Notes on Relativity and Cosmology - Physics Department, UCSB

5.3. HOMEWORK PROBLEMS 137
A
B
C
The rockets are not rotating and have no relative motion. Rocket A is
equidistant from B and C (i.e., it is the same distance from each). Rocket
A sends out a light pulse and, when the rockets B and C receive this pulse,
each starts its engine. (The things sticking out of the rockets in the picture
above are the antennas that emit and receive the light pulse.) Rockets B
and C are identical, and their guidance computers are programmed in
exactly the same way. As a result, as reckoned by A, rockets B and C will
have the same velocity at every instant of time and so remain a constant
distance apart (again as measured by A). Rocket A remains in the same
inertial frame the entire time. Rockets A, B, and C carry clocks of identical
construction, all of which reach t = 0 when the light signal arrives at B
and C.
(a) Using the reference frame of Rocket A, draw a spacetime diagram
showing the worldlines of rockets B and C.
(b) Suppose that, at some time after rockets B and C begin to accelerate,
an observer in the inertial rocket (A) takes readings of the clocks in
rockets B and C. As usual, A does this by using various friends with
the same reference or by otherwise ensuring that light travel time is
not an issue. How do the clocks A, B, and C compare? (Just say
which is running fastest, slowest, etc. I don’t need a quantitative
answer.)
(c) What happens if an observer in rocket B compares the clocks? Hint:
Recall that since the speed of B (relative to A) continues to increase,
the associated time dilation factor will not be constant. As a result,
the answer (at least for comparing B’s clock to A’s) will depend on
when B makes this comparison. For simplicity, I suggest you think
about what happens at a very late time, long after B has passed A,
when the relative speed of A and B is nearly the speed of light.
(d) What happens if an observer in rocket C compares the clocks? Suppose
that B and C begin very close together and consider only what
happens at a very late time. Note: A complete analysis covering all
cases is more complicated. It turns out that whether C sees A’s clock
run faster or B’s clock run faster depends on the initial separation
between B and C.
(e) [Optional] Does the answer to D depend on whether rockets A and C
start off close together or far apart?
Preliminary Comments for problems 5 and 6: The problems below
will give you a feel for how to answer more complicated questions in special