THE OCKET EXPEIENT. «Hoogenous» gravitational field Let s assue, fig., that we have a body of ass Μ and radius. fig. As it is known, the gravitational field of ass Μ (both in ters of geoetry and dynaics) can never be hoogenous, fig.. In certain cases, however, the gravitational field of ass Μ, ay be considered «locally» very nearly hoogenous. Thus, e.g. at a distance h near the surface of ass Μ, to which the following relation applies: h 0 () The gravitational field of ass Μ ay be considered «locally» hoogenous in the sense that, its dynaic lines are parallel and its intensity g is constant fro the surface of ass Μ, to height h. Thus, in a celestial body of ass Μ and radius = 6000 k the gravitational field at a height h = 00 fro its surface, ay be considered very nearly hoogenous, because based on relation () we have:
h 0 = 6 6 0 =,6 0 5 h i.e., 0 A NOTABLE OBSEVATION As observed, the hoogenous field of a body of ass Μ and radius, only relates to the radius and never the ass Μ of this body.. Free fall of a body in a hoogenous gravitational field Let s assue, fig. that we have a body of ass Μ and radius. fig. At a height h, we place a point ass. We now allow asses and Μ to ove freely under the influence of the force of universal attraction. In this case, ass will fall freely towards ass Μ and ass Μ will fall freely towards ass. Let s say that, at the oent of collision of the two asses and Μ, is the velocity of ass and V is the velocity of ass Μ, relative to an inertial observer O.
Thus, in the syste of the two bodies, by applying the principle of conservation of energy and the principle of conservation of oentu, we have: G + h = + V = V G () Where G is the constant of universal attraction and the nubers h,,,v are positive. Solving the equation syste (), relative to and V, we have: = V= h G (+ h) + h G (+ h) + (.) elations (.) are of great iportance, in ters of physics, as they express the free fall of ass in the non-hoogenous gravitational field of ass Μ. We now place ass at a height h, to which the following relation applies: h 0 (.) in this case, the gravitational field of ass Μ fro its surface and up to height h, can be considered very nearly hoogenous, with constant intensity g, as entioned above. Thus, fro relation (.), we have: h 0 h + + 0 + h + h (.3) Based on relation (.3), relations (.), yield (approxiately):
= G V= h + G h + (.4) Because, however, the intensity g of the hoogenous gravitational field of ass, fro its surface to height h, is considered constant, we have: g= G (.5) Thus, based on relation (.5), relations (.4), yield (approxiately): = gh + (.6) V= gh + As observed in relations (.6) velocity is a factor of ass and velocity V is a factor of ass Μ, obviously taking sizes g and h as constant. elations (.6) are of great iportant, in ters of physics, as they express the free fall of ass in the non-hoogenous gravitational field of ass Μ. In relations (.6), if we agree that ass is uch saller than ass Μ, i.e.: 0 the relations (.6) yield: = gh V = 0 (.7) elations (.7), are the well-known relations of Eleental echanics.
As observed, relations (.7) are independent of the ass of the body, which falls freely in the gravitational field of ass Μ. Obviously, relations (.7) expresses Galileo s well-know, erroneous law on the free fall of bodies. As observed, the conclusion of relations (.) and (.6) is in total contrast with Galileo s law on the free fall of bodies, which clais that: The velocity of bodies falling freely in the gravitational field of a ass Μ is independent fro the ass of the falling body. In other words, all bodies fall with the sae velocity in the gravitational field of a ass Μ, regardless of how large or sall their ass ay be. Obviously, what Galileo (and Einstein) clais is a big istake, because: LAW: The velocity of bodies falling freely in the gravitational field of a ass Μ, is always a function of their ass, whether the gravitational field of ass Μ is non-hoogenous, relations (.), or it is considered hoogenous (at a sall distance h near the surface of ass Μ, relations (.6)). Only equal asses fall with the sae velocity through the above gravitational fields (non-hoogenous or hoogenous), and never, under any circustances, unequal asses. In other words, therefore, believing that, «all bodies (regardless of their ass), fall with the sae velocity» in a hoogenous gravitational field is a grave istake. NOTE: elations (.6) also result fro relations: g ( + h) = V = + V + g (.8) by applying the principle of conservation of energy and the principle of conservation of oentu to the syste of the two bodies Μ of fig.. Solving the equation syste (.8) relative to and V, yields relations (.6). At this point, we need to stress thatin a syste of two bodies, fig. relations (.8) are only valid when: h i.e. when ass is in the hoogenous field of ass Μ) and obviously relations (.8) are never valid for just any h. That is a very «sensitive issue», which we ust bear in ind when referring to relations (.8). 0 ECAPITULATION Following what we looked at above, in a syste of two bodies and taking e.g. ass, as a point ass, fig., we have:. elations (.), apply to any asses and Μ and to any h.. elations (.6) are only valid when: h 0
i.e., when ass is within the hoogenous field of ass Μ. 3. elations (.7) are only valid when: h 0 and 0 i.e., when ass is within the hoogenous field of ass Μ and only when ass is uch saller that ass Μ, as, e.g. is the case in Galileo s experient (the leaning tower of Pisa). 3. The Proble Let s assue that we have a rocket S, within which there is an astronaut Ο. Thus, we tell astronaut Ο: Your rocket S will ove in one of the following phases, i.e.: PHASE Ι: Your rocket S will perfor unifor accelerated otion with constant acceleration γ, relative to an inertial observer O, fig. 3(a), or PHASE ΙΙ: Your rocker S will be placed otionless on the surface of a celestial body of ass Μ and radius. In this case, you will consider the gravitational field within your chaber as hoogenous and of constant intensity g, fig. 3 (b). fig.3 Next, we assign the astronaut Ο with the following proble: Can you, fro within your chaber, and by conducting various echanics experients, prove in which of the above two phases (Phase I or Phase II) your rocket is?
