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Day 5.2: Time Dilation
1) You have been placed in a rocket traveling at 99% c and cannot look out the window. List all the tests that
you could do to tell that you are traveling so fast.
There is nothing you can do. Galileo made this point over 300 years ago.
2) You are moving at ½ c toward a star. How fast does the light from the star travel toward you?
A) 0.5 c
B) 1.5 c
C) c
D) something else
An answer of 0.5 c makes sense but only if you are moving away from the star.
An answer of 1.5 c makes sense. However, this is not what light does.
An answer of c makes sense, because Maxwell’s equations predict this speed for electromagnetic
radiation and they don’t specify any reference frame. This is what convinced Einstein – not the M-M
experiment. There is simpler experimental evidence today. Neutral pions can be produced moving at
speeds over 0.99975 c. They decay by emitting two photons. These photons travel at c.
3) What are the two postulates of Special Relativity?
The two postulates are that all inertial frames are equivalent- equally valid (question #1) and that the
speed of light in a vacuum is invariant (question #2). Einstein took Galileo’s’ principle of relativity and
extended it to include light. He started with the assumption that light was invariant and then explored
what the implications of this were. All of the counter-intuitive predictions of special relativity come from
these two postulates.
4) Look at Al’s Relativistic Adventure. http://www.onestick.com/relativity/
Al reviews how velocities are relative and the two postulates. Stop once it starts on space contraction.
a) What would you notice if your time slowed down? Nothing.
b) What would you notice if space got squished? Nothing.
c) Al’s rocket is moving past his mom with a speed of 0.8 c. Al has a ‘light clock’ with a photon
bouncing back and forth between two mirrors, perpendicular to the velocity of the rocket. His mom
sees the light travel a longer distance. She will measure that the photon has a larger
A) speed
B) frequency
C) wavelength
D) time per tick
The speed of the light must be the same. The larger distance means there must be a larger time.
d) Suppose that Al has an ordinary clock next to his light clock. His mother will see it ticking
A) more slowly
B) faster
C) normally
D) it depends
This clock must be ticking the same as the light clock - more slowly. If they were different you would
have a way to tell which frame is actually moving. This means that it is time itself that is changed, not the
clock. Time flows more slowly in a frame that moves relative to you.
e) His mom has an identical light clock beside her. Al will see her clock ticking
A) more slowly
B) faster
C) normally
D) it depends
The laws of physics are identical in an inertial frame. There is no experiment that will distinguish between
the two. This means each frame should see the light clock in the other frame slowed down.
5) The invariant speed of light means that time slows down and time intervals are not the same. Use
Pythagoras’s theorem and the diagram below to derive the formula for the ratio between the two times
t/t’, where t’ is the time in the frame at rest with the clock. This ratio is called gamma, .
The F frame sees light travel a distance ct. The F’ frame sees light travel a distance ct’.
Using Pythagoras’s theorem we have
c2 t2 = v2 t2 + c2t’2
2
Solving for t , we have
t2 = c2t’2/(c2 – v2) = t’2/(1 – v2/c2)
The times differ by a factor
 = 1/(1 – v2/c2) ½
ct
ct'
vt
6) Gamma is a measure of how strong the relativistic effects are. Calculate gamma when v =
i) 0
ii) 3.00 x 107 m/s
iii) 0.995 c
iv) c
i) 1
ii) 1.005
iii) 10.0
iv) undefined
th
Notice how, even at 1/10 of the speed of light gamma is only 0.5%.
The last value suggests that you cannot go at the speed of light.
7) We don’t have rockets that go anywhere close to the speed of light. However, there are particles that do.
Muons are formed at the top of the atmosphere by cosmic rays and race toward the Earth at 0.995 c.
a) The atmosphere is 10 km thick. How long does it take the muons to pass through according to us?
t = 10, 000/(0.995 x 3.00 x 108 ) = 3.35 x 10-5 s.
b) Muons at rest have a half-life of 2.2 x 10-6 s. How many half-lives pass during the descent?
