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Day 5.4:
Twin Paradox
1) Brenda Travels at 4/5 c to a distant star and then returns. Her twin brother Ali stays on Earth. When she
gets back she is no longer the same age as Ali! Sketch the trip in the spacetime diagram to the right.
a) Sketch the trip in the spacetime diagram to the right.
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Brenda’s path is a diagonal line with a slope of 5/4 and then another
with a slope of -5/4.
b) Doesn’t relativity say their ages should be the same?
A) Yes, all frames are equivalent.
B) Yes, all inertial frames are equivalent.
C) No, only inertial frames are equivalent.
Relativity says that all inertial frames are equivalent. Brenda accelerates.
c) You are in a closed room. What tests could you do to determine if you were accelerating?
There are lots of tests for acceleration – none for velocity. You feel accelerations. You feel normal on a
subway train until it slows down, speeds up or turns. A pendulum would slant away from vertical if you
are accelerating horizontally. A spring pendulum would be pulled down if you were accelerating
vertically. Most phones have accelerometers in them. If the acceleration is small, like the rotation of the
Earth, then you need more sensitive tests like a Foucault pendulum. In the last unit we will see that tests
for acceleration and for gravity have a lot in common.
d) If Ali ages by 10 years, by how much does Brenda age?
= 5/3, 10*3/5 = 6 years
d) When does Ali pick up the extra time?
Halfway through on the way out, Ali says it is 5 years Brenda says it is 3.
When Brenda says it is 3, Ali says it is 3*3/5 = 1.8 years.
They both claim that the other clock is slower on the way out and on the way back.
During the turnaround, Brenda switches frames and has the three different time axes show in green. Her
time doesn’t change but Ali picks up and extra 10 – 2x 1.8 = 6.4 years.
e) What would happen if Brenda stopped at the star? She would be
A) 6 years younger B) 4 years younger C) 2 years younger D) the same age
When she stops, she accelerates and changes into Ali’s frame except that her clock still reads 3 years but
his reads 5 years. During the deceleration he gained 3.2 years. At this point, she could stay there and they
could send each other videos or journals of what they did. Brenda would have stories about the three years
she spent and Ali would have stories about his five years of adventures.
f) The GPS satellites are moving relative to us at 3874 m/s. This speed that
A) we think their clocks are slow and the GPS thinks our clocks are slow.
B) we think their clocks are slow and the GPS thinks our clocks are fast.
C) nothing because the speed too small to matter
D) nothing because gravity has a stronger effect.
The effects are small but they matter if you want accuracy to the nearest metre.
Gravity does have a stronger effect and in the opposite direction. However, both effects matter.
The GPS satellite is accelerating, so it is acting like Brenda. Less time passes on the GPS.
2) Brenda travels at ½ c to and from a star that is 2.5 light-years away.
a) Sketch the trip on a spacetime diagram. Slopes of 2 and -2.
b) How long must Ali wait on Earth for her to get back? t = d/v =5/½ = 10 years
c) How much time passes for Brenda? Calculate gamma.10 * 0.87 = 8.7 years
d) Ali sends a message every year. How many messages does she get? Draw them.
He will send nine. The tenth would be sent just as she arrives.
e) How many does she get on the trip out? back?
She only gets 3 on the way out and 6 on the way back.
f) She sends a message to Ali every year. How many does Ali get?
She send one after 1, 2, 3, 4, 5, 6, 7 and 8 years have passed.
g) Ali and Brenda agree on their relative speed but not on the time or distance for the trip. What is the
proper distance and the proper time? Ali measures a larger time and distance, but they are larger by the same
amount, so the speed is the same. Brenda’s time is the ‘proper time’ because the two events – saying
goodbye at (0, 0) and getting to the star can be measured by a clock at rest in her frame. Her measurement
just involves time and not space. You can see that on the diagram because the two points are on her t’ axis. In
contrast, the two points for Ali involve changing position as well as time. The proper time is always the
shortest time that any frame measures. In this case 4.35 light-years. Ali’s measurement of the distance to the
star is the ‘proper’ distance because it is purely spatial. It can be measured with a ruler at rest in his frame.
This can be seen by a line drawn from the Earth to the star at any time in his frame. The proper distance is
always the largest distance measured by any frame.
3) Periodic signals carried by light (like the messages in the last question) show a Doppler shift if the source
is moving relative to the receiver.
a) The formula for the relativistic Doppler shift is fr/fs = sqrt((1-v/c)/1+v/c). This formula is different from
the Doppler shift for sound. Why is it different? Hint: How are the speeds different?
The difference is caused by the fact that the speed of light is the same in all frames, whereas the speed of a
sound wave is relative to its medium.
b) Why do astronomers think that
there was a Big Bang? Use the data
to calculate when it occurred.
The astronomers look at the spectra of
the star light. In this light there are
dark bands characteristic of the
various elements in the stars’
atmosphere. They are all Doppler
shifted to lower frequencies, also
known as red-shifted. The farther
away, the faster they are moving. For
Hydra, t = d/v = 19 billion years.
This assumes that the speed has been
constant. There are reasons to believe
that this is not quite the case and the
present value is 13.8 billion years.
4) Do Brenda and Ali’s Train Adventures
Part 1: The Lightning Strikes
a) Ali and Brenda are in the middle of their trains. The trains are traveling on adjacent tracks. Ali’s train
is 30 m long. How long is that in ns? 100 ns.
b) Mark the x and t axes with a scale of 5 squares representing 50 ns. This is Ali’s frame. At t = 0 ns Ali
is in the middle of his train at x = 0. Draw Ali’s train by highlighting a line from x = -50 ns to x = 50 ns.
c) Brenda is moving at 3/5 c relative to Ali. Calibrate Brenda’s frame appropriately. At t’ = 0 ns she is
in the middle of her train opposite Ali at x’ = 0. Draw her train by highlighting a line in a different color.
d) At t = 0, lightning bolts hit the front and rear of Ali’s train, leaving burn marks. They also burn the
front and rear of Brenda’s train. Draw the light from the bolt moving in the positive and negative
directions until they meet at Ali. When does Ali see the two flashes? 50 ns
e) Follow the light rays until they meet up with Brenda on the t’ axis. When do they reach Brenda?
Label this. ~ 25 ns for the right, ~ 100 ns for the left.
f) When does Brenda say the lightning strikes occur? Both occurred at t = 0 ns for Ali, but Brenda says
that they were not simultaneous. She says that they occurred at t’ = +/- 40 ns
g) How long must her train be? It must be larger than Ali’s to get hit by the same bolts. Looks like 2 x
66 ns = 132 ns long = 39.6 m. Calculated, it is 30 m x 5/4 = 37.5 ns. This is just like the Space Fight and
the Ladder in the barn Problems.
Part 2: The Fireworks
a) On another occasion, they are again in the middle of their trains. At t = t’ = 0 fireworks are let off at
the middle of the trains. Draw firework rays leaving the center of the two trains and spreading out. Draw
Ali’s train when the light hits the ends of his train. Label the two events A1 and A2. At what time do these
events occur? 50 ns according to Ali and 25 ns and 100 ns according to Brenda
b) Locate where the light reaches the ends of Brenda’s train. Draw the train where this occurs. Label the
two events B1 and B2. How much time did this take according to Brenda? How much time did this take
according to Brenda? 50 ns according to Brenda and 25 ns and 100 ns according to Ali
c) Draw the positions of Ali’s train and Brenda’s trains just as the ends pull away from each other. Mark
the position of the ends as E3. How long did it take for the engines to meet until the ends separated? 2 x
170 ns = 340 ns for Ali and Brenda
Textbook:
p. 579 # 1 - 11