Notes on Relativity and Cosmology - Physics Department, UCSB

3.5. TIME DILATION 69
ct us
vt us
L
The length of each side of the triangle is marked on the diagram above. Here,
L is the length of her rod and t us is the time (as measured by us) that it takes
the light to move from one end of the stick to the other. To compute two of the
lengths, we have used the fact that, in our reference frame, the light moves at
speed c while our friend moves at speed v.
The interesting question, of course, is just how long is this time t us . We know
that the light takes 1 second to travel between the tips of the rod as measured in
our friend’s reference frame, but what about in ours? It turns out that we can
calculate the answer by considering the length of the path traced out by the light
pulse (the hypotenuse of the triangle above). Using the Pythagorean theorem,
the distance that we measure the light to travel is √ (vt us ) 2 + L 2 . However, we
know that it covers this distance in a time t us at speed c. Therefore, we have
c 2 t 2 us = v2 t 2 us + L2 , (3.1)
or,
L 2 /c 2 = t 2 us − (v/c) 2 t 2 us = (1 − [v/c] 2 )t 2 us. (3.2)
L
Thus, we measure a time t us = √ between when the light leaves
c 1−(v/c) 2
one mirror and when it hits the next! This is in contrast to the time t friend =
L/c = 1second measured by our friend between these same to events. Since this
will be true for each tick of our friend’s clock, we can conclude that:
Between any two events where our friend’s clock ticks, the time t us that we
measure is related to the time t friend measured by our friend by through
t us =
t friend
√
1 − (v/c)
2 . (3.3)
Finally, we have learned how to label another line on our diagram above: