Home
Wirenet Communications
Contact us

Ether and
Magnetic field

Galileo and
Einstein
are wrong

Equivalence
Principle

Ether and
Equivalence
Principle

Proof  for
the advance
of Mercury's
perihelion


Open
Letter

 

The
Electro
gravitational
Theory I

The
Electro
gravitational
Theory II

The
Electro
gravitational
Theory III

The
Electro
gravitational
Theory IV

The
Electro
gravitational
Theory V

Generalised
Geometry

Mathematics
of degree

Video 01

Video 02


Ether
and
Light

 


Experiment 21
Experimental
Verification

 


Experiment 22
Experimental
Verification

 


The mistakes
of Einstein

 


Spherical
Shell
Problem
 

Recapitu-
lation


TECHNOLOGY
Fusion:
The “ZEUS”
machine

 


CERN/OPERA
IKARUS
TSOLKAS

 

SPHERICAL SHELL PROBLEM (SSP)

SUMMARY

The spherical shall problem (SSP) is a very simple and very important experiment of Classical Physics. The problem (SSP) is a special (and simple) case of the “three bodies” problem.
By solving this problem, which is done exclusively on the basis of Newtonian Mechanics, and assuming Newtonós axiom on the equality of the inertial mass mi and the gravitational mass mg, (mi = mg), we formulate the fundamental law of the free fall of bodies.
Based on the fundamental law of the free fall of bodies, we can prove that:
Gravitational fields never have the remarkable property of imparting all bodies with the same acceleration, regardless of their mass and their material constitution, as Galileo and Einstein mistakenly claim.
See: http://www.tsolkas.gr/forums/tga6.jpg (Book: THE PRINCIPLE OF RELATIVITY, by A. EINSTEIN, H.A. LORENTZ, H. WEYL, H. MINKOWSKI).
Simply put, based on the fundamental law of the free fall of bodies, we can prove that:
1) The results of Galileoós experiment (the tower of Pisa experiment) are not theoretically correct, but empirically and approximately.
2) The “equivalence principle” (strong and weak) of the General Theory of Relativity is a completely false principle of Physics.

SOLVING THE SPHERICAL SHELL PROBLEM (SSP)

Let us assume, Fig. 1, that we have a homogenous spherical shell, with a mass of m1 and a radius of Ro.

Fig. 1

At the centre of the spherical shell m1, we place a point mass m2.
At a distance h from the centre of the spherical shell m1, is a body of mass M.
Note: Each one of masses m1
, m2, M is considered a totally solid body (i.e. not altered by the gravitational forces exercised between them).
Obviously, the system of the two masses m1 and m2 (i.e. of the spherical shell m1 and the point mass m2) does not behave as a totally solid body of mass m1 + m2.
We now perform, Fig. 1, our experiment in two phases (Phase I and Phase II), as follows:
PHASE I: During this phase, and at the moment to = 0 , the point mass m2 is at the centre of the spherical shell m1 and the centre of the spherical shell m1 is at a distance h from the mass M. Also, during this phase (to = 0), the spherical shell m1, the point mass m2 and the mass M are motionless as to the inertial reference system S.
In order words, at to = 0, we have §1 = 0, §2 = 0 and V = 0 , where §1 is the velocity of mass m1, §2 is the velocity of point mass m2 and V is the velocity of mass ╠. In order words,
Velocities §1 = 0, §2 = 0, V = 0 obviously refer to the inertial reference system S.

PHASE ╔╔: During this phase, we allow, from a height h, the spherical shell m1 (with the point mass m2, which is at its centre) to move freely under the influence of the force of universal attraction exercised between masses m1, m2 and ╠.
Let us assume now that after a time t1 > 0 of free fall, the velocity of mass m1 is §1 > 0, the velocity of point mass m2 is §2 > 0 and the velocity of mass ╠ is V > 0.

After what we discussed above, the critical question that emerges is this:
QUESTION: During the performance of the spherical shell experiment, Fig. 1, and at the moment t1 > 0 (Phase ╔╔) of free fall, will the point mass m2 remain at the centre of the spherical shell m1 where it was originally (to = 0) placed, or will it move from the centre of the spherical shell m1?
In accordance with the proof (which follows), the answer to that question is:
In the spherical shell problem, Fig. 1:
  1. When the two masses m1 and m2 are equal (m1 = m2), then, at the moment t1 > 0 the point mass m2 will be at the centre of the spherical shell m1.
  2.Conversely, if the two masses m1 and m2 are not equal ( m1 m2), then, at the moment t1 > 0 the point mass m2 will move from the centre of the spherical shell m1.

PROOF

Let us assume, Fig.1, that at the moment t1 > 0 the point mass m2 is with the spherical shell m1 and the distance of mass m1 (i.e. the centre of the spherical shell m1 ) from the centre of mass ╠ is h1 and the distance of point mass m2 from the centre of mass ╠ is h2.
If we now apply to the system of the three masses m1, m2, M at the moment t1 > 0:
     1. The principle of conservation of energy, and
     2. The principle of conservation of
momentum,
we have relations:

In relations (1) the numbers §1, §2, V, h1, h2, h are all considered positive numbers.
Also, relations (1), can also be expressed as follows:

Posing now:

and  ┬ = ╠V
relations (2) can be expressed as:

In relations (3) ┴ and ┬ are obviously positive numbers, i.e. ┴ > 0 and ┬ > 0.
Now, if we solve the system of equations (3), in terms of §1 and §2, we have:

In relations (4) and (4.1), it is obviously:

where,

From now on, relations (4) will be referred to as the basic relations of the spherical shell problem (SSP).
As mentioned above, in relations (4), ╩ can only have one of two values, i.e.:

Note: As we will see below:

  • For m1<m2, relations (4) are acceptable and relations (4.1) are rejected.
  • For m1>m2, relations (4.1) are acceptable and relations (4) are rejected.
    Where, in cases (a) and (b) in relations (4) and (4.1), K>0.
  • For m1=m2, relations (4) or relations (4.1) are acceptable. Where, in case (c) in relations (4) and (4.1), K=0.

