7. Einstein’s hypotheses:
1. The laws of nature are equally valid
in every inertial reference frame.
Including
Maxwell’s 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.
8. All observers see light flashes go
by them with the same speed
c
v
Both guys see the light flash
travel with velocity = c
No matter how fast
the guy on the rocket
is moving!!
9. Even when the light flash is
traveling in an opposite direction
c
v
Both guys see the light flash
travel past with velocity = c
10. Gunfight viewed by observer at rest
Bang
! Bang
!
He sees both shots
fired simultaneously
14. Viewed by a moving observer
Bang
!
Bang
!
He sees cowgirl shoot
1st & cowboy shoot later
15. 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
19. Same events, different observers
x
y
x
t
(x1,t1)
x
(x2,t2)
x1 x2
x’
y’
x1’
(x1’,t1’)
y’
x’
x1’ x2’
(x2’,t2’)
t’ t’
Prior to Einstein, everyone agreed
the distance between events depends
upon the observer, but not the time.
dist’
dist
20. Time is the 4th dimension
Einstein discovered that there is no
“absolute” time, it too depends upon
the state of motion of the observer
Newton
Space
&
Time
Einstein
Space-Time
completely
different
concepts
2 different aspects
of the same thing
21. How are the times seen
by 2 different
observers related?
We can figure this out with
simple HS-level math
( + a little effort)
22. Catch ball on a rocket ship
w=4m
t=1s
v= =4m/s
w
t
Event 1: boy throws the ball
Event 2: girl catches the ball
23. Seen from earth
w=4m
v0t=3m
v= = 5m/s
d
t
t=1s
V0=3m/s
V0=3m/s
Location of the 2
events is different
Elapsed time is
the same
The ball appears
to travel faster
24. Flash a light on a rocket ship
w
t0
c=
w
t0
Event 1: boy flashes the light
Event 2: light flash reaches the girl
25. Seen from earth
w
vt
c= =
d
t
t=?
V
V
Speed has to
Be the same
Dist is longer
Time must be
longer
(vt)2+w2
t
26. How is t related to t0?
c = (vt)2+w2
t
t= time on Earth clock
c =
w
t0
t0 = time on moving clock
ct = (vt)2+w2
(ct)2 = (vt)2+w2
ct0 = w
(ct)2 = (vt)2+(ct0)2 (ct)2-(vt)2= (ct0)2 (c2-v2)t2= c2t0
2
t2 = t0
2
c2
c2 – v2
t2 = t0
2
1
1 – v2/c2
t = t0
1
1 – v2/c2
this is called g
t = g t0
27. Properties of g = 1
1 – v2/c2
1
1 – (0.01c)2/c2
g =
Suppose v = 0.01c (i.e. 1% of c)
1
1 – (0.01)2c2/c2
=
1
1 – (0.01)2
g =
1
1 – 0.0001
= 1
0.9999
=
g = 1.00005
28. Properties of g = (cont’d)
1
1 – v2/c2
1
1 – (0.1c)2/c2
g =
Suppose v = 0.1c (i.e. 10% of c)
1
1 – (0.1)2c2/c2
=
1
1 – (0.1)2
g =
1
1 – 0.01
= 1
0.99
=
g = 1.005
29. Let’s make a chart
v g =1/(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
30. Other values of g =
1
1 – v2/c2
1
1 – (0.5c)2/c2
g =
Suppose v = 0.5c (i.e. 50% of c)
1
1 – (0.5)2c2/c2
=
1
1 – (0.5)2
g =
1
1 – (0.25)
= 1
0.75
=
g = 1.15
32. Other values of g =
1
1 – v2/c2
1
1 – (0.6c)2/c2
g =
Suppose v = 0.6c (i.e. 60% of c)
1
1 – (0.6)2c2/c2
=
1
1 – (0.6)2
g =
1
1 – (0.36)
= 1
0.64
=
g = 1.25
33. 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
34. Other values of g =
1
1 – v2/c2
1
1 – (0.8c)2/c2
g =
Suppose v = 0.8c (i.e. 80% of c)
1
1 – (0.8)2c2/c2
=
1
1 – (0.8)2
g =
1
1 – (0.64)
= 1
0.36
=
g = 1.67
35. 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
36. Other values of g =
1
1 – v2/c2
1
1 – (0.9c)2/c2
g =
Suppose v = 0.9c (i.e.90% of c)
1
1 – (0.9)2c2/c2
=
1
1 – (0.9)2
g =
1
1 – 0.81
= 1
0.19
=
g = 2.29
37. 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
38. Other values of g =
1
1 – v2/c2
1
1 – (0.99c)2/c2
g =
Suppose v = 0.99c (i.e.99% of c)
1
1 – (0.99)2c2/c2
=
1
1 – (0.99)2
g =
1
1 – 0.98
= 1
0.02
=
g = 7.07
39. 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
40. Other values of g =
1
1 – v2/c2
1
1 – (c)2/c2
g =
Suppose v = c
1
1 – c2/c2
=
1
1 – 12
g =
1
0
= 1
0
=
g = Infinity!!!
