Einstein's Theory of Relativity 
 Versus Classical Mechanics
  Paul Marmet
       "It follows from the theory of relativity that mass and energy are both different manifestations of the same thing -a somewhat unfamiliar conception for the average man.  . . . the mass and energy in fact were equivalent." Albert Einstein
          "The Quotable Einstein", Princeton University Press, Princeton New Jersey (1996), also in the Einstein film produced by Nova Television, 1979
        We must note that the equivalence of mass and energy is different from the principle of mass-energy conservation, which is not applied in Einstein's relativity (see Straumann)
( Last checked 2012/07/02 - The estate of Paul Marmet )

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Chapter One
The Physical Reality of Length Contraction.
1.1 - Introduction.
        In this first chapter, we will show that it is possible to establish links between quantum mechanics and mass-energy conservation. These links will help us calculate the interatomic distances in molecules and in crystals as a function of their gravitational potential. We will show that the natural interatomic distance calculated using quantum mechanics leads to the length contraction (or dilation) predicted by relativity. This result will be obtained here without using the hypothesis of the constancy of the velocity of light. It will appear instead as a consequence of quantum mechanics when mass-energy conservation is taken into account.
        Since length contraction appears as a consequence of quantum mechanical calculations, the physical reality of those predictions can be verified experimentally. We will show that the results of the most precise quantum mechanical experiments prove that the change of length is real. Two different experiments which have been found to give sufficient accuracy to verify this change of length will be described in detail. We will show that the dimensions of matter really change naturally depending on its location in a gravitational potential.

1.2 - Mass-Energy Conservation at Macroscopic Scale.
        The most reliable principle in physics seems to be the principle of mass-energy conservation: mass can be transformed into energy and vice versa. Without this principle, one would be able to create mass or energy from nothing. We do not believe that absolute creation from nothing is possible.  Surprizingly, most scientists do not know that Einstein's general relativity is not compatible with the principle of mass-energy conservation [Ref]
        In order to understand the fundamental implications related to mass-energy conservation, let us consider the following example. Suppose momentarily that the Earth is not moving around the Sun, but has been pushed away with a powerful rocket and has reached interstellar space at location P (see figure 1.1). It now has a negligible residual velocity with respect to the Sun and except for the fact that the Sun has faded away, everything appears the same. The Earth is still made of about 1050 atoms, its center contains iron, is surrounded by oceans, deserts, cities and the atmosphere is the same. The planet is still populated by about the same five billion people.

Figure 1.1

        Let us assume that after a while, the planet starts falling slowly from P toward the Sun. Due to the solar attraction, the Earth accelerates until it reaches the distance of 150 million kilometers (from the Sun) corresponding to its normal orbit. At that moment, one can calculate that the Earth has reached a velocity of 42 km/s. This velocity is too large for the Earth to be in a stable orbit around the Sun as it is normally. It must be reduced to 30 km/s, the velocity for a stable orbit. The Earth must be slowed down.
        It is decided that the velocity of the Earth can be reduced with the help of a strong rope attached to a group of stars at the center of our galaxy. The force produced by the rope will generate energy at the center of the galaxy while the Earth is slowed down to the desired velocity for a stable orbit around the Sun.
        Knowing that the Earth has a mass of 5.97×1024 kg, it is easy to calculate the amount of work transferred to the center of the galaxy. It corresponds to slowing down the Earth from 42 km/s to 30 km/s. This represents an amount of work equal to 2.6×1033 joules. Therefore the Earth must get rid of 2.6×1033 joules to go back to its normal orbit and the center of the galaxy must absorb that same amount of energy. The rope used to slow down the Earth could then run a generator located at the center of the galaxy to produce 2.6×1033 joules of energy.
        However, due to the principle of mass-energy conservation, the energy carried out to the center of the galaxy to slow down the Earth can be transformed into mass. Using the relation E = mc2, we find that the mass corresponding to 2.61033 joules of energy is equal to 2.9×1016 kg. This means that 29 billions of millions of kilograms of mass have been transferred from the Earth to the center of the galaxy through the rope. This mass-energy is a very small fraction of the Earth’s mass but it must be coming from the Earth and received at the center of the galaxy.
        After the re-establishment of the Earth’s orbit at one astronomical unit from the Sun, the inhabitants of the Earth find nothing changed. Other than the neighboring Sun, no difference can be noticed compared with when the Earth, still made of its initial 1050 atoms, was away from the Sun. The question is: How can the Earth not lose one single atom or molecule while 29 billions of millions of kilograms of mass have been lost and received at the center of the galaxy? There is only one logical answer. Since each atom on Earth was submitted to the force of the rope, each atom has lost mass in a proportion of approximately one part per one hundred million.
        Note that this situation is equivalent to the formation of a hydrogen atom. When a proton and an electron come together to form a hydrogen atom, energy is released in the form of light. This light corresponds to the work transferred to the center of the galaxy in our problem.

