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
upload
2009/11/7 )
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 2.6
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 (see reference):
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 (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 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
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