshman!

THE FUNDAMENTAL N-BODY LAW

LAW: A closed N-body system (), which under the influence of its internal gravitational forces moves at a time t (t>0) from a state Α over to another state Β, ( ) then, the interval run by each body, its speed, acceleration, momentum, etc are at all times a function of this body’s mass.

The above law is verified:
1. In the two-body problem (Ν=2), such as in the following cases:
   a. In the free-fall of a body within the gravitational field of another one.
   b. In the movement of double stars.
   c. In the movement of a planet around the Sun, etc.

2. In the three-body problem (Ν=3), such as in the following cases:
   a. The problem of Fig.1, according to the two proofs referred to above.
   b. The three-body (planet) system, e.g. Mercury-Venus-Earth, etc

3. In the N-body problem, such as in the following cases:
   a. The solar system (Ν=10)
   b. The movement of stars inside a galaxy, etc.

As it can be observed, Galileo’s Pisa Experiment is a partial case of the fundamental N-body law stated above.
Thus, assuming that Galileo let e.g. two bodies of different mass fall freely from the top of the Tower of Pisa, then Ν=3 in the above fundamental law (the third body is the Earth).
If, for example, Galileo let four bodies of different mass fall freely in the same way, then Ν=5 in the above fundamental law.
Also, if he let four bodies of different mass fall freely in the same way, then Ν=101 in the above fundamental law, and so forth…..

Therefore, according to the fundamental N-body law, in all the above cases the bodies will fall towards the Earth at a different speed, acceleration, momentum, etc, and they will never fall at the same speed as Galileo wrongly asserts.
Consequently, after everything stated above, it is quite clear that the Galileo’s “law of free fall” is wrong and apparently the “equivalence principle” (weak and strong equivalence principle) of the General Theory of Relativity is also a totally erroneous principle of physics.

THE “NON-EQUIVALENCE PRINCIPLE” OF MASSES AND FIELDS

As is well known, in the “Strong Equivalance Principle” of the General Relativity Theory, Einstein claims that:
 “All bodies, regardless of their mass, fall in the same way through the gravitational field of another body”.

Essentially, the “Strong Equivalence Principle” is Galileo’s experiment (the leaning tower of Pisa) on an astronomical scale.
Thus, according to the “Strong Equivalence Principle” if we let a small aluminium sphere (a), the Moon (b) and the Earth (c) fall freely through the gravitational field of a distant star M, then all three bodies fall in the same way. In other words, they fall at the same velocity or acceleration. (Forgive me, but I personally find this very funny!) (See Clifford M. Will, Was Einstein Right?, http://www.tsolkas.gr/forums/777.jpg http://www.tsolkas.gr/forums/888.jpg).
After presenting the above, we shall now prove that everything claimed by Einstein in the “Strong Equivalence Principle” (and the weak equivalence principle) is completely false.
The proof that we will employ is very simply and is as follows:

Proof

Let’s take, for example, the simple case of the problem of 2-bodies. Let’s assume (fig 1.4) that we have an inertial observer O’ and two masses m₁ and Μ.
Mass m₁ is considered a point mass, while mass M is a uniform sphere with a radius of R. I addition, both masses m₁ and M are considered totally solid bodies.
We now place mass m₁, at a distance h over the surface of mass M.

fig. (1.4)

At (t=0) we will allow mass m₁ to fall freely from height h into the gravitational field of mass M.
The two masses m₁ and M, attracted to each other by the force of universal attraction, will collide after a time period (t > 0), with mass m₁ moving towards mass M, and mass M towards mass m₁ .
Specifically, point mass m₁ will hit the surface of mass M with a velocity of υ₁ and mass M will hit the point mass m₁ with a velocity of V.
Obviously, the two velocities υ₁ and V will have opposite directions.
If we now apply to the system of masses m₁-Μ the principle of conservation of energy and the principle of conservation of momentum when mass m₁ is at height h, (t = 0) and then when the two masses collide (t > 0), we have:

In relations (R.17.5), the numbers υ₁ , V, h, and R are considered positive numbers and G is the constant of universal attraction.
Solving the equation system (R.17.5) relative to υ₁ and V, we have:

Where υ₁ and V are the velocities of masses m₁ and Μ at the moment of collision, in terms of the inertial observer O’ .
Obviously, the velocity υ₁’ of mass m₁ , in terms of an observer O who is on mass M, is:

υ₁’ = υ₁  +  V

PLEASE NOTE: In relations (A), if, for example, we allow for the following:
a. Height h to be much smaller, in relation to radius R, i.e. approximately R+hR και
b. Mass m₁ to be much smaller, in relation to mass M, i.e. approximately ,
Then, from relations (A), we have::

relative to the inertial observer  O’, and

relative to observer O, who is on mass M. .
Where  is the intensity of the gravitational field of mass M on its surface.

