Slide #1:

Time Dilation


Here we take an approach complementary to that in the book...one that does not 
use the spacetime diagrams in the same way but nevertheless obtains the same results.

Consider an observer with a light clock who observes the time for a tick of an 
identical light clock which moves relative to her with velocity V.


The following figure shows the set up:

<img src="timedil1.GIF"> 				

The Pythagorean theorem suffices to express the needed relation between the
time it takes to go from the bottom to the top mirror, the speed of the mirror with
respect to O, and the vertical distance the light must travel:

<img src="timedil2.GIF"> 				 

Solving for tdu  we find 

<img src="timedil3.GIF"> 
---------------------------------------------------------------------------------------
Slide #2:
					 
Similarly, for the second half of the tick:

<img src="timedil4.GIF">				 

Solving for tud, we find

<img src="timedil5.GIF">				 

Thus, the total time for a single tick as observed by O is:

<img src="timedil6.GIF">			 

Now, observer O¹ would claim that the time for one tick on her light clock would be

<img src="timedil7.GIF">	
---------------------------------------------------------------------------------------

Slide #3		 

Postulate II demands that c = c¹.  This
implies that 

<img src="timedil8.GIF">			 


Thus moving clocks run slow...time is streched or dilated.  
This implies that there is no absolute time since each observer
will measure a local time interval between events which is different
from any other inertial observers with differing velocities.

---------------------------------------------------------------------------------------

Slide #4

Length Contraction


Consider again two observers¹ account of a single tick of a light clock, except 
now we lay the clock on its side and analyze the distance between the two mirrors.

<img src="timedil9.GIF"> 

We now analize the single tick of this clock, taking into account the time dilation 
effect as needed.

For the trip from the left mirror to the right we have:

<img src="timedil10.GIF">			 

For the trip from the right mirror to the left we have:

<img src="timedil11.GIF">

				 
From the time dilation effect, we know that

<img src="timedil12.GIF">
 
and  

<img src="timedil13.GIF">

---------------------------------------------------------------------------------------

Slide #5



 
Then, we find


<img src="timedil14.GIF">

 
and

<img src="timedil15.GIF">
 

Using these,


<img src="timedil16.GIF">
 

But c = c¹ from Postulate II, so

 
 <img src="timedil17.GIF">
 
				
Thus the distance between the two ends of the clock laying on its side depends 
on the relative velocity of O and O¹. 


Hence lengths are not indpendent of motion.