Announcements
• Reading for Wednesday: Review Chapter 1
(up to 1.13).
• HW2 due Wednesday noon.
• HW3 posted, Solutions to HW1 on D2L
Today’s class
• More about time
• Twins paradox
• Length contraction
Last time
• Discussed consequences of the speed of
light being the same in all intertial
reference frames
• Simultaneity and order of events may
depend on you frame of reference
• Contradiction between Einstein and
common sense (breakdown of
causality)???
• Time dilation
‘Evidence’ for time Dilation
Mirror
h
Lucy
Lucy measures the time interval: Δt
(Not a big surprise!)
v
= 2h/c
Evidence for ‘Time Dilation’
Let’s look at it from the tracks:
algebra
c2(Δt’/2)2=h2+v2(Δt’/2)2
Mirror
c2(Δt’/2)2-v2(Δt’/2)2=h2
v
h
v·
Note: This experiment requires two observers.
Time Dilation
Time Dilation: Two observers (moving relative to each
other) can measure different durations between two
events.
Lucy
v
h
Ethel
Lucy measures:
Δt’ = 2h/c
Here: Δt’ is the proper time
Ethel and Ricky:
Δt = γ 2h/c , with
Δt’ =(2h)2/(c2-v2)=
Δt’/
2
(2h) 2
(2h ) 2
=
c2
€
h
(Δt’)2(c2-v2)=(2h)2
= 2 γ2
2&
#
c
v
c 2 %1 − 2 (
Ricky
Ethel
$ c '
Ethel and Ricky measure the time interval:
2h
1
But Lucy
Δt' = γ , γ ≡
measured
2
€
c
v
Δt = 2h/c !! So Δt’ =γΔt
1−
Ricky
Ethel
c2(Δt’)2-v2(Δt’)2=(2h)2
h
Ricky
Which statement is true?
a) Time dilation means that clocks in the moving
frame run slow, but time itself goes at the same
“speed” in the moving frame and in the
stationary frame relative to the observer. (i.e.
she will see the same tree grow at the same
rate whether or not the tree is on the train or on
the ground)
b) Time dilation means that observer on the
ground sees that time itself is slower for things
on the train. (the tree on the train will grow
slower)
Which statement is true?
a) Time dilation means that clocks in the moving
frame run slow, but time itself goes at the same
“speed” in the moving frame and in the
stationary frame relative to the observer. (i.e.
she will see the same tree grow at the same
rate whether or not the tree is on the train or on
the ground)
b) Time dilation means that observer on the
ground sees that time itself is slower for things
on the train. (the tree on the train will grow
slower)
Why?
a) Because there is no other way to explain
Michelson-Morley experiment
b) Because our “speed of time” depends on
how fast we are going
c) Because time dilation is a postulate of the
theory of relativity
d) It is convenient to define time this way
Why?
a) Because there is no other way to explain
Michelson-Morley experiment
b) Because our “speed of time” depends on
how fast we are going
c) Because time dilation is a postulate of the
theory of relativity
d) It is convenient to define time this way
Let’s have some fun!
The Twin ‘Paradox’
A little journey
A little journey
Vicki
Carol
v = 0.8c
γ = 5/3
Vicki
Carol
v = 0.8c
γ = 5/3
8 light-years (measured by Vicki)
8 light-years (measured by Vicki)
Carol and Vicki are identical twins. While Vicki stays on
Earth, Carol departs for the star Sirius, 8 light-years
away, traveling at a speed v = 0.8 c (Note γ = 5/3).
According to Vicki, how long does the trip take? (Assume
Earth and Sirius are not moving relative to each other.)
a) 6 years
b) 8 years
c) 10 years
Vicki knows that this trip takes Carol 10 years.
How long is that trip according to Carol?
d) 16.67 years
a) 6 years
A little journey
b) 8 years
c) 10 years
d) 16.67 years
A little journey
v
Vicki
Carol
v
Why? Because Carol’s clock is present at both events:
Event 1: Carol leaves Earth.
Event 2: Carol arrives at Sirius.
Time between these two
events is the proper time
Vicki
Carol
Upon arriving at Sirius, Carol immediately turns around and
heads home at v = 0.8c. When she returns, she has aged
12 years, while Vicki has aged 20 years!
So if Δt is Carol’s proper time between these events, and
Δt’ is the time in the Earth-Sirius system, we have:
Δt =
Remember: Proper time ≤ Δt’
Δt' 10years
=
= 6years
in any other frame!
γ
5 /3
Proper time
€
Vicki
Carol
But wait....
From Carol’s point of view, it was Vicki who was moving.
So we should expect:
In Vicki’s frame (the Earth), Carol ages 12 years while Vicki
ages 20.
In Carol’s frame (the spacecraft), Vicki ages 12 years while
Carol ages 20.
And they can’t both be right!
Resolution: Carol has to turn around
(not an inertial frame!)