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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 are
willing to accept it.
A rocket ship is heading toward you at ½ c.
How fast does the light from its headlamps
travel toward you?
a) c
b) 1.5 c
c) 0.5 c
d) 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
A
B
C
D
x
Which line represents light?
t
A
B
C
D
x
Which line represents an impossible motion?
t
A
B
C
D
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?
A
t
B
C
t
t’
t
t’
x
t’
x
x
A speed of 3/5 c has a slope of 5/3.
t
t’
x
Which is the correct x’ axis?
A
t
B
t
t’
C
t
t’
t
t’
x’
x
x’
x’
x
x
Hint: What is true about light in all frames?
A
t
B
t
t’
C
t
t’
t
t’
x’
x
x’
x’
x
x
The two axes must be symmetric about
the path of light because light must have
the same speed in all frames
t
t’
Dt’
x’
Dx’
x
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?
The pod travels at ½ c relative to the rocket.
Which line is the pod’s?
A
t
t’
B
C
D
x’
x
The pod must cover one unit of
space in two units of time.
A
t
t’
B
C
D
x’
x
How fast is it moving relative to the Earth?
t
t’
x’
x
How fast is it moving relative to the Earth?
It travels at 4/5 c
t
t’
x’
x
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
This line
marks
simultaneous
later times.
t’
t
x’
x
The x axis
marks all
the places
where t = 0.
This line marks equal or simultaneous times t’,
in the other frame, F’.
t
t’
x’
The x’ axis
marks all the
points where
x t’ = 0.
Suppose lightning strikes twice as shown.
Did the strikes happen at the same time?
t
t’
x’
x
Simultaneity is relative.
Consider the point where the two gridlines cross.
Does t = t’?
Hint: Does t’ = t”?
They can’t be
the same.
Let’s say that
t = g t’
t
t”
t’
x’
x
We can find the formula for g by considering a
‘light’ clock, which sends light up and down
between mirrors once every time t.
Suppose that this clock is on a rocket ship
moving at a speed v to the right.
We observe that the light travels farther in our
frame. To keep c constant, it must also take more
time, c = Dx/Dt. Therefore, our time interval is
larger than the rocket’s. Their clock runs slower.
How much more slowly can be found by
applying Pythagoras’ theorem to the diagram
below. Write the equation and solve for t.
ct
ct’
vt
c2 t2 = v2 t2 + c2 t’2
t = 1/ 1 – v2/c2
(c2 - v2 ) t2 = c2 t’2
t’
g= 1/ 1 – v2/c2
This formula does not depend on the fact that we
used a ‘light’ clock. Anything that ‘ticks’ will do.
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
What is g, if the relative speed
of the two frames is 3/5 c?
g= 1/ 1 – v2/c2
t
t”
t’
x’
x
g= 5/4
What is t’, if t = 10?
t
10
t’
8
x’
x
What is t” ?
t” = 4/5 * 8
= 6.4
10
t”
8
x’
x
t’ is shorter but it ‘looks’ longer.
t
10
t’
8
x’
x
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.
The first part looks like this. During this
half, Ali sees Brenda’s clock run slowly
and Brenda sees Ali’s clock run slowly.
Ali
t
Brenda
t’
x’
x
The second part looks like this. During the
second half they each see the other
person’s time run slower.
Ali
t
Brenda
t’
x’
x
Suppose Ali ages 10 years during the trip.
How much did Brenda age?
What is v?
v is 4/5 c.
t
What is g?
10
g is 5/3.
What is
the time
in the
middle of
the trip?
5
t’
x’
3
Brenda only
aged 6 years
during the trip.
x
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
t’
x’
Brenda turns around.
Her frame is not
inertial andx therefore
not equivalent.
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 4 years?
t
t’
x’
x
During the turnaround.
t
t’
x’
x
This is Brenda’s time at the start of the turn…
t
t’
x’
x
... in the middle …
t
t’
x’
x
… and at the end.
t
t’
x’
x
Almost no time passed for Brenda during the
turnaround, while a lot passed for Ali.
t
t’
x’
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 Shift
Brenda goes on another long voyage.
t
t’
x
Ali sends Brenda a message every year. How many
messages does she receive on the way out, back?
t
t’
x
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.
t
t’
x
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 only fewer years
have passed for her, so Ali receives fewer.
t
t’
x
Reality Check #5:
Physicists know
that the universe
is expanding
because the
frequency of light
from galaxies is
Doppler shifted.
The galaxies are
moving so fast, that
astronomers must
use the formula for
the relativistic
Doppler shift, fo =
fe
(1-v/c)/(1+v/c)
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.