RESEARCH ON THE SPHERICAL SHELL PROBLEM

In the spherical shell problem, let us take, e.g. relations (4) of the basic relations (4) and (4.1).
Relations (4) yield:

resulting in:

Based, therefore, on relation (8), we can now formulate the following fundamental law:

LAW ╔

In relation (8):

  • When ╩ = 0, then §1 = §2 and, conversely, when §1 = §2, then ╩ = 0.
  • When ╩ > 0, then §1 > §2 and, conversely, when §1 > §2, then ╩ > 0.

Law ╔, as described above, plays a very important part in the research on the spherical shell problem, and especially in terms of basic relations (4) and (4.1).

╔. EQUAL MASSES

In the spherical shell problem, Fig. 1, and in the event that the masses m1 and m2 are equal (m1 = m2), we will prove that:
In relations (4), when the masses m1 and m2 are given, (m1 = m2), then §1 ≠ §2 can never apply.

Proof

According to the problem, since when masses m1 and m2 are given (m1 = m2) §1 §2 can never apply, it means (correspondingly), that:
When masses m1 and m2 are given

m1 = m2 (9)

then the only relation that applies is:

§1 = §2 (10)

because §1 ≠ §2 and §1 = §2 are mutually exclusive, (only one or the other can apply).
Hypothesis: Let us assume that relation (10) applies, then from relations (9) and (10), we have:

m1§1 = m2§2 (11)

m1§1 – m2§2 = 0 (12)

In addition, from relations (4), we have:

And relations (13) yield:

In addition, from relations (12) and (14) we have:

Because, according the data of the problem, m1 = m2, then, based on relation (9), relation (15) yields:

Based on relation (16), relation (8) yields:

Relation (17) corroborates our hypothesis 1 = §2).
Subsequently, our hypothesis is correct.
Our problem, therefore, has been proven.
We have proven that:
In relations (4), when masses m1 and m2 are given (m1 = m2), then §1 §2 can never apply, because we have proven that relation §1 = §2 always applies.
Therefore, the following relation will always apply:

where, in relation (18) m1 and m2, (m1 = m2) are the data of the problem, and §1, §21 = §2) are the conclusion.

b) Conversely: In relations (4), when the velocities §1 and §2 are given (§1 = §2), then m1 ≠m2 can never apply.

Proof

According to the problem, since when velocities §1 and §2, 1 = §2) are given, m1 m2 can never apply, this means (correspondingly) that:
When velocities §1 and §2 are given:

§1 = §2 (18.1)

then the only relation that applies is:

m1 = m2 (18.2)

because m1 ≠ m2 and m1 = m2 are mutually exclusive, (only one or the other can apply).
Hypothesis: Let us assume that relation (18.2) applies, then from relations (18.1) and (18.2), we have:

m1§1 = m2§2             (18.3)
 m1§1 – m2§2 = 0         (18.4)

But because relations (4) yield:

Then (as in case (a) above), relations (18.4) and (19), yield:

Ion addition, since according to the data of the problem §1 = §2 , relation (8) yields:

╩ = 0 (21)

Based on relation (21), relation (20) yields:

Relation (22) corroborates our hypothesis (m1 = m2).
Therefore our hypothesis is correct.
Which means that our problem has been proven.
We have proven that:
In relations (4), when velocities §1 and §2 1 = §2) are given, then m1 m2 can never apply, because we have proven that relation m1 = m2 always applies.
Therefore, the following relation will always apply:

where, in relation (23) §1 and §2, 1 = §2) are the data of the problem and m1, m2 (m1=m2) are the conclusion. 

After what we discussed above, we can draw the following basic conclusion.

CONCLUSION ╔

In the spherical shell problem, Fig. 1, and in relations (4):

    a. When the masses m1 and m2 are given (m1 = m2), then the velocities §1 and §2 are always equal (§1 = §2) and, conversely,
    b. When the velocities §1 and §2 are given 1 = §2) then masses m1 and m2 are always equal (m1 = m2).
    From conclusions (a) and (b) results the relation:

    when, in relation (24), m1 = m2 and §1 = §2.
    Therefore, in accordance with relation (23.1), it emerges that:
    c. When the masses m1 and m2 are given (m1 ≠ m2) then the velocities §1 and §2 are always §1 ≠ §2 and, conversely,
    d. When the velocities §1 and §2 are given1 ≠ §2), then masses m1 and m2 are always m1 ≠ m2.

Based on the above conclusion, we can now formulate the following law:


LAW ╔╔

In the spherical shell problem, when the two masses m1 and m2 are equal (m1 = m2), then:
During the fall for of the equal masses m1 and m2 within the field of gravity of mass ╠, then both masses m1 and m2 always fall with the same velocities §1 and §2, (§1 = §2) as per the inertial reference system S.

m1 = m2    ,   §1 = §2

 

    ╔╔. UNEQUAL MASSES

    According to conclusion I (paragraph (c)) as above, when m1 ≠ m2, then §1 ≠ §2.
    In the spherical shell problem, Fig. 1, we will prove that:
    When masses m1 and m2 are given (m1 < m2), then §1 < §2 can never apply.

