Experiment 13


	THE ROCKET EXPERIMENT 1. «Homogenous» gravitational field Let’s assume, fig.1, that we have a body of mass Μ and radius R.



	fig. 1 As it is known, the gravitational field of mass Μ (both in terms of geometry and dynamics) can never be homogenous, fig. 1. In certain cases, however, the gravitational field of mass Μ, may be considered «locally» very nearly homogenous. Thus, e.g. at a distance h near the surface of mass Μ, to which the following relation applies:

The gravitational field of mass Μ may be considered «locally» homogenous in the sense that, its dynamic lines are parallel and its intensity g is constant from the surface of mass Μ, to height h.
Thus, in a celestial body of mass Μ and radius R = 6000 km the gravitational field at a height h = 100 m from its surface, may be considered very nearly homogenous, because based on relation (1) we have:


	i.e.,

A NOTABLE OBSERVATION

As observed, the homogenous field of a body of mass Μ and radius R, only relates to the radius R and never the mass Μ of this body.

2. Free fall of a body in a homogenous gravitational field

Let’s assume, fig. 2 that we have a body of mass Μ and radius R.

fig. 2

At a height h, we place a point mass m.
We now allow masses m and Μ to move freely under the influence of the force of universal attraction.
In this case, mass m will fall freely towards mass Μ and mass Μ will fall freely towards mass m.
Let’s say that, at the moment of collision of the two masses m and Μ, υ is the velocity of mass m and V is the velocity of mass Μ , relative to an inertial observer .
Thus, in the system of the two bodies m – M, by applying the principle of conservation of energy and the principle of conservation of momentum, we have:

Where G is the constant of universal attraction and the numbers h, R, υ ,V are positive.
Solving the equation system (2), relative to υ and V, we have:

Relations (2.1) are of great importance, in terms of physics, as they express the free fall of mass m in the non-homogenous gravitational field of mass Μ.

We now place mass m at a height h, to which the following relation applies:

in this case, the gravitational field of mass Μ from its surface and up to height h, can be considered very nearly homogenous, with constant intensity g, as mentioned above.

Thus, from relation (2.2), we have:

Based on relation (2.3), relations (2.1), yield (approximately):

Because, however, the intensity g of the homogenous gravitational field of mass M, from its surface to height h, is considered constant, we have:

Thus, based on relation (2.5), relations (2.4), yield (approximately):

As observed in relations (2.6) velocity υ is a factor of mass m and velocity V is a factor of mass Μ, obviously taking sizes g and h as constant.
Relations (2.6) are of great important, in terms of physics, as they express the free fall of mass m in the non-homogenous gravitational field of mass Μ.
In relations (2.6), if we agree that mass m is much smaller than mass Μ, i.e.:

the relations (2.6) yield:

Relations (2.7), are the well-known relations of Elemental Mechanics.
As observed, relations (2.7) are independent of the mass m of the body, which falls freely in the gravitational field of mass Μ.
Obviously, relations (2.7) expresses Galileo’s well-know, erroneous law on the free fall of bodies.
As observed, the conclusion of relations (2.1) and (2.6) is in total contrast with Galileo’s law on the free fall of bodies, which claims that:
The velocity of bodies falling freely in the gravitational field of a mass Μ is independent from the mass of the falling body. In other words, all bodies fall with the same velocity in the gravitational field of a mass Μ , regardless of how large or small their mass may be.
Obviously, what Galileo (and Einstein) claims is a big mistake, because:
LAW: The velocity of bodies falling freely in the gravitational field of a mass Μ, is always a function of their mass, whether the gravitational field of mass Μ is non-homogenous, relations (2.1), or it is considered homogenous (at a small distance h near the surface of mass Μ, relations (2.6)).
Only equal masses fall with the same velocity through the above gravitational fields (non-homogenous or homogenous), and never, under any circumstances, unequal masses.
In other words, therefore, believing that, «all bodies (regardless of their mass), fall with the same velocity» in a homogenous gravitational field is a grave mistake.
NOTE: Relations (2.6) also result from relations:




	by applying the principle of conservation of energy and the principle of conservation of momentum to the system of the two bodies m – Μ of fig. 2. Solving the equation system (2.8) relative to υ and V, yields relations (2.6). At this point, we need to stress thatin a system of two bodies m – M, fig. 2 relations (2.8) are only valid when:



		i.e. when mass m is in the homogenous field of mass Μ) and obviously relations (2.8) are never valid for just any h. That is a very «sensitive issue», which we must bear in mind when referring to relations (2.8). RECAPITULATION Following what we looked at above, in a system of two bodies m – M and taking e.g. mass m, as a point mass, fig.2, we have: 1. Relations (2.1), apply to any masses m and Μ and to any h. 2. Relations (2.6) are only valid when:


	i.e., when mass m is within the homogenous field of mass Μ. 3. Relations (2.7) are only valid when:


	and

i.e., when mass m is within the homogenous field of mass Μ and only when mass m is much smaller that mass Μ , as, e.g. is the case in Galileo’s experiment (the leaning tower of Pisa).

3. The Problem

Let’s assume that we have a rocket S, within which there is an astronaut Ο.
Thus, we tell astronaut Ο:
Your rocket S will move in one of the following phases, i.e.:
PHASE Ι: Your rocket S will perform uniform accelerated motion with constant acceleration γ, relative to an inertial observer , fig. 3(a), or
PHASE ΙΙ: Your rocker S will be placed motionless on the surface of a celestial body of mass Μ and radius R.
In this case, you will consider the gravitational field within your chamber as homogenous and of constant intensity g, fig. 3 (b).

fig.3

Next, we assign the astronaut Ο with the following problem:
Can you, from within your chamber, and by conducting various Mechanics experiments, prove in which of the above two phases (Phase I or Phase II) your rocket is?
That is the problem assigned to astronaut Ο.

4. The astronaut’s experiments

Astronaut Ο, in order o prove which of the two phases (Phase I or Phase II) his rocket is in, acts as follows:

a. The device for measuring velocity

To begin with, astronaut Ο places near the floor CD of his chamber, a device Ε that radiates e.g. two parallel laser beams L₁ and L₂. Those parallel laser beams are at a small distance d from one another, and are parallel to the floor CD of the chamber. Beams L₁ and L₂ are connected to e.g. an oscilloscope, which acts as a chronometer.
Also, beams L₁ and L₂ are interrupted during the fall e.g. of a mass m, and the oscilloscope (chronometer) records the time taken by mass m to cross the distance d the two parallel beams L₁ and L₂.
In this way, astronaut Ο measures the velocity υ with which a mass m, which was released from the top AB of his chamber, falls onto the floor CD. Velocity υ is calculated using relation:

where t is the time recorded by the oscilloscope (chronometer), when mass m crossed the known distance d between the two parallel Laser beams L₁ and L₂.

b. Performing the experiments

Astronaut Ο, after installing the device Ε for measuring velocity, as described above, now performs the following experiments:
To begin with, he releases, from the top AB of his chamber, a mass m₁, which falls freely to the floor CD of his chamber and, using device E (in accordance with relation (a)), measures the velocity of the fall of mass m₁.
He then repeats the same experiment with a different mass m₂, (m₁<m₂) and measures, once again using device E, the velocity υ₂ with which mass m₂ falls on the floor CD of his chamber.
In other words, astronaut Ο allows masses m₁ and m₂ to fall separately (first mass m₁ and the repeats the process with mass m₂) and does not allow masses m₁ and m₂ to fall simultaneously from the top of his chamber.
Now astronaut Ο compares the velocities υ₁ and υ₂ of masses m₁ and m₂ and concludes that:

Α. If velocities υ₁ and υ₂ are equal, then his rocket S is in Phase Ι.
Astronaut Ο bases this conclusion on the fundamental property of accelerating reference systems to give all bodies (regardless of their mass) the same acceleration and, subsequently, velocity.

