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Time dilation

Consider two inertial systems moving with constant velocity v against each other. At time t = 0 both systems should coincide. One system will be considered as the rest system of an observer, it has coordinates (t,x). The other systems, with coordinates (t´,x´),  moves away with constant velocity v and at time dt it has a distance of v * dt from the origin of the (t,x) system (see the picture below). The distance v * dt is measured by the observer in the (t,x) system.

There should exist a time t < 0 (maybe -10 hours, that means 10 hours ago) where both systems had no move against each other and both systems had synchronized clocks. The system (t',x') then was accelerated on velocity v against the system (t,x), so that both systems coincide at time t = 0, but have constant velocity v against each other. It is important that the accelaration can only be recognized in the system (t',x'). An observer moving with the system (t',x') feels this accelaration, an observer in the rest system (t,x) feels nothing. Therefore both systems are not symmetric.

This fact is essential to avoid a paradox.

The time dt will be measured in the system (t,x), the time d will be measured in the system (t´,x´).

During the time dt the system (t',x') moves away to a distance v * dt.

In the system (t',x') only the time d  < dt will be measured (see the picture below).

The observer in (t,x) will say that the clock which is connected with the (t´,x´) system and does not move against this system moves away with the velocity v. He may say that the time measured by this clock is stretched (dilated) against the time measured by himself in the system (t,x). This effect is named time dilation. A reasoning for the time delation is given by the Lorentz-transformation.

Time delation means, that the observer in (t´,x´) needs fewer time to reach a far away object than calculated by the observer in (t,x), the time in (t´,x´) is stretched.

But the observer in (t´,x´) will measure the same velocity v between the two systems. In his system he is not moving, for him the other system is moving away with velocity v. He is only measuring the time d  < dt , but v * d < v * dt. This will raise a contradiction, if not another effect will be considered, the Lorentz contraction.


 
 

The observer in (t',x') measures a length contraction for distances measured by the observer in the system (t,x).  As a consequence all distances in the universe appear shorter for the observer moving away with the system (t',x').

This length contraction is also a consequence of the lorentz contraction.

For explaing this contraction it is important how the observer in (t´,x´) measures the incoming signals. Imagine two objects in the system (t,x) at distance l which send light signals.  The spatial dimension of this two objects shall be neglected. The incoming signals will be collected by the observer in (t´,x´) at a distance l´.
The observer in (t´,x´) will measure a distance l´ < l if he uses signals that income at the same time t´.

But this is only possible if the outgoing signals have been sent at different times t1 and t2.




However, this interpretation gives some difficulties. What is the length of l in this picture measured by the observer in (t,x)?
The length l is sending signals out to different times t1 and t2, the measuring l´ gives a numerical value smaller than l.

To see this, consider the movement from the observers point of view, who does move with his system (t´,x´). For this observer it is his rest system.

Now this system will be named (t,x), the former system (t,x) moves away with velocity v, now named (t´,x´).

Let l´ = x2´ - x1´ the distance between the two objects considered above, in the system now moving away,
sending signals at time t1´ and t2´, so that the incoming signals can be measured at the same time t.

(now the former rest system is moving, therefore the former x values are transformed to x´values - this corresponds with a change of view).

Then the Lorentz transformation gives the following data:

From this the following relation can be calculated:

That means, the length x2 - x1 is shorter than the length x2´ - x1´, independent from the times, when the signals have been sent.
This result explains the length contraction.


Going back to the original example, the length contraction for distances in the universe appears only in the system which has been accelarated to a velocity v. In the other system the time delation works.

Light is moving with velocity v = c. This would imply d  = 0, therefore light quants have no proper time, but they also have no rest mass. Elementary particles with a rest mass greater then zero have a proper time which is going slower if they reach velocities approximating v = c.

This explains the observation: Elementary particles with a limited life time therefore live longer if they are moved quickly around a circle in a labor.

d  is the inner clock of an elementar particle with a rest mass, it measures it's proper time.


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