    Proof

    According to the problem, since when masses m1 and m2 are given (m1 < m2) §1 < §2 can never apply, this means (correspondingly) that:
    When masses m1 and m2 are given,

    m1 < m2       (24.1)

    then the only relation that applies is:

    §1 > §2              (24.2)

    because §1 < §and §1 > §2 are mutually exclusive, (only one or the other can apply).
    Hypothesis: Let us assume that relation (24.2) applies.
    If we solve the system of equations (3) for m1 and m2, we have:

    Since, according to the data of the problem:

    m1 < m2 (26)

    then relations (25) and (26) yield:

    In addition, since according to conclusion ╔, when m1 ≠ m2 then §1 ≠ §2, this means that:
    In relations (25), when m1 < m2 then:

    §1 ≠ § (27.1)

    Therefore, relation (27), (which results from relations (25)) will always be §1 ≠ §2, i.e.:

    §1 - §> 0 (27.2)

    according to the hypothesis.
    Based, therefore, on relation (27.2), relation (27) yields:

    (┬§1 – ┴) §1 > (┴ – ┬§21
       ┬(§12 + §22) > ┴(§1 + §2) (27.3)

    In relation (27.3), if we replace ┴ and ┬ as these result from relations (3), we have:

    In relation (27.4), since, according to the data of the problem:

    m1 < m2
    i.e.:  m2 – m1 > 0 (27.5)

    relation (27.4) is verified only when:

    §1 – §2 > 0
    i.e.:    §1 > §2 (27.6)

    Relation (27.6) corroborates our hypothesis §1 > §2 .
    Therefore, our hypothesis is correct.
    Which means our problem has been proven.  
    We have proven that:
    In the spherical shell problem, Fig. 1, when the masses m1 and m2 are given (m1 < m2) then §1 < §2 can never apply, because we have proven that relation §1 > §2 always applies.
    Therefore, the following relations will always apply:

    where, in relations (27.7) m1 < m2 are the data of the problem and §12 are the conclusion.
    Conversely: Based on relations (21) and in the same way we described above, we can prove that:
    When the data is §1 > §2, then relation m1 > m2 can never apply.
    Therefore, relation m1 < m2 must apply, because m1 > m2 and m1 < m2 are mutually exclusive.
    In that case, we are lead back to the same relation:

    1 – §2) (m2 – m1) > 0 (a)

    where the data in relation (a) are §1 > §2, i.e. §1 – §2 > 0.

    SUMMARY

      1. In accordance with the above proof:
      When the data is m1 < m2, then §1 > §2 and conversely, when the data is §1 > §2, then m1 < m2, i.e. the following relation applies:
      2. Therefore, according to relation (b):
      When the data given is m1 > m2, then §1 < §2 will apply and, conversely, when the data given is §1 < §2, then m1 > m2, i.e. the following relation applies:

    CONCLUSION ╔╔

    In the spherical shell problem, Fig. 1:

      a. When m1 < m2, then §1 > §2 (in accordance with relation (b), as above).
      b. When m1 > m2, then §1 < §2 (in accordance with relation (c), as above).

    In other words, as demonstrated by the above cases (a) and (b), a higher velocity corresponds to the smaller mass, and a lower velocity corresponds to the larger mass.

    After what we described above, we can now formulate the following basic law.

    LAW ╔╔╔

    In the spherical shell problem, Fig. 1, when the two masses m1 and m2 are unequal (m1 < m2), then:
    During the free fall of these unequal masses m1 and m2 within the field of gravity of mass ╠, the smaller mass m1 falls with a higher velocity §1, while, conversely, the larger mass m2, falls with a lower velocity §2, (§1 > §2), as per the inertial reference system S.

    m1 < m2    ,       §1 > §2

      A notable observation

      As we can see, laws I, II, II and conclusion I, cited above, are very important conclusions in the research on the spherical shell problem, Fig. 1.
      Laws ╔, ╔╔, ╔╔╔ apply for every moment t1 (t1 > t0 = 0) of the free fall of masses m1 and m2, within the field of gravity of mass ╠.
      Note
      : Following the above, in basic relations (4) and (4.1), is it plain to see that:
      In the spherical shell problem, Fig. 1:

        a. When masses m1 and m2 are m1 < m2, then relations (4) apply and relations (4.1) are rejected.
        In this case, in relations (4), m1 < m2, §1 > §2 and ╩ > 0.
        b. When masses m1 and m2 are m1 > m2, then relations (4.1) apply and relations (4), are rejected.
        In this case, in relations (4.1), m1 > m2, §1 < §2 and ╩ > 0.
        c. When masses m1 and m2 are m1 = m2, then either relations (4) or relations (4.1) apply.
        In this case, in relations (4) or relations (4.1), m1 = m2, §1 = §2 and ╩ = 0.
        d. Finally, we need to underline that:
        We are led to these same laws ╔, ╔╔, ╔╔╔ and the same conclusion ╔ if, in basic relations (4) and (4.1) instead of relations (4) that we have used in this project, we use relations (4.1). The method is exactly the same, as applied above for relations (4).

      Conclusion

      In the spherical shell problem, Fig. 1, when m1 < m2 based on relation (27.4) the velocities §1 and §2 result from relations (4), where (in relations (4)) m1 < m2, §1 > §2 and ╩ > 0. Obviously, for m1 < m2 based on relation (27.4) relations (4.1) are rejected, because relations (4.1) for ╩ > 0, yield:

        i.e.: §1 < §2

        which is in contrast to relation (27.4).

        SUMMARY II

        As described above, the reasoning (the steps) that we used to solve the spherical shell problem, Fig. 1, are:

        ╔. EQUAL MASSES

        Step 1: In accordance with the problem, if the data given is m1 = m2 then §1§2 can never apply.

        Proof

        Therefore, if the data given is m1 = m2, then §1 = §2 must always apply, as we have proven, i.e.:

        Step 2: Conversely: In accordance with the problem, if the data given is §1 = §2 then m1m2 can never apply.

        Proof

        Therefore, if the data given is §1 = §2 then m1 = m2 must always apply, as we have proven, i.e.:

        Conclusion

        Therefore, from relations (┴.1) and (┴.2), we have:

        ╔╔. UNEQUAL MASSES

        Step 3: Therefore, based on relation (┴.3) it emerges that:

        Step 4: In accordance with the problem and based on relation (┴.4), when m1< m2, then §1 < §2 can never apply.

        Proof

        Therefore, if the data given is m1< m2, then §1 > §2 must always apply, as we have proven, i.e.:

        Step 5: Conversely: In accordance with the problem and based on relation (┴.4), when §1 > §2, then m1 > m2 can never apply.

