Notes on Relativity and Cosmology - Physics Department, UCSB

4.5. HOMEWORK PROBLEMS 119
is easiest to use Alphonse’s frame of reference). Now, suppose that
Alphonse emits one light ray every year (according to his own proper
time). Draw these light rays on the diagram. How many of these
light rays does Gaston see on his way out? [Hint: You should be able
to read this off of your graph.] How many does Gaston see on his way
back? What does this tell you about what Gaston actually sees if he
watches Alphonse through a telescope?
(b) Now let’s figure out what Alphonse sees. Draw another spacetime
diagram for the trip in some inertial frame (again, Alphonse’s frame
is the easiest one to use), but this time suppose that Gaston emits
one light ray every year (according to his own proper time – you may
have to calculate this). What does this diagram tell you about what
Alphonse sees if he watches Gaston through a telescope during the
trip??
4-5. In relativity, it is always nice to look at things from several different reference
frames. Let’s look at the same trip of Gaston to Alpha Centauri
and back, with Alphonse staying at home. This time, though, draw the
spacetime diagram using the inertial reference frame that Gaston had on
his outward trip. Using this frame of reference, calculate the total proper
time that elapses for both Alphonse and Gaston between the event where
Gaston leaves Alphonse and the event where they rejoin.
4-6. How about some practice working with hyperbolic trig functions? Recall
that
sinhθ = eθ − e −θ
,
2
cosh θ = eθ + e −θ
,
2
and
.
tanhθ = sinhθ
coshθ
(a) Verify that cosh 2 θ − sinh 2 θ = 1.
(b) Verify that
tanhθ 1 + tanhθ 2
1 + tanhθ 1 tanhθ 2
= tanh(θ 1 + θ 2 ),
so that the law of composition of velocities from last week just means
that “boost parameters add.”