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Einstein’s theory of special relativity makes some very bizarre and counter-intuitive predictions.
Anything that violates common sense like this must be strongly supported by logic and evidence before we accept it.
A rocket ship is heading toward you at ½ c. How fast does the light from its headlamps travel toward you?
a) cb) 1.5 cc) 0.5 cd) something else
The speed of light is c in all frames.
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 are no tests that you can do. All inertial frames are equivalent.
1) The speed of light is c in all frames.
2) All inertial frames are equivalent.
All the strange predictions of Special Relativity come from these two postulates and can be developed by examining the geometry of spacetime diagrams.
Spacetime diagrams can help you to visualize relativity. They are similar to scale diagrams and
freebody diagrams.
A spacetime diagram has time vertical.The axes are calibrated in years and light-
years.
t
x
Which line represents an object at rest?
t
xA B C D
Which line represents light?
t
xA B C D
Which line represents an impossible motion?
t
xA B C D
t
x
We want a line for an object moving at 3/5 c.
This will be the t’ axis for the ‘moving’ frame. Which one is correct?
x
t’t
x
x
t’
t
x
t
x
t’
x
A B C
A speed of 3/5 c has a slope of 5/3.
x
t’t
Which is the correct x’ axis?
x
t’t
x
x
x’
t
x
t
xx’
t
x
A B C
t’ t’
x’
Hint: What is true about light in all frames?
x
t’t
x
x
x’
t
x
t
xx’
t
x
A B C
t’ t’
x’
x
t’t
The two axes must be symmetric about the path of light because light
must have the same speed in all frames
x’
x’
t’
The Cosmic Speed Limit
Suppose you launch a pod at ½ c from a rocket traveling at ½ c relative to the Earth.
How fast will it go?
x
t’t
The pod travels at ½ c relative to the rocket. Which line is the pod’s?
x’
A
B
C
D
x
t’t
x’
The pod must cover one unit of space in two units of time.
A
B
C
D
x
t’t
How fast is it moving relative to the Earth?
x’
x
t’t
It travels at 4/5 c
x’
How fast is it moving relative to the Earth?
Reality Check #1: The cyclotron at Triumf can form pions moving at 0.96 c which decay by emitting muons and neutrinos.
Many of these emitted particles go faster than 0.96 c, but none go faster than light.
Reality Check #2: At CERN, neutral pions were accelerated to 0.99975 c. When these pions decayed, they emitted light.
All the light emitted by the pions travelled
at c.
Time Slows Down
x
t’t
x’
This line marks simultaneous later times.
The x axis marks all the places where t = 0.
x
t’t
x’
The x’ axis marks all the points where t’ = 0.
This line marks equal or simultaneous times t’, in the other frame, F’.
x
t’t
x’
Consider the point where the ‘stationary’ frame’s time hits the t’ axis. We can’t assume that t = t’. Let’s assume for now that t = t’.
x
t’
x’
Consider the point where the ‘moving’ frame’s time hits the t axis. We’ll label the two times as t1 and t2. All frames are equivalent, so t’ = t1.
t2
t1
x
t’
x’
How are t1 and t2 related?
t2
t1
t2 = t’
t2 = t1
t’ = t1
t2 = t1 What does that tell you about ?What does that tell you about
time?
x
t’t
x’
is greater than one.
t”
Each observer sees the other ‘moving’ observer with their time ticking more slowly.
The time t’ is shorter even though on the diagram it ‘looks’ longer.
x
t’t
x’
If you were on a rocket ship and sent a signal to Earth every time your heart beat, doctors on Earth would say that your pulse
was slow.
If the Earth doctor sent you a signal every time that her heart beat, you would say that her
pulse was
a) faster b) normal c) slower
Hint: All inertial frames are equivalent.
a) faster b) normal c) slower
Reality Check#3: Muons at CERN were accelerated to high speeds and lived 20 times longer than normal.
The Twin Paradox
Brenda goes off 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.
x
t
The first part looks like this. During this half, Ali sees Brenda’s clock run slowly and Brenda sees Ali’s clock
run slowly.
t’
x’
Ali Brenda
x
t
The second part looks like this. During the second half they each see the other person’s time run slower.
t’
x’
Ali Brenda
x
t
If Brenda always sees Ali’s time ticking more slowly and Ali always sees Brenda’s time
ticking more slowly. Why don’t they age by the same amount?
t’
x’
Brenda turns around. Her frame is not inertial and therefore not equivalent.
x
t
If Brenda always sees Ali’s time ticking more slowly and Ali always sees Brenda’s time
ticking more slowly. When does Ali pick up the extra years?
t’
x’
x
t
During the turnaround.
t’
x’
x
t
This is Brenda’s time at the start of the turn…
t’
x’
x
t
... in the middle …
t’
x’
x
t
… and at the end.
t’
x’
x
t
Almost no time passed for Brenda during the turnaround, while a lot passed for Ali.
t’
x’
Reality Check #4: GPS satellites are constantly turning around.
This makes their time run slower than ours
by 8.99 x 10-11s every second.
If the GPS failed to adjust for the different times, then they would be out by 2.5 km
after one day!
The Doppler ShiftBrenda goes on another long voyage.
x
t’t
Ali sends Brenda a message every year. How many messages does she receive on the way
out, back?
x
t’t
The frequency is greater on the way back when the ship is moving toward the signals. This is similar to the Doppler shift for sound.
x
t’t
Brenda also sends Ali messages every year. How many will he receive compared
to Brenda?
A) the same number B) fewer
C) more
x
She sends fewer because fewer years have passed for her, so Ali receives
fewer.
x
t’t
Reality Check #5: Physicists know that the
universe is expanding
because the frequency of
light from galaxies is
Doppler shifted.
This animation shows how things would look to an observer moving to the right. It shows the colour changes due to the relativistic Doppler shift and the distortion of space.