1.3 - Mass-Energy Conservation at a Microscopic Scale.
        The experiment described above takes place at a macroscopic scale. Each individual atom loses mass because a force interacts on all atoms when the Earth decelerates in the Sun's gravitational potential. It is normally assumed that atoms have a constant mass. For example we learn that the mass of the hydrogen atom is mo = 1.6727406×10-27 kg. Can we have hydrogen atoms with less or more mass? From the thought experiment of section 1.2, we see that the principle of mass-energy conservation requires a transformation of mass into energy on each atom forming the Earth, since each of them has contributed to generate energy transmitted to the center of the galaxy.
        Let us study the following experiment. We first consider that an individual hydrogen atom is placed on a table on the first floor of a house in the gravitational field of the Earth, as shown on figure 1.2. The hydrogen atom is then attached to a fine (weightless) thread so that the atom can be lowered down slowly to the basement of the house, while the experimenter remains on the first floor. When the atom is lowered down, its weight produces a force F in the thread. That force is measured by the experimenter on the first floor. It is given by:

F = mo 1.1 

Figure 1.2

        The slow descent of the atom attached to the thread is stopped every time a measurement is made, which means that the kinetic energy is zero at the moment of the measurement. When the atom has traveled a vertical distance Dh, the observer on the first floor observes that the energy DE produced by the atom and transmitted through the thread to the first floor is:

DE = FD 1.2 
        The work extracted from the descent of the atom is positive when the final position of the atom is under the first floor (Dh is positive). Then, according to the principle of mass-energy conservation, the energy produced at the first floor by the descent of the atom in the basement can be transformed into mass according to the relationship (see reference):
E = mc2 1.3 
        The important point that must be retained about equation 1.3 is that the energy E is proportional to the mass, independently of the fact that it just happens that the numerical value of the constant of proportionality is equal to the square of the velocity of light. From equations 1.1, 1.2 and 1.3, the amount of mass Dmf generated at the first floor by the descent is:
        This amount of mass (or energy) carried by the thread is generated by the weight of the atom which slowly moves down to the basement. When the hydrogen atom lies on the table, its mass is mo. However, during its descent, it produces work (corresponding to the mass Dmf generated at the first floor). The initial mass mo of the particle is now transferred into the mass-energy Dmf generated at the first floor by the falling particle, plus the remaining mass mb of the particle now in the basement.  Using equation 1.4, we find:
       According to the principle of mass-energy conservation, the mass of the hydrogen atom in the basement is now different from its initial mass mo on the first floor. It is slightly smaller than mo and is now equal to mb. Any variation of g with height is negligible and can be taken (with g) into account in equations 1.4 and 1.5.
        Of course, the relative change of mass Dmf/mo is extremely small. (It was equally small in the case of the Earth falling back to its normal orbit, as seen above in section 1.2.) The change of mass given by equation 1.5 is so small that it cannot be verified using a weighing scale. However, this reduction of mass must exist, otherwise, mass-energy would be created from nothing. We will see below that this change of mass has actually been measured.
        It was quite arbitrary for us to assume that the initial mass of hydrogen on the first floor is mo. Physical tables do not mention all the experimental conditions in which an atom is measured. Furthermore, the accuracy of this value is quite insufficient now to detect Dmf (equation 1.5). A change of altitude of one meter near the Earth’s surface gives a relative change of mass of the order of 10-16. Masses are not known with such an accuracy.
        At this point, we must recall that in the above reasoning, we have made a choice between the principle of mass-energy conservation and the concept of absolute identical mass in all frames. It is illogical to accept both principles simultaneously since they are not compatible. We have chosen to rely on the principle of mass-energy conservation which is equivalent to not believing in "absolute creation from nothing" as defined in section 1.2. We must realize that without mass-energy conservation not much of physics remains. Physics becomes magic.

