"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
-------------------------------------------------------
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 is
possible.
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 2.6 1033 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:
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:
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
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:
|
|
1.4 |
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:
|
|
1.5 |
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:
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:
|
|
1.7 |
where
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:
|
|
1.9 |
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:
|
|
1.10 |
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:
|
|
1.11 |
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:
|
|
1.12 |
From equation 1.12, we can find the relationship between the change of
electron mass and the change of energy:
|
|
1.13 |
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:
|
|
1.14 |
where ao is a function of the electron mass
me, given by:
|
|
1.15 |
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:
By
differentiation of equation 1.16, we find:
|
|
1.17 |
Differentiation of equation 1.14 gives:
|
|
1.18 |
Combining equations 1.11, 1.13, 1.17 and 1.18, we get:
|
|
1.19 |
Since these quantities are extremely small but finite, we can
write:
|
|
1.20 |
From equation 1.7, we have:
|
|
1.21 |
Equations 1.20 and 1.21 give:
|
|
1.22 |
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:
|
|
1.23 |
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. 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 Versot 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 |
