PHIR11: UNIT-III
SPECIAL THEORY OF RELATIVITY
Dr. AWNISH KUMAR TRIPATHI
1
Theory of Relativity

General Theory of Relativity (1915) : massive
objects cause a distortion in space-time, which is
felt as gravity.
 Gravitational lensing
 Changes in the orbit of Mercury
 Frame-dragging of space-time around
rotating bodies
 Gravitational redshift
 Gravitational waves
2
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Special Theory of Relativity (1905)
3
Galilean-Newtonian Mechanics
4
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
GTR
STR
G-N
5
6
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Time evolution in the Theory of Mechanics
7
UNIT-III
Special Theory of Relativity
Topics to be covered:







Impact of Theory of Relativity in daily Life
The Michelson-Morley experiment
Relativistic transformations
Length contraction
It was found that there was no
displacement of the interference
Time dilation
fringes, so that the result of the
experiment was negative and
Variation of mass with velocity
would, therefore, show that there is
still a difficulty in the theory itself…
mass-energy equivalence
- Albert Michelson, 1907
8
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Impact of Einstein's Theory of Relativity
in daily Life

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

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Electromagnetism
Global Positioning System
Gold's yellow color
Gold doesn't corrode easily
Mercury is a liquid
Old (Cathode ray based) TV
Light
Nuclear Power
Sunlight
Our existence
https://interestingengineering.com/10-ways-can-see-einsteins-theory-relativity-real-life-keyword-theoryrelativity
https://www.livescience.com/58245-theory-of-relativity-in-real-life.html
9
Newtonian (Classical) Relativity
Assumption:


It is assumed that Newton’s laws of motion must
be measured with respect to (relative to) some
reference frame.
A reference frame is basically a coordinate
system where the origin is either fixed with the
system or sometimes with the observer
depending on the convenience.
10
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Inertial Reference Frame


A reference frame is called an inertial frame
if Newton laws are valid in that frame.
Such a frame is established when a body, not
subjected to net external forces, is observed
to move in rectilinear motion at constant
velocity.
11
Newtonian Principle of Relativity



If Newton’s laws are valid in one reference
frame, then they are also valid in another
reference frame moving at a uniform
velocity relative to the first system.
This is referred to as the Newtonian
principle of relativity or Galilean
invariance/relativity.
The laws of mechanics are independent on
the state of movement in a straight line at
constant velocity.
12
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Inertial Frames K and K’



K is at rest and K’ is moving with constant velocity along x
direction.
Axes are parallel.
K and K’ are said to be INERTIAL COORDINATE SYSTEMS.
13
The Galilean Transformation
For a point P


In system K: P = (x, y, z, t)
In system K’: P = (x’, y’, z’, t’)
P
x’
x
14
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Conditions of the Galilean-Newtonian
Transformation



Parallel axes, NO change in y and z coordinates.
K’ has a constant relative velocity in the x-direction
with respect to K
ux’ = ux – v
ax’ = ax
uy’ = uy
ay’ = ay
u z’ = u z
a z’ = a z
Time (t) for all observers is a Fundamental invariant,
i.e., it remain the same for all inertial observers.
15
The Inverse Galilean-Newtonian Transformation
Step 1. Replace
with
.
Step 2. Replace “primed” quantities with
“unprimed” and “unprimed” with “primed.”
ux= ux’ + v
ax= ax’
uy= uy ’
ay= ay ’
u z= u z ’
a z= a z’
Different observers (in O and O’) will assign different
velocities because of their relative motion, but will
measure the same acceleration !
16
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Consequences:
Length intervals and time intervals are the same for all
inertial observers of the same events.

Two meter sticks (same length) would be of the same length
when compared in relative motion to one another;
 the
distance between two points, A and B, measured at a given
instant, is the same for each observer:
xB’ – xA’ = xB - xA

if clocks are calibrated and synchronized when at rest, their
readings and rates will agree even if they are in relative
motion to one another.
 It
follows from these transformations that the time interval between
two events, P and Q, is the same for each observer:
tP’ – tQ’ = tP – tQ
17
Consequences: Galilean-Newtonian Relativity
 The acceleration of a particle is the same in all reference
frames which move relative to one another with constant
velocity.
 Since in classical physics, the mass is unaffected by the
motion of the reference frame, the product of mass and
acceleration (ma) will be the same for all inertial observers.
 Since F = ma is the definition of force, this will also be the
same for each inertial observer.
Newton’s laws of motion and the equations of
motion of a particle are exactly the same in all
inertial frames  the laws of mechanics are the
same in all inertial frames.
18
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Transition to Modern Relativity
Although Newton’s laws of motion had the same form
under the Galilean transformation, Maxwell’s equations
(ELECTROMAGTIC THEORY) did not.

