1) General derivation of time dilation.-----[ Return home ]
It has been derived that for a simple clock where longitudinal waves have been used, that the frequency inside the resonator decreases according to the factor: .
But, if we take into account the fact that in the case of a real clock, the distance between the mirrors must decrease according to the factor: , due to the theoretical contraction of the electron generalized to matter, the frequency of the clock must decrease (in absolute motion) according to:.
This result has also been obtained inside the whole wave structure of the electron.
Thus, there is no obstacle to generalise the phenomenon, because the frequency inside the electron follows this law, and also because any attempt of rigorous contruction of a clock with electromagnetic waves lead to this remarkable equation.
An exemple will be given here for an electromagnetic cavity. The method used will be also valid for other cavities.
The same way, we have good reasons to think that when a new hydrogen atom will be build from the present theory, that it will be exact when the spectral line frequencies will decrease according to the time dilation law.
Especially, if we see that the frequency of the electron at rest appears in the Balmer-Rydberg constant in the following well-known formula:.
B. Exemple of calculation of the classical time dilation of an electromagnetic cavity.
Due to the complexity of the calculation, only one exemple is given, but the reader may experience himself other derivations for other cavities using nearly the same method.
Here, we use a electromagnetic cavity made of piece of a rectangular waveguide (closed at the ends) orientated in the direction of motion.
In this case, with a well chosen frequency, the electromagnetic field for a semi-travelling wave along the waveguide is made of two tranverse electromagnetic waves (TEM) whose interference yields the mode . The name " semi-travelling " has been used for the mode, because it is not a pure travelling wave which has nodes on the walls of the waveguide like any standing wave.
Nevertheless, each TEM wave is a pure travelling wave with a Poynting vector making an angle with the axis of the waveguide and which follows the equation: where d is the width of the guide, f the frequency.
We suppose, obviously, that the cavity is fully transparent for the ether and thus that the inner waves must be considered as propagating into vacuum without to be carring along by the cavity.
Now, we consider the cavity in resonance and in motion with two semi-travelling waves (mode)in the positive and negative way along the axis of motion (X-axis).
For the mode propagating in the negative way of the X-axis of the rest frame, its TEM waves of frequency and with the Poynting angle are going to reflect on the left closing of the cavity with a change of direction (now positive way), angle and of frequency due to the classical Doppler effect.
According to formulas already established, the classical Doppler effect obeys the following equations: and with the mirror angle: where the original angles have been changed to be in agreement with the figure below showing (for simplicity) only one TEM wave for each mode semi-travelling in the negative and positive (reflected wave) ways.
With one of the Doppler formulas, we obtain the following incredible result: which means that the reflected TEM waves have the good direction and the good frequency to propagate in the positive way in order to build the allowed mode at that frequency.
This strange result means that the criterions of the propagation of the modes in the waveguides are intimately related to the laws of the classical Doppler effect. I think, that there is something there of very interesting which needs to think about it.
More as the reflection on the right mirror obeys the same Doppler relations including
and , we may conclude that the departure frequency will be regenerated and that a steady state wave structure is possible, even if the cavity is in motion.
An other important fact to point out is that, in the present calculation, the frequency of the cavity at rest may not be any one, because the number of wavelengths along a round-trip would not be an integer number.
In such a physical condition, even if allowed by the wave equation, the inner frequency would not be well-defined and would lead to erroneous mathematical results. Thus, we consider the cavity at the resonance frequency at rest and in motion.
In fact, as it is a cavity, the number of wavelengths along the round-trip in the cavity at is implicitely of two and the frequency at rest of: where L is the length of the cavity (see waveguide theory).
But now, for the frequency inside the cavity in motion (Galileo's frame in motion), we are going to see that it is unique. To show that, we consider one TEM wave of the mode propagating in the positive way whose equation is: . But now, if we look at the wave equation in the moving frame using the Galileo'transformations (x'=x-vt, t'=t), we see that the equation becomes: where the frequency is: . For the mode propagating in the negative way, the obtained frequency is: and is the same as for the positive way according to one of the Doppler formulas.
It is now possible to derive the mode velocities (positive and negative) in the rest frame using this unique frequency in the moving frame:.
If we consider the positive mode, the sum of its TEM waves which are is:
shows that the velocity is: . The same way, the velocity for the negative mode is: .
For the positive way, as we have: , we obtain the second degree equation in cos(a): with the unique positive solution: .
In the negative way, the unique solution is:.
Thus, as the wavelength connected to the positive way is: and: in the negative way, the number of wavelengths along the round-trip is:. Here the reader must note that the Lorentz'contraction has been applied.
As this number may not have changed during the acceleration of the cavity (like for clock1 elsewhere), and also because the cavity must stay in resonance, it must be still the same as at rest, with the same integer number of 2: .
The last equation has been obtained with v=0, and the number 2 in using the resonance frequency at rest for Fo (see above).
After a difficult algebraic calculation, the equality between the number of wavelengths in motion and at rest, become very simple:.
This formula would have been without using the Lorentz'contraction of the waveguide.
Even if the present calculation has not bee easy, it shows that time dilation is a general phenomenon.
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