and the Principle of Equivalence.
Physics Department, University of Ottawa, Ottawa,
2012/03/17 - The estate
Paul Marmet )
an analysis of Einstein's principle of equivalence between
gravitational acceleration and its consequences on general
It is shown that the simple application of that principle to
in the Sun's gravitational potential leads to an equation which is
compatible with the one predicting the deflection of light by the
Therefore, the principle of equivalence is not compatible with
general relativity is based on the principle of equivalence:
"In an arbitrary gravitational field no
can distinguish a freely falling nonrotating system (local
from a uniformly moving system in the absence of a
principle, the gravitational acceleration g due to universal
between masses is equivalent to an inertial acceleration a
(= g) due to a change of velocity. This principle is considered to
of the foundations of general
it should be tested. There exists a direct test to determine
principle of equivalence is compatible with general relativity.
it has never been performed. In this article, we will describe
and then use it to test the principle of equivalence.
his principle of equivalence using a gedanken (thought)
he compares the trajectory of a particle in a gravitational
acceleration g) with its trajectory in the absence of a field,
respect to an accelerated observer whose acceleration a
is equal to g. This is illustrated on figure 1.
a particle "p" enters an elevator located in zero
field at time t = 0 when the vertical components of velocity of
and of the elevator are both zero. The elevator's upward
given by a rocket placed under it producing a force F (shown by
arrows on figure 1A). Because of that force F, the elevator and
accelerate following Newton's law:
is the mass of the elevator (including the observer's mass) and
is its acceleration given by:
time interval DtA,
the particle hits the opposite wall. It has traveled a vertical
relative to the moving elevator. However, it has obviously
absolute vertical distance of zero since there is no gravitational
(the gravitational field caused by the elevator and the observer
now a similar elevator located at rest on Earth as illustrated on
1B. The same particle "p" enters this elevator. After a
when the particle hits the opposite wall of the elevator, it has
a vertical distance DhB.
According to Einstein’s principle of equivalence, both experiments
be undistinguishable and the relative distances must be the same
this gedanken experiment is not easily transformed into a
test which has never been done. Nobody has ever made a serious
test directly the principle of equivalence in this way,
this is the basis of modern physics. Consequently, the principle
has been accepted as a dogma.
will see that one can use the sum of many similar gedanken
and integrate the result for a continuously varying
in order to simulate the well studied problem of photons
gravitational field of the Sun. This comparison gives us the
at last, to verify directly Einstein's principle of equivalence
predictions and observational data.
of Light According to Einstein's General Relativity.
general relativity predicts
that light passing at
a distance r from the center of a star of mass M must be
angle d according to the equation:
= 6.67 ´ 10-11
is the Cavendish constant of gravitation, c = 2.998 ´
108 m/s is the velocity of light
= 1.989 ´ 1030
kg is the mass of the Sun. Einstein’s equation gives a numerical
rs = 6.96 ´ 108
m is the radius of the Sun.
This result is
known and is expected to be quite reliable.
of the Fundamental Experiment.
the experiment described on figure 1A can be used as a test of
of equivalence using light deflection by a gravitational field.
now consider this experiment in detail and show that the sum of
number of the same experiment, with different and continuously
values of the gravitational field, corresponds to the one in
is deflected by the solar gravitational field. According to
principle of equivalence, the same relative deflection
whether there is a gravitational field acting on light or
is no gravitational field but an equal (but inertial)
to the Sun (see figure 2).
illustrated on figure 2 is the same as the one on figure 1.
gravitational field is constant in figure 1 but variable in
deflection is strongly exaggerated in figure 2 and the x
is discussed separately in section 5) of the displacement of the
2A, the Sun is represented surrounded by its gravitational
a particle crosses the gravitational field, it is accelerated
Sun according to the gravitational intensity at its position.
acceleration of the moving particle is determined by the mass
the gravitational field (here the Sun) and the distance of the
from that source, all particles at a given distance receive the
of velocity (acceleration), independently of their mass.
all particles have the same trajectory in a gravitational field
of their mass.
principle of equivalence, let us substitute the gravitational
the Sun by a suitable change of velocity of the Sun (as on
now to figure 2B, an observer must measure the same relative
between the particle and the Sun as when the particle is
gravity. The equivalence between inertial and gravitational
can be complied at any distance from the Sun by giving it an
in the direction of the particle. This acceleration varies as a
of the location of the moving particle with respect to the Sun.
particle is located at a variable distance in the gravitational
the Sun, the acceleration given to the Sun must always be equal
gravitational acceleration the particle would feel if there were
the acceleration that must be applied to the Sun in order to
same relative motion in figures 2A and 2B according to
of equivalence. The interesting point is that nobody can
the correct trajectory of the moving particle since all
photons) must travel in a straight line in the absence of
on figure 2B, the photon (as well as any particle) must move in
line. According to Einstein's principle of equivalence, both
on figures 2A and 2B) are undistinguishable and must lead to
results. Whatever the result is, let us calculate the relative
produced between the particle and the Sun assuming the
principle of equivalence.