        Proof

        Therefore, if the data given is §1 > §2, then m1 < m2 must always apply, as we have proven, i.e.:

        Conclusion

        Therefore, from relations (┴.5) and (┴.6) we have:

        Step 6: Therefore, based on relation (┴.7) it emerges that:

        Proof

        Relation (┴.8) can be proven very easily, by the same method we used to prove relation (┴.7), as above.

        These are, therefore, the (6) steps we used to solve the spherical shell problem, Fig. 1, this very important problem of Physics.

        A SECOND SOLUTION TO THE SPHERICAL SHELL PROBLEM

        (An important physics problem, for University Professors, students, etc, who still think that the Theory of Relativity is correct!!!)

        An interesting, second solution to the spherical shell problem, Fig. 1, is the following:
        Given the masses m1, m2, (m1 < m2) and the distance h work out the three functions:

          which give us functions of the time t, the velocities §1, §2, V, the masses m1, m2 , in terms of the inertial reference system S, Fig. 1.
          Note: We assume that ratio k, i.e.:

          ňis relatively small.

          This second solution, i.e. working out the functions (27.1), is a very interesting solution to the spherical shell problem, Fig. 1.
          Obviously, this second solution must be in agreement with law III, as above, and is also proof of this law.

          Problem
          Free fall of the Moon to Earth

          According to Newtonian Mechanics:
          Let us assume that the Earth and the Moon revolve around their axis and around the Sun and are motionless as per an inertial reference system S.
          Let us also assume, Fig. 1, that the Moon has the form of a spherical shell with mass  and radius Ro.
          At the centre of the Moon (i.e. the spherical shell m1) we place an aluminium sphere with mass m2 (similar to the size of a football).
          We now allow the Moon m1 and the aluminium sphere m2 to fall freely from a distance h, within the field of gravity of the Earth M (where h is the distance between the Earth and the Moon), Fig. 1. (see, fig. a).

          We are looking for:
          1. After how much time t1 (from the beginning to = 0 of the free fall of masses  and , within the field of gravity of the Earth ╠) will the distance ho (between the centre c΄ of the Moon  and c the Earth ╠) be ;
          2. At that moment t1, what is the value of velocities , , V masses , , and ╠ in terms of the inertial reference system S?
          Given: The mass of the Moon, m1= 7,347.1022 kg
                    
          The mass of the aluminium sphere, m2 = 10 kg
                    
          The mass of the Earth, M = 5,973.1024 kg
                    
          The distance between Earth and Moon, h = 384.000 km
                     The radius of the Moon, Ro = 1.738 km

          Question

          Which Physics Professor, student, etc, will give us the correct answer to this very important Physics problem?

          A notable observation

          Based on the first and second solution to the spherical shell problem described above, it is now proven, clearly and beyond any doubt, that:
          a. Gravitational fields never have the remarkable property of imparting all bodies (regardless of their mass) with the same acceleration, as Einstein mistakenly claims, according to the “equivalence principle”.
          b.
          The źequivalence principle╗ (Strong and weak) is a totally false theory of Physics, as demonstrated at http://www.tsolkas.gr/forums/tga5.jpg, http://www.tsolkas.gr/forums/tga4.jpg and http://www.tsolkas.gr/forums/tga6.jpg , mentioned above.

          FREE FALL OF UNEQUAL MASSES

          1. The momentum of masses m1 and m2
          Based on relations (4) and in accordance with law III as above, we have proven that:
          In the spherical shell problem, Fig. 1, the smaller mass m1, (m1 < m2) falls with higher velocity §1 (compared to the larger mass m2), which falls with lower velocity §2, ( §12), within the field of gravity of mass ╠.
          In other words, during the free fall of the unequal masses m1 and m2, (m1< m2) the velocity §1 of mass m1 is always higher than the velocity §2 of mass m1, (§1> §2).
          Conversely, with the momentum J1=m1 §1 of mass m1 and J2=m2 §2 of mass m2 that is not the case.
          Specifically, with regard to momentums, J1 and J2, the following happens:
          From the beginning t0 = 0 of the free fall of the unequal masses m1 and m2, (m1 < m2), until a moment tp, (tp > t0 = 0) the momentum J1 is lesser that the momentum J2, i.e.:

            At the moment tp momentums J1 and J2 are equal, i.e.:

            After the moment tp momentum J1 is always greater than momentum J2, i.e.:

              Note: The fact that, in relation (28) m1§1 < m2§2 and not, e.g.  m1§1 > m2§2, (see, in detail at the beginning of the link “Galileo and Einstein are wrong” at the site, www.tsolkas.gr.)

              Therefore, the formulation of the various relations, which result from the above cases (a), (b), (c) is as follows:
              1. In accordance with relation (28):

                  and since, in accordance with law III:

                  relations (31) and (32) yield:

                  Obviously, relation (33) is valid from the beginning t0 = 0 of the free fall of masses m1 and m2, (m1< m2) until the moment tp.

                    2. In accordance with relation (29):

                  and since, in accordance with law III:

                    relations (34) and (35) yield:

                    Obviously, relation (36) is valid only at the moment tp.
                         3. In accordance with relation (30):

                      and since, in accordance with law III:

                      relations (37) and (38) yield:

                      Obviously, relation (39) is valid after the moment tp.

                      The moment tp will be known as the moment tp of equal momentum. Relations (33), (36), (39), will henceforth be known as basic relations of momentum of the spherical shell problem for unequal masses m1 and m2, (m1< m2).
                      In particular, relation (33) will be known as the first constant relation of unequal momentum and relation (39) will be known as the second constant relation of unequal momentum. In contrast, relation (36) will be known as momentary relation of equal momentum.
                      In the spherical shell problem Fig.1, we also observe that the motion of the unequal masses m1 and m2, (m1< m2) during their free fall within the field of gravity of mass M is obviously accelerated and, specifically, non-uniformly accelerated, in accordance with relations (4).
                      Finally, since throughout the free fall the velocity §1 of the smaller mass m1, (m1< m2) is always higher than the velocity §2 of the larger mass m2, this means that:

                      In Fig. 2, the velocity §1 of mass m1, corresponds to curve (§΄1) and the velocity §2 of mass m2, corresponds to curve (§΄2).