1.4 - Mass Loss of the Electron.
        There is a way to measure experimentally the mass difference between a hydrogen atom in the basement and one on the first floor. In equation 1.5, we see that a mass Dmf appears and increases when the atom moves down in the gravitational field. Due to mass-energy conservation, the mass mb of the atom moving down decreases by the same amount, that is:

Dmb=Dmf 1.6
        Since the hydrogen atom has lost a part of its mass due to the change of gravitational potential energy, we must expect (according to equation 1.5) that the electron as well as the proton in the atom have individually lost the same relative mass. Let us calculate the relative change of mass of the electron (Dme/me) and of the proton inside the hydrogen atom due to its change of height.  From equations 1.5 and 1.6, we have:
Dme=Dmb 1.8 
        When Dh is a few meters, equation 1.7 gives a relative change of mass of the order of 10-16. Consequently, the first order term gives an excellent approximation. Let us use:
        The electron mass me (as well as the proton mass) is not constant and decreases continuously when the atom is moving down. Equation 1.7 shows that independently of the mass of the particle, the relative change of mass is the same. This means that for the same change of altitude, the relative change of mass of the electron is the same as for the proton.
        Due to the principle of mass-energy conservation, we must conclude that a hydrogen atom at rest has a less massive electron and a less massive proton at a lower altitude than at a higher altitude. The mass of an electron and of a proton can be tested very accurately in atomic physics. Quantum physics shows us how to calculate the exact structure of the hydrogen atom as a function of the electron and proton mass. From that, one can calculate the Bohr radius of an atom having a different mass. Fortunately, the Bohr radius can also be measured with extreme accuracy experimentally.

1.5 - Change of the Radius of the Electron Orbit.
        It is shown in textbooks how quantum physics predicts the radius of the orbit of the electron in hydrogen for a given electronic state. This is given by the well known Bohr equation:

where rn is the radius of the Bohr orbit of the electron with principal quantum number n, me is the mass of the electron (actually, me is the reduced mass, but it is approximately the same as the electron mass), h is the Planck constant (= 2p), k is the Coulomb constant (1/4peo), e is the electronic charge and Z is the number of charges in the nucleus (Z = 1 corresponds to atomic hydrogen). Furthermore when we choose n = 1 and Z = 1, rn becomes ao, which is called the Bohr radius. The Bohr radius is 5.291772×10-11 m at the Earth's surface (for the case of R¥ for which the nucleus is very massive). Equation 1.10 illustrates a simple principle. It illustrates the fact that the circumference of the electron orbit is exactly equal to (or any multiple of) the de Broglie wavelength of the electron orbiting the nucleus.
        Since, as we have seen above, the electron mass me changes with its position in a gravitational potential, let us calculate (using Bohr's equation) the change of radius rn caused by that change of electron mass. This is given by the partial derivative of rn with respect to me. From equation 1.10 we find:
        Equation 1.11 shows that any relative decrease of electron mass is equal to the same relative increase of the radius of the electron orbit. According to the principle of mass-energy conservation, the electron mass decreases when brought to a lower gravitational potential. Consequently, quantum physics (Bohr's equation) shows that the radius of the electron orbit in hydrogen must increase when the atom is at a lower altitude. Using equation 1.10, quantum physics gives us the possibility to predict the size of the electron orbit rn in an atom for different values of electron mass. Let us study the change of size of the electron orbit as a function of the altitude where the particle is located in a gravitational field.

1.6 - Change of Energy of Electronic States.
        Since it has been observed and accepted that the laws of quantum physics are invariant in any frame of reference, let us calculate the energy states of atoms having an electron (and a proton) with a different mass. The consequences of the change of proton mass are easily calculated since the energy levels depend only on the reduced mass of the electron-proton system. In the Bohr equation, we take me as the reduced mass. This does not produce any relevant difference in the problem here.
        The binding energy between the electron and the proton is a function of the electrostatic potential between the nucleus and the electron. Quantum physics teaches that the energy En of the nth state as a function of the electron mass is:

        From equation 1.12, we can find the relationship between the change of electron mass and the change of energy:
        The Bohr radius ao is the average radius of the electron orbit for n = 1. According to quantum physics the energy of state n is:
        where ao is a function of the electron mass me, given by:
        We know that the energy of electronic states of atoms can be measured very accurately in spectroscopy from the light emitted during the transition between any two states En and En'. Extremely accurate results can also be obtained in some nuclear reactions with the help of Mössbauer spectroscopy.
        The frequency nn of the radiation emitted as a function of the energy En of level n is given by:
En = hnn 1.16 
        By differentiation of equation 1.16, we find:
        Differentiation of equation 1.14 gives:
        Combining equations 1.11, 1.13, 1.17 and 1.18, we get:
        Since these quantities are extremely small but finite, we can write:
        From equation 1.7, we have:
        Equations 1.20 and 1.21 give:
       Equation 1.22 shows that the relative change of size of the Bohr radius Dao/ao is equal to -gDh/c2.
        This shows that following the laws of quantum physics, a change of electron mass due to a change of gravitational potential (which results necessarily from the principle of mass-energy conservation) produces a physical change of the Bohr radius.
        We must notice here that using the relativistic correction given by Dirac's mathematics is irrelevant and does not solve this problem. Relativistic quantum mechanics introduces a relativistic correction due to the electron velocity with respect to the center of mass of the atom. The change in electron mass brought by the relativistic correction implied in this chapter is due to the gravitational potential originating from outside the proton-electron system. It is not due to any internal velocity within the atom. The use of the relativistic Dirac equation is not related to calculating how the Bohr radius changes between its value in the initial gravitational potential and its value in the final gravitational potential.

1.7 - Experimental Measurements of Length Dilation in a Gravitational Potential.
        A measurement proving that there is a change of the Bohr radius due to the change of gravitational potential has already been made. The difference of energy for an atom corresponding to its change of size is observed as a red shift of its spectroscopic lines. The change of mass can be applied quite generally to any particle or subatomic particle in physics placed in a gravitational potential. It can also be applied to astronomical bodies like planets and galaxies since it relies on the principle of mass-energy conservation which is always valid.

1.7.1 - Pound and Rebka's Experiment.
        A spectroscopic measurement of the highest precision has been reported by Pound and Rebka [1] in 1964 with an improved result by Pound and Snider in 1965. Since we have seen that the change of ao corresponds to a change of energy of spectroscopic levels, let us examine Pound and Rebka's experiment. They used Mössbauer spectroscopy to measure the red shift of 14.4 keV gamma rays from Fe57. The emitter and the absorber were placed at rest at the bottom and top of a tower of 22.5 meters at Harvard University.
        The consequence of the gravitational potential on the particles is such that their mass is lower at the bottom than at the top of the tower. Therefore an electron in an atom located at the base of the tower has a larger Bohr radius than an electron located 22.5 meters above, as given by equation 1.22. The same equation also shows that electrons orbiting with a larger radius have less energy and emit photons with longer wavelengths.
        Pound and Rebka reported that the measured red shift agrees within one percent with the equation:

        Not only is the change of energy predicted by relativity and verified experimentally by Pound and Rebka (equation 1.23) numerically compatible with the change of energy predicted by the conservation of mass-energy, but the predicted relativistic equation is mathematically identical to the one predicting the increase of Bohr's radius (equation 1.22). Since the red shift measured corresponds exactly to the change of the Bohr radius existing between the source and the detector, we see that it cannot be attributed to an absolute increase of energy of the photon during its trip in the gravitational field.
        This result is exactly the one that proves that matter at the base of the tower is dilated with respect to matter at the top. It is clear that the Bohr radius has actually changed as expected which means that the physical length has really changed. Therefore, this phenomenon is not space dilation. The real physical dilation of matter is observed because electrons (as well as all particles) have a lower mass at the bottom of the tower which gives them a longer de Broglie wavelength. Space dilation is not compatible with a rational interpretation of modern physics. A rational interpretation has already been presented [3].
        The equilibrium distance between particles is now increased because the Bohr radius has increased. When atoms are brought to a different gravitational potential, the electron and proton must reach a new distance equilibrium as required by quantum physics in equation 1.12. Quantum physics and the principle of mass-energy conservation lead to a real physical contraction or dilation. This solution solves the mysterious description of space contraction in relativity without involving any new hypothesis or new logic. Length contraction or dilation is real and is demonstrated here as the result of actual experiments. Let us also note that this length dilation is done without producing any internal mechanical stress in solid material. Finally, if the source were above the detector, we would observe a blue shift proving that the Bohr radius in matter above the detector has decreased with respect to the Bohr radius in matter at lower altitude. One can conclude that Pound and Rebka's experiment has shown that matter is contracted or dilated when it is moved to a different gravitational potential.

