Notes on Relativity and Cosmology - Physics Department, UCSB

5.3. HOMEWORK PROBLEMS 135
that the speed of the back at A B is greater than that of the front at C F .
B
F
t = t
0
Z
B
B
D
F
t = - t
0
A
B
Front of Rocket
C
F
A
F
Back of
Rocket
Thus, relative to the inertial frame in which the diagram is drawn, the back of
the rocket experiences more time dilation in the interval (−t 0 , t 0 ) and it’s clock
runs more slowly. Thus, the proper time along the back’s worldline between
events A B and B B is less than the proper time along the front’s worldline
between events C F and D F . ⋆⋆ We now combine this with the fact that the
proper time along the front’s worldline between A F and B F is even greater than
that between C F and D F . Thus, we see that the front clock records much more
proper time between A F and B F than does the back clock between A B and B B .
5.3 Homework Problems
5-1. Suppose that you are in a (small) rocket and that you make the following
trip: You start in a rocket in our solar system (at rest with respect to
the Sun). You then point your rocket toward the center of the galaxy and
accelerate uniformly for ten years of your (i.e., proper) time with a proper
acceleration of 1g = 10m/s 2 . Then, you decelerate uniformly for ten years
(of proper time) at 1g, so that you are again at rest relative to the sun:
(a) Show that 1g is very close to 1light − year/year 2 . Use this value for
g in the problems below. [Note: This part is just a unit-conversion
problem.]
(b) Draw a spacetime diagram showing your worldline (and that of the
Sun) in the Sun’s reference frame.