The Principle of
Equivalence.
10.1 - Introduction.
Among numerous
postulates, Einstein proposed the equivalence principle which states that
no experiment can distinguish the acceleration due to gravity from the
inertial acceleration due to a change of velocity. To illustrate that
principle, Einstein used thought experiments involving elevators. He
compared different phenomena related to accelerations observed inside an
elevator. He purposely limited the range of observations to the frame of
the elevator, excluding other predictable consequences that should
logically take place inside other frames. The principle of equivalence
being a postulate, the reasons for which Einstein did not take into
account the motion of his own frame were not explained.
In physics as in logic,
a principle is valid only when it is coherent with all the facts. An
exception always disproves the rule. It is surprising to read how the
equivalence principle has been generally accepted while it is so easy to
prove that it is not coherent with the behavior of bodies located in other
frames, as we will see below.
10.2 - Deflection of Light in an
Elevator Moving at Constant Velocity.
Experiments describing
a constant relative transverse velocity between a source and an elevator
are generally ignored. Let us consider a horizontal parallel beam of light
(or particles, as on figure 10.1) projected on an elevator (of negligible
mass) moving upward at a constant velocity v with respect to the source.
The experiment takes place in outer space far away from any gravitational
field.
Because momentum must be conserved, the beam of light must move in a
straight line. On figure 10.1, the dotted line inside the elevator shows
where the photons can be detected with respect to the moving elevator at
different times. The relative location of the photons with respect to the
elevator moving at a constant velocity v is:
 |
10.1 |
This problem of
constant velocity is simple but rarely considered. Obviously, the beam
will not appear to move horizontally for the observer inside the elevator.
However, as seen on the external frame, the beam of particles travels
horizontally. This shows that the relative transverse velocity between the
source and the elevator is measurable.
Figure 10.1
10.3 - Inertial versus
Gravitational Acceleration of Masses.
Before considering the
problem of photons moving with respect to an accelerated frame, let us
study a mass m moving horizontally. The mass
enters an elevator which has an upward acceleration a in outer space at the moment its vertical velocity
with respect to the source of the mass is zero. The elevator is
accelerated by a rocket placed under it to produce a force F (shown by
upward arrows on figure 10.2A). Due to that force F, the elevator (and the
observer) accelerates following Newton's law:
where M is the mass of
the elevator (including the observer's mass) and a is its acceleration given by:
 |
10.3 |
After a time interval Dt, the mass will hit the opposite wall. It will have
traveled a vertical distance DhA
relative to the moving elevator. Obviously, the mass will have traveled an
absolute vertical distance of zero since there is no gravitational
field.
Figure
10.2A
Figure 10.2B
Let us
consider a similar elevator located at rest on Earth as illustrated on
figure 10.2B. The Earth's gravitational field accelerates the mass m toward the Earth's center. After a time interval
Dt, when the mass hits the opposite wall of the
elevator, it will have traveled an absolute vertical distance DhB.
In the experiment
described on figure 10.2A, the mass m is
completely free of any field and any force and therefore cannot gain any
absolute energy when the floor of the elevator approaches it. An atomic
clock bound to that free mass m will maintain a
constant rate since no acceleration (therefore no energy) is given to the
electrons or particles of the atomic clock. However, the elevator with the
observer will gain kinetic energy (and therefore mass) due to the momentum
transferred by the rocket. The observer's clock located on the floor of
the elevator will slow down (absolute time) due to its increase of
velocity in free space as given in equation 3.9. Consequently, the
observer using the moving clock will observe a relative blue shift on
light emitted from mass m. Let us note that the
Doppler effect is considered separately and has not been taken into
account.
In the experiment
described on figure 10.2B, the elevator and the observer cannot gain any
energy as a function of time since no work is produced on them. Neither
the potential of the observer nor its velocity change. Therefore, the
atomic clock of the stationary observer will keep giving a constant rate
as a function of time. However, the clock on the falling mass will slow
down for two reasons (independent of the Doppler effect): First, because
of its increase of velocity (equation 3.10) and second, because of its
decrease of potential energy (equation 1.22). Consequently, the observer
standing in the elevator will observe a red shift on light emitted by the
falling mass m.