Attention! Based on relations (A.1), it appears that the free fall of bodies dropped from a height h is independent of their mass!
That, however, is a grave error, because relations (A.1) express an approximate result (with minor or major deviations from the true values of υ₁ and V) and in under no circumstances do they express the natural reality, as expressed in relations (A).
For this reason, relations (A.1) must never be employed in strictly theoretical problems, such as the case of Galileo’s law on the “Free Fall of Bodies” and especially the formulation of the “Equivalence Principle” of the General Relativity Theory.
Because if we do employ relations (A.1) in a purely theoretical problem, such as “The Equivalence Principle”, it is absolutely certain we will be led to incorrect theoretical conclusions (which was the case with Einstein in the formulation of “The Equivalence Principle”).
We can obviously apply relations (A.1) for educational purposes only in Elemental Mechanics but never in Theoretical Mechanics in purely theoretical problems.
In addition, in our problem fig (1.4) , we repeat our experiment with another mass m₂, (m₂>m₁) without altering the values of M, R, and h.

As a result, in this case relations (A) are:

where υ₂ and V’ are the velocities of masses m₂ and Μ at the moment of collision, relative to the inertial observer O’.

Relation (R.17.6) expresses the strict law on “the free fall of bodies”. In relation (R.17.6), where m₂ > m₁ it follows that  υ₁>υ₂ .
In other words, in our problem (fig 1.4) we observe that larger masses fall with a lesser velocity (compared to smaller masses) within the gravitational field of mass M!
Obviously, when masses m₁ and m₂ are equal, then from relation (R.17.6) it results that velocities υ₁ and υ₂ are equal as well (υ₁=υ₂₎.
In addition, from the first relation (A) it emerges that velocity υ₁ is always function of mass m₁ , and from the first relation (B) that velocity υ₂ is always function of mass m₂. The same certainly applies to velocities V and V’ , , which are function of mass M.
Consequently, from our observations above, the following two fundamental questions emerge:
Question 1. In the law on “the free fall of bodies”, on the basis of which mathematical
                     relation does Galileo prove that “all bodies, regardless of their mass,
                     fall in the same way”?
Question 2. In the “Equivalence Principle”, i.e. the “strong Equivalence Principle”,
                   on the basis of which mathematical relation does Einstein prove that
                   “all bodies, regardless of their mass, fall in the same way” within the gravitational
                   field of another body, as in the example of the aluminium sphere (a), the Moon
                   (b) and the Earth (c), mentioned above?
Bear in mind that in the above example, the Earth (c) is 6^.10²⁴ times bigger (!!!) than the aluminium sphere (a), assuming that the aluminium sphere (a) weights, e.g., 1 kg.
Meaning that, if we were to repeat the experiment (under the same conditions) in three separate stages, i.e.:
Stage 1: With the Aluminium sphere (a) and distant star M
Stage 2: with the Moon (b) and distant star M
Stage 3: With the Earth (c) and distant star M,
Then, in all three stages (first, second and third), the Aluminium sphere (a), the Moon (b) and the Earth (c) will fall with the same velocity υ₁ onto the surface of distant star M, relative to an inertial observer Ο’;
This is obviously a grave mistake on Einstein’s part, when he makes such claims, i.e. in the “Strong Equivalence Principle”.

I am sorry, but everything Einstein claims in the “strong Equivalence Principle” (as per the above example) is not simply arbitrary and antiscientific but, above all, laughable…

a. What answer do the relativists give to the above two questions?
b. Can the relativists show us the mathematical relations employed in the above two questions (1) and (2)?

After examining the above, we are led to the following basic conclusion:

At this point, we need to stress that relations (A), relations (B) and relation (R.17.6) are complete, because they are strict expressions, in terms of quality and quantity, of the natural reality of the problem of the two bodies of fig. (1.4).
In the above relations, if we make certain allowances, i.e. that mass m₁ is much smaller than mass Μ,

and mass m₂ is much smaller than mass M,

limiting, in this way, the general shape of our problem), then relation (R.17.6) yields:

and, based on relations (R.17.7) and (R.17.8), the following results from relation (R.17.9):

But even so, with the allowances (R.17.7) and (R.17.8) that we made, it emerges from relation (R.17.10) that υ₁υ₂ (with minor or major deviations) and never υ₁=υ₂, as Galileo claims in the law on “the free fall of bodies” and Einstein in the “Equivalence Principle”. (Their claim that that “all bodies, regardless of their mass, fall in the same way”). Based on the above, we can now formulate the following basic law:

LAW: In the problem of 2 – bodies, as this is given in fig. (1.4), and based on relation (R.17.6):

Velocities υ₁ and υ₂ and equal (υ₁=υ₂)only when masses m₁ and m₂ are equal (m₁=m₂).
In any other case (with any assumptions or allowances), it is always υ₁ υ₂ and never υ₁=υ₂.

EXAMPLE

From the top of the Tower of Pisa, we release a mass m₁=1kgr made from cotton wool..
We then repeat the experiment with a mass m₂=10¹⁸kgr , consisting of the material of a neutron star.
If υ₁ is the velocity of mass m₁ as it falls onto the surface of the Earth and υ₂ is the velocity of mass m₂ (as it falls onto the surface of the Earth), we are seeking::
How much greater the velocity with which the cotton wool mass falls onto the surface of the Earth is, compared to the velocity of the neutron star mass.
The mass of the Earth is taken as M=6^.10²⁴kgr and masses m₁ and m₂ are considered point masses.