                      Fig. 2

                      When, during the free fall of masses m1 and m2, (m1< m2)  within the field of gravity of mass ╠, it occurs, at a moment tp, that the ratio of velocities is equal to the ratio of masses  , Fig. 2 in accordance with case (b) of relation (29), the momentums J1=m1§1 of mass m1 and J2=m2§2 of mass m2 will be equal J1= J2, (m1§1=m2§2) and the momentary relation (36 will apply.
                      Obviously, before the moment tp the first constant relation (33) will apply, and after the moment tp the second constant relation (39) will apply.
                      It is plain to see that everything we discussed above and which is illustrated in Fig. 2 is a very interesting conclusion of the spherical shell.

                      2. The kinetic energy of masses m1 and m2
                      In the same way that we approached the momentums J1 and J2 of unequal masses m1 and m2, (m1 < m2), we will now approach the kinetic energies ╩1 and 2 of those masses.
                       Thus, the kinetic energy 1 of mass m1, will be:

                      and the kinetic energy 2 of mass m2, is:

                      1. In the spherical shell problem, Fig. 1, during the free fall of the unequal masses m1 and m2 , (m1 < m2) at a moment tk, Fig. 3.

                        Fig. 3

                        the kinetic energy 1 of mass m1 will be equal to the kinetic energy 2 of mass m2, i.e.:

                          and since, in accordance with law III:

                          Relations (42) and (43) yield:

                          Relation (44) will be known as the momentary relation of equal kinetic energies of masses m1 and m2.
                          In addition, the moment tk will be known as moment tk of equal kinetic energies of masses m1 and m2.

                          2. The time period, Fig. 3, from the beginning t0 = 0 of the free fall of masses m1 and m2 until the moment tk will be:

                              and since, in accordance with law III:

                              Relations (45) and (46) yield:

                              Relation (47) will be known as the first constant relation of unequal kinetic energies of masses m1 and m2.

                               3. The time period, Fig. 3, after the moment tk will be:

                                and since, in accordance with law II:

                              Relations (48) and (49) yield:

                              Relation (50) will be known as the second constant relation of unequal kinetic energies of masses m1 and m2.

                              Everything that we discussed above is illustrated in Fig. 3.

                              NOTE: We need to stress, at this point, that in the spherical shell problem, Fig. 1, the moment tp in general of equal momentums, Fig. 2, does not coincide with the moment tk of equal kinetic energies of masses m1 and m2, Fig. 3.

                              Following what we discussed above, a (difficult) problems arises:

                              PROBLEM

                              In the spherical shell problem, Fig. 1, what are the minimum masses m1, m2, ╠ and the minimum height h from which the two unequal masses m1 and m2, (m1 < m2) can fall freely within the field of gravity of mass ╠, so that we have a moment tp of equal momentums and a moment tk of equal kinetic energies?

                              Example

                              From what we discussed above, we observe that in free fall lighter bodies fall with higher velocity than heavier bodies, within the field of gravity of a mass ╠.
                              Thus, if (e.g. in a void) we simultaneously release from a height h over the surface of the Earth a feather and a tank, and allow them to fall freely, then the feather will reach the surface of the Earth before the tank!!!

                              Conversely (as is well known), Galileo, Newton, Einstein and many contemporary Physicists claim that the feather and the tank will reach the surface of the Earth at the same time.
                              However, what these Physicists claim is obviously a very big mistake, as proven by the solution of the spherical shell problem, as above.

                              SPECIAL CASE WHEN m1 , m2 , << ╠, (m1 < m2 )

                              In the spherical shell problem, let us now examine the special case where mass m1 of the spherical shell and the point mass m2, are much smaller compared to mass ╠, Fig. 1, i.e.:

                              In this based, based on the second of relations (3), we have:

                              or

                              Therefore, relation (53) based on relations (51), yields:

                              In other words, in our case, based on relation (54) mass ╠, is considered, by close approximation, motionless, i.e. itós velocity V is:

                              V = 0

                              in terms of the inertial reference system S, Fig. 1.

                              Let us assume now, Fig. 4, that we let (to = 0) the spherical shell m1 and the point mass m2 (which is at the centre of the spherical shell m1) fall freely and together (simultaneously) from a height h within the field of gravity of the motionless (V = 0) mass ╠.

                              Fig. 4

                              Let us also assume that, at the moment t1 > 0 (of the free fall of masses m1 and m2), in terms of the inertial reference system S, we have:

                                1. Vcm = the velocity of the centre of mass of masses m1 and m2
                                  §1 = the velocity of mass m1
                                  §2 = the velocity of mass m2
                                2. h΄c = the distance of the centre of mass of masses m1 and m2 from the upper limit of height h.
                                1 = the distance of mass m1
                                2 = the distance of mass m2

                                where:

                                If we now apply the principle of conservation of energy at the moment t1 > 0 of the free fall, we have:

                                  a. For the centre of mass of masses m1 and m2:

                                  Relation (56) yields:

                                  As we can see, in relation (57) the velocity Vcm of the centre of mass of masses m1 and m2 , is independent of masses m1 and m2, i.e. valid for all values of masses m1 and m2.
                                  b. Similarly, for mass m1, we have:

                                    Relation (58) yields:

                                    As we can see, in relation (59) the velocity §1 of mass m1 at the moment t1 > 0 of its free fall, is independent of mass m1 , i.e. valid for all values of mass m1.

                                    c. Similarly, for mass m2, we have:

                                      Relation (60) yields:

                                        As we can see, in relation (61) the velocity §2 of mass m2 at the moment t1 > 0 of its free fall, is independent of mass m2, i.e. valid for all values of mass m2 .