The Doppler
contribution to the shift of frequency is identical in figures 10.2A and
10.2B (if a = g). Its amplitude is much more
important than the one due to the change of internal mass. However it can
be subtracted out to show the difference explained above.
We see that the
principle of mass-energy conservation implies that there is a fundamental
difference between an inertial acceleration and an acceleration due to
gravity since the consequences of each acceleration are just opposite. In
the case of inertial acceleration (figure 10.2A) the clock located on the
apparently falling mass will run faster than the observer's
clock because of the slowing down of the observer's clock. On the
contrary, in the case of gravitational acceleration (figure 10.2B), the
falling clock will run more slowly than the observer's clock. One must
conclude that the physical properties of the gravitational acceleration
are different from the ones of inertial acceleration which means that the
gravitational acceleration is not equivalent to the inertial
acceleration. 10.4 - Bremsstrahlung Due to
Inertial and Gravitational Accelerations.
To illustrate the
difference between inertial and gravitational accelerations, let us
consider another thought experiment in which electric charges are placed
in a gravitational field. One or more electrons are deposited on a
stationary insulator in the Earth's normal gravitational field. This is
static electricity. It is well known that Maxwell's equations predict that
any accelerated electric charge must emit radiation called bremsstrahlung.
According to Einstein's principle of equivalence, charges at rest in the
Earth's gravitational field should emit bremsstrahlung because of the
gravitational acceleration. However, no experiment has ever detected the
emission of bremsstrahlung due to the gravitational acceleration of static
electricity. The emission of radiation due to gravitational acceleration
has been overlooked.
There is a way to prove
that charges submitted to a gravitational acceleration do not emit
bremsstrahlung. The principle of mass-energy conservation requires that
energy must be given to an electric charge in order to compensate for the
electromagnetic energy emitted during its acceleration. Let us try to
identify the origin of the energy responsible for the bremsstrahlung
predicted by Maxwell's equations and Einstein's principle of
equivalence.
If bremsstrahlung is
emitted when electric charges are submitted to gravity, there must be an
energetic mechanism available to compensate for the energy lost by
radiation. That continuous emission of radiation due to gravitational
acceleration must necessarily extract energy from a source. Therefore,
after a long period of time, the accumulated loss of energy in the source
will be more easily detectable than the weak bremsstrahlung emitted. In
the case of individual electrons stationary in a gravitational field, the
only source of energy available is their mass. Consequently, the electron
mass should decrease as a function of time to compensate for the
electromagnetic energy bound to be emitted. If the electron mass decreases
when standing in a gravitational field, one should eventually find
electrons with different masses depending on the time they have been
submitted to the Earth's gravitational acceleration.
However, it is observed
that electrons maintain their full integrity and do not lose any mass
while standing in a gravitational field. All electrons have the same mass.
Due to the principle of mass-energy conservation, the absence of any
source of energy shows that no bremsstrahlung can be emitted from
gravitationally accelerated electrically charged particles. However, in
the case of inertial acceleration, the mechanical energy required is well
identified and compensates for the electromagnetic energy emitted as
bremsstrahlung.
These considerations
show again that gravitational acceleration is different from inertial
acceleration. Bremsstrahlung is emitted only when submitted to inertial
acceleration. Since Einstein's general relativity is based on Maxwell's
equations and the principle of equivalence, we must reexamine Einstein's
predictions.
10.5 - Behavior of
Light.
10.5.1 - Light Path in an
Accelerated Elevator.
Let us now consider the
experiment described in section 10.3 but using light instead of masses
(figure 10.3A). Due to the conservation of momentum, light keeps moving in
a straight line (as on figure 10.1) and takes a time interval Dt to go across the elevator. Because of the elevator's
increasing upward velocity, during the time interval Dt, light seems to travel a vertical distance Dh:
 |
10.4 |
Therefore, as
illustrated on figure 10.3A, for the accelerated observer, the beam of
light will appear to follow a curve and will hit the
opposite wall at a distance Dh below the entrance
height.
Let us assume that the
acceleration due to the rocket produces a change of velocity dv/dt equal
to g = 9.8 m/s which is the gravitational acceleration on Earth. Observer
A will feel that the upward force of the floor produces the
same downward path on the photon as for a massive particle accelerated in
the Earth's gravitational field (figures 10.2A and 10.2B).