SOLUTION

From relation (R.17.6) we have:

Substituting the known values in the above equation, we have:

Thus, the velocity υ₁ of the cotton wool mass is 1,000000083 times greater than the velocity of  υ₂ of the neutron star mass.
As we can see, the cotton wool mass m₁=1kgr falls to the surface of the Earth with a greater velocity than the neutron star mass  m₂=10¹⁸kgr !!!
That is a nice, tidy conclusion, which proves that Galileo was wrong!!!

FIELDS OF GRAVITATIONAL AND INERTIAL FORCES

LAW: All bodies, regardless of their mass, “fall” or move in exactly the same way (however great their mass may be) only through inertial force fields and never through gravitational force fields.

In other words, inertial force fields and gravitational force fields have huge qualitative and quantitative differences and they are never equivalent to one another.
Specifically, the basic differences between gravitational force fields and inertial force fields are as follows:

1. Gravitational fields are objective, meaning they exist for all observers (inertial or non-inertial), without exception.
2. Conversely, inertial force fields are perceptual, meaning they only exist for observers who are within the accelerating or decelerating reference system (S) and for no other observer.
3. Gravitational fields can never be uniform fields within a space (Ω), i.e. their intensity vectors cannot be parallel to one another, or have the same speed (>O) and direction.
4. Conversely, inertial force fields may be uniform fields (e.g. a reference system (S) moving with linear, accelerating motion).
5. In a gravitational field (e.g. spherical symmetry), the intensity vectors of the gravitational field are always convergent, in relation to the centre of the mass that creates the gravitational field.
6. Conversely, in any inertial force field, the field’s intensity vectors are always divergent or parallel and never convergent, in relation to the centre of the curvature of the reference system (S) of the inertial forces. In other words, the intensity vectors are always centrifugal, in relation to the observer, who is within the reference system (S).
7. The causes that create gravitational fields (i.e. the masses) always influence one another dynamically (i.e. based on the laws of Newton).
8. Conversely, the causes that create inertial force fields never influence one other dynamically (i.e. based on the laws of Newton).
9. A gravitational field can never (not in a single point of its area, nor locally, nor macroscopically) be equivalent to an inertial or non-inertial reference system.
This conclusion is known as the “Non-equivalence Principle of fields”.
10. The inertial mass m_i is always equal to the gravitational mass m_g and the two masses m_i and m_g are never equivalent to one another.
This conclusion is known as the “Non-equivalence Principle of masses”.
11. The “Non-equivalence Principle of fields” and the “Non-equivalence Principle of masses” always co-exist (they are irrevocably tied to one another) and are known collectively as the “Non-equivalence Principle”.
12. The “Non-equivalence Principle” (in terms of physics) expresses the exact opposite of what is expressed by the “Equivalence Principle” of the General Relativity Theory.

INERTIAL AND GRAVITATIONAL MASS

A COMMON MISTAKE MANY PHYSICISTS MAKE IN TERMS OF “VERIFYING” THE EQUIVALENCE PRINCIPLE OF THE GENERAL RELATIVITY THEORY.

As is well known, the fact that a body’s inertial mass m_i and its gravitational mass m_g are equal, i.e.

is a law formulated for the first time by Newton.
This law (in terms of what we currently know as Physics) has been proven true through Nature, and verified by very accurate experiments, e.g. R. Eotvos, L. Sauthern, P. Zeeman, etc.
Unfortunately, however, the equality m_i = m_g, is often used (incorrectly) as “proof” of the Equivalence Principle of the General Relativity Theory.
Attention! This is where confusion arises, and the “sensitive” point of our problem, as, specifically:

1. The equality m_i = m_g does not necessarily mean that masses m_i and m_g are equivalent. .
2. Conversely, however, the equivalence of masses m_i and m_g does mean that masses  m_i and m_g, are equal, i.e. m_i = m_g.

Because, however (as we previously proved), the Equivalence Principle of the General Relativity Theory is false, it follows that the equivalence of masses m_i and m_g does not exist in Nature.
Consequently, as case (2) above is not valid, case (1) is valid by necessity (and has been verified through experiments, as mentioned above), meaning that m_i = m_g is always valid, while masses m_i and m_g are never equivalent.
Unfortunately, many physicists confuse and equate (mistakenly) the two concepts of equality and equivalence of masses m_i and m_g.
That is obviously a grave mistake of the part of those physicists, because the equality and the equivalence of the masses m_i and m_g, are completely different “things” and should never be confused or taken as one and the same.

Indeed, the equality m_i = m_g (which does, in fact, exist in Nature) must never be used to purportedly “prove” the “Equivalence Principle” of the General Relativity Theory, which is completely false, as we proved earlier.

CONCLUSION

After the above discussion, we conclude that:

1. Galileo’s law on the “free fall of bodies” is false.
2. The “Equivalence Principle” (both strong and weak) of the General Relativity Theory is a false principle of Physics.
3. A body’s inertial mass  m_i is always equal to its gravitational mass m_g, (m_i = m_g) and the two masses m_i and m_g are never equivalent  to one another.

