Special Theory of Relativity

1
Special Theory of Relativity
Albert Einstein

• Micro-world Marco-world
• Lect 11

2
watching a light flash go by
v
c
2kk
The man on earth sees c ? ( agrees with
Maxwell)
3
watching a light flash go by
v
c
If the man on the rocket sees c-v, he disagrees
with Maxwell
4
watching a light flash go by
It doesnt works for this guy,
v
c
If Maxwells theory works for this guy,
or vice versa
5
Do Maxwells Eqns only work in one reference
frame?
If so, this would be the rest frame of the
luminiferous Aether.
6
If so, the speed of light should change
throughout the year
upstream, light moves slower
downstream, light moves faster
Aether wind
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Michelson-Morley
No aether wind detected 1907 Nobel Prize
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Einsteins hypotheses
1. The laws of nature are equally valid in every
inertial reference frame.
Including Maxwells eqns
2. The speed of light in empty space is same
for all inertial observers, regard- less of
their velocity or the velocity of the source
of light.
9
All observers see light flashes go by them with
the same speed
v
No matter how fast the guy on the rocket is
moving!!
c
Both guys see the light flash travel with
velocity c
10
Even when the light flash is traveling in an
opposite direction
v
c
Both guys see the light flash travel past with
velocity c
11
Gunfight viewed by observer at rest
He sees both shots fired simultaneously
Bang!
Bang!
12
Viewed by a moving observer
13
Viewed by a moving observer
He sees cowboy shoot 1st cowgirl shoot later
Bang!
Bang!
14
Viewed by an observer in theopposite direction
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Viewed by a moving observer
He sees cowgirl shoot 1st cowboy shoot later
Bang!
Bang!
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Time depends of state of motion of the observer!!

• Events that occur simultaneously according to one
  observer can occur at different times for other
  observers

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Light clock
18
Seen from the ground
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Events
y
(x2,t2)
(x1,t1)
x
x
x1
x
x2
t
20
Same events, different observers
Prior to Einstein, everyone agreed the distance
between events depends upon the observer, but
not the time.
y
y
y
(x2,t2)
(x1,t1)
x
x
(x1,t1)
(x2,t2)
t
t
x1
x1
x2
dist
x
x
x1
x
x2
t
dist
21
Time is the 4th dimension
Einstein discovered that there is no absolute
time, it too depends upon the state of motion of
the observer
Einstein Space-Time
Newton Space Time
completely different concepts
2 different aspects of the same thing
22
How are the times seen by 2 different observers
related?
We can figure this out with simple HS-level math
( a little effort)
23
Catch ball on a rocket ship
Event 2 girl catches the ball
wt
v 4m/s
w4m
t1s
Event 1 boy throws the ball
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Seen from earth
V03m/s
V03m/s
Location of the 2 events is different Elapsed
time is the same The ball appears to travel
faster
d?(3m)2(4m)2 5m
w4m
v0t3m
dt
v 5m/s
t1s
25
Flash a light on a rocket ship
Event 2 light flash reaches the girl
wt0
c
w
t0
Event 1 boy flashes the light
26
Seen from earth
V
V
Speed has to Be the same Dist is longer Time
must be longer
d?(vt)2w2
w
vt
dt
c ?
(vt)2w2 t
t?
27
How is t related to t0?
t time on Earth clock
t0 time on moving clock
wt0
c ?
(vt)2w2 t
c
ct ?(vt)2w2
ct0 w
(ct)2 (vt)2w2
(ct)2 (vt)2(ct0)2
? (ct)2-(vt)2 (ct0)2
? (c2-v2)t2 c2t02
1 1 v2/c2
c2 c2 v2
? t2 t02
? t2 t02
1 ?1 v2/c2
? t t0
? t g t0
this is called g
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Properties of g
1 ?1 v2/c2
Suppose v 0.01c (i.e. 1 of c)
1 ?1 (0.01c)2/c2
1 ?1 (0.01)2c2/c2
g