However, the
accelerated observer A gains velocity and energy due to his increase of
velocity. Therefore, his clock will slow down between the time light
enters the elevator and the time light reaches the opposite wall of the
elevator. Consequently, even if we do not take into account the Doppler
blue shift due to the increase of relative velocity of the observer with
respect to light, the observer detecting the apparently deflected light
will measure an apparent increase of its frequency (blue shift) because of
the absolute slowing down of his clock.
10.5.2 - Light Path in a
Gravitational Field.
Let us assume
momentarily that the equivalence principle is valid. Therefore, with
respect to observer B on figure 10.3B, light entering the room
horizontally would be deflected as illustrated. This hypothesis implies
that light is attracted by gravity. However, to be valid, we must verify
that such an hypothesis is compatible with mass-energy conservation. If
light is deflected, let us calculate the energy relationship caused by
that deflection.
Let us call F the
hypothetical gravitational force on a photon in the direction of the
gravitational acceleration. During its passage across the elevator, we
assume that the photon is deflected on a distance Dh in the direction of the force F, as shown on figure
10.3B. Mass-energy conservation requires that a displacement Dh in the same direction of a force F gives an increase
of energy DW equal to:
The photon affected by
the gravitational force F will then reach the opposite wall with an energy
increase of DW at a distance Dh below its initial height. We have seen that the
absolute photon energy is proportional to its absolute frequency.
Therefore the photon should gain absolute energy and frequency (blue
shift) and this should be seen by observer B.
However we have seen in
chapter one that the absolute energy of a photon moving downward does not
increase. The Mössbauer experiment shows that there are local changes of
clock rate at different altitudes but the absolute energy of the photon
does not change. An absolute change of photon energy in a gravitational
field is contrary to mass-energy conservation. Consequently:
From equation 10.5,
since F and Dh have to be in the same direction,
the only way to produce a deflection (Dh ¹ 0) with DW = 0 is to have
Dh different from zero when F = 0. This means a
deflection of photons when there is no force acting on them. This is
contrary to Newton's second law on inertia.
Consequently, to be
compatible with mass-energy conservation, there is either no deflection or
no force (which leads to no deflection anyway). The curved trajectory on
figure 10.3B is erroneous, light must move in a straight line in a
gravitational field. We have then:
| DW = 0, F = 0 and Dh = 0 |
10.7 |
An observer located
straight in front of the entrance aperture on figure 10.3B will observe
the beam reaching him at that location without any change of frequency. We
conclude that light is apparently deflected with respect to
an accelerated observer with an inertial acceleration as illustrated in
figure 10.3A. However, as given in equation 10.7, light cannot be
deflected by gravity because of mass-energy and momentum conservation. We
must conclude again that Einstein's equivalence principle is erroneous
which means that the behavior of light is perceived differently by
observers subjected to gravitational acceleration and inertial
acceleration.
It has been claimed in
the past that such a deflection (by a gravitational field) has been
measured experimentally during solar eclipses. The reliability of such
results are generally claimed only by those who have never read seriously
the original articles describing those experiments. The report given in
appendix
II gives a shocking proof of the weakness of the experiment. A
small deflection of starlight by a gravitational field has been predicted
by Einstein. However, it has never been seriously proved
experimentally.
10.5.3 - The Equivalence
Principle and Light Deflection.
It has been well
recognized that the deflection of light rays is closely related to the
equivalence principle discussed above. According to the paper "The
Equivalence Principle with Light Rays"[1]:
Since the equivalence
between inertial and gravitational acceleration assumed by Einstein is
erroneous as shown above in several independent ways, it is not surprising
that its consequence (light deflection) is also erroneous.
It is well
known that Einstein predicted in 1911 that light should be deflected due
to the solar gravitational field. In fact, this prediction is almost
identical to the one given by Soldner in 1801 using Newton's law. This
demonstration can be understood easily. In classical mechanics, the amount
of deviation of any massive object passing near the Sun at velocity v is
totally independent of the mass of the object. Also, it was
assumed that the velocity of light c could be treated as any velocity v.
