and the Principle of Equivalence.

Paul Marmet,
Physics Department, University of Ottawa, Ottawa, Canada, K1N 6N5
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( Last checked 2012/03/17 - The estate 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 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, 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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