That is the proble assigned to astronaut Ο. 4. The astronaut s experients 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 easuring velocity To begin with, astronaut Ο places near the floor CD of his chaber, a device Ε that radiates e.g. two parallel laser beas L and L. Those parallel laser beas are at a sall distance d fro one another, and are parallel to the floor CD of the chaber. Beas L and L are connected to e.g. an oscilloscope, which acts as a chronoeter. Also, beas L and L are interrupted during the fall e.g. of a ass, and the oscilloscope (chronoeter) records the tie taken by ass to cross the distance d the two parallel beas L and L. In this way, astronaut Ο easures the velocity with which a ass, which was released fro the top AB of his chaber, falls onto the floor CD. Velocity is calculated using relation: d = (a) t where t is the tie recorded by the oscilloscope (chronoeter), when ass crossed the known distance d between the two parallel Laser beas L and L. b. Perforing the experients Astronaut Ο, after installing the device Ε for easuring velocity, as described above, now perfors the following experients: To begin with, he releases, fro the top AB of his chaber, a ass, which falls freely to the floor CD of his chaber and, using device E (in accordance with relation (a)), easures the velocity of the fall of ass. He then repeats the sae experient with a different ass, ( < ) and easures, once again using device E, the velocity with which ass falls on the floor CD of his chaber. In other words, astronaut Ο allows asses and to fall separately (first ass and the repeats the process with ass ) and does not allow asses and to fall siultaneously fro the top of his chaber. Now astronaut Ο copares the velocities and of asses and and concludes that: Α. If velocities and are equal, then his rocket S is in Phase Ι. Astronaut Ο bases this conclusion on the fundaental property of accelerating reference systes to give all bodies (regardless of their ass) the sae acceleration and, subsequently, velocity. Β. Conversely, if velocities and are unequal, and if, specifically >, the rocket S is in Phase II. Astronaut Ο bases this concussion on the first of relations (.6). Subsequently, conclusions Α and Β, as detailed above, are astronaut O s answer to the proble we set hi. 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:
Ι. The body of ass Μ and radius entioned above, ay be, for exaple:. A balloon full of air, of a radius e.g. =6000 k.. A sphere, ade of cork or iron, of a radius e.g. =000 k. 3. A sall asteroid, the oon, the Earth, a planet, etc, etc. II. The asses and used by astronaut Ο in the experients he conducts within his chaber ay be, for exaple:. A sall sphere ade of cork, of diaeter e.g. D= c.. An iron sphere, of diaeter e.g. D=50 c. 3. A sphere, consisting of the aterial of a neutron star (the density of which is known to be d= 0 5 kg/c 3 ), of diaeter e.g. D=0 c, etc, etc. III. The height h of the astronaut s chaber is sall, easuring a few etres, e.g. h = 0 (saller or larger) so that the gravitational field of ass Μ within the astronaut s chaber can be considered very nearly hoogenous (for the above nuerical exaple Ι), in accordance with the «notable observation» entioned 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 proble we set hi, 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 dynaoeter D at the top of y chaber, I can calculate the acceleration γ with which y rocket oves, relative to the inertial observer O, i.e.: F γ= (3) where F is the reading of dynaoeter D and is a known ass attached to the end of the dynaoeter s spring. Since the inertial force field within the chaber is hoogenous, with constant intensity g (g =γ). b. If, however (says astronaut Ο) I reach conclusion (b.b), and because the gravitational field within y chaber is hoogenous, with constant intensity g, then I will act as follows: As, with the help of device Ε, I have found out the velocities and of asses and, that fall towards the floor CD of y chaber, I will obviously also know their ratio k, i.e.: = k (4) Thus, fro the first of relations (.6) the velocity of ass, is:
gh = (5) + and the velocity of ass, is: gh = (6) + Fro relations (5) and (6), because <, it follows that >. Dividing now by eber relations (5) and (6), we have: + = (7) + And based on relation (4), relation (7) yields: k = (8) k In relation (8), because,, k are known, astronaut Ο also knows the ass Μ of the celestial body, where his rocket S is docked. Also, fro the first of relations (.6), we have = gh + or gh = (9) + Now, by replacing the Μ in relation (9) with that of relation (8), we have: ( k ) = + h k g (0) In relation (0) factors, h,, k are known.