                                        Note: In cases (a), (b), (c) above, we assume that at the moment t1 > 0, the point mass m2 is always within the spherical shell m1.

                                        Also, as we know, the velocity Vcm of the centre of mass of masses m1 and m2 at the moment t1 > 0 is:

                                        Relation (62) yields:

                                        Substituting now in relation (63) Vcm , §1, §2 that result from relations (57), (59), (61) we have:

                                        As we can see in relation (64) the coefficients of m1 and m2, are independent of masses m1 and m2. In other words, the coefficient (within the brackets) of m1 always returns the same value for all values of masses m1 and m2 at the moment t1 > 0 of the free fall of masses m1 and m2, with the field of gravity of the motionless (V = 0) mass ╠.

                                        The same applies to the coefficient (within the brackets) of m2.

                                        Consequently, since (64) at the moment t1 > 0 must be valid for all values of masses m1 and m2 (relation (64)) becomes an identity and we have:

                                        Therefore, in order to verify the identity (65) for all values of m1 and m2 it must be:

                                        Thus, from the first of relations (66), we have:

                                        hc = h1 (67)

                                        and from the second of relations (66), we have:

                                        hc = h2 (68)

                                        Relations (67) and (68) yield:

                                        hc = h1 = h2 (69)

                                        According to relation (69), relations (57), (59), (61) yield:

                                        From relation (70) it arises that: 1) The centre of mass of the spherical shell m1 and the point mass m2 2) The spherical shell m1 and 3) The point mass m2, fall with the same velocity, within the field of gravity of the motionless (V = 0) mass ╠.

                                        In addition, relations (69) and (55) yield:

                                        Based on relation (71) we observe that, from the upper limit of height h: 1) The distance c of the centre of mass of masses m1 (of the spherical shell) and the point mass m2) The distance 1 of mass m1 (of the centre of the spherical shell) and 3) The distance h΄2 of the point mass m2, are always equal at the moment t1 > 0 of the free fall of the spherical shell m1 and the point mass m2, within the field of gravity the motionless (V = 0) mass ╠.
                                        After, therefore, what we discussed above, and according to relations (70) and (71), we draw the following basic conclusion.

                                        Conclusion

                                        In the case of Fig. 4 and during t1 > 0 of the free fall of mass m1 (of the spherical shell) and the point mass m2, from height h within the field of gravity of the motionless (V = 0) mass ╠, ( m1, m2 << M and m1 < m2) the point mass m2 will always remain at the centre of the spherical shell m1, where it was originally placed at the moment to=0.

                                        As we can see, the above conclusion can be applied to Galileoós experiment (the tower of Pisa experiment), fore the two masses m1 and m2 (m1, m2 << M and m1 < m2 ) that Galileo allowed to fall freely from the top of the tower of Pisa, where ╠, is the mass of the Earth.
                                        NOTE:
                                        In theory, based on relation:

                                        and relation (33):

                                        relation (70), i.e.:

                                        §1 = §2

                                        is not, in reality, §1 = §2, but:

                                        §1 §2

                                        i.e.:  §1 - §2 = ─§ → 0      (73)
                                        This fact (that ─§ → 0), was technically (experimentally) impossible for Galileo to identity with the technical resources he had.
                                        That is the reason why Galileo was deceived and led to an incorrect formulation of the results of the experiment of the free fall of bodies (the tower of Pisa experiment).

                                        Therefore, if the results of the Tower of Pisa experiment are examined in strict theoretical terms, it is proven that they are completely incorrect in accordance with relations  (72) and (73). Simply put, the results of the Tower of Pisa experiment are empirical and approximate and are not valid in theoretical terms. 

                                        THE “TOWER OF PISA” EXPERIMENT
                                        ON AN ASTEROID

                                        Let us assume that the Tower of Pisa was built on an asteroid with a diameter, e.g. of D = 50 m and that Galileo released from the top of that tower e.g. a cotton wool sphere with a diameter of d1=10 cm and a sphere with a diameter of d2=10 cm, which consists of the material of a neutron star.
                                        In this case, will the two spheres each the surface of the asteroid at the same time, as with the Tower of Pisa experiment that Galileo performed on Earth?

                                        The answer to that question is negative.
                                        In the case of the “asteroid experiment” , according to relation (72), i.e.:

                                        the sphere made of cotton wool will be the first to reach the surface of the asteroid, followed by the sphere that consists of the material of a neutron star.
                                        In the “asteroid experiment” described above, Galileoós fallacy and Einsteinós mistake (regarding the “remarkable” property of gravitational fields), as cited in the beginning (in the project summary) and as we will demonstrate next, are completely clear.
                                        Unfortunately, Galileoós fallacy and Einsteinós mistake led contemporary Physics down the wrong path.

                                        THE FUNDAMENTAL LAW OF THE FREE FALL OF BODIES

                                        In recapitulating, the fundamental law of the free fall of bodies is formulated as follows: 

                                        Einsteinós big error!

                                        In the General Theory of Relativity Einstein claims that: 
                                        “Gravitational fields have the remarkable property of imparting all bodies with the same acceleration (irrespective of their mass and material constitution) at all times”.

                                        See:

                                        However, according to the law of the free fall of bodies, everything that Einstein maintains is utterly mistaken for the following reasons:
                                        In nature, the above “remarkable” property (to which Einstein refers in the case of gravitational fields) is exhibited only by fields of inertial forces and by no other field of forces (e.g. gravitational, electrical, magnetic, etc.). 
                                        This is where Einstein greatly erred. In fact, this is one of his many serious errors in formulating the Theory of Relativity.

                                        More specifically, looking at the text of the books (1) and (2) cited above, Einsteinós error is very much apparent --especially in book (2), where Einstein claims that the aluminum sphere and the Earth fall at the same velocity!!! In the gravitational field of a star having a mass M.
                                        According to the fundamental law of the free fall of bodies and the example of the “asteroid experiment” mentioned above, Einstein was largely mistaken.

