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Time Dilation
The aim here is to imagine asking two observers, who are in relative motion, to make measurements of the time between the same two events, using identical clocks.
The two events will be:  
1. a very brief pulse of light is produced and  
2. that pulse of light hits a mirror.
   
The observers will be called A and B, not very imaginative I know but what did you expect?  
(You might have already met them in the relativity of simultaneity page.)  
     
  Observer A holds the flash-light and mirror (note that this means that A has zero velocity relative to the apparatus)
   
   
  A holds the apparatus such that the line between flash-light and mirror is at 90° to the direction of the relative motion of A and B.
   
  This diagram shows the situation from A's point of view.
   
  The speed of light (the same for all inertial observers) is c.
  If the time between the two events, as measured by A, is to then we have
 
  so
 
   
We will now consider the same situation from B's point of view.  
   
This diagram shows the trajectory of the light pulse from B's point of view (in B's frame of reference, if you prefer).
 
We will suppose that, at the instant when the light pulse is produced, B and A are very close together.
 
Notice that we are here assuming that the speed of B relative to A is a significant fraction of the speed of light.
 
As B sees things, the mirror moves (to his/her right) during the time that the light is in motion.
 
So B concludes that the light moved further to get to the mirror.
 
   
   
In the time taken for the light to reach the mirror, A and B have moved a distance x  relative to each other.
 
For B, the distance moved by the light to reach the mirror is
so B will say that the time between the two events, t, is
also, as the speed of A relative to B is v we have
From the above we can write
In this equation we see the relation between the two times as measured by A and B.
 
Rearranging this, we obtain the more convenient expression  
 
If we imagine A to repeatedly send a new flash of light towards the mirror each time a reflected pulse returns, then we can say that A has a sort of “light beam clock”.  
We can further imagine that A uses this to drive ordinary mechanical clock hands which could be observed (probably with the assistance of a telescope!) by B as they race past each other.  
   
The conclusion from our thought experiment is that B would see A’s clock running slowly compared with any similar clock carried by (and therefore at rest relative to) him/her-self.  
   
However, note that A would, of course, think the same thing about a clock carried by B.  
   
This is where the term time dilation comes from.  
It refers to the “stretching out” of time intervals as indicated by clocks which are in motion relative to the observer.  
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