UNIT I
RELATIVISTIC MECHANICS
Lecture 1:
Frame of reference, Inertial & non-inertial frames, Galilean transformations
INTRODUCTION:
The old theory of classical mechanics was based on Newton’s law. Initially it was believed that second
law is universally applicable but with new experimental evidences in the beginning of 20 th century, it
was revealed that this law is applicable only for the particle moving with low velocity and fails when
applied to the particle moving with velocity comparable to velocity of light.
This failure of old theory, led to the development of THEORY OF RELATIVITY by Albert Einstein in
1905. According to this theory, everything in universe is relative and nothing is absolute. Einstein told
that old theory is the limiting case of new theory.
This new theory of relativity is further divided into two categories as General Theory and Special
Theory respectively. General theory is applied to the cases of relative motion and to accelerated
systems with respect to one another. Whereas the Special theory is applicable with observers which
are either in relative motion in a straight line at a constant speed or at rest.
Ques: What is frame of reference? Define inertial and non-inertial frames of reference.
Ans: Frame of Reference
The frame of reference is a coordinate system with respect to which we can predict the position and
motion of any object at different instants of time.
Frames of reference are of two types-
1. Inertial Frames of Reference: Inertial frames of reference are those frames of reference in
which Newton’s law of inertia and other laws of motion hold good. These frames are either at
rest or move with a constant velocity with respect to each other. Any object with net force zero
on it, will be un-accelerated with respect to these frames of reference.
Example: A train is in rest or in uniform motion.
2. Non-inertial Frames of Reference: Non-inertial frames of reference are those frames of
reference in which Newton’s law of inertia and other laws of motion do not hold. These frames
are accelerated. Any object with net force zero on it, will be accelerated with respect to non-
inertial frames of reference.
Example: A train is accelerated or retarded motion, Rotating merry-go-round.
Ques: Is Earth an inertial or non-inertial frame of reference? Discuss.
Ans: Earth is a non-inertial frame of reference because it is an accelerated frame of reference. Earth is
rotating not only about its own axis but also orbiting around the sun. Due to its orbital motion it
1
,continuously changes its velocity and direction of motion. Due to change in velocity, it is an
accelerated frame of reference.
The value of acceleration is very small (0.006 m/s2) and can be neglected. So, for practical
purposes, earth is regarded as an inertial frame of reference.
Ques: Describe Galilean transformation equations. What were the limitations of these
transformation equations?
Ans : Galilean-Newtonian transformation equations:
Let S & S’ be two frames of reference such that frame S’ is moving along positive X axis with v velocity
relative to frame S. If (x, y, z, t) and (x’, y’, z’,t’) are the coordinates of the same event in frame S and S’.
According to classical mechanics, the measurement in x direction made in S’ will be smaller than that
made in S by the amount vt, which is the distance moved by S’ in direction x, i.e.
x’ = x – vt ……(1)
there is no relative motion in y and z directions, so
y’ = y …...(2)
and z’ = z ..…(3)
in classical mechanics,
t’ = t . . . . . (4)
eqns. (1) to (4) are Galilean –Newtonian transformation equations.
Limitation of Galilean –Newtonian transformation equations
If we apply the concept of relativity (i.e. v = c) in equation (1) of Galilean equations, then in frame S’
the observed velocity would be c’ = c – v. which is the violation of the idea of relativity. So this requires
the modification in Galilean Transformation equations.
Lecture 2:
Ques: What is ether hypothesis?
Ans: Ether hypothesis:
The physicists of 19th century assumed that entire space of universe including vacuum is filled by a
hypothetical light transmitting medium called ether: which is rigid, invisible, massless, perfectly
transparent, high elasticity and negligible density. All bodies including earth move freely through this
hypothetical medium without disturbing it.
Ques: What was the objective of Michelson-Morley experiment? Discuss in detail.
2
,Ans: Objective of Michelson-Morley experiment was to detect the motion of earth with respect to
ether.
In 1887 Michelson & Morley performed an experiment. A simplified plan of experiment is shown below
containing a monochromatic source S. A partially silvered glass plate P is inclined at 45𝑜 to the incident
beam. Two mirrors 𝑀1 and 𝑀2 are placed at equal distance 𝑙 from glass plate. Telescope T is used to
see interference.
