between Einstein's General Relativity
and the Principle of Equivalence.
Physics Department, University of Ottawa, Ottawa,
Canada, K1N 6N5
checked 2018/01/15 -
The estate of Paul Marmet )
paper reports an analysis of Einstein's principle of equivalence
between inertial and gravitational acceleration and its
consequences on general relativity. It is shown that the simple
application of that principle to photons moving in the Sun's
gravitational potential leads to an equation which is not
compatible with the one predicting the deflection of light by the
Sun. Therefore, the principle of equivalence is not compatible
with general relativity.
Einstein's general relativity is based on the principle of
"In an arbitrary gravitational field no
local experiment can distinguish a freely falling
nonrotating system (local inertial system) from a
uniformly moving system in the absence of a gravitational
to this principle, the gravitational acceleration g due to
universal attraction between masses is equivalent to an inertial
acceleration a (= g) due to a change
of velocity. This principle is considered to be one of the
foundations of general relativity.
Obviously, it should be tested. There exists a direct test to
determine whether the principle of equivalence is compatible with
general relativity. Unfortunately, it has never been performed. In
this article, we will describe this method and then use it to test
the principle of equivalence.
2 - Illustration of the Principle of Equivalence.
Einstein illustrates his principle of equivalence using a
gedanken (thought) experiment in which he compares the
trajectory of a particle in a gravitational field (gravitational
acceleration g) with its trajectory in the absence of a field,
but with respect to an accelerated observer whose acceleration a is equal to g. This is illustrated on
figure 1A, a particle "p" enters an elevator located in
zero gravitational field at time t = 0 when the vertical
components of velocity of the particle and of the elevator are
both zero. The elevator's upward inertial acceleration a is given by a rocket placed under it
producing a force F (shown by upward arrows on figure 1A).
Because of that force F, the elevator and the observer
accelerate following Newton's law:
is the mass of the elevator (including the observer's mass) and a is its acceleration given by:
time interval DtA, the particle hits the opposite wall. It has traveled a
vertical distance DhA relative to the moving elevator. However, it has
obviously traveled an absolute vertical distance of zero since
there is no gravitational field (the gravitational field caused by
the elevator and the observer is negligible).
consider now a similar elevator located at rest on Earth as
illustrated on figure 1B. The same particle "p" enters this
elevator. After a time interval DtB, when the particle hits the opposite wall of the
elevator, it has traveled a vertical distance DhB. According to Einsteinís principle of equivalence, both
experiments must be undistinguishable and the relative distances
must be the same (i.e. DhA= DhB).
Unfortunately, this gedanken experiment is not easily transformed
into a practical experimental test which has never been done.
Nobody has ever made a serious effort to test directly the
principle of equivalence in this way, even if this is the basis of
modern physics. Consequently, the principle of equivalence has
been accepted as a dogma.
we will see that one can use the sum of many similar gedanken
experiments and integrate the result for a continuously varying
gravitational field, in order to simulate the well studied
problem of photons deviated by the gravitational field of the
Sun. This comparison gives us the possibility, at last, to
verify directly Einstein's principle of equivalence with
mathematical predictions and observational data.
Deflection of Light According to Einstein's General
Einsteinís general relativity predicts
that light passing at a distance r from the center of a star of
mass M must be deflected by an angle d
according to the equation:
= 6.67 ī 10-11 Nm2kg-2 is the Cavendish constant of gravitation, c = 2.998 ī 108 m/s is the velocity of light and M = 1.989 ī 1030 kg
is the mass of the Sun. Einsteinís equation gives a numerical
deflection equal to:
where rs = 6.96 ī 108 m is the radius of the Sun.
This result is
well known and is expected to be quite reliable.
Integration of the Fundamental Experiment.
that the experiment described on figure 1A can be used as a test
of the principle of equivalence using light deflection by a
gravitational field. Let us now consider this experiment in
detail and show that the sum of a large number of the same
experiment, with different and continuously changing values of
the gravitational field, corresponds to the one in which light
is deflected by the solar gravitational field. According to
Einstein's principle of equivalence, the same relative
deflection must appear whether there is a gravitational field
acting on light or whether there is no gravitational field but
an equal (but inertial) acceleration given to the Sun (see
problem illustrated on figure 2 is the same as the one on figure
1. However, the gravitational field is constant in figure 1 but
variable in figure 2. The deflection is strongly exaggerated in
figure 2 and the x component (which is discussed separately in
section 5) of the displacement of the Sun is not shown.
figure 2A, the Sun is represented surrounded by its
gravitational field. When a particle crosses the gravitational
field, it is accelerated toward the Sun according to the
gravitational intensity at its position. Since the acceleration
of the moving particle is determined by the mass generating the
gravitational field (here the Sun) and the distance of the
particle from that source, all particles at a given distance
receive the same change of velocity (acceleration), independently
of their mass. Consequently, all particles have the same
trajectory in a gravitational field independently of
Applying Einsteinís principle of equivalence, let us substitute
the gravitational field around the Sun by a suitable change of
velocity of the Sun (as on figure 1). Going now to figure 2B, an
observer must measure the same relative motion between
the particle and the Sun as when the particle is submitted to
solar gravity. The equivalence between inertial and
gravitational acceleration can be complied at any distance from
the Sun by giving it an acceleration in the direction of the
particle. This acceleration varies as a function of the location
of the moving particle with respect to the Sun. Since the
particle is located at a variable distance in the gravitational
field of the Sun, the acceleration given to the Sun must always
be equal to the gravitational acceleration the particle would
feel if there were a gravitational field.
calculate the acceleration that must be applied to the Sun in
order to obtain the same relative motion in figures 2A
and 2B according to the principle of equivalence. The
interesting point is that nobody can disagree about the correct
trajectory of the moving particle since all particles (including
photons) must travel in a straight line in the absence of
gravity. Consequently, on figure 2B, the photon (as well as any
particle) must move in a straight line. According to Einstein's
principle of equivalence, both problems (illustrated on figures
2A and 2B) are undistinguishable and must lead to identical
results. Whatever the result is, let us calculate the relative
deflection produced between the particle and the Sun assuming
the correctness of the principle of equivalence.
figure 3, a moving photon travels from left to right at a
constant velocity c. At time t, the angle between the particle
and the Sun is equal to q(t).
mathematics will be simplified here because we know that the angle
of deflection d is extremely small. We
will therefore neglect the change of minimum distance of the
particle to the Sun rm with the
position of the particle in the direction x.
principle of equivalence requires that the inertial acceleration a applied to the Sun be equal to the
gravitational acceleration g where the photon is located. Since
acceleration a (Sun) given to the Sun
consider that the origin of the coordinates (figure 3) of the
photon is the location where the photon is at minimum distance rm from the Sun. When the photon moves away from this
central location, the force on the Sun decreases as a function of
from equation 7 into equation 6 gives:
components of that acceleration of the Sun. The transverse
(upward) component ay is:
Equation 8 in equation 9 gives:
represents the transverse component of acceleration ay that
must be given to the Sun. During the full passage of the photon from - to +, the acceleration of
the Sun is varied continuously in order to compensate exactly for
the variable distance r from the Sun and the angle q of the force. Using Newtonís law, we know
that at every location, the change of velocity Dvy is equal to the acceleration multiplied by the time. The
total change of velocity Dvy along the y-axis is:
photon as a function of time. The photon moves on the axis in
straight line at velocity c (since now, the gravitational force no
longer exists). Therefore when we set x=0 at t=0, the relative
location of the photon, as a function of time is:
calculate the angle q as a function of
time. On figure 3, we find:
The derivative of equation 14 is:
equation 11. When the photon travels from x = - to x = +, the angle q passes from -p
/2 to +p /2. We have:
Equation 17 gives the final relative velocity Dvybetween
the photon and the Sun, after the passage of the photon, when the
acceleration of the Sun was at all time identical to the
gravitational acceleration, the photon would have felt at a given
location, as explained previously. According to the principle of
equivalence, equation 17 is valid either when the photon is
attracted by the gravitational field or when the Sun is
accelerated and the particle travels in straight line (without any
passage of the photon, the relative transverse velocity of the
photon with respect to the Sun makes an angle d
given by the relationship:
(tand = d
since d is extremely small). This
relative deflection between light and the Sun is the result
expected to be compatible with the principle of equivalence.
Equations 17 and 18 give:
gives the only solution compatible with Einsteinís principle of
equivalence and is the direct consequence of Einsteinís gedanken
experiment illustrated on figure 1. Consequently, if the principle
of equivalence is valid, the deflection of light by the
gravitational field of the Sun must be as given by equation 19.
However, equation 19 gives a deflection d
at the limb of the Sun (rm= rs) equal to:
Axial Component of Velocity (Dvx).
figure 2, we see that when a particle approaches the Sun, in order
to satisfy the principle of equivalence, one must accelerate the
Sun in the x direction. However, when the particle recedes from
the neighborhood of the Sun, the inverse phenomenon takes place.
We can see that the change of velocity Dvx that must be given to the Sun when the particle
approaches it, is in the opposite direction of the change of
velocity given to the Sun when the particles recedes from it.
Therefore, the total velocity given to the Sun is zero in this
axis. Consequently, the x-axial acceleration of the particle (if
any) has no global effect on the apparent deflection of the
particle since Dvx is zero.
Therefore, the principle of equivalence predicts no deflection
related to the axial acceleration of the particle.
Consequences of the Principle of Equivalence.
must conclude that if Einsteinís principle of equivalence is
valid, equation 19 shows that the deflection must be d = 0.875≤. We
know (equations 3 and 4) that Einsteinís general relativity
claims that the deflection is twice this amount (d =1.75≤).
Therefore, Einsteinís prediction is not compatible with
his own principle of equivalence. Einsteinís theory is
deflection of light by the Sun has been tested in several
experiments measuring either the deflection of visible light of
the delay of radio signals traveling in the Sun's gravitational
potential. A critical analysis  of the
results has been investigated. It shows that there is an acute
need for more reliable experiments.
The Three Options.
consider three possibilities.
1) The deflection is 1.75≤.
result is not compatible with Einsteinís principle of
equivalence, as shown above.
result is compatible with the predictions of general relativity.
generally implied in general relativity, this result it is not
compatible with the principle of mass-energy
2) The deflection is 0.875≤.
hypothesis is compatible neither with Einsteinís general
relativity nor with claimed observations during eclipses or with
is however compatible with the principle of equivalence.
generally implied in general relativity, this result it is not
compatible with the principle of mass-energy
3) There is no deflection at all.
result is perfectly compatible with the
principle of mass-energy conservation.
result is not compatible with the predictions of general
incoherence reported here must be seriously reconsidered . Furthermore, it has been shown that the
experimental data gathered during solar eclipses or using radio
signals do not possess sufficient reliability to prove any
deflection of light by the solar gravity.
author wishes to acknowledge the collaboration of Christine
Couture and the personal encouragement and financial
contribution of Mr. Bruce Richardson which helped to pursue this
 Norbert Straumann, General
Relativity and Relativistic Astrophysics, Springer-Verlag,
Berlin, 1991, 459 pages.
 Paul Marmet,Einsteinís Theory of
Relativity versus Classical Mechanics. Newton
Physics Books, Ogilvie Rd. Gloucester, Ontario, Canada, K1J 7N4,
 P. Marmet, C. Couture, Relativistic Deflection of
Light Near the Sun Using Radio Signals. Physics Essays March issue
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