                                        BASIC CONCLUSION  

                                        As demonstrated in the previous chapter and according to relation (70):  

                                        The special case (referred to above), where e.g. two bodies in free fall, with small masses m1 and m2, happen coincidentally to fall at the same velocity §1 and §212)  inside the gravitational field of a large mass M (m1,m2<<M), does not signify in any case that:  
                                        Gravitational fields have the remarkable property of imparting all bodies unexceptionally with same velocity (regardless of their mass), as Einstein erroneously maintains in his equivalence principle (See the above two pictures (1) and (2)).   Gravitational fields never have this “remarkable” property as Einsteinós asserts.
                                        Reiteration: In nature, this “remarkable” property is exhibited only by fields of inertial forces and by no other field of forces (e.g. gravitational, electrical, magnetic, etc.).   

                                        Obviously, in formulating the General Theory of Relativity, Einstein made one of his greatest errors.

                                        The principle of equivalence and the spherical shell problem (SSP)

                                        Let us assume (Fig, 1) that spherical shell m1 is a spherical elevator in which there is an observer (¤). At time to=0 the distance between the spherical elevator m1 and a mass is h. Also, at time to=0 the velocity §1 of the spherical elevator m1 and the velocity V of mass ╠, relative to the inertial reference frame S are §1=0 and V = 0 respectively.
                                        The observer (¤) at a moment to=0 places a point mass m2, (m1<m2) in the centre of the spherical shell m1.
                                        At some point, we let the spherical elevator m1 (in which point mass m2 and observer (¤) are found) fall freely from a height h, inside the gravitational field of mass ╠.
                                        In this case (according to the fundamental law of the free fall of bodies referred to above) at the time t1>0 of the free fall of the spherical elevator m1 and point mass m2 inside the gravitational field of mass ╠ relative to the inertial reference frame S, the following will be observed:
                                        The velocity §1 of the spherical elevator will be §1>0 and the velocity §2 of point mass will be §2>0, where:

                                        §21 (75)

                                        Consequently, on the basis of relation (75), at the time t1>0 of the free fall, point mass m2 will move from the centre of spherical elevator m1 where it was originally placed at a moment to=0.
                                        Therefore, since from relation (75) it transpires that §21, the observer (¤) inside the spherical elevator m1 (which is in free fall inside the gravitational field of mass M) will observe the following:
                                        Point mass m2 will move from the centre of the spherical elevator m1 and will continuously ascend towards the elevatorós top (“roof”) during the free fall of the spherical elevator m1 and point mass m2 inside the gravitational field of mass M.
                                        As is well known, the above conclusion is completely opposed to the “equivalence principle” of the General Theory of Relativity for the following reason:
                                        According to the “equivalence principle”, Einstein claims that:
                                        During the free fall of the spherical elevator m1 inside the gravitational field of mass M, the observer (O) found inside the spherical elevator will observe that point mass m2 remains at all times at the centre of the spherical elevator m1..

                                        However, according to the fundamental law of the free fall of bodies Einsteinós conclusion is wrong.
                                        Therefore, after everything discussed above, we come to the following basic conclusion:

                                        Conclusion

                                        On the basis of the fundamental law of the free fall of bodies, it is demonstrated beyond any doubt that the “equivalence principle” (strong and weak) of the General Theory of Relativity is a completely false principle of Physics. Consequently, the Theory of Relativity should also be regarded as an utterly erroneous theory of Physics.   

                                        Two big mistakes many physicists make!

                                        Mistake 1: One mistake still made by a great many physicists (Einstein included) is the following: 
                                        Let us assume that there is a body of mass m falling freely inside the gravitational field of a mass M, and at a moment t the distance between the two masses m and is r.
                                        In this case, according to the law of universal attraction and the fundamental law of Newtonian Mechanics, the following relation will apply:

                                        where mi and mg are the inertial and gravitational mass of mass m respectively.
                                        a, is the acceleration of the body during a moment t > 0 of its free fall inside the gravitational field of mass M, and
                                        G
                                        is the universal gravitational constant.
                                        In relation (76), since mi=mg (according to Newtonós axiom) we have:

                                        a = g (77)

                                        where g is the intensity of the gravitational field of mass M at a distance r, namely:

                                        So, on the basis of relation (77), we observe that the acceleration a of the body falling freely inside the gravitational field of mass M is independent of that bodyós mass m.
                                        Consequently, physicists mistakenly claim that:
                                        Based on relation (77), since the acceleration a of bodies falling freely inside the gravitational field of mass M is independent of these bodiesó mass, all bodies (heavy or light) will fall with the same acceleration a inside the gravitational field of mass M.
                                        Moreover, all these physicists invoke Galileoós experiment (the tower of Pisa experiment) for the experimental verification of relation (77). Thus, the following question now emerges: 
                                        On the basis of relation (77), which is the big mistake that all these physicists make?
                                        The answer to this question is:
                                        By using relation (77) alone, the problem of the free fall of bodies is solved in the wrong way, because according to this relation the solution of the problem is not complete and therefore is wrong.
                                        Conversely, the solution of the problem of the free fall of bodies (e.g. two bodies m1 and m2) inside the gravitational field of a mass M is integral and accurate only when we apply to the closed system of the three bodies m1 – m2 – M:

                                        1. The principle of conservation of energy,
                                        2. The principle of conservation of momentum,
                                        3. Newtonós axiom mi=mg, and
                                        4. Taking obviously into account the free fall (movement) of mass M towards the two masses m1 and m2 (as was exactly the case with the solution of the spherical shell problem).

                                        The above prerequisites (1), (2), (3), (4) must be taken into account for the correct solution of our problem. Conversely, these prerequisites were never considered by the various physicists (Einstein included) who dealt with this problem.
                                        This is, therefore, the big mistake that many physicists made (and continue to make) in solving the spherical shell problem (SSP).