COMMENT

Once more, I would like inform all physicists (who believe that the Relativity Theory is correct) that: that are mistaken, and are doing nothing more than maintaining the “mummy” of Relativity Theory, a theory which has been dead since the moment of its birth, in the year 1905!
I am certain that the Physicists of the future will laugh at this ridiculous theory of Physics, the “Relativity Theory”!

CONCLUSIONS

   Ι. GALILEO

   As it is known, Galileo (based on his historical experiment conducted from the Leaning Tower of Pisa) asserts that all bodies, irrespective of their mass, fall at the same velocity in the gravitational field of a mass M (such as the Earth).
   However, –and according to what has been demonstrated above– what Galileo asserts is utterly wrong, for instance when it comes to unequal masses m₁ and m₂ which are in free-fall in the gravitational field of mass M,either the gravitational field of mass M is non-uniform or uniform (Fig. 1). 

   ΙΙ. EINSTEIN

   a) As it is well-known, in formulating the “Equivalence Principle” of the General Theory of Relativity, Einstein relied on the accuracy of the results of Galileo’s experiment.
   Nevertheless, according to everything proven above, Galileo erred in his assertions, and consequently so did Einstein when he formulated the “Equivalence Principle”.
   Specifically where the Equivalence Principle is concerned, Einstein by resting on the erroneous results of Galileo’s experiment asserts that the gravitational fields are equivalent to the fields of inertial forces, in the sense that gravitational fields cause all bodies, irrespective of their mass, to undergo thesame acceleration.
   Yet, what Einstein asserts is utterly wrong, since only the fields of inertial forces have this property and not gravitational fields, as demonstrated earlier.
   b) Moreover, Einstein’s Equivalence Principle asserts the following: 
   “A frame of reference, once in free fall in the gravitational field of a mass M, is locally equivalent to an inertial frame of reference”.
   However, Einstein’s assertion is once again wrong for the following reason:
   Let us assume that spherical shell m₁(Fig. 2) is an “elevator” free-falling in the gravitational field of mass M.   

   Then, if an observer Ο΄ who is inside the “elevator” m₁ places at a time t=0 a point mass m₂ (m₁≠m₂) at the center of the spherical shell (elevator), mass m₂ after a time t>0 will not remain at the center of the spherical shell (as Einstein holds) but will move from its center, as proven earlier (Fig. 1).

Conclusion

   Therefore, the “Equivalence Principle” of the General Theory of Relativity fails to apply not merely “locally” but to any single point at the center of the spherical shell (elevator), as Einstein erroneously asserts based on his “elevator” thought experiments that led him to the formulation of the “EquivalencePrinciple” of the General Theory of Relativity.
   Consequently (as proven above), because of the invalidity of the “Equivalence Principle” of the General Theory of Relativity, the inertial mass Μ_i of a body is never equivalent to gravitational mass Μ_g.
   On the other hand, a body’s inertial mass Μ_i and its gravitational mass Μ_g are always equal, i.e. Μ_i=M_g, these two masses are never equivalent to each other as Einstein erroneously asserts in the “Equivalence Principle” of the General Theory of Relativity.
   Thus, after everything analyzed above, we reach the conclusion that Galileo and Einstein (the latter relied on the experiment of the former) drew and formulated totally wrong conclusions in respect of free-fall, especially in the case of “Galileo’s experiment” on which the “Equivalence Principle” of the General Theory of Relativity was predicated.
   Therefore (as demonstrated above), since the “Equivalence Principle” of the General Theory of Relativity is not valid, then the Theory of Relativity itself should be deemed as an utterly erroneous theory of Physics, failing to represent physical reality under all circumstances.

CRITIQUE

 1. Critique of relation γ=g in Galileo’s experiment

   As it is well-known, Einstein’s General Theory of Relativity postulates the following:
   “Gravitational fields have the remarkable property of imparting the same acceleration to all bodies independent of their material composition and amount of mass”.
   (See THE PRINCIPLE OF RELATIVITY, BY A. EINSTEIN – H.A LORENTZ – H. MINKOWSKI – H. WEYL, Page 114 DOVER PUBLICATIONS, INC).
   Einstein based this conclusion on the all-too-familiar Galileo’s experiment (the experiment conducted from the Leaning Tower of Pisa).
   All the above generate the following question:
   Are Einstein’s assertions true or false?
   The answer is that what Einstein asserts is false, for the following reasons:
   With regard to the problem (Fig. 1) discussed earlier, it has been proven that:
   a) In relation to an inert observer Ο, Fig.1 the two masses m₁ and m₂ (m₁< m₂), fall at different velocities υ₁ and υ₂, (υ₁>υ₂) in the gravitational field of mass Μ. (See  “The fundamental law of the free-fall”, relation (29)).
   b) Conversely, Einstein by accepting in this case the conclusion of Galileo’s experiment as true, asserts that:
   In relation to inert observer Ο, Fig. 1, the two masses m₁ and m₂ fall at the same velocity, i.e. υ₁ =υ₂ at all times.
   Yet, which of these two conclusions (a) and (b) is true?   
   Conclusion (a) or conclusion (b)?
   The answer to this question is the following: Conclusion (a) is true for the reasons explained below:
   Einstein’s major error lies in that he did not apply
   1. The “principle of conservation of energy”, and
   2. The“principle of conservation of momentum”
in order to find the accurate solution to the problem encountered in the system of the three masses m₁,m₂ and M; instead,  for masses m₁and m₂ he employed only Newton’s  first and second law , maintaining that: 
   For a body of any mass m which is in free fall inside the gravitational field of mass M, the following relation applies:

and Einstein continues......
Because force F which causes the body to move is its weight W, then:

and because:

 relations (33) and (34) yield:

   Relation (37) (as it is well known), expresses the conclusion of Galileo’s experiment.
   Einstein erred in his interpretation of the resulting relation (37), namely by asserting that:
   “Because mass m of a body does not exist in relation (37), then acceleration γ is the same for all bodies that are free-falling in the gravitational field of a mass M, independent of these bodies’ mass”.
   This is exactly why Einstein mentions the remarkable property of gravitational fields referred to above, which he bases on the conclusion of Galileo’s experiment expressed by relation (37).
   However, here lies Einstein’s great error, because the accurate interpretation of relation (37) is the following:

   Let us assume that there is a body of mass m which is in free fall in the gravitational field of a mass M.  
   Moreover, let us assume that during time t = t₁>0, body m lies at a distance from mass M.
   Then according to Newton’s first and second law, during time t₁ its acceleration will be:

   As it can be observed in relation (39), mass m of the body does not exist.
   However, this does not signify that all bodies, independent of their mass, fall freely in the gravitational field of mass M with the same acceleration.
   The explanation to that is very simple, because:
   As it can be observed in relation (39), acceleration γ_t is a function of distance  h_t, that is:

   However, according to the “two-body problem”, distance h_t relative to an inert observer Ο is always a function of masses m, M, of distance h (initially separating the two masses m and M at time t = 0, where their velocities equaled 0) and of time t, that is:

Therefore, relations (39), (40) and (41) yield:

   Thus, the above lead to the following conclusion:
The value of acceleration  γ_t of a body which is in free fall in the gravitational field of mass M at time   t = t₁ is given by relation (39),  where is the distance between the two masses m and M at time t = t₁.
   This acceleration γ_t  is never independent of mass m of the body (as Einstein erroneously asserts on the basis of relation (37)), but is always a function of mass m of the falling body, of mass M, of the initial distance h and of time t, according to relation (42). 
   At this point it should be noted that during the free fall of body m in the gravitational field of mass M, distance h_t between the two masses m and Μ constantly shortens, due to the universal force of attraction exerted by one mass over the other, a fact that was never taken into account during the performance of the Galileo experiment.
   Consequently, the above conclusion opposes the one of Galileo’s experiment expressed by relation (37).
   Therefore, gravitational fields do not have the remarkable property of imparting the same acceleration to all bodies independent of their mass, as Einstein falsely holds.  
   This property is observed exclusively in fields of inert  forces and absolutely not in any other field of forces in Nature.
   Hence Einstein’s major error!!!

2. Critique of Galileo’s experiment

   As it is well known, the conclusion of Galileo’s experiment (at the Leaning Tower of Pisa) which is interpreted based on relation (37) and states that “all bodies fall at the same rate (velocity), irrespective of their mass,in the gravitational field of mass M” is utterly false for the reasons given below:
   In the first place, Galileo’s experiment is a problem involving three bodies (special case); See problem in Fig. 1.
   A direct consequence of  the resolution of this problem, is that, because mass M of the Earth is much greater than masses m₁ and m₂, (m₁<m₂) of the bodies that Galileo dropped from the top of the Tower of Pisa, these two masses m₁ and m₂ struck the ground with a slight difference in velocity Δυ, (Δυ → 0).
   (See, “Fundamental law of free-fall” relations (29)).
   As it was natural, Galielo was not in a position to measure this slight difference in velocity Δυ → 0.
   This is why he reached the all-too-familiar conclusion (inreality and he was mistaken) that “all bodies independent of their mass fall towards the Earth at the same rate”!!!
   Galileo’s error can be immediately perceived through the following simple example:
   Let us assume that Galileo carried out his experiment by using a very small asteroid of mass M and letting a sphere of  mass m₁ and diameter 10 cm made of cork, and a sphere of mass m₂ and diameter 10 cm made from the material of a white dwarf drop freely from a height h. Would these two bodies fall on the surface of the asteroid with the same velocity?   
   The answer to this question is NO. Because according to the fundamental law of free-fall, the sphere of cork will strike the surface of the asteroid sooner that the sphere of the white dwarf!!!
   Consequently, Galileo’s experiment is theoretically solved (and this is the accurate method) solely as problem of three (3) bodies m₁, m₂ and Μ (See problem in Fig 1), and not with the plain, false method applied by Einstein that led him to relation (37) as explained above.
   So, this is Einstein’s major error in interpreting Galileo’s experiment!!!

