Notes on Relativity and Cosmology - Physics Department, UCSB

78 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
c) Mark any two timelike related events on your diagram and label them
both ‘T.’
3-4. You and your friend are again inertial observers and this time your relative
velocity is 1 2c. Also, suppose that both of your watches read t = 0 at the
instant that you pass each other. At the event where your worldlines cross,
a firecracker explodes (or a light bulb is turned on and off very quickly, or
something else happens to create a brief burst of light).
a) Draw a spacetime diagram in your reference frame showing your friend’s
worldline and the outgoing light from the explosion.
b) Let events A and B be the events on the left and right light rays
respectively that are one second (as determined by your own reference
frame) after the explosion. Label these two events on your diagram.
c) Let event C be the the event on the left light ray that is simultaneous
with B as determined by your friend. Mark this event on the diagram.
3-5. As in problem (3-4), you and your friend are inertial observers with relative
velocity 1 2c and your watches both “tick” t = 0 at the instant that you
pass each other. For this problem, it is important that you use a large
scale to draw the spacetime diagram so that t = 1 is far from the origin.
a) Draw the two worldlines on a spacetime diagram in your frame of reference,
and draw and label (i) the event where your watch “ticks” t = 1
and (ii) the event where your friend’s watch “ticks” t = 1. At what
time is this second event in your reference frame? Did you draw it in
the right place?
b) On the same diagram, sketch the line of events which are at t = 1 second
as determined by your system of reference. This is your t = 1 line of
simultaneity.
c) On the same diagram, sketch the line of events which are at t = 1 second
as determined by your friend’s system of reference. (Hint: At what time
in your system of reference does his watch tick t = 1?) This is your
friend’s t = 1 line of simultaneity.
d) Consider the event where your friend’s t = 1 line of simultaneity crosses
your worldline. What time does your friend assign to this event? What
time do you assign to this event?
e) Consider the event where your t = 1 line of simultaneity crosses your
friend’s worldline. What time do you assign to this event? What time
does your friend assign to this event?
3-6. Suppose that you are in an airplane and that you are watching another
airplane fly in the opposite direction. Your relative velocity is roughly
600m/s. Calculate the size of the time dilation effect you would observe if
you measured the ticking of a clock in the other airplane. If that clock ticks