Figure: Michelson Morley Setup
Two mirrors are at equal distance l from glass plate. If apparatus is at rest then the two rays
will take equal time to return to plate P but apparatus is moving with earth with respect to ether with
velocity v (say). Let earth is moving in direction (PM2) of incident light then the time difference
between two rays will be calculated as:
if c is the velocity of light in direction of arm PM2, then the relative velocity will be (c - v) while on
returning it will become (c + v). If t1 is the time taken in complete journey then
𝑙 𝑙 2𝑙𝑐 2𝑙 1
t1 = (𝑐−𝑣) + (𝑐+𝑣) = = ( ) ……. …… (1)
(𝑐 2 − 𝑣 2 ) 𝑐 1− 𝑣2
𝑐2
The second part of the beam moving towards arm PM1 with velocity c. If t’is the time taken by the
beam in going from plate to mirror then the distance travelled will be ct’. Also in time t’ the mirror
travel the distance vt’. In right angle triangle PM1M1’ ,
Considering figure, {PM1 =l, M1M1’ = vt’, PM1’ = ct’ }
(PM1’)2 = (PM1)2 + (M1M1’)2,
i.e. (ct’)2 = l2 + (vt’)2
c2t’2 – v2t’2 = l2
t’2(c2 – v2) = l2
𝑙 𝑙
t’ = (𝑐 2− 𝑣2)1/2 = 2 1/2
𝑣
𝑐(1− 2 )
𝑐
Now if t2 is the total time in journey PM1’P’ then
3
, 𝟐𝒍 𝟏
t2 = 2t’ = ……(2)
𝒄
𝒗𝟐
√(𝟏− 𝟐 )
𝒄
If Δt is the time difference of two beams interfering at point P’, then
1
−1 −
2𝑙 2𝑙 2𝑙 𝑣2 𝑣2 2
Δt = t1 – t2 = 𝑣2
- 1 = [(1 − ) − (1 − ) ]
𝑐(1− 2 ) 𝑐 𝑐2 𝑐2
𝑐 𝑣2 2
𝑐(1− 2 )
𝑐
Using binomial expansion and neglecting higher terms
2𝑙 𝑣2 𝑣2
Δt = 𝑐 [(1 + + … … ) − (1 + … … )]
𝑐2 2𝑐 2
2𝑙 1 𝑣 2
Δt = [2 𝑐 2 ]
𝑐
𝑙𝑣 2
Δt = ----------(3)
𝑐3
The corresponding path difference will be
𝑙𝑣 2
δ = c Δt = c 𝑐 3
𝑙𝑣 2
δ= ----------(4)
𝑐2
If v is made zero, then δ would become zero.
We know that if path difference between two interfering waves changes by λ, then a shift of one fringe
appears. Thus, if N fringes are shifted then
𝛿 𝑙𝑣 2
ΔN = 𝜆 = 𝑐 2𝜆 .
In actual experiment, the apparatus was placed on a block of stone floated on mercury, was rotated by
2𝑙𝑣 2
90°, to introduce a path difference of same amount in opposite direction. Hence a shift of was
𝑐 2𝜆
expected.
To get the observable shift, experimental values are:
l = 11meter, v (velocity of earth) = 3x104 m/sec, λ = 5.5x10-7 m and c = 3x108 m/sec,
the expected shift is
2
2 X 11 X (3 X 104 )
ΔN = (3 X 108)2 X 5.5 X 10−7 = 0.4
The physicists were surprised to see that there was no shift in the fringes. The experiment was
repeated various times and at various places but no shift was observed. Thus, the motion of earth
through the ether could not be experimentally detected.
Ques: How were the negative results of Michelson- Morley experiment interpreted?
Ans: Explaination of negative result: Many explanations were given to explain negative result. Few of
them are:
1. Ether-Drag Hypothesis: According to this hypothesis, the moving earth drags ether with it. So
there is no relative motion between earth and ether. Hence no fringe shift is obtained.
2. Fitzgerald-Lorentz Contraction Hypothesis: Fitzgerald and Lorentz gives an hypothesis that all
material bodies moving through ether are contracted in direction of motion by a
𝑣2
factor√(1 − 𝑐 2 ). This contraction equalizes time t1 and t2 and no shift was observed.
3. Light Velocity Hypothesis (or Constancy of velocity of light): According to Einstein, velocity of
light is invariant in every inertial frame of reference. In other words velocity of light is constant
in free space.
4
RELATIVISTIC MECHANICS
Lecture 1:
Frame of reference, Inertial & non-inertial frames, Galilean transformations
INTRODUCTION:
The old theory of classical mechanics was based on Newton’s law. Initially it was believed that second
law is universally applicable but with new experimental evidences in the beginning of 20 th century, it
was revealed that this law is applicable only for the particle moving with low velocity and fails when
applied to the particle moving with velocity comparable to velocity of light.
This failure of old theory, led to the development of THEORY OF RELATIVITY by Albert Einstein in
1905. According to this theory, everything in universe is relative and nothing is absolute. Einstein told
that old theory is the limiting case of new theory.
This new theory of relativity is further divided into two categories as General Theory and Special
Theory respectively. General theory is applied to the cases of relative motion and to accelerated
systems with respect to one another. Whereas the Special theory is applicable with observers which
are either in relative motion in a straight line at a constant speed or at rest.
Ques: What is frame of reference? Define inertial and non-inertial frames of reference.
Ans: Frame of Reference
The frame of reference is a coordinate system with respect to which we can predict the position and
motion of any object at different instants of time.
Frames of reference are of two types-
1. Inertial Frames of Reference: Inertial frames of reference are those frames of reference in
which Newton’s law of inertia and other laws of motion hold good. These frames are either at
rest or move with a constant velocity with respect to each other. Any object with net force zero
on it, will be un-accelerated with respect to these frames of reference.
Example: A train is in rest or in uniform motion.
2. Non-inertial Frames of Reference: Non-inertial frames of reference are those frames of
reference in which Newton’s law of inertia and other laws of motion do not hold. These frames
are accelerated. Any object with net force zero on it, will be accelerated with respect to non-
inertial frames of reference.
Example: A train is accelerated or retarded motion, Rotating merry-go-round.
Ques: Is Earth an inertial or non-inertial frame of reference? Discuss.
Ans: Earth is a non-inertial frame of reference because it is an accelerated frame of reference. Earth is
rotating not only about its own axis but also orbiting around the sun. Due to its orbital motion it
1
,continuously changes its velocity and direction of motion. Due to change in velocity, it is an
accelerated frame of reference.
The value of acceleration is very small (0.006 m/s2) and can be neglected. So, for practical
purposes, earth is regarded as an inertial frame of reference.
Ques: Describe Galilean transformation equations. What were the limitations of these
transformation equations?
Ans : Galilean-Newtonian transformation equations:
Let S & S’ be two frames of reference such that frame S’ is moving along positive X axis with v velocity
relative to frame S. If (x, y, z, t) and (x’, y’, z’,t’) are the coordinates of the same event in frame S and S’.
According to classical mechanics, the measurement in x direction made in S’ will be smaller than that
made in S by the amount vt, which is the distance moved by S’ in direction x, i.e.
x’ = x – vt ……(1)
there is no relative motion in y and z directions, so
y’ = y …...(2)
and z’ = z ..…(3)
in classical mechanics,
t’ = t . . . . . (4)
eqns. (1) to (4) are Galilean –Newtonian transformation equations.
Limitation of Galilean –Newtonian transformation equations
If we apply the concept of relativity (i.e. v = c) in equation (1) of Galilean equations, then in frame S’
the observed velocity would be c’ = c – v. which is the violation of the idea of relativity. So this requires
the modification in Galilean Transformation equations.
Lecture 2:
Ques: What is ether hypothesis?
Ans: Ether hypothesis:
The physicists of 19th century assumed that entire space of universe including vacuum is filled by a
hypothetical light transmitting medium called ether: which is rigid, invisible, massless, perfectly
transparent, high elasticity and negligible density. All bodies including earth move freely through this
hypothetical medium without disturbing it.
Ques: What was the objective of Michelson-Morley experiment? Discuss in detail.
2
,Ans: Objective of Michelson-Morley experiment was to detect the motion of earth with respect to
ether.
In 1887 Michelson & Morley performed an experiment. A simplified plan of experiment is shown below
containing a monochromatic source S. A partially silvered glass plate P is inclined at 45𝑜 to the incident
beam. Two mirrors 𝑀1 and 𝑀2 are placed at equal distance 𝑙 from glass plate. Telescope T is used to
see interference.
Figure: Michelson Morley Setup
Two mirrors are at equal distance l from glass plate. If apparatus is at rest then the two rays
will take equal time to return to plate P but apparatus is moving with earth with respect to ether with
velocity v (say). Let earth is moving in direction (PM2) of incident light then the time difference
between two rays will be calculated as:
if c is the velocity of light in direction of arm PM2, then the relative velocity will be (c - v) while on
returning it will become (c + v). If t1 is the time taken in complete journey then
𝑙 𝑙 2𝑙𝑐 2𝑙 1
t1 = (𝑐−𝑣) + (𝑐+𝑣) = = ( ) ……. …… (1)
(𝑐 2 − 𝑣 2 ) 𝑐 1− 𝑣2
𝑐2
The second part of the beam moving towards arm PM1 with velocity c. If t’is the time taken by the
beam in going from plate to mirror then the distance travelled will be ct’. Also in time t’ the mirror
travel the distance vt’. In right angle triangle PM1M1’ ,
Considering figure, {PM1 =l, M1M1’ = vt’, PM1’ = ct’ }
(PM1’)2 = (PM1)2 + (M1M1’)2,
i.e. (ct’)2 = l2 + (vt’)2
c2t’2 – v2t’2 = l2
t’2(c2 – v2) = l2
𝑙 𝑙
t’ = (𝑐 2− 𝑣2)1/2 = 2 1/2
𝑣
𝑐(1− 2 )
𝑐
Now if t2 is the total time in journey PM1’P’ then
3
, 𝟐𝒍 𝟏
t2 = 2t’ = ……(2)
𝒄
𝒗𝟐
√(𝟏− 𝟐 )
𝒄
If Δt is the time difference of two beams interfering at point P’, then
1
−1 −
2𝑙 2𝑙 2𝑙 𝑣2 𝑣2 2
Δt = t1 – t2 = 𝑣2
- 1 = [(1 − ) − (1 − ) ]
𝑐(1− 2 ) 𝑐 𝑐2 𝑐2
𝑐 𝑣2 2
𝑐(1− 2 )
𝑐
Using binomial expansion and neglecting higher terms
2𝑙 𝑣2 𝑣2
Δt = 𝑐 [(1 + + … … ) − (1 + … … )]