3. Critique of the “Equivalence Principle”

   As it is well known, the axiom that a body’s inertial and gravitational mass are completely equivalent was formulated in Newton’s era and its validity was thereafter proven through a variety of high precision experiments (carried out by Newton, Eötvös, Dicke,...).
   At this point, it is essential to note the following:
   a) If the equivalence of inertial and gravitational masses applies, then equivalence between their masses should apply, as well.
   b) Conversely, if the inertial and gravitational masses of any body are identical , this implies that these masses shall not necessarily be equivalent.
   In other words, in nature there may be equivalence between the inertial and gravitational mass of any object (as is the case) but the latter does not necessarily signify equivalence between these masses, as Einstein asserts.
   Because the equivalence between inertial and gravitational masses presupposes and necessitates equivalence between the fields of inertial and gravitational forces, even “locally” or at least in some point in space.
   Let us follow now Einstein’s reasoning in his attempt to formulate the equivalence principle:
   Einstein asserts the following:
   1. Based on Galileo’s experiment (Relation (37)) it is demonstrated that gravitational fields have the remarkable property of imparting the same acceleration to all bodies independent of their mass”.
   However, (as it is well known) the same property in encountered in the fields of gravitational forces. Einstein continues by suggesting that:
   2. Therefore, since these two fields share the same property, then an observer A found, for example, in an accelerating vehicle cannot “locally” distinguish by experiment whether the forces acting inside his vehicle are gravitational or inertial forces. From a physical point of view, this signifies that “locally’ a field of gravitational forces is equivalent to a field of inertial forces. Einstein continues:
   3. If these two fields are equivalent to each other, then their respective masses that are subject to the forces of these two fields should also be equivalent, that is, inertial mass m_i of one body should be equivalent to this body’s gravitational mass m_g. Einstein continues: 
   4. If, however, these two masses m_i and m_g (inertial and gravitational mass) are equivalent to each other, then they must necessarily be equal to one another. Einstein continues: 
   5. If, however, these two masses m_i and m_g (inertial and gravitational mass) are equal to each other, then the latter should be experimentally proven.
   Indeed, in Nature there is equality between a body’s inertial and gravitational mass in accordance with the above-mentioned experiments.
   Therefore, (according to Einstein) assuming that the “equivalence principle” is true, the following conclusion is drawn:
   a) While according to classical Physics there is equality between a body’s inertial and gravitational mass, the General Theory of Relativity holds that there is equivalence between them, and    
   b) While according to classical Physics the fields of inertial and gravitational forces are different fields of forces, under the General Theory of Relativity these fields are equivalent to one another.
   However, the question that is being raised is the following:
   Are Einstein’s above assertions true?
   The answer is no, and the reasons are the following:
   To begin with, from a logical point of view Einstein’s reasoning is correct. 
   Yet, his initial and fundamental reasoning (1), on which all his other reasonings rest, is erroneous because the fields of gravitational and inertial forces are not always equivalent to one another. 
   For we should bear in mind that equivalence between these fields would require that the conclusion of Galileo’s experiment be accurate, i.e. relation (37).
   However, according to the fundamental law of free-fall (relation (37)), the conclusion of Galileo’s experiment (relation (29)) is wrong.
   Thus, Einstein’s fundamental reasoning (1) is totally erroneous.
   Consequently, since the basis for reasoning (1) is inaccurate then all his other reasonings (2), (3), (4), (5) are pointless and have no meaning at all.
   Therefore, after everything explained above it is clearly demonstrated that the “equivalence principle” is false.
   Furthermore, worth noting at this point is a characteristic example that helps readers understand Einstein’s major error where the “equivalence principle” is concerned».
   The example is given below:
   Einstein’s “equivalence principle” and in particular his “strong equivalence principle” states the following:
   “If we let (a) a sphere made of aluminum,  (b) the Moon and (c) the Earth fall freely in the gravitational field of a distant body of mass M, then these three bodies (a), (b), (c) will fall at the same rate (i.e. velocity)!!!
   (See “Was Einstein right?” (Clifford M. Will, chapter 7, pict. 7.1)).
   Obviously, Einstein erred greatly in this assertion because the above cannot possible apply in nature based on the established laws of Physics.
   Consequently, what Einstein suggests in the “equivalence principle” is utterly erroneous.
   More specifically all that are very simple, non-scientific beliefs which do not correspond to nor express in any way physical reality.
   Therefore, the “equivalence principle” –weak equivalence principle (WEP) and the strong equivalence principle (SEP) – is a totally erroneous theory of Physics.   
   Hence Einstein’s major error!!!

THE LAWS OF FREE-FALL

   The following three laws apply when relative to an inertial observer O, the two masses m₁ and m₂ (m₁ < m₂) fall freely at the same time from a height h in the gravitational field of mass M:

Law 1: i. The smaller mass m₁ falls at a greater velocity υ₁ and the larger mass m₂ falls at a smaller
               velocity υ₂, (υ₂ < υ₁).

ii. Fraction υ₂ / υ₁ of velocities υ₂ and υ₁ always obeys the following relation:

iii. When masses m₁ and m₂ are equal (m₁ = m₂), then velocities υ₁ and υ₂ are also the same, i.e. υ₂ = υ₁.

