This document summarizes key concepts from Einstein's theory of relativity, including: 1) Moving clocks run slower than stationary clocks, as time depends on the observer's frame of reference (time dilation). 2) Moving objects appear shorter in the direction of motion, an effect called length contraction. 3) The mass of a moving object increases relative to an observer, known as relativistic mass increase. 4) The twin paradox explains that in a scenario where one twin travels in a rocket close to light speed while the other stays on Earth, the traveling twin ages less upon returning due to the effects of time dilation.

1. THEORY OF RELATIVITY
DINESH. V
Assistant Professor
HOD of Physics
Govt. First Grade College, Sagar -577401
E-mail: dinu.v18@gmail.com

2. Theory of Relativity
Physics 100
Chapt 18
Albert
Einstein

3. watching a light flash go by
c
v
The man on earth sees c =
(& agrees with Maxwell)
2k
k

4. watching a light flash go by
c
v
If the man on the rocket sees c-v,
he disagrees with Maxwell

5. Do Maxwell’s 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 chan
ge throughout the year
upstream,
light moves
slower
downstream,
light moves
faster “Aether wind”

7. Michelson-Morley
No aether wind detected: 1907 Nobel Prize

8. 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.

9. All observers see light flashes go b
y them with the same speed
c
v
Both guys see the light flash
travel with velocity = c
No matter how fast t
he guy on the rocket
is moving!!

10. Even when the light flash is travelin
g in an opposite direction
c
v
Both guys see the light flash
travel past with velocity = c

11. Gunfight viewed by observer at rest
Bang
! Bang
!
He sees both shots
fired simultaneously

12. Viewed by a moving observer

13. Viewed by a moving observer
Bang
! Bang
!
He sees cowboy shoot
1st & cowgirl shoot later

14. Viewed by an observer in the
opposite direction

15. Viewed by a moving observer
Bang
!Bang
!
He sees cowgirl shoot
1st & cowboy shoot later

16. Time depends of state of motion
of the observer!!
Events that occur simultaneously
according to one observer can occur a
t different times for other observ
ers

17. Light clock

18. Seen from the ground

19. Events
x
y
x
t
(x1,t1)
x
(x2,t2)
x1 x2

20. 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

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
Newton
Space
&
Time
Einstein
Space-Time
completely
different
concepts
2 different aspects
of the same thing

22. How are the times seen
by 2 different observe
rs related?
We can figure this out with
simple HS-level math
( + a little effort)

23. 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

24. Seen from earth
w=4m
v0t=3m
v= = 5m/s
d
tt=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

25. 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

26. Seen from earth
w
vt
c= =d
tt=?
V
V
Speed has to
Be the same
Dist is longer
Time must be
longer
(vt)2+w2
t

27. 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
2c2
c2 – v2
 t2 = t0
21
1 – v2/c2
 t = t0
1
1 – v2/c2
this is called g
 t = g t0

28. Properties of g = 1
1 – v2/c2
1
1 – (0.01c)2/c2g =
Suppose v = 0.01c (i.e. 1% of c)
1
1 – (0.01)2c2/c2
=
1
1 – (0.01)2g =
1
1 – 0.0001
= 1
0.9999
=
g = 1.00005

29. Properties of g = (cont’d)1
1 – v2/c2
1
1 – (0.1c)2/c2g =
Suppose v = 0.1c (i.e. 10% of c)
1
1 – (0.1)2c2/c2
=
1
1 – (0.1)2g =
1
1 – 0.01
= 1
0.99
=
g = 1.005

30. Let’s make a chart
v g =1/(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005

31. Other values of g = 1
1 – v2/c2
1
1 – (0.5c)2/c2g =
Suppose v = 0.5c (i.e. 50% of c)
1
1 – (0.5)2c2/c2
=
1
1 – (0.5)2g =
1
1 – (0.25)
= 1
0.75
=
g = 1.15

32. Enter into chart
v g =1/(1-v2/c2)
0.01 c 1.00005
0.1 c 1.005
0.5c 1.15

33. Other values of g = 1
1 – v2/c2
1
1 – (0.6c)2/c2g =
Suppose v = 0.6c (i.e. 60% of c)
1
1 – (0.6)2c2/c2
=
1
1 – (0.6)2g =
1
1 – (0.36)
= 1
0.64
=
g = 1.25

34. 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

35. Other values of g = 1
1 – v2/c2
1
1 – (0.8c)2/c2g =
Suppose v = 0.8c (i.e. 80% of c)
1
1 – (0.8)2c2/c2
=
1
1 – (0.8)2g =
1
1 – (0.64)
= 1
0.36
=
g = 1.67