Subsequently, fro relation (0), astronaut Ο also knows the intensity g of the hoogenous gravitational field of ass Μ, where his rocket is S is docked. Finally, fro the relation: g= G () We have: G = () g Where G is the constant of universal attraction. Substituting, now, in relation () the Μ and g provided by relations (8) and (0) results in: ( k ) ( )( k ) h = G (3) In relation (3) factors G, h,,,, k are known. Subsequently, on the basis of relation (3), astronaut Ο also knows the radius of ass Μ, where his rocket S is docked. Following what we discussed above, regarding the rocket experient, we are led to this basic conclusion: CONCLUSION Ι Under no circustances can a hoogenous gravitational field, with intensity g, be equivalent to a hoogenous inertial force field, with acceleration γ, (γ= g). In other words: An observer Ο, who is within chaber S (and by conducting various echanics experients), can easily ascertain: whether his chaber is perforing unifor accelerated otion with constant acceleration γ, relative to an inertial observer O or whether his chaber is otionless in a hoogenous gravitational field with constant intensity g, of a ass Μ, which is outside his chaber. Thus, according to the above conclusion, the «equivalence principle» of the General Theory of elativity is copletely erroneous. ANOTHE ISTAKE BY EINSTEIN... As is well known, in the «equivalence principle» (weak equivalence principle), Einstein clais that: «A reference syste S (e.g. an elevator), falling freely in the gravitational field of a ass Μ, is locally equivalent to an inertial reference syste». Einstein s clai, however, is wrong, because: Let s assue, fig. 4, that we have a spherical shell (e.g. a spherical elevator), of ass and radius, falling freely fro
a height h, in the gravitational field of a ass Μ. An observer Ο, who is within the elevator, places a ass, ( < ) at he centre Κ of the spherical elevator chaber. fig. 4 Now, if what Einstein clais above, fig. 4, on the «equivalence principle» were correct, then, during the elevator s free fall in the gravitational field of ass Μ, ass would reain at the centre Κ of the elevator chaber (as would be the case if the elevator were an inertial reference syste S, away fro 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 ass of the spherical elevator falls at a greater velocity than the velocity of ass, relative to an inertial observer O. Subsequently, observer Ο, ο who is within the elevator, will observe ass oving fro the centre Κ towards the «ceiling» of the chaber during the elevator s free fall in the gravitational field of ass Μ. After exaining the above, we are led to the following basic conclusion:
CONCLUSION II A reference syste S, falling freely in the non-hoogenous or hoogenous gravitational field of a ass Μ can never be locally equivalent to an inertial reference syste. In other words: An observer Ο, who is within a reference syste S, which is falling freely in the gravitational field of a ass, by conducting various echanics experients (such as, e.g. the ass placed at the centre K of the spherical elevator, as described above) can easily ascertain whether his reference syste is falling freely in the gravitational field of a ass, or whether his reference syste S is an inertial reference syste, away fro gravitational fields. Specifically, in ters of our experient above:. If ass reains at the centre Κ of the spherical elevator chaber, the observer Ο, who is within the elevator, will know that his chaber is an inertial reference syste, away fro gravitational fields.. If ass oves fro the centre Κ of the spherical elevator chaber, the observer Ο, who is within the elevator, will know that his chaber is a reference syste S, falling freely in the gravitational field of a ass Μ, which is outside his chaber. Thus, in accordance wit the above, the «equivalence principle» of the General Theory of elativity ust be rejected as copletely erroneous. A QUESTION ON THE «EQUIVALENCE PINCIPLE»... It is well known that, in the «strong equivalence principle», Einstein accepts that: «All bodies (regardless of their ass) fall with the sae velocity in the gravitational field of a ass», (See http://www.tsolkas.gr/forus/777.jpg), book:clifford. Will, Was Einstein right?. QUESTION: Can «Einstein» (the relativists) show us the atheatical relation that the above conclusion is based on? Which is that atheatical relation? Let «Einstein» (the relativists) present us with this atheatical relation, so that we can see and judge it or ourselves. ECAPITULATION Following all that we exained 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 copletely erroneous theory of Physics. Subsequently, the General Theory of elativity (which, as we know, is based on the «equivalence principle») ust also be considered as a copletely erroneous theory of Physics. Christos A. Tsolkas tsolkas@otenet.gr