In 1905, Albert Einstein proposed a fundamental
connection between space and time and that Newton’s
laws are only an approximation.

19
The Need for Ether




It was assumed that like sound wave, even Electromagnetic
waves require a medium to travel through.
The wave nature of light suggested that there existed a
propagation medium called the luminiferous ether or just ether
having below listed properties.

Ether had to have such a low density that the planets could move through it
without loss of energy.

It also had to have an elasticity to support the high velocity of light waves.
Let’s call the “ether” frame O, and assume it to be an inertial
one in which the observer measures the speed of light to be
exactly c.
In a frame O’ moving at a constant speed v with respect to the
ether frame, an observer would measure a different speed for the
light pulse, ranging from c + v to c - v, depending on the
direction of relative motion, according to the Galilean velocity
transformation.
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
20
Whereas: Maxwell’s Equations


In Maxwell’s theory the speed of light, in terms of the
permeability and permittivity of free space, was given by
Thus the velocity of light between even in moving systems
must be a constant.
21
ETHER: An Absolute Reference System


Ether was proposed as an absolute reference
system in which the speed of light was the
constant c and from which other measurements
could be made.
The Michelson-Morley experiment was an
attempt to show the existence of ether and to
figure out Earth’s relatives movement through
(with respect to) the ether so that Maxwell’s
equations could be corrected for this effect.
22
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
23
The details of Michelson-Morley experiment at the
end as Annexure.
24
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Michelson’s Conclusion



He concluded that the hypothesis of existence of
the stationary ether must be incorrect.
After several repeats and refinements with
assistance from Edward Morley (1893-1923),
both Michelson and Morley concluded again a
null result.
Finally it was established that, ether does not
seem to exist!
25
Einstein’s Postulates

Albert Einstein (1879–1955) was only two years
old when Michelson reported his first null
measurement for the existence of the ether.

At the age of 16, Einstein began thinking about
the form of Maxwell’s equations in moving
inertial systems.

In 1905, at the age of 26, he published his
startling proposal about the principle of
relativity, which he believed to be fundamental.
26
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Einstein’s Two Postulates
With the belief that Maxwell’s equations must be
valid in all inertial frames, Einstein proposed the
following postulates:
1) The principle of relativity: The laws of
physics are the same in all inertial systems.
There is no way to detect absolute motion, and
no preferred inertial system exists.
2) The constancy of the speed of light:
Observers in all inertial systems measure the
same value for the speed of light in a vacuum.
27
Re-evaluation of Time

In Newtonian physics we previously assumed
that t = t’


Thus with “synchronized” clocks, events in K and
K’ can be considered simultaneous
Einstein realized that each system must have
its own observers with their own clocks
and meter sticks

Thus events, happening at the same instant of
time in both K and K’, may not be simultaneous in
both K and K’.
28
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Problem of Simultaneity
Frank at rest is equidistant from events A and B:
A
−1 m
B
+1 m
0
Frank “sees” both flashbulbs go off
simultaneously.
29
The Problem of Simultaneity
Frank, moving to the right with speed v,
observes events A and B in different order:
−1 m
A
c-v
0
c+v
+1 m
B
Frank “sees” event B, then A.
30
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
We thus observe…

Two events that are simultaneous in one
reference frame (K) are not necessarily
simultaneous in another reference frame (K’)
moving with respect to the first frame.

This suggests that each coordinate system
has its own observers with “clocks” that are
synchronized…
31
Synchronization of Clocks
Step 1: Place observers with clocks
throughout a given system.
Step 2: In that system bring all the clocks
together at one location.
Step 3: Compare the clock readings.

If all of the clocks agree, then the clocks
are said to be synchronized.
32
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
A method to synchronize…

One way is to have one clock at the origin set
to t = 0 and advance each clock by a time
(d/c) with d being the distance of the clock
from the origin.