1 ?1 0.0001
1 ?1 (0.01)2
1 ?0.9999


g
g 1.00005
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Properties of g (contd)
1 ?1 v2/c2
Suppose v 0.1c (i.e. 10 of c)
1 ?1 (0.1c)2/c2
1 ?1 (0.1)2c2/c2
g

1 ?1 0.01
1 ?1 (0.1)2
1 ?0.99


g
g 1.005
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Lets make a chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005







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Other values of g
1 ?1 v2/c2
Suppose v 0.5c (i.e. 50 of c)
1 ?1 (0.5c)2/c2
1 ?1 (0.5)2c2/c2
g

1 ?1 (0.25)
1 ?1 (0.5)2
1 ?0.75


g
g 1.15
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Enter into chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15






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Other values of g
1 ?1 v2/c2
Suppose v 0.6c (i.e. 60 of c)
1 ?1 (0.6c)2/c2
1 ?1 (0.6)2c2/c2
g

1 ?1 (0.36)
1 ?1 (0.6)2
1 ?0.64


g
g 1.25
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Back to the chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25





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Other values of g
1 ?1 v2/c2
Suppose v 0.8c (i.e. 80 of c)
1 ?1 (0.8c)2/c2
1 ?1 (0.8)2c2/c2
g

1 ?1 (0.64)
1 ?1 (0.8)2
1 ?0.36


g
g 1.67
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Enter into the chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25
0.8c 1.67




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Other values of g
1 ?1 v2/c2
Suppose v 0.9c (i.e.90 of c)
1 ?1 (0.9c)2/c2
1 ?1 (0.9)2c2/c2
g

1 ?1 0.81
1 ?1 (0.9)2
1 ?0.19


g
g 2.29
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update chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25
0.8c 1.67
0.9c 2.29



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Other values of g
1 ?1 v2/c2
Suppose v 0.99c (i.e.99 of c)
1 ?1 (0.99c)2/c2
1 ?1 (0.99)2c2/c2
g

1 ?1 0.98
1 ?1 (0.99)2
1 ?0.02


g
g 7.07
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Enter into chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25
0.8c 1.67
0.9c 2.29
0.99c 7.07


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Other values of g
1 ?1 v2/c2
Suppose v c
1 ?1 (c)2/c2
1 ?1 c2/c2
g

1 ?0
1 ?1 12
1 0


g
g ?
Infinity!!!
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update chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25
0.8c 1.67
0.9c 2.29
0.99c 7.07
1.00c ?

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Other values of g
1 ?1 v2/c2
Suppose v 1.1c
1 ?1 (1.1c)2/c2
1 ?1 (1.1)2c2/c2
g

1 ?1-1.21
1 ?1 (1.1)2
1 ?-0.21


g
g ???
Imaginary number!!!
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Complete the chart
v g 1/?(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15
0.6c 1.25
0.8c 1.67
0.9c 2.29
0.99c 7.07
1.00c ?
Larger than c Imaginary number
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Plot results
?
1 ?1 v2/c2
g
Never-never land
x
x
x
x
x
vc
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Moving clocks run slower
v
t0
1 ?1 v2/c2
t t0
t
t g t0
g gt1 ? t gt t0
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Length contraction
v
L0
timet L0 vt
Shorter!
man on rocket
Time t0 t/g Length vt0
vt/g
L0/g
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Moving objects appear shorter
Length measured when object is at rest
L L0/g
g gt1 ? L lt L0
V0.1c
V0.86c
V0.99c
V0.9999c
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Length contraction
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mass
change in v time
Fm0a
m0
t0
a
timet0
m0
Ft0 m0
change in v
Ft0 change in v
m0
mass increases!!
g Ft0 change in v
Ft change in v
g m0
m

m g m0
tgt0
by a factor g
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Relativistic mass increase
m0 mass of an object when it is at rest
? rest mass
mass of a moving object increases
g
as v?c, m??
m g m0
as an object moves faster, it gets harder
harder to accelerate
by the g factor
vc
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summary