The principle of equivalence implies the equivalence between the
inertially accelerated elevator (figure 10.3A) and the gravitationally
accelerated photon on figure 10.3B. Due to the force on the elevator and
on observer A, the photon hits the opposite wall of the elevator after the
elevator has moved up the distance Dh. This
apparent deviation, which corresponds to 0.87" near the solar limb (see
Figure 10.3A) is clearly the one required in the case of inertial
acceleration of the elevator. Assuming the principle of equivalence, the
same value should be found in general relativity. However, Einstein's
general relativity predicts a deviation of 1.74". That is twice as much as
drawn on figure 10.3A).
The reliability of the
apparent deviation illustrated on figure 10.3A is so great that one cannot
believe that this amount of deflection could be doubled to satisfy
Einstein's predictions of general relativity and the principle of
equivalence. Einstein's claim is astonishing. If the opposite wall of that
elevator on figure 10.3A is open, that double amount of deflection means
that there would be an absolute deflection in a rest frame, even in the
absence of any gravitational field. There is no logical way to explain
that the amplitude of the deviation in the elevator submitted to inertial
acceleration could be twice as much as the one illustrated on figure
10.3A, since the crossing light is subjected to no interaction in a space
having zero field and the observer A has certainly moved only
through the distance Dh. One cannot
claim that light is deflected just because there exists an observer. If
that doubled amount of deviation cannot exist in the case of inertial
acceleration, it cannot exist either in a gravitational field without
contradicting the equivalence principle on which is based the theory
leading to the deviation of light in a gravitational field (1.74").
Consequently,
Einstein's prediction giving a deviation of 1.74" is self contradictory
and cannot be compatible with the principle of equivalence.
10.6 - Gravitational
Lenses.
There are several
consequences to the fact that light is not deviated in a gravitational
field. The deviation of light by a gravitational field gave birth to the
claim that rings in space are caused by the focusing of light coming from
remote sources by the gravitational mass of intervening galaxies. This
explanation is certainly erroneous since light is not deviated by a
gravitational field.
These rings can be
explained more logically by the presence of large quantities of ions
moving in the magnetic field of a galaxy. It is well known that ions
spread naturally into rings in a magnetic field. This is a rational
interpretation of a phenomenon that has been erroneously interpreted as
Einstein's rings.
10.7 - Attracting Force between
Parallel Beams of Charged Particles.
We have seen in section
10.4 that electrical phenomena can be used to demonstrate that
gravitational acceleration is different from inertial acceleration. To end
this chapter, we will give an example using electricity disproving the
principle of reciprocity (for another proof, see section 3.9).
In elementary
physics, Ampere's law teaches how to calculate the force between two
parallel straight conductors carrying currents in the same direction. We
learn that a force F between parallel conductors spaced by a distance
Dx is induced because the current i' in the
second conductor passes in the magnetic field generated by the current i
in the first conductor. The force F by unit of length (in MKS units)
is:
 |
10.8 |
That force is so well
recognized in physics that it was used "as the basis of the definition of
the ampere in the MKS system" [2].
The force between these conductors is attractive when the currents are in
the same direction and repulsive when the currents are in opposite
directions.
With the modern
development of accelerators and intense beams of charged particles, the
electric conductor is no longer necessary to observe this phenomenon and
the interaction of independent electric charges in the magnetic field
generated by comoving electric charges has been observed directly. In
fact, the magnetic field produced by comoving electric charges produces a
focusing that reduces the dispersion of the beam of particles. One can
clearly observe particles all having the same velocity in a parallel beam
attracting each other due to the magnetic field produced by the velocity
of the neighboring charges.
Let us now consider an
observer moving with that beam of particles. In his frame of reference,
the particles appear stationary with respect to him. Then, no magnetic
field is produced. Using Einstein's principle of reciprocity within that
moving frame, the charged particles should repel each other according to
the electrostatic repulsion of charges having the same polarity. However,
they attract each other as calculated above and observed experimentally.
This is clearly not acceptable. Other experiments involving Maxwell's
equations exist which are not compatible with the reciprocity principle.
However, the ones described above suffice to disprove this
principle.
10.8 - References.
[1]
http://altair.syr.edu:2024/lightcone/equivalence.html
[2] F. W. Sears, Principles of
Physics, Addison-Wesley, p. 267, 1946
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