10.1 - Introduction.
Einstein's Theory of Relativity
Classical Mechanicsby Paul Marmet
( Last checked 2012/07/02 - The estate of Paul Marmet )
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Chapter Ten The Principle of Equivalence.
10.2 - Deflection of
Light in an Elevator Moving at Constant Velocity.
Experiments describing a constant relative transverse velocity between a source and an elevator are generally ignored. Let us consider a horizontal parallel beam of light (or particles, as on figure 10.1) projected on an elevator (of negligible mass) moving upward at a constant velocity v with respect to the source. The experiment takes place in outer space far away from any gravitational field.
Because momentum must be conserved, the beam of light must move in a straight line. On figure 10.1, the dotted line inside the elevator shows where the photons can be detected with respect to the moving elevator at different times. The relative location of the photons with respect to the elevator moving at a constant velocity v is:
10.3 - Inertial versus
Gravitational Acceleration of Masses.
Before considering the problem of photons moving with respect to an accelerated frame, let us study a mass m moving horizontally. The mass enters an elevator which has an upward acceleration a in outer space at the moment its vertical velocity with respect to the source of the mass is zero. The elevator is accelerated by a rocket placed under it to produce a force F (shown by upward arrows on figure 10.2A). Due to that force F, the elevator (and the observer) accelerates following Newton's law:
|F = Ma||10.2|
Figure 10.2A Figure 10.2B
consider a similar elevator located at rest on Earth as
illustrated on figure 10.2B. The Earth's gravitational field
accelerates the mass m toward the
Earth's center. After a time interval Dt,
when the mass hits the opposite wall of the elevator, it will
have traveled an absolute vertical distance DhB.
In the experiment described on figure 10.2A, the mass m is completely free of any field and any force and therefore cannot gain any absolute energy when the floor of the elevator approaches it. An atomic clock bound to that free mass m will maintain a constant rate since no acceleration (therefore no energy) is given to the electrons or particles of the atomic clock. However, the elevator with the observer will gain kinetic energy (and therefore mass) due to the momentum transferred by the rocket. The observer's clock located on the floor of the elevator will slow down (absolute time) due to its increase of velocity in free space as given in equation 3.9. Consequently, the observer using the moving clock will observe a relative blue shift on light emitted from mass m. Let us note that the Doppler effect is considered separately and has not been taken into account.
In the experiment described on figure 10.2B, the elevator and the observer cannot gain any energy as a function of time since no work is produced on them. Neither the potential of the observer nor its velocity change. Therefore, the atomic clock of the stationary observer will keep giving a constant rate as a function of time. However, the clock on the falling mass will slow down for two reasons (independent of the Doppler effect): First, because of its increase of velocity (equation 3.10) and second, because of its decrease of potential energy (equation 1.22). Consequently, the observer standing in the elevator will observe a red shift on light emitted by the falling mass m.
The Doppler contribution to the shift of frequency is identical in figures 10.2A and 10.2B (if a = g). Its amplitude is much more important than the one due to the change of internal mass. However it can be subtracted out to show the difference explained above.
We see that the principle of mass-energy conservation implies that there is a fundamental difference between an inertial acceleration and an acceleration due to gravity since the consequences of each acceleration are just opposite. In the case of inertial acceleration (figure 10.2A) the clock located on the apparently falling mass will run faster than the observer's clock because of the slowing down of the observer's clock. On the contrary, in the case of gravitational acceleration (figure 10.2B), the falling clock will run more slowly than the observer's clock. One must conclude that the physical properties of the gravitational acceleration are different from the ones of inertial acceleration which means that the gravitational acceleration is not equivalent to the inertial acceleration.
10.4 - Bremsstrahlung
Due to Inertial and Gravitational Accelerations.
To illustrate the difference between inertial and gravitational accelerations, let us consider another thought experiment in which electric charges are placed in a gravitational field. One or more electrons are deposited on a stationary insulator in the Earth's normal gravitational field. This is static electricity. It is well known that Maxwell's equations predict that any accelerated electric charge must emit radiation called bremsstrahlung. According to Einstein's principle of equivalence, charges at rest in the Earth's gravitational field should emit bremsstrahlung because of the gravitational acceleration. However, no experiment has ever detected the emission of bremsstrahlung due to the gravitational acceleration of static electricity. The emission of radiation due to gravitational acceleration has been overlooked.
There is a way to prove that charges submitted to a gravitational acceleration do not emit bremsstrahlung. The principle of mass-energy conservation requires that energy must be given to an electric charge in order to compensate for the electromagnetic energy emitted during its acceleration. Let us try to identify the origin of the energy responsible for the bremsstrahlung predicted by Maxwell's equations and Einstein's principle of equivalence.
