Notes on Relativity and Cosmology - Physics Department, UCSB

3.8. HOMEWORK PROBLEMS 77
that moves with the train at all times, even slowing down and stopping as the
train slows down and stops. However, we do not know enough to do this yet
since such a frame is not inertial. We will get to accelerating reference frames
in chapter 5.
3.8 Homework Problems
3-1. Use your knowledge of geometry (and/or trigonometry) to show that the
angle α between the worldline of an inertial observer and the lightcone
(drawn in an inertial frame using units in which light rays travel at 45
degrees) is the same as the angle β between that observer’s line of simultaneity
and the lightcone. (Hint: Many people find it easier to use
trigonometry to solve this problem than to use geometry.)
x = 0
f
x =0 us
t = const
f
green line of
simultaneity
C
Us
α
B
β
t
us
t =0
us
=+1sec
3-2. Suppose that you and your friend are inertial observers (that is, that both
reference frames are inertial). Suppose that two events, A and B, are
simultaneous in your own reference frame. Draw two spacetime diagrams
in your reference frame. Include your friend’s worldline in both. For the
first, arrange the relative velocity of you and your friend so that event A
occurs before event B in your friend’s reference frame. For the second,
arrange it so that event B occurs first in your friend’s frame. In both
cases, include one of your friend’s lines of simultaneity on the diagram.
3-3. Draw a spacetime diagram in an inertial reference frame.
a) Mark any two spacelike related events on your diagram and label them
both ‘S.’
b) Mark any two lightlike related events on your diagram and label them
both ‘L.’