                                        Mistake 2: Another big mistake that physicists have made and continue to do so is the following:
                                        Newtonós axiom on the equality of the inertial mass mi and gravitational mass mg (mi=mg) is regarded as accurate and has been experimentally verified with great precision by means of various experiments. This axiom was used in the solution of the spherical shell problem (SSP) referred to above. 
                                        Unfortunately, many physicists confuse the equality mi=mg with the equivalence between mi and mg ( as claimed by Einstein in the equivalence principle of the General Theory of Relativity).
                                        This is a big mistake on the part of physicists for the following reason:
                                        Under no circumstances does the equality mi=mg in Newtonós axiom signify an equivalence between mi and mg as physicists wrongly claim. This is therefore the big mistake of these physicists.
                                        Simply put, the experimental verification of Newtonós axiom does not imply an experimental verification of an equivalence between mi and mg, as is (unfortunately) asserted by many scientists.
                                        This is apparently a very delicate point, and caution is required when we refer to the equality mi=mg and the equivalence between mi and mg.
                                        From the point of view of physics, the equality of and equivalence between mi and mg are two totally different concepts and should never be confused.
                                        Specifically, whereas in classical Physics the equality mi=mg is an axiom (Newtonós axiom), in the theory of Relativity this particular axiom (mi=mg) is demonstrated on the basis of the General Theory of Relativity. 
                                        Yet, since (according to the fundamental law of the free fall of bodies) it is proven that the “equivalence principle” of the General theory of Relativity is an utterly erroneous principle of Physics, the following occurs:
                                        A bodyós inertial mass mi is at all times equal to its gravitational mass mg (mi=mg) (as Newton accurately maintains), and these two masses mi and mg are never equivalent (as Einstein mistakenly claims) as per the “equivalence principle”
                                        .
                                        So this is Einsteinós second big mistake –and of many other physicistós also–, as regards the equality of and equivalence between a bodyós inertial and gravitational mass mi and mg.

                                        EPILOGUE

                                        The Theory of Relativity (Special and General) is unquestionably a completely false Theory of Physics.
                                        Therefore, a well-disposed physics researcher who studies carefully this site (www.tsolkas.gr) will wonder about the following: 

                                        Five Questions

                                        Question 1: Why isnót a simple experiment like the Cedarholm – Townes experiment (1959) carried out on a moving vehicle (e.g. train, aircraft, satellite, etc), so that it can be established once and for all whether ether exists in nature?
                                        What is, therefore, the reason for not performing the Cedarholm – Townes experiment on a moving vehicle?

                                        Question 2: Why isnót experiment 21 (See www.tsolkas.gr) reperfomed, since it is a very simple and low-cost experiment which proves that the Special Theory of Relativity is completely false?
                                        What is, therefore, the reason for not performing experiment 21?

                                        Question 3: Why has experiment - 14 been kept quiet (See www.tsolkas.gr), since it demonstrates theoretically (in a very simple manner and without any cost) that the “principle of equivalence” of the General Theory of Relativity is an utterly erroneous principle of physics?
                                        What is, therefore, the reason for keeping quiet about this very important experiment?

                                        Question 4: Why is the real cause of the advance of Mercuryós perihelion (43΄΄per century), i.e. the Sunós revolution around the centre of mass of the Solar system, being kept quiet?
                                        Because this advance (of the planetsó perihelia) is not attributed to the curvature of time-space around the Sun as Einstein wrongly maintains (See ˇ˘´ www.tsolkas.gr).
                                        Why is this very significant point (purposely) suppressed?

                                        Question 5: Why is the spherical shell problem (discussed above) being kept quiet, since it proves in a very simple manner that : 1) the “principle of equivalence” of the General Theory of Relativity is a completely false principle of Physics, and 2) gravitational fields do not have the “remarkable” property of imparting the same acceleration to all bodies (irrespective of the latterós mass), as Einstein erroneously claims?
                                        What is, therefore, the reason for keeping quiet about the spherical shell problem?

                                        What answers have physics professors given to Questions (1), (2), (3), (4), (5)?
                                        Is there some “expediency” for not providing an answer to the above five questions? After everything discussed in this paper (and on www.tsolkas.gr in general), we come to the following important conclusion:

                                        Conclusion

                                        Modern physics should be rid once and for all of the “pseudo-science” of the Theory of Relativity.
                                        This theory should no longer be taught in universities for the simple reason that it is false and does not reflect natural reality.
                                        If however, the “pseudo-science” of the Theory of Relativity continues to be taught in universities, this will be the biggest shame in the history of Physics.

                                        Lastly,  I have a more general question to pose. In closing up.

                                         QUESTION

                                        Could it be that physics is currently controlled and directed by various “Centers” (universities, professors, journals, etc) towards an anti-scientific path that is taking us to a new Middle Ages?

                                        I am posing this question because this inexplicable (and perhaps intentional) silence and non performance of the above-mentioned experiments probably lend truth to it. 

                                        I HAVE A QUESTION...

                                        As is well-known, tremendous amounts of money have been spent on the Gravity Probe b experiment, and eventually this experiment failed!
                                        This compels me to ask all these Physics “Centers” the following question:
                                        Why not spend just a small amount of money to carry out two simple experiments, namely: 

                                        1. The Cedarholm – Townes (1959) experiment on a moving vehicle (e.g. train, aircraft, satellite, etc), and
                                        2. Experiment – 21 (See www.tsolkas.gr), in order to establish once and for all the accuracy or fallacy of the Theory of Relativity.

                                        I repeat, why donót these Physics “Centers” explain to us the reason for not conducting these two experiments?
                                        Is there some “expediency” behind this, --so that the foundations of contemporary physics wonót utterly shaken? 

                                        I believe this is a very rational question.
                                        This is where my paper ends. Time will tell whether I am right or not.

                                        Copyright 2010: Christos A. Tsolkas    Christos A. Tsolkas, August 23, 2010

                                        ©  Copyright 2001 Tsolkas Christos.  Web design by Wirenet Communications