Β. Conversely, if velocities υ₁ and υ₂ are unequal, and if, specifically υ₁ > υ₂, the rocket S is in Phase II.
Astronaut Ο bases this concussion on the first of relations (2.6).

Subsequently, conclusions Α and Β, as detailed above, are astronaut O’s answer to the problem we set him.
Obviously, this answer provided by astronaut Ο is correct and in accordance with reality, regarding the Phase (Phase Ι or Phase II) in which rocket S actually is.

NOTE:

The body of mass Μ and radius R mentioned above, may be, for example:
1. A balloon full of air, of a radius e.g. R=6000 km.
2. A sphere, made of cork or iron, of a radius e.g. R=2000 km.
3. A small asteroid, the Moon, the Earth, a planet, etc, etc.
The masses m₁ and m₂ used by astronaut Ο in the experiments he conducts within his chamber may be, for example:
1. A small sphere made of cork, of diameter e.g. D=1 cm.
2. An iron sphere, of diameter e.g. D=50 cm.
3. A sphere, consisting of the material of a neutron star (the density of which is known to be ), of diameter e.g. D=10 cm, etc, etc.
The height h of the astronaut’s chamber is small, measuring a few metres, e.g. h = 10 m (smaller or larger) so that the gravitational field of mass Μ within the astronaut’s chamber can be considered very nearly homogenous (for the above numerical example Ι), in accordance with the «notable observation» mentioned at the beginning of this project.

5. Various other conclusions drawn by the astronaut

Astronaut O, following the above correct answer he gave to the problem we set him, now proceeds to other conclusions regarding the Phase (Phase I or Phase II) his rocket S is in.
Astronaut Ο follows the thought process below:
Astronaut Ο says:

a. In accordance with the above, if I reach conclusion (b.A) then, by placing a dynamometer D at the top of my chamber, I can calculate the acceleration γ with which my rocket moves, relative to the inertial observer O’, i.e.:

where F is the reading of dynamometer D and m is a known mass attached to the end of the dynamometer’s spring.
Since the inertial force field within the chamber is homogenous, with constant intensity , g’ (g’=γ).
b. If, however (says astronaut Ο) I reach conclusion (b.B), and because the gravitational field within my chamber is homogenous, with constant intensity g, then I will act as follows:
As, with the help of device Ε, I have found out the velocities υ₁ and υ₂ of masses m₁ and m₂, that fall towards the floor CD of my chamber, I will obviously also know their ratio k, i.e.:

Thus, from the first of relations (2.6) the velocity υ₁ of mass m₁ , is:

and the velocity υ₂ of mass m₂, is:

From relations (5) and (6), because m₁<m₂, it follows that υ₁ > υ₂.

Dividing now by member relations (5) and (6), we have:




	And based on relation (4), relation (7) yields:

In relation (8), because m₁, m₂, k are known, astronaut Ο also knows the mass Μ of the celestial body, where his rocket S is docked.

Also, from the first of relations (2.6), we have


	or


	Now, by replacing the Μ in relation (9) with that of relation (8), we have:

In relation (10) factors υ₁, h, m₁, k are known.
Subsequently, from relation (10), astronaut Ο also knows the intensity g of the homogenous gravitational field of mass Μ, where his rocket is S is docked.
Finally, from the relation:

We have:

Where G is the constant of universal attraction.
Substituting, now, in relation (12) the Μ and g provided by relations (8) and (10) results in:

In relation (13) factors G, h, υ₁, m₁, m₂, k are known. Subsequently, on the basis of relation (13), astronaut Ο also knows the radius R of mass Μ, where his rocket S is docked.
Following what we discussed above, regarding the rocket experiment, we are led to this basic conclusion:

CONCLUSION Ι

Under no circumstances can a homogenous gravitational field, with intensity g, be equivalent to a homogenous inertial force field, with acceleration γ, (γ= g).