41. 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
42. Other values of g =
1
1 – v2/c2
1
1 – (1.1c)2/c2
g =
Suppose v = 1.1c
1
1 – (1.1)2c2/c2
=
1
1 – (1.1)2
g =
1
1-1.21
= 1
-0.21
=
g = ??? Imaginary number!!!
43. 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
50. Relativistic mass increase
m0 = mass of an object when it
is at rest “rest mass”
m = g m0
mass of a moving
object increases
by the g factor
as vc, m
as an object moves
faster, it gets
harder & harder
to accelerate
g
v=c
51. summary
• Moving clocks run slow
• Moving objects appear shorter
• Moving object’s mass increases
53. Twin paradox
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
a-centauri
54. Light year
distance light travels in 1 year
dist = v x time
1cyr = 3x108m/s x 3.2x107 s
= 9.6 x 1015 m
We will just use cyr units
& not worry about meters
= c yr
55. Time on the boy’s clock
tout =
d0
v
4.3 cyr
0.9c
= = 4.8 yrs
According to the boy
& his clock on Earth:
tback =
d0
v
4.3 cyr
0.9c
= = 4.8 yrs
ttotal = tout+tback = 9.6yrs
56. What does the boy see on her
clock?
tout =
tout
g
4.8 yrs
2.3
= = 2.1 yrs
According to the boy
her clock runs slower
tback =
tback
g
4.8 yr
2.3
= = 2.1 yrs
ttotal = tout+tback = 4.2yrs
57. So, according to the boy:
his clock her clock
out: 4.8yrs 2.1yrs
back: 4.8yrs 2.1yrs
total: 9.6yrs 4.2yrs
58. But, according to the
girl, the boy’s clock is
moving &, so, it must be
running slower
tout =
tout
g
2.1 yrs
2.3
= = 0.9 yrs
According to her, the
boy’s clock on Earth says:
tback =
tback
g
2.1 yrs
2.3
= = 0.9 yrs
ttotal = tout+tback = 1.8yrs
59. Her clock advances 4.2 yrs
& she sees his clock advance
only 1.8 yrs,
She should think he has aged
less than her!!
60. As seen by him
Events in the boy’s life:
As seen by her
She leaves
She arrives
& starts turn
Finishes turn
& heads home
She returns
4.8 yrs
4.8 yrs
short time
9.6+ yrs
0.9 yrs
????
0.9 yrs
1.8 + ??? yrs
61. turning around as seen by her
He sees her
start to turn
He sees her
finish turning
According to her, these
2 events occur very,very
far apart from each other
Time interval between 2 events depends
on the state of motion of the observer
62. Gunfight viewed by observer at rest
Bang
! Bang
!
He sees both shots
fired simultaneously
64. Viewed by a moving observer
Bang
! Bang
!
He sees cowboy shoot
1st & cowgirl shoot later
65. as seen by him
In fact, ???? = 7.8+ years
as seen by her
She leaves
She arrives
& starts turn
Finishes turn
& heads home
She returns
4.8 yrs
4.8 yrs
short time
9.6+ yrs
0.9 yrs
???
0.9 yrs
1.8 + ???yrs
7.8+ yrs
9.6+ yrs
66. No paradox: both twins agree
The twin that
“turned around”
is younger
67. Ladder & Barn Door paradox
1m
2m
???
ladder
Stan & Ollie puzzle over how
to get a 2m long ladder thru
a 1m wide barn door
68. Ollie remembers Phys 100 & the
theory of relativity
1m
2m
ladder
Stan, pick up
the ladder &
run very fast
tree
69. View from Ollie’s ref. frame
1m
2m/g
Push,
Stan!
V=0.9c
(g=2.3)
Ollie Stan
70. View from Stan’s ref. frame
2m
1m/g
V=0.9c
(g=2.3)
Ollie Stan
But it
doesn’t fit,
Ollie!!
71. If Stan pushes both ends of the
ladder simultaneously, Ollie sees the
two ends move at different times:
1m
Too
soon
Stan!
V=0.9c
(g=2.3)
Ollie Stan
Stan
Too late
Stan!
74. status
Einstein’s 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.