Law 2: i. Momentum J₁ of the smaller mass m₁ is less than momentum J₂ of the larger mass m₂, hence
               the relation:

ii. When masses m₁ and m₂ are equal (m₁ = m₂), then the momentums are also the same, that is, J₁ = J₂ ή m₁ υ₁ = m₂ υ₂.

Law 3: According to the “two body problem”:
             i. The acceleration  of a body having mass m and falling freely from a height h in the
             gravitational field of mass M, at time t₁ is given by the following relation:

where,   the distance between masses m and Μ at time t₁.
ii. Acceleration of the free-falling body is always a function of its mass m, of mass M, of height h and of time t, hence the relation:

iii. This acceleration is never independent of the body’s mass m, as Galileo erroneously asserts

These are therefore the three laws of free fall.

DETECTOR OF GRAVITATIONAL FIELDS

A spherical shell of mass m₁, within which there is another mass m₂ (m₁ ≠ m₂)  (Fig. 2), will be called “detector of gravitational fields”.

Analytically:

(a) If mass m₂ is found at the center of the spherical shell m₁, then the detector of
     gravitational fields will be in its “constant” state and will be symbolized as G+, (Fig. 2a).
(b) Conversely, if mass m₂ is not found at the center of spherical shell m₁, then the detector
     of gravitational fields will be in its “inconstant” state and will be symbolized as G-,
     (Fig. 2b).

The great importance of the “detector of gravitational fields” lies in that it fully explains the concept of “Locally”, which the General Theory of Relativity uses in its various experiments (such as the well-known elevator experiment).

Fig. 2

Note: If in Fig. 2 mass m₁ consists of a light material, e.g. aluminum, wood, etc, and mass m₂ consists of the material of a white dwarf, black hole, etc, then the above detector will be called “detector of gravitational fields of a thick mass”.

CASE 1: Let us assume (Fig. 3) that there is an inertial frame of reference (S).
An observer O who stands in this inertial frame of reference, places the detector of gravitational fields in its “constant” state G+ at a point P.
Then (t = 0), he lets the detector of gravitational fields move freely .
After a time t > 0, the detector of gravitational fields will be at the exact same point P in its “constant” state G+.

Fig. 3

CASE 2: Lets us assume (Fig. 4), that an elevator (S) falls freely in the gravitational field of a celestial body of mass M₀ (earth, moon, asteroid, etc.). An observer O who stands in the elevator places at a point P the detector of gravitational fields in its “constant” state G+.
Then (t = 0), the observer lets the detector of gravitational fields move freely.
According to the “three-body problem” (m₁, m₂, M₀), (m₁ ≠ m₂) referred to above, after a time t > 0, the detector of gravitational fields will be in the elevator in its “inconstant” state G-.

Fig. 4

CASE 3: Let us assume (fig.5) that an elevator cab (S) moves with acceleration γ away from the gravitational fields. An observer O who is found in the cab places at a point P the detector of gravitational fields in its “constant” state G+.
Then (t = 0) the observer lets the detector of gravitational fields move freely.
After a time t > 0, the detector of gravitational fields will be at another point P΄ again in its “constant” state G+.

Fig. 5

CASE 4: Let us assume (Fig. 6) that an elevator cab (S) is at rest at the top of a tall building or a pillar of height h from the surface of a celestial body of mass M₀ (earth, moon, asteroid, etc). An observer O who is found in the cab places at a point P the detector of gravitational fields in its “constant” state G+ (Fig. 6a).
Then (t = 0) the observer lets the detector of gravitational fields move freely.
According to the “three-body problem” (m₁, m₂, M₀), (m₁ ≠ m₂) referred to above, after a time t > 0, the detector of gravitational fields will be at another point P΄ in its “inconstant” state G- (Fig. 6b).

Fig. 6

RECAPITULATION

According to the three fundamental laws and the way of operation of the D.G.F. (DETECTOR OF GRAVITATIONAL FIELDS) that we mentioned above, we are now able to formulate the following law:

 THE FUNDAMENTAL LAW OF SEPARATION OF GRAVITITATIONAL FIELDS AND FIELDS OF INERTIAL FORCES

LAW:

An observer O which is within a frame of reference (S), O.XYZ that moves:
a.    In any kind of movement on a plain field or in space (eg. rectilinear accelerating, circular, spiral etc), away from the gravitational field of a mass M, or
b.    Falls freely within the gravitational field of the mass M, or
c.    Is idle at a constant height h, above the mass M, or
d.    Is an inertial frame of reference
(see above cases)

 Then, the observer O, using a D.G.F, can easily know “locally” and particularly for each point Pi (X_i, Y_i, Z_i), i = 1, 2, 3… of the frame of reference O.XYZ whether the forces exercised on the point mass, which is located at the center P_i  of the D.G.F. are due to a gravitational  field (gravitational forces), or they are due to a field of inertial forces (inertial forces) or the  frame  of reference O.XYZ  is an inertial frame of reference.

This foundational law that we mentioned above is fully contrary to the “Equivalence Principle” of the General Theory of Relativity, which (the Equivalence Principle), according to this law, is proven to be a wrong principle of Physics.