Allow each of these clocks to begin timing when a
light signal arrives from the origin.
t=0
t = d/c
d
t = d/c
d
33
The Lorentz Transformations
The special set of linear transformations that:
1) preserve the constancy of the speed of light
(c) between inertial observers;
and,
2) account for the problem of simultaneity
between these observers
known as the Lorentz transformation equations
34
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Lorentz Transformation Equations
Hendrik Antoon Lorentz
35
Lorentz Transformation Equations
A more symmetric form:
36
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Properties of γ
Recall β = v/c < 1 for all observers.
equals 1 only when v = 0.
1)
2)
Graph of β vs γ : (note v ≠ c)
37
Derivation of Lorentz Transformations equations




Use the fixed system K and the moving system K’
At t = 0 the origins and axes of both systems are
coincident with system K’ moving to the right along the x
axis.
A flashbulb goes off at the origins when t = 0.
According to postulate 2, the speed of light will be c in
both systems and the wavefronts observed in both
systems must be spherical.
K
K’
38
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Derivation
Spherical wavefronts in K:
Spherical wavefronts in K’:
Note: These are not preserved in the classical
transformations with
39
Derivation
1) Let x’ = (x – vt) so that x =
(x’ + vt’)
2) By Einstein’s first postulate:
3) The wavefront along the x, x’- axis must satisfy:
x = ct and x’ = ct’
4) Thus ct’ =
(ct – vt) and ct = (ct’ + vt’)
5) Solving the first one above for t’ and substituting into
the second...
40
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Derivation
Gives the following result:
from which we derive:
41
Finding a Transformation for t ’
Recalling x’ = (x – vt) substitute into x = (x’ + vt)
and solving for t ’ we obtain:
which may be written in terms of β (= v/c):
42
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Thus the complete set of Transformation equations
Galilean-Newtonian
Transformation
Lorentz
Transformation
Inverse-Lorentz
Transformation
Inverse
 = v/c
43
Remarks
1)
If v << c, i.e., β ≈ 0 and ≈ 1, we see these
equations reduce to the familiar Galilean
transformation.
2)
Space and time are now not separated.
3)
For non-imaginary transformations, the frame
velocity cannot exceed c.
44
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Time Dilation and Length Contraction
Consequences of the Lorentz Transformation:

Time Dilation:
Clocks in K’ run slow with respect to
stationary clocks in K.

Length Contraction:
Lengths in K’ are contracted with respect to
the same lengths stationary in K.
45
Time Dilation
To understand time dilation the idea of
proper time must be understood:

The term proper time,T0, is the time
difference between two events occurring at
the same position in a system as measured
by a clock at that position.
Same location
46
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Time Dilation
Improper Time
Beginning and ending of the event occur at
different positions
47
Time Dilation
Frank’s clock is at the same position in system K when the
sparkler is lit in (a) and when it goes out in (b). Mary, in the
moving system K’, is beside the sparkler at (a). Melinda then
moves into the position where and when the sparkler
extinguishes at (b). Thus, Melinda, at the new position,
measures the time in system K’ when the sparkler goes out in
(b).
48
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
According to Mary and Melinda…

Mary and Melinda measure the two times for the
sparkler to be lit and to go out in system K’ as times
t’1 and t’2 so that by the Lorentz transformation:

Note here that Frank records x2 – x1 = 0 in K with
a proper time: T0 = t2 – t1 or
with T ’ = t ’2 – t ’1
49
Time Dilation
1. T ’ > T0 or the time measured between two
events at different positions is greater than the
time between the same events at one position:
time dilation.
2. The events do not occur at the same space
and time coordinates in the two system
3. System K requires 1 clock and K’ requires 2
clocks.
4. A moving clock slows down.
50
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Length Contraction
To understand length contraction the idea of
proper length must be understood:


Let an observer in each system K and K’
have a meter stick at rest in their own system
such that each measure the same length at
rest.
The length as measured at rest is called the
proper length.
51
What Frank and Mary see…
Each observer lays the stick down along his or her
respective x axis, putting the left end at xℓ (or x’ℓ)
and the right end at xr (or x’r).


