Physics Department, University of Ottawa, Ottawa, Canada, K1N 6N5

<><><><><><><><><><><><>

( Last
checked 2018/01/15 -
The estate of Paul Marmet )

Return to: List of Papers on the WebGo to: Frequently Asked Questions

** 1-
Introduction.**

Einstein's general relativity is based on the principle of
equivalence[1]:

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."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."

**
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 |

2 |

Let us consider now a similar elevator located at rest on Earth as illustrated on figure 1B. The same particle "

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

However, we will see that

** 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 |

4 |

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 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 |

6 |

7 |

8 |

9 |

10 |

11 |

12 |

13 |

14 |

15 |

16 |

17 |

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 |

19 |

20 |

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

** 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.

<><><><><><><><><><><><>

Return to: Top of page

Return to: List of papers on the web

Information: About the author

July 1999