36. 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

37. Other values of g = 1
1 – v2/c2
1
1 – (0.9c)2/c2g =
Suppose v = 0.9c (i.e.90% of c)
1
1 – (0.9)2c2/c2
=
1
1 – (0.9)2g =
1
1 – 0.81
= 1
0.19
=
g = 2.29

38. 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

39. Other values of g = 1
1 – v2/c2
1
1 – (0.99c)2/c2g =
Suppose v = 0.99c (i.e.99% of c)
1
1 – (0.99)2c2/c2
=
1
1 – (0.99)2g =
1
1 – 0.98
= 1
0.02
=
g = 7.07

40. 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

41. Other values of g = 1
1 – v2/c2
1
1 – (c)2/c2g =
Suppose v = c
1
1 – c2/c2
=
1
1 – 12g =
1
0
= 1
0
=
g =  Infinity!!!

42. 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 

43. Other values of g = 1
1 – v2/c2
1
1 – (1.1c)2/c2g =
Suppose v = 1.1c
1
1 – (1.1)2c2/c2
=
1
1 – (1.1)2g =
1
1-1.21
= 1
-0.21
=
g = ??? Imaginary number!!!

44. 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

45. Plot results:
1
1 – v2/c2
g =
v=c
x x
x
x
x

Never-neverland

46. Moving clocks run slower
t
t0
t = g t0
t = t0
1
1 – v2/c2
v
g >1  t > t0

47. Length contraction
man on
rocket
Time = t0 =t/g
Length = vt0
L0
v
time=t
L0 = vt
=vt/g =L0/g

48. Moving objects appear shorter
L = L0/g
g >1  L < L0
Length measured when
object is at rest
V=0.1cV=0.86cV=0.99cV=0.9999c

49. Length contraction

50. mass:
m0
a
F=m0a
change in v
time
Ft0
m0
change in v =
time=t0
t=gt0
Ft
change in v
m =
= m0
mass
increases!!
t0
= gm0
Ft0
change in v
m0 =
gFt0
change in v=
m = g m0
by a factor g

51. 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 vc, m
as an object moves
faster, it gets
harder & harder
to accelerate
g
v=c

52. summary
• Moving clocks run slow
• Moving objects appear shorter
• Moving object’s mass increases

53. Plot results:
1
1 – v2/c2
g =
v=c
x x
x
x
x

Never-neverland

54. Twin paradox
Twin brother
& sister
She will travel to
a-centauri (a near-
by star on a special
rocket ship v = 0.9cHe will stay home
& study Phys 100
a-centauri

55. 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

56. 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

57. What does the boy see on her cloc
k?
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

58. So, according to the boy:
his clock her clock
out: 4.8yrs 2.1yrs
back: 4.8yrs 2.1yrs
total: 9.6yrs 4.2yrs

59. But, according to the gi
rl, the boy’s clock is mov
ing &, so, it must be run
ning 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

60. Her clock advances 4.2 yrs
& she sees his clock advance
only 1.8 yrs,
She should think he has aged l
ess than her!!

61. 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

62. 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

63. Gunfight viewed by observer at rest
Bang
! Bang
!
He sees both shots
fired simultaneously

64. Viewed by a moving observer

65. Viewed by a moving observer
Bang
! Bang
!
He sees cowboy shoot
1st & cowgirl shoot later

66. 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

67. No paradox: both twins agree
The twin that
“turned around”
is younger

68. Ladder & Barn Door paradox
1m
2m
???
ladder
Stan & Ollie puzzle over how
to get a 2m long ladder thru
a 1m wide barn door

69. Ollie remembers Phys 100 & the
theory of relativity
1m
2m
ladder
Stan, pick up
the ladder &
run very fast
tree

70. View from Ollie’s ref. frame
1m
2m/g
Push, St
an!
V=0.9c
(g=2.3)Ollie Stan

71. View from Stan’s ref. frame
2m
1m/g
V=0.9c
(g=2.3)
Ollie Stan
But it does
n’t fit, Olli
e!!

72. If Stan pushes both ends of the
ladder simultaneously, Ollie sees the
two ends move at different times:
1mToo soo
n Stan!
V=0.9c
(g=2.3)Ollie StanStan
Too late
Stan!

73. Fermilab proton accelerator
2km
V=0.9999995c
g=1000

74. Stanford electron accelerator
g=100,000
v=0.99999999995 c

75. status
Einstein’s theory of “special relativity” has be
en carefully tested in many very precise ex
periments and found to be valid.
Time is truly the 4th dimension of space & ti
me.

76. test
g=29.3