Thus, in system K, Frank measures his stick to be:
L0 = xr – xℓ
Similarly, in system K’, Mary measures her stick at
rest to be:
L’0 = x’r – x’ℓ
52
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
What Frank and Mary measure

Frank in his rest frame measures the moving length in
Mary’s frame moving with velocity.

Thus using the Lorentz transformations Frank measures
the length of the stick in K’ as:
Where both ends of the stick must be measured
simultaneously, i.e, tr = tℓ
Here Mary’s proper length is L’0 = x’r – x’ℓ
and Frank’s measured length is L = xr – xℓ
53
Frank’s measurement
So Frank measures the moving length as L
given by
but since both Mary and Frank in their
respective frames measure L’0 = L0
and L0 > L, i.e. the moving stick shrinks.
54
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Addition of Velocities
Taking differentials of the Lorentz
transformation, relative velocities may be
calculated:
55
So that…
defining velocities as: ux = dx/dt, uy = dy/dt,
u’x = dx’/dt’, etc. it is easily shown that:
With similar relations for uy and uz:
56
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Lorentz Velocity Transformations
In addition to the previous relations, the Lorentz velocity
transformations for u’x, u’y , and u’z can be obtained by
switching primed and unprimed and changing v to –v:
Lorentz Velocity
Transformation
57
Relativistic Mass and Momentum

The rest mass of an object is its inertial mass :
m0 

However, the mass of an object as defined by:
m
F
a
Etotal
c2
will be larger if the object is moving with respect to a given frame of
reference since the object will also have kinetic energy.

Einstein suggested that the relativistic energy of an object would be
given by:
2
Etotal 
m0 c
v2
1 2
c
Etotal   m0 c 2
58
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Relativistic mass
Etotal
m 2 
c
m0
1
2
v
c2
  m0
Relativistic Momentum
The momentum of an object is also relativistic:
p  mv 
m0 v
1
2
v
c2
  m0 v
59
Relativistic Energy

Due to the new idea of relativistic mass, we must
now redefine the concepts of work and energy.

Therefore, we modify Newton’s second law to include
our new definition of linear momentum, and force
becomes:
60
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Relativistic Kinetic Energy

To derive an expression for the relativistic energy
we start with the work-energy theorem
W  K  Fdx (Work done)

W  

dP
dx
dt
Using the chain rule repeatedly this can rewritten as
W 
dP
vdv
dv

m0 v
d 

vdv

2
2
dv  1  v c 
0


v
(Work done accelerating an
object from rest some
velocity)
61
v

0

m0 v
d 

 vdv W 
dv  1  v 2 c 2 


 m0
 1  v
 W  m0
v
2
c
2

3
dv
v
 1  u c 
3
2


W  m0 c 1  u c
2
Dr. AWNISH KUMAR TRIPATHI
m0 v
 1  v
0
2
c
2

3
dv
2
Integrating by substitution:
2

m0  1  u c 2


1
2 
2c 2

v
du m0

2v
2
2
du
 1  u c 
2
3
2
u

1
2

  m0 c 2 1  u c 2

0

2


1
2
 m0 c 2
SPECIAL THEORY OF RELATIVITY

1
u
2
0
62

W  m0 c 2 1  u c 2
Therefore W 

1
2
 m0 c 2
m0 c 2
1  v
2
c2

1
2
 m0 c 2
However this equal to K
therefore the Relativistic Kinetic energy is
K
m0c 2
1  v
2
c2

1
2
 m0c 2

The velocity independent term ( m0 c 2 ) is the rest
energy – the energy an object contains when it is at
rest.

At low speeds v << c we can write the kinetic energy as

K  m0 c 2 1  v 2 c 2
K  m c 1 
 12
Using a Taylor expansion we get,
2
0

1
2
v
2


1

c  v c
2
3
8
2

2 2
63

5
16
v
2
c

2 3
 
 ...  1
Taking the first 2 terms of the expansion


K  m0 c 2 1  12 v 2 c 2  1
So that
K  12 m0 v 2 (classical expression for the Kinetic energy)

The expression for the relativistic kinetic energy
m0 c 2
1 v2 c2
 K  m0 c or mc 2  K  m0 c 2
2
where E  mc is the total relativistic energy

If the object also has potential energy it can be
shown that
mc 2  K  V  m0 c 2
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
64
Relativistic and Classical Kinetic Energies
65
Relativistic Energy





The relationship E  mc is just Einstein’s mass-energy
equivalence equation, which shows that mass is a form of
energy.
An important fact of this relationship is that the relativistic
mass is a direct measure of the total energy content of an
object.
It shows that even a small mass corresponds to an
enormous amount of energy. This is the foundation of
nuclear physics.
Anything that increases the energy in an object will
increase its relativistic mass.
In certain cases the momentum or energy is known
instead of the speed.
2
66
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
m02 c 4
Note that E  m c 
1 v2 c2
2

2
4

 m 2 c 4 1  v 2 c 2  m02 c 4
 m 2 c 4  E 2  m 2 c 2 v 2  m02 c 4
Since
P  mv
E 2  P 2 c 2  m02 c 4
When the object is at rest, p

 0 so that E  mc2
67
 0 so that E  mc2

When the object is at rest, p

For particles having zero mass (photons) we see
that E  pc
the expected relationship relating
energy and momentum for photons and neutrinos
which travel at the speed of light.

When the object is at rest, p

For particles having zero mass (photons) we see
that E  pc
the expected relationship relating
energy and momentum for photons and neutrinos
which travel at the speed of light.
 0 so that E  mc2
68
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
ANNEXURE
MICHELSON-MORLEY EXPERIMENT
Dr. AWNISH KUMAR TRIPATHI
69
The Need for Ether




In early 19 century, It was assumed that like sound wave, even
Electromagnetic waves require a medium to travel through.
The wave nature of light suggested that there existed a
propagation medium called the luminiferous ether or just ether.

Ether had to have such a low density that the planets could move
through it without loss of energy.

It also had to have an elasticity to support the high velocity of light
waves.
Let’s call the “ether” frame O, and assume it to be an inertial
one in which the observer measures the speed of light to be
exactly c.
In a frame O’ moving at a constant speed v with respect to the
ether frame, an observer O’ would measure a different speed for
the light pulse, ranging from c + v to c - v, depending on the
direction of relative motion, according to the Galilean velocity
transformation.
70
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Maxwell’s Equations


In Maxwell’s theory the speed of light, in terms of the
permeability and permittivity of free space, was given by
Thus the velocity of light between moving systems must
be a constant.
71
ETHER: An Absolute Reference System


Ether was proposed as an absolute reference system in
which the speed of light was the constant c and from
which other measurements could be made.
The Michelson-Morley experiment was an attempt to
show the existence of ether and to figure out Earth’s
relatives movement through (with respect to) the ether
so that Maxwell’s equations could be corrected for this
effect. .
1. Before understanding Michelson-Morley experiment, we
have to understand Michelson interferometer.
2. Before understanding Michelson interferometer we
have to understand INTERFERENCE because of
Division of Amplitude.
72
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Change of Phase in Reflection
String Analogy
Point to remember: An EM wave undergoes a phase
change of 180° upon reflection from a medium that has a
higher refractive index than the one in which it is traveling.
73
n1
nf
n2
C
t
t
D
C
for the first two reflected
beams
A
D
i
B
t
A
B Optical path difference
Path difference 
 n f [AB  BC]  n1 (AD)
AB  BC  d /cos t
AD  AC sin i  2d tan t
d
nf
sin t
n1
  2n f dcost For air n1=1
Condition for maxima
dn f cos t  (2m  1)
f
4
m  0, 1, 2,...
Condition for minima
dn f cos t  2m
f
4
m  0, 1, 2,...
74
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Interference in Thin Films
• A wave traveling from a medium
of index of refraction of n1
towards a medium with index of
refraction of n2 undergoes a 180°
phase change upon reflection if n2
> n1 and no phase change if n2 <
n1.
• The wavelength of light  n in a
medium with index of refraction n
is given by,  n =  / n.
For constructive
interference
For destructive
interference
1

2t   m  n
2

2nt  m
1

2nt   m  
2

m = 0,1,2,…
m = 0,1,2,…
75
What is a typical Michelson’s Interferometer ?
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Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Michelson’s Interferometer: Working principle
S
Monochromatic light source
S1’
Beam splitter partially silvered at the back side
S2’
Beam compensator

M1’
M2
M1
Fixed mirror and can be tilted
M2
Movable mirror
S’
Image of S because of Beam splitter
M1’
Image of M1 because of Beam splitter
S2’ Image of S’ because of mirror M 2
S
S1’ Image of S’ because of mirror M 1’
M1
Interference fringe pattern at
constant angles 
S’
Screen
77
Non-localized Interference pattern
78
S 1'
SG＝S'G
S'M 2 =M 2S'2
2d
S 2'

ΔL  (L  2d)2  R 2  L2  R 2
L  2 Ld / L  R  2d cos 
2
2
The condition of bright fringe is
M 1'
M2
d
Since L>>d, expression becomes
G
L
S
S'
2dcos =k
k=0,1,2,
R
E
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
P
M1
Interferogram demonstration
79
2d cos   k
Maxwell’s Equations


In Maxwell’s theory the speed of light, in
terms of the permeability and permittivity of
free space, was given by
Thus the velocity of light between moving
systems must be a constant.
80
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Michelson-Morley Experiment

Albert Michelson (1852–1931) was the first
U.S. citizen to receive the Nobel Prize for
Physics (1907), and built an extremely
precise device called an interferometer
(Michelson interferometer) to measure the
minute phase difference between two light
waves traveling in mutually orthogonal
directions.
81
The Michelson Interferometer
1. AC is parallel to the
motion of the Earth
inducing an “ether wind”
2. Light from source S is
split by mirror A and
travels to mirrors C and D
in mutually perpendicular
directions
3. After reflection the
beams recombine at A
slightly out of phase due
to the “ether wind” as
viewed by telescope E.
82
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
83
Typical interferometer fringe pattern
expected when the system is rotated by 90°
NEVER OBSERVED !!!!
84
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Analysis
Assuming the Galilean Transformation
Time t1 from A to C and back:
Time t2 from A to D and back:
So that the change in time is:
85
The Analysis (continued)
Upon rotating the apparatus 90o, the optical path lengths
ℓ1 and ℓ2 are interchanged producing a different change
in time: (note the change in denominators)
86
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Analysis (continued)
Thus a time difference between rotations is given by:
and upon a binomial expansion, assuming
v/c << 1, this reduces to
87
Results

Using the Earth’s orbital speed as:
V = 3 × 104 m/s
together with
ℓ1 ≈ ℓ2 = 1.2 m
So that the time difference becomes
Δt’ − Δt ≈ v2(ℓ1 + ℓ2)/c3 = 8 × 10−17 s


The (Δt’ − Δt ) will introduce a path difference of c(Δt’ − Δt )
which will lead to a phase difference 2 c(Δt’ − Δt)/ between
ray AC and AB.
Although a very small number, it was within the experimental
range of measurement for light waves.
88
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
Michelson’s Conclusion




Michelson noted that he should be able to detect
a phase shift of light due to the time difference
between path lengths but found none.
He thus concluded that the hypothesis of the
stationary ether must be incorrect.
After several repeats and refinements with
assistance from Edward Morley (1893-1923),
again a null result.
Thus, ether does not seem to exist!
89
Possible Explanations

Many explanations were proposed but the
most popular was the ether drag hypothesis.


This hypothesis suggested that the Earth
somehow “dragged” the ether along as it rotates
on its axis and revolves about the sun.
This was contradicted by stellar abberation
wherein telescopes had to be tilted to observe
starlight due to the Earth’s motion. If ether was
dragged along, this tilting would not exist.
90
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
The Lorentz-FitzGerald Contraction

Another hypothesis proposed independently by
both H. A. Lorentz and G. F. FitzGerald suggested
that the length ℓ1, in the direction of the motion
should be contracted by a factor of
…thus making the path lengths equal to account for
the zero phase shift.

This, however, was an ad hoc assumption that
could not be experimentally tested (at that time
but later proved).
91
Length contracted for the
moving muon, it’s own life
time just 2.2 micro seconds
Life time of the muon delayed
for observer on Earth so that it
can travel the whole distance
as observed from Earth
Great thing about special
relativity is that one can
always take two viewpoints,
moving with the experiment,
watching the experiment
move past, the observations
need to be consistent in both
cases
They move with about 98 % the speed of light
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY
92
END
93
Dr. AWNISH KUMAR TRIPATHI
SPECIAL THEORY OF RELATIVITY