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 2017/01/15 - The estate of Paul Marmet
)
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 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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