• Moving clocks run slow
• Moving objects appear shorter
• Moving objects mass increases

By a factor of g
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Plot results
?
1 ?1 v2/c2
g
Never-never land
x
x
x
x
x
vc
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Twin paradox
a-centauri
4.3 light years
Twin brother sister
She will travel to a-centauri (a near- by star on
a special rocket ship v 0.9c
He will stay home study Phys 100
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Light year
distance light travels in 1 year
dist v x time
c yr
1cyr 3x108m/s x 3.2x107 s
9.6 x 1015 m
We will just use cyr units not worry about
meters
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Time on the boys clock
d04.3 cyr
v0.9c
v0.9c
According to the boy his clock on Earth
d0 v
4.3 cyr 0.9c
tout

4.8 yrs
d0 v
4.3 cyr 0.9c
tback

4.8 yrs
ttotal touttback
9.6yrs
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What does the boy see on her clock?
d4.3 cyr
v0.9c
v0.9c
According to the boy her clock runs slower
tout g
4.8 yrs 2.3
tout

2.1 yrs
tback g
4.8 yr 2.3
tback

2.1 yrs
ttotal touttback
4.2yrs
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So, according to the boy
d4.3 cyr
v0.9c
v0.9c
his clock her clock
out 4.8yrs 2.1yrs
back 4.8yrs 2.1yrs

• She ages
• less

total 9.6yrs 4.2yrs
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But, according to the girl, the boys clock is
moving , so, it must be running slower
v0.9c
According to her, the boys clock on Earth
says
tout g
2.1 yrs 2.3
tout

0.9 yrs
tback g
2.1 yrs 2.3
tback

0.9 yrs
v0.9c
ttotal touttback
1.8yrs
60
Her clock advances 4.2 yrs she sees his clock
advanceonly 1.8 yrs,
A contradiction??

• She should think he has aged less than her!!

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Events in the boys life
As seen by her
As seen by him
She leaves
0.9 yrs
4.8 yrs
She arrives starts turn
short time
????
Finishes turn heads home
0.9 yrs
4.8 yrs
9.6 yrs
1.8 ??? yrs
She returns
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turning around as seen by her
According to her, these 2 events occur
very,very far apart from each other
He sees her finish turning
He sees her start to turn
Time interval between 2 events depends on the
state of motion of the observer
63
Gunfight viewed by observer at rest
He sees both shots fired simultaneously
Bang!
Bang!
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Viewed by a moving observer
65
Viewed by a moving observer
He sees cowboy shoot 1st cowgirl shoot later
Bang!
Bang!
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In fact, ???? 7.8 years
as seen by him
as seen by her
She leaves
0.9 yrs
4.8 yrs
She arrives starts turn
7.8 yrs
short time
???
Finishes turn heads home
0.9 yrs
4.8 yrs
9.6 yrs
1.8 ???yrs
9.6 yrs
She returns
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No paradox both twins agree
The twin that turned around is younger
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Ladder Barn Door paradox
Stan Ollie puzzle over how to get a 2m long
ladder thru a 1m wide barn door
???
1m
2m
ladder
69
Ollie remembers Phys 100 thetheory of
relativity
Stan, pick up the ladder run very fast
1m
tree
2m
ladder
70
View from Ollies ref. frame
Push, Stan!
1m
2m/g
V0.9c (g2.3)
Stan
Ollie
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View from Stans ref. frame
But it doesnt fit, Ollie!!
1m/g
V0.9c (g2.3)
2m
Ollie
Stan
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If Stan pushes both ends of theladder
simultaneously, Ollie sees the two ends move at
different times
Too late Stan!
Too soon Stan!
1m
clunk
clank
V0.9c (g2.3)
Stan
Ollie
Stan
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Fermilab proton accelerator
V0.9999995c
g1000
2km
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Stanford electron accelerator
v0.99999999995 c
3km
g100,000
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status

• Einsteins theory of special relativity has
  been carefully tested in many very precise
  experiments and found to be valid.
• Time is truly the 4th dimension of space time.

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test
g29.3