Summarizing all the above, we come to the conclusion that:
The “Equivalence Principle” of the General Theory of Relativity is wrong and consequently:

1. The fields of the gravitational and inertial forces are never of the same nature (equivalent) as Einstein claims, but they are two completely different fields and these fields are different in nature (not equivalent).
2. The gravitational mass Mg of a body M is always equal to the inertial mass Mi, namely Mg = Mi, and these two masses Mi and Mg are neverequivalent to one another, as Einstein wrongly claims. 

TWO QUESTIONS PUT TO “EINSTEIN”

Α. GRAVITATIONAL AND INERTIAL MASS

  Let us assume that Ο.ΧYZ (Fig.1) is an inertial frame of reference and (Ο) is an observer standing in it.
   Also, m₀ and Μ₀, (m₀ < M₀) are two masses which at time t=0 are at rest and lie at a distance h from each other.
   We now let the two masses m₀ and M₀ move freely under the influence of the universal force of attraction.
   After a time t > 0, the distance between the two masses m₀ and M₀ will be h΄, (h΄< h) and their velocities v and V respectively, (“two-body problem”).

   As it is well-known, according to the Theory of Relativity, when a mass moves relative to an inertial observer (Ο), then this mass increases in relation to the rest mass it had in the inertial frame of reference.
   Therefore, in the case shown in Fig. 1, relative to the inertial observer (Ο) and at time t > 0, the two rest masses m₀ and M₀ will increase (mass = function of velocity) and the following relations will apply:

                               m = m₀ f (v)             (1)

                               M = M₀ f (V)             (2)

where, m > m₀ and M > M₀.

   Moreover, as it is well-known, according to the Theory of Relativity, when a mass moves under the influence of a gravitational field, then the inertial and gravitational masses are equivalent (according to the “equivalence principle”) and consequently since they are equivalent they will also be equal.
   Thus, on the basis of the above, relative to the inertial observer (Ο) and at time t > 0, the following relations will apply for the two masses m and Μ in Relations (1) and (2):

                               m_i = m_g                 (3)

                               M_i = M_g                 (4)

where, m_i, M_i are the inertial masses of masses m and Μ respectively, and
         m_g, M_g are the gravitational masses of masses m and Μ respectively.
   Therefore, after everything presented above, we are putting to “Einstein” the following question:

   QUESTION I: In the example of Fig. 1 (“two-body problem”), let us consider e.g. the smaller mass m. Then:
   Relative to the inertial observer (Ο) and at time t > 0, on the basis of the axiomatic foundation and the laws of the Theory of Relativity, and taking into account the fact that the mass increases with velocity (Relation (1)), in this case:

a. By which mathematical formula A of the Theory of Relativity is inertial mass m_i expressed, relative to inertial observer (Ο)?
b. By which mathematical formula Β of the Theory of Relativity is gravitational mass m_g expressed, relative to inertial observer (Ο)?
c. How does relation m_i = m_g (as the Theory of Relativity maintains) relative to inertial observer (Ο) result from the above two mathematical formulas A and B?

We would very much like “Einstein’s” answer to these questions.

Β. GALILEO’S EXPERIMENT

   As it is well-known, in formulating the “Equivalence Principle” of the General Theory of Relativity, Einstein accepts the results of Galileo’s experiment as utterly correct (the Tower of Pisa experiment).
   Thus, in this case, we are putting the following question to “Einstein”:

   QUESTION II: Based on the axiomatic foundation and the laws of the Theory of Relativity and taking into account the fact that the mass increases with velocity, can “Einstein” prove to us that the result of Galileo’s experiment is accurate?
   That is, can “Einstein” prove (Fig. 2) that:

   Relative to an inertial frame of reference O.XYZ, two unequal masses (with rest masses) m₀ and m₀΄, (m₀ < m₀΄) which are let simultaneously to fall freely from this height h in the gravitational field of a third mass Μ₀, (taking into account the fact that these masses increase with velocity during their free-fall), do these two masses m₀ and m₀΄ fall always at the same velocity?
   Therefore, where is the full mathematical proof –according to the axiomatic foundation and the laws of the Theory of Relativity– of Galileo’s experiment accuracy?

We would very much like “Einstein’s” answer to this question, also.

COMMENT

   Can “Einstein” provide an answer to the questions posed above?
   From my point of view, NO!
   Consequently, since Einstein cannot answer the above two questions, then the Theory of Relativity is an erroneous Theory of Physics due to its inconsistence and many internal contradictions, as stated above. 

NOTE: Any physicist, mathematician, university who/which objects to everything analyzed above, is welcome to publish his/her/its objections, so that a constructive dialogue can be established. Because undoubtedly dialogue provides the basis for scientific research.

EPILOGUE

Enough energy, time and money have been wasted on the Theory of Relativity so far.
It is no use wasting any more, because: 
The Theory of Relativity has been proven to be an utterly erroneous Theory of Physics!!!

Copyright 2006: Christos A. Tsolkas                                              Christos Α. Τsolkas
                                                                                                             October 2006

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