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
modified:
2009/11/7 )
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 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).
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
DtB,
when the particle hits the opposite wall of the elevator, it has
traveled
a vertical distance DhB.
According to Einstein’s 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.
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
Nm2kg-2
is the Cavendish constant of gravitation, c = 2.998 ´
108 m/s is the velocity of light and
M
= 1.989 ´ 1030
kg is the mass of the Sun. Einstein’s equation gives a numerical
deflection
equal to:
 |
4 |
where
rs = 6.96 ´ 108
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 rm 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 rm
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 ay
is:
 |
9 |
Equation
8
in
equation
9
gives:
 |
10 |
According
to
the
principle
of
equivalence,
equation 10 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 Newton’s 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:
 |
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 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
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 (rm= rs)
equal to:
 |
20 |
5 -
Axial Component
of Velocity (Dvx).
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
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.
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 self-contradictory.
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 mass-energy 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 mass-energy conservation.
3) There is no deflection at all.
a) This
result
is perfectly compatible[2] with
the principle of
mass-energy 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, Springer-Verlag, Berlin, 1991, 459
pages.
[2] Paul Marmet,Einstein’s
Theory of Relativity versus Classical Mechanics. Newton Physics
Books, 2401 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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