1.7.2 - The Solar Red Shift.
        Other experiments also show the reality of length contraction or dilation. For example, the atoms at the surface of the Sun have been measured to show exactly the gravitational dilation due to the decrease of mass of the electrons in the solar gravitational potential. The gravitational potential at the Sun's surface is well known. As shown above, it is a change of electron mass in the hydrogen atom due to the gravitational potential that produces a change of the Bohr radius. It is that change of Bohr's radius that produces a change of energy between different atomic states. Brault [2] has reported such a change of energy between atomic states. It corresponds exactly to the change of Bohr's radius caused by the gravitational potential. The atoms on the Sun emit light at a different frequency because the electrons are lighter on the solar surface than on Earth, exactly as required by the principle of mass-energy conservation. The change of electron mass on the Sun produces displaced spectral lines toward longer wavelengths as given by equation 1.22 (see other reference). Since quantum physics is valid on the solar surface, we can understand that the electrons have less mass due to the solar gravitational potential. This leads to an increase of the Bohr radius for the atoms located on the solar surface which leads to atomic transitions having less energy, as observed experimentally.
        The Mössbauer experiment as well as the solar red shift experiment prove that atoms are really dilated physically. This means that the physical length of objects actually changes. We also find that not only do protons and electrons lose mass in a gravitational potential but so do nuclear particles in the nucleus of Fe57, as observed in the Mössbauer experiment of Pound and Rebka.

1.8 - The Crucial Influence of the Electron Mass on the Fundamental Laws of Relativity.
        Macroscopic matter is formed by an arrangement of atoms. In molecular physics, we learn that quantum physics predicts that interatomic distances are proportional to the Bohr radius. Those distances are calculated as a function of the Bohr radius. According to quantum physics, a smaller Bohr radius will lead to a smaller interatomic distance between atoms in molecular hydrogen. The interatomic distance in molecules is known to be a function of the Bohr radius just as the interatomic distance in a crystalline structure is proportional to the Bohr radius. This means that since the Bohr radius changes with the intensity of the gravitational potential, the size of molecules and crystals also changes in the same proportion. This is true even in the case of large organic molecules. Therefore the size of all biological matter is proportional to the Bohr radius. This point is explained in more details in appendix I.
        Because the size of macroscopic matter changes with the gravitational potential, the original length of the standard meter transferred to a location having a different gravitational potential will also change. To be more specific, mass-energy conservation requires that the standard meter made of platinum-iridium alloy becomes shorter if we move it to the top of a mountain. Furthermore, due to the increase of electron mass, an atomic clock will increase its frequency by the same ratio when it is moved to the top of the same mountain. However, since the velocity of light (or any other velocity) is the ratio between these two units, it will not change at the top of the mountain with respect to any frame of reference. This point will be discussed later. Because the relative changes of length and clock rate are equal, they will be undetectable when simply using proper values within a frame of reference. All matter, including human bodies, composed of atoms and molecules will change in the same proportion since the intermolecular distance depends on the Bohr radius and consequently on the electron mass which is reduced when located in a gravitational potential.
        It is important to notice that length dilation or contraction is predicted and explained here without using the relativistic Lorentz equations nor the constancy of the velocity of light. Consequently, we must consider now that we have demonstrated experimentally (using Pound and Rebka's results) the physical change of length of an object in a gravitational potential. More demonstrations will be given in the following chapters.
        The experiments reported here showing length dilation use atoms that are at rest. They are solely related to the potential energy. We will see that the problems of kinetic energy and velocities require new considerations in the next chapters.

1.9 - References.

[1] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company San Francisco. page 1056. See also: Pound R. V. and G. A. Rebka, Apparent Weight of Photons, Phys. Rev. Lett., 4, 337 1964. See also: Pound R. V. and Snider, J.L. Effect of gravity on Nuclear Resonance, Phys. Rev. B, 140, 788-803, 1965. This has been measured in a rocket experiment by Vessot and Levine (1976) with an accuracy of 2 x 10-4.

[2] J. W. Brault, The Gravitational Redshift in the Solar Spectrum, Doctoral dissertation, Princeton University, 1962. Also Gravitational Redshift in Solar Lines, Bull. Amer. Phys. Soc., 8, 28, 1963.

[3] P. Marmet, Absurdities in Modern Physics: A Solution, ISBN 0-921272-15-4 Les Éditions du Nordir, c/o R. Yergeau, 165 Waller, Ottawa, Ontario K1N 6N5, 144p. 1993.

1.10 - Symbols and Variables.

DE energy produced by the atom and transmitted to the first floor
Dh distance travelled by the atom
Dmb amount of mass lost by the atom
Dme amount of mass lost by the electron
Dmf amount of mass generated on the first floor
En energy of the hydrogen atom in state n
F weight of the atom
mo mass of the atom on the table
nn frequency of the radiation emitted corresponding to En
rn radius of the orbit of the electron in hydrogen in state n
Z number of charges in the nucleus

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