In other words:
An observer Ο, who is within chamber S (and by conducting various Mechanics experiments), can easily ascertain: whether his chamber is performing uniform accelerated motion with constant acceleration γ, relative to an inertial observer or whether his chamber is motionless in a homogenous gravitational field with constant intensity g, of a mass Μ, which is outside his chamber.
Thus, according to the above conclusion, the «equivalence principle» of the General Theory of Relativity is completely erroneous.

ANOTHER MISTAKE BY EINSTEIN...

As is well known, in the «equivalence principle» (weak equivalence principle), Einstein claims that:
«A reference system S (e.g. an elevator), falling freely in the gravitational field of a mass Μ, is locally equivalent to an inertial reference system».
Einstein’s claim, however, is wrong, because: Let’s assume, fig. 4, that we have a spherical shell (e.g. a spherical elevator), of mass m₁ and radius R, falling freely from a height h, in the gravitational field of a mass Μ. An observer Ο, who is within the elevator, places a mass m₂, (m₁<m₂) at he centre Κ of the spherical elevator chamber.

fig. 4

Now, if what Einstein claims above, fig. 4, on the «equivalence principle» were correct, then, during the elevator’s free fall in the gravitational field of mass Μ, mass m₂ would remain at the centre Κ of the elevator chamber (as would be the case if the elevator were an inertial reference system S, away from gravitational fields).
As we have proven in a previous chapter of our project, however (See, www.tsolkas.gr, link: «Galileo and Einstein are wrong») the mass m₁ of the spherical elevator falls at a greater velocity υ₁ than the velocity υ₂ of mass m₂ , relative to an inertial observer O’.
Subsequently, observer Ο, ο who is within the elevator, will observe mass m₂moving from the centre Κ towards the «ceiling» of the chamber during the elevator’s free fall in the gravitational field of mass Μ.
After examining the above, we are led to the following basic conclusion:

CONCLUSION II

A reference system S, falling freely in the non-homogenous or homogenous gravitational field of a mass Μ can never be locally equivalent to an inertial reference system.

In other words:
An observer Ο, who is within a reference system S, which is falling freely in the gravitational field of a mass M, by conducting various Mechanics experiments (such as, e.g. the mass m₂placed at the centre K of the spherical elevator, as described above) can easily ascertain whether his reference system is falling freely in the gravitational field of a mass M, or whether his reference system is an inertial reference system, away from gravitational fields.
Specifically, in terms of our experiment above:

If mass m₂ remains at the centre Κ of the spherical elevator chamber, the observer Ο, who is within the elevator, will know that his chamber is an inertial reference system, away from gravitational fields.
If mass m₂ moves from the centre Κ of the spherical elevator chamber, the observer Ο, who is within the elevator, will know that his chamber is a reference system S, falling freely in the gravitational field of a mass Μ, which is outside his chamber.

Thus, in accordance wit the above, the «equivalence principle» of the General Theory of Relativity must be rejected as completely erroneous.

A QUESTION ON THE «EQUIVALENCE PRINCIPLE»...

It is well known that, in the «strong equivalence principle», Einstein accepts that: «All bodies (regardless of their mass) fall with the same velocity in the gravitational field of a mass M», (See http://www.tsolkas.gr/forums/777.jpg), book:Clifford M. Will, “Was Einstein right?”.
QUESTION: Can «Einstein» (the relativists) show us the mathematical relation that the above conclusion is based on?
Which is that mathematical relation?
Let «Einstein» (the relativists) present us with this mathematical relation, so that we can see and judge it or ourselves.

RECAPITULATION

Following all that we examined in our project, and on the basis on Conclusion I and Conclusion II, it is clearly and undoubtedly proven that the «equivalence principle» (weak and strong equivalence principle) is a completely erroneous theory of Physics.
Subsequently, the General Theory of Relativity (which, as we know, is based on the «equivalence principle») must also be considered as a completely erroneous theory of Physics.