If bremsstrahlung is emitted when electric charges are submitted to gravity, there must be an energetic mechanism available to compensate for the energy lost by radiation. That continuous emission of radiation due to gravitational acceleration must necessarily extract energy from a source. Therefore, after a long period of time, the accumulated loss of energy in the source will be more easily detectable than the weak bremsstrahlung emitted. In the case of individual electrons stationary in a gravitational field, the only source of energy available is their mass. Consequently, the electron mass should decrease as a function of time to compensate for the electromagnetic energy bound to be emitted. If the electron mass decreases when standing in a gravitational field, one should eventually find electrons with different masses depending on the time they have been submitted to the Earth's gravitational acceleration.
However, it is observed that electrons maintain their full integrity and do not lose any mass while standing in a gravitational field. All electrons have the same mass. Due to the principle of mass-energy conservation, the absence of any source of energy shows that no bremsstrahlung can be emitted from gravitationally accelerated electrically charged particles. However, in the case of inertial acceleration, the mechanical energy required is well identified and compensates for the electromagnetic energy emitted as bremsstrahlung.
These considerations show again that gravitational acceleration is different from inertial acceleration. Bremsstrahlung is emitted only when submitted to inertial acceleration. Since Einstein's general relativity is based on Maxwell's equations and the principle of equivalence, we must reexamine Einstein's predictions.
10.5 - Behavior of Light.
10.5.1 - Light Path
in an Accelerated Elevator.
Let us now consider the experiment described in section 10.3 but using light instead of masses (figure 10.3A). Due to the conservation of momentum, light keeps moving in a straight line (as on figure 10.1) and takes a time interval Dt to go across the elevator. Because of the elevator's increasing upward velocity, during the time interval Dt, light seems to travel a vertical distance Dh:
Let us assume that the acceleration due to the rocket produces a change of velocity dv/dt equal to g = 9.8 m/s which is the gravitational acceleration on Earth. Observer A will feel that the upward force of the floor produces the same downward path on the photon as for a massive particle accelerated in the Earth's gravitational field (figures 10.2A and 10.2B).
Figure 10.3A Figure 10.3B
10.5.2 - Light Path
in a Gravitational Field.
Let us assume momentarily that the equivalence principle is valid. Therefore, with respect to observer B on figure 10.3B, light entering the room horizontally would be deflected as illustrated. This hypothesis implies that light is attracted by gravity. However, to be valid, we must verify that such an hypothesis is compatible with mass-energy conservation. If light is deflected, let us calculate the energy relationship caused by that deflection.
Let us call F the hypothetical gravitational force on a photon in the direction of the gravitational acceleration. During its passage across the elevator, we assume that the photon is deflected on a distance Dh in the direction of the force F, as shown on figure 10.3B. Mass-energy conservation requires that a displacement Dh in the same direction of a force F gives an increase of energy DW equal to:
|DW = FDh||10.5|
|DW = 0||10.6|
|DW = 0, F = 0 and Dh = 0||10.7|
10.5.3 - The
Equivalence Principle and Light Deflection.
It has been well recognized that the deflection of light rays is closely related to the equivalence principle discussed above. According to the paper "The Equivalence Principle with Light Rays":
Since the equivalence between inertial and gravitational acceleration assumed by Einstein is erroneous as shown above in several independent ways, it is not surprising that its consequence (light deflection) is also erroneous.
"This [the equivalence principle] led Einstein to predict that light is bent by a gravitational field around the Sun"
10.6 - Gravitational
There are several consequences to the fact that light is not deviated in a gravitational field. The deviation of light by a gravitational field gave birth to the claim that rings in space are caused by the focusing of light coming from remote sources by the gravitational mass of intervening galaxies. This explanation is certainly erroneous since light is not deviated by a gravitational field.
These rings can be explained more logically by the presence of large quantities of ions moving in the magnetic field of a galaxy. It is well known that ions spread naturally into rings in a magnetic field. This is a rational interpretation of a phenomenon that has been erroneously interpreted as Einstein's rings.
10.7 - Attracting Force
between Parallel Beams of Charged Particles.
We have seen in section 10.4 that electrical phenomena can be used to demonstrate that gravitational acceleration is different from inertial acceleration. To end this chapter, we will give an example using electricity disproving the principle of reciprocity (for another proof, see section 3.9).
In elementary physics, Ampere's law teaches how to calculate the force between two parallel straight conductors carrying currents in the same direction. We learn that a force F between parallel conductors spaced by a distance Dx is induced because the current i' in the second conductor passes in the magnetic field generated by the current i in the first conductor. The force F by unit of length (in MKS units) is:
10.8 - References.
 F. W. Sears, Principles of Physics, Addison-Wesley, p. 267, 1946
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