3, a moving photon travels from left to right at a constant
At time t, the angle between the particle and the Sun is equal
will be simplified here because we know that the angle of
is extremely small. We will therefore neglect the change of
of the particle to the Sun rm
position of the particle in the direction x.
of equivalence requires that the inertial acceleration a
applied to the Sun be equal to the gravitational acceleration g
photon is located. Since
acceleration a (Sun) given to the
consider that the origin of the coordinates (figure 3) of the
is the location where the photon is at minimum distance rm
from the Sun. When the photon moves away from this central
force on the Sun decreases as a function of q
from equation 7 into equation 6 gives:
components of that acceleration
of the Sun. The transverse (upward) component ay
component of acceleration ay
that must be given to the Sun. During
the full passage of the photon from -
to +, the
Sun is varied continuously in order to compensate exactly for the
distance r from the Sun and the angle q
force. Using Newton’s law, we know that at every location, the
velocity Dvy is
equal to the acceleration multiplied by the time. The total change
along the y-axis is:
photon as a function of time.
The photon moves on the axis in straight line at velocity c (since
the gravitational force no longer exists). Therefore when we set
t=0, the relative location of the photon, as a function of time
calculate the angle q as a function of
On figure 3, we find:
equation 11. When the photon
from x = - to x = +,
the angle q passes from -p
/2 to +p /2. We have:
photon and the Sun, after the passage of the photon, when the
of the Sun was at all time identical to the gravitational
the photon would have felt at a given location, as explained
According to the principle of equivalence, equation 17 is valid
when the photon is attracted by the gravitational field or when
is accelerated and the particle travels in straight line (without
passage of the photon, the relative transverse velocity of the
respect to the Sun makes an angle d
(tand = d
is extremely small). This relative deflection between light and
is the result expected to be compatible with the principle of
Equations 17 and 18 give:
gives the only solution compatible with Einstein’s principle of
and is the direct consequence of Einstein’s gedanken experiment
on figure 1. Consequently, if the principle of equivalence is
deflection of light by the gravitational field of the Sun must be
by equation 19. However, equation 19 gives a deflection
at the limb of the Sun (rm= rs)
of Velocity (Dvx).
2, we see that when a particle approaches the Sun, in order to
the principle of equivalence, one must accelerate the Sun in the x
However, when the particle recedes from the neighborhood of the
inverse phenomenon takes place. We can see that the change of
that must be given to the Sun when the particle approaches it, is
opposite direction of the change of velocity given to the Sun when
particles recedes from it. Therefore, the total velocity given to
is zero in this axis. Consequently, the x-axial acceleration of
(if any) has no global effect on the apparent deflection of the
Dvx is zero. Therefore, the
principle of equivalence predicts no deflection related to the
of the particle.
of the Principle of Equivalence.
that if Einstein’s principle of equivalence is valid, equation
that the deflection must be d =
We know (equations 3 and 4) that Einstein’s general relativity
the deflection is twice this amount (d
Therefore, Einstein’s prediction is not compatible with
principle of equivalence. Einstein’s theory is
of light by the Sun has been tested in several experiments
the deflection of visible light of the delay of radio signals
in the Sun's gravitational potential. A critical analysis 
of the results has been investigated. It shows that there is an
for more reliable experiments.
1) The deflection is 1.75².
is not compatible with Einstein’s principle of equivalence, as
is compatible with the predictions of general relativity.
implied in general relativity, this result it is not
the principle of mass-energy conservation.
2) The deflection is 0.875².
is compatible neither with Einstein’s general relativity nor
observations during eclipses or with radio signals.
compatible with the principle of equivalence.
implied in general relativity, this result it is not
the principle of mass-energy conservation.
3) There is no deflection at all.
is perfectly compatible with
the principle of
is not compatible with the predictions of general relativity.
reported here must be seriously reconsidered .
it has been shown that the experimental data gathered during
or using radio signals do not possess sufficient reliability to
deflection of light by the solar gravity.
wishes to acknowledge the collaboration of Christine Couture and
encouragement and financial contribution of Mr. Bruce Richardson
helped to pursue this research work.
 Norbert Straumann,
and Relativistic Astrophysics, Springer-Verlag, Berlin,
 Paul Marmet,Einstein’s
Theory of Relativity versus Classical Mechanics.
Books, Ogilvie Rd. Gloucester, Ontario, Canada, K1J 7N4, 1997.
 P. Marmet, C. Couture, Relativistic
Deflection of Light Near the Sun Using Radio Signals.
March issue 1999.
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