Description
Problems
1. Space travel: You are the head of an expedition to some planet X orbiting the closest
star to us, Proxima Centauri. You know that the distance to X from Earth is 4.22
light-years and that the spaceship is going to travel with constant speed v =
√
0.9999c.
(a) (5 pts) How long the travel is going to take from the perspective of an observer
staying on Earth ?
(b) (10 pts) You suspect that for an observer on the spaceship the clock is going to
run differently. You want to know for sure, so you could take an adequate amount
of supplies for the trip, but not more than necessary, since the storage space in
the spaceship is limited. How long the travel is going to take from the perspective
of an observer on the spaceship ?
2. Alien rocket ship race: As you safely arrive on planet X after the space trip and
start studying everyday life of the alien population there, you attend a rocket ship race.
There, among other things, you witness the following event. Rocket ships A and B
move on the space track in the same direction on a straight line separated by distance
l = 2.26829 · 108 m with a speed 40
41 c. Ship A is leading. At the moment when ship
A is at point a and ship B at point b, ship A breaks down and instantly slows down
to speed 9
41 c, while ship B maintains its original speed. After a short moment ship B
catches up with ship A and they collide. Thinking about this event you wonder about
the following (for simplicity, take c = 3 · 108 m/s):
(a) (8 pts) In your reference frame, how much time passed from the moment of breakdown to the moment the rockets collided ?
(b) (11 pts) How fast was rocket ship B approaching in A’s frame? How fast was A
approaching in B’s frame ?
(c) (11 pts) You wonder if the crew of either ship could avoid the collision. How much
time elapsed from the moment when ship A was at point a to the collision from
the perspective of ship A ? How much time elapsed from the moment when ship
B was at point b to the collision from the perspective of ship B ?
3. Lorentz transformations: As you explore the planet X and see that the laws of
physics work there exactly in the same way as they do on Earth, you run into alien
graduate students who are surprised by your great expertise and ask you to help them
with the following problem. A reference frame K0
is moving with respect to a frame
K with velocity V along the x axis. The axes of K0 are parallel to the axes of K. A
clock at rest in K0 at point (x
0
0
, y0
0
, z0
0
) shows time t
0
0
at the moment when it passes by
a clock which is at rest in the frame K at point (x0, y0, z0) and shows time t0.
(a) (15 pts) Write down the Lorentz transformations that relate the space and time
coordinates between the two frames.
4. Boost in arbitrary direction: Later on, you run into a mean alien professor who
doubts that the Earth civilization possesses enough knowledge of special relativity to
make a trip from Earth to planet X on its own. He gives you the following problem as
a test. Given a four vector x
µ = (1, 3, 2, 4) m (with x
0 = ct) in a frame K, consider
how it is observed in a frame K0 which moves with velocity ~v0 = (0.1, 0.7, 0.3)c with
respect to K.
(a) (20 pts) Calculate the four vector x
0µ observed in the frame K0
. First, derive the
symbolic result for general x
µ = (x
0
, ~x) and ~v0 from the formula for a boost along
a coordinate axis. Then plug in the specific values above.
(b) (10 pts) Determine the rapidity of the Lorentz boost from K to K0
.
(c) (10 pts) Is the four vector time-like, space-like or light-like ?
