Notes on Relativity and Cosmology - Physics Department, UCSB

3.8. HOMEWORK PROBLEMS 83
(e) Use the results of part d to show that
v B =
v A + v BA
1 + v A v BA /c2, (3.4)
or, in units where c = 1,
v B = v A + v BA
1 + v A v BA
. (3.5)
3-11. What happens in the relativistic addition of velocities formula (derived
in problem (10) when the two velocities are both very small? Try it for
v BA = .01c and v A = .01c. How much does the relativistic result differ
from the Newtonian result (1.2) in this case?
3-12. What happens in the relativistic addition of velocities formula (derived in
problem (10) when one of the velocities (say, v BA ) is the speed of light??
3-13. The most common use of relativity in every day life involves what is called
the ‘Doppler effect.’ This is an effect in which the frequency (= rate at
which crests pass by you) of a wave depends on the reference frame. There
is also a Doppler effect in Newtonian physics, but Newtonian physics would
predict a different amount of the effect. The Doppler effect is used in
various devices such as ‘Doppler radar’ and police radar guns to measure
the speed of storm systems or cars. For this problem, derive the formula
for the relativistic Doppler effect
τ R =
√
c + v
c − v τ S. (3.6)
where v is the velocity of the source (S) away from the receiver (R). Here,
I have expressed the formula using the period τ (the time between wave
crests) as measured by each observer.
Hint: A good way to figure this out is to think about a strobe light carried
by the source which emits light pulses at regular intervals τ S . Then we
can ask what the time interval τ R is (as measured by the receiver) between
the events where the light pulses are received. If we then think about each
of the strobe pulses as tracking a wave crest, this will give us the Doppler
effect formula.