Incompatibility
between
Einstein's
General Relativity
and the Principle of Equivalence.
Paul Marmet,
Physics Department, University of Ottawa, Ottawa,
Canada,
K1N 6N5
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( Last
checked
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of
Paul Marmet )
Abstract.
This
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.
1
Introduction.
Einstein's
general relativity is based on the principle of equivalence[1]:
"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 field."
According
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[1] 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 1.
Figure 1
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:

1 
where
M
is the mass of the elevator (including the observer's mass) and
a
is its acceleration given by:

2 
After
a
time interval Dt_{A},
the particle hits the opposite wall. It has traveled a vertical
distance
Dh_{A}
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).
Let us
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
Dt_{B},
when the particle hits the opposite wall of the elevator, it has
traveled
a vertical distance Dh_{B}.
According to Einstein’s principle of equivalence, both experiments
must
be undistinguishable and the relative distances must be the same
(i.e.
Dh_{A}=
Dh_{B}).
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.
However,
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.
3 
Deflection
of Light According to Einstein's General Relativity.
Einstein’s
general relativity predicts[1]
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:

3 
where
G
= 6.67 ´ 10^{11}
Nm^{2}kg^{2}
is the Cavendish constant of gravitation, c = 2.998 ´
10^{8} m/s is the velocity of light
and
M
= 1.989 ´ 10^{30}
kg is the mass of the Sun. Einstein’s equation gives a numerical
deflection
equal to:

4 
where
r_{s} = 6.96 ´ 10^{8}
m is the radius of the Sun.
This result is
well
known and is expected to be quite reliable.
4 
Integration
of the Fundamental Experiment.
We see
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 figure 2).
Figure 2
The
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.
In
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 their mass.
Applying
Einstein’s
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.
Let us
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.
In
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).
Figure 3
The
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 r_{m}
with
the
position of the particle in the direction x.
The
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

5 
the
inertial
acceleration a (Sun) given to the
Sun
must be:

6 
Let
us
consider that the origin of the coordinates (figure 3) of the
photon
is the location where the photon is at minimum distance r_{m}
from the Sun. When the photon moves away from this central
location,
the
force on the Sun decreases as a function of q
where:

7 
Substituting
r
from equation 7 into equation 6 gives:

8 
Let
us
consider
separately
the
x
and
y
components of that acceleration
of the Sun. The transverse (upward) component a_{y}
is:

9 
Equation
8
in
equation
9
gives:

10 
According
to
the
principle
of
equivalence,
equation
10
represents the
transverse
component of acceleration a_{y}
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 Newton’s law, we know that at every location, the
change
of
velocity Dv_{y }is
equal to the acceleration multiplied by the time. The total change
of
velocity
Dv_{y}
along the yaxis is:

11 
Let
us
calculate
now
the
distance
of
the
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:

12 
Let
us
calculate the angle q as a function of
time.
On figure 3, we find:

13 
which
gives:

14 
The
derivative
of
equation
14
is:

15 
Let
us
substitute
equations
15
and
10
in
equation 11. When the photon
travels
from x =  to x = +,
the angle q passes from p
/2 to +p /2. We have:

16 
which
gives:

17 
Equation
17
gives
the
final
relative
velocity
Dv_{y}between
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
gravitational
field).
After the
passage of the photon, the relative transverse velocity of the
photon
with
respect to the Sun makes an angle d
given
by
the relationship:

18 
(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:

19 
Equation
19
gives the only solution compatible with Einstein’s principle of
equivalence
and is the direct consequence of Einstein’s 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 (r_{m}= r_{s})
equal to:

20 
5 
Axial Component
of Velocity (Dv_{x}).
From
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
Dv_{x}
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 xaxial acceleration of
the
particle
(if any) has no global effect on the apparent deflection of the
particle
since
Dv_{x} is zero. Therefore, the
principle of equivalence predicts no deflection related to the
axial
acceleration
of the particle.
6 
Consequences
of the Principle of Equivalence.
One
must
conclude
that if Einstein’s principle of equivalence is valid, equation
19 shows
that the deflection must be d =
0.875².
We know (equations 3 and 4) that Einstein’s general relativity
claims
that
the deflection is twice this amount (d
=1.75²).
Therefore, Einstein’s prediction is not compatible with
his own
principle of equivalence. Einstein’s theory is
selfcontradictory.
The
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 [3]
of the results has been investigated. It shows that there is an
acute
need
for more reliable experiments.
7 
The
Three Options.
Let us
consider
three possibilities.
1) The deflection is 1.75².
a) This
result
is not compatible with Einstein’s principle of equivalence, as
shown
above.
b) This
result
is compatible with the predictions of general relativity.
c) As
generally
implied in general relativity, this result it is not
compatible[2] with
the principle^{ }of massenergy conservation.
2) The deflection is 0.875².
a) This
hypothesis
is compatible neither with Einstein’s general relativity nor
with
claimed
observations during eclipses or with radio signals.
b) It
is
however
compatible with the principle of equivalence.
c) As
generally
implied in general relativity, this result it is not
compatible[2] with
the principle^{ }of massenergy conservation.
3) There is no deflection at all.
a) This
result
is perfectly compatible[2] with
the principle^{ }of
massenergy conservation.
b) This
result
is not compatible with the predictions of general relativity.
The
incoherence
reported here must be seriously reconsidered [2].
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[3].
8 
Acknowledgments.
The
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 research work.
9
References.
[1] Norbert Straumann,
General Relativity
and Relativistic Astrophysics, SpringerVerlag, Berlin,
1991, 459
pages.
[2] Paul Marmet,Einstein’s
Theory of Relativity versus Classical Mechanics.
Newton Physics
Books, Ogilvie Rd. Gloucester, Ontario, Canada, K1J 7N4, 1997.
[3] P. Marmet, C. Couture, Relativistic
Deflection of Light Near the Sun Using Radio Signals.
Physics Essays
March issue 1999.
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