3.35 x 10-5 s/2.2 x 10-6 s = 15
c) What fraction of the muons should reach the Earth, ignoring relativity?
(½)15 ~ one millionth
d) Relativity says that we should see the muon’s time running ten times slower than ours. What fraction of
the muons will reach the Earth? The muons’ time is passing 10 times slower in our frame, so the half-life
is ten times longer and so only 1.5 half lives pass. In between ½ and ¼ will make it. This is what happens.
e) How thick is the atmosphere in the muons frame?
A) 10 km
B) 100 km
C) 1 km
D) 1/10 km
We agree on the relative speed. If the time is 1/10th, the distance must be as well. v = x/t = x/t
8) The GPS satellites are moving relative to us and the relativistic effects are just one part in a billion. Why
do these tiny effects matter? Watch The GPS and Relativity
http://www.youtube.com/watch?v=zQdIjwoi-u4
There are two reasons. The GPS multiplies time by a very large number - the speed of light. A tiny error
becomes a medium one. The error accumulates. There are 24 x 3600 = 86,400 seconds in a day. The
medium sized error is multiplied by that and results in an error of ~10 km a day. (10-9 x 105 x 105 ~ 10 km)
9) The GPS satellites move at a speed of 3874 m/s.
a) Calculate .
Most calculators will give an answer of 1 because they can’t handle that many digits.
b) When v is much less than c, gamma can be approximated by 1 + v2/2c2. Calculate .
This is the first two terms of the binomial expansion which they may have dealt with in math. They
still won’t be able to calculate the full expression with their calculators, but with a little nudging they
can realize that they can calculate the difference from 1, namely v2/2c2. This is 8.348 x 10-11.
c) How much time passes on the satellite when one second passes on Earth?
The satellite experiences one second minus the difference above.
d) How much less time passes on the satellite when one day passes on Earth?
8.348 x 10-11 x 86,400 s = 7.21 x 10-6 s. It still looks negligible.
e) If this time difference wasn’t corrected, how far off would the calculated distance be?
7.21 x 10-6 x 2.9979 x 108 m/s = 2.16 km. Note, this is smaller than the 11 km mentioned in the video
because it is just the effect from special relativity and does not include the more significant effect from
general relativity.
10) Watch GPS and Nuclear Detection http://www.youtube.com/watch?v=ky4RgRvVDoA Why did the
engineers and physicists disagree?
The engineers didn’t believe that general relativity would have an effect. At this point special relativity
was extremely well tested – but general relativity wasn’t. In particular atomic clocks were flown on
airplanes and the effect measured. Since then, general relativity has been tested many times and confirmed
to extreme precision.
Textbook: 11.2 (p. 569-572)
p. 573 # 1-4
Extra:
A GPS satellite sends a signal with a time of 9:00:27.723 119 038 (i.e. 9 am and 27. 723 119 038 seconds.)
The signal is received at a time of 9:00:27.790 502 104 according to the receiver.
a) How long did it take the signal to travel from the satellite to the receiver?
Subtracting the two times gives 0.067 383 066
b) How far is the receiver from the satellite? c = 2.99792458 x 108 m/s.
Multiplying the time by c gives 20 200 935 m. Note that eight digits are significant. This precision is
physically important because you want to know where you are within metres not ten’s of metres. The
satellites are 20 200 000 m above the Earth’s surface, and this distance is only slightly larger. This
means it is almost directly overhead.
The GPS calculates the distance from a satellite by multiplying the time the signal takes to travel from the
satellite to the receiver. Let the time interval be 0.068503387 s.
a) Calculate the distance using just the first three digits for time and speed. 2.055 x 107 m.
b) Calculate the distance using all of the digits.2.0536798.8 x 107 m.
c) What is the difference between the answers? Is it significant? Explain.
13,201.2 m. You will be wrong by over 13 km. That is significant because you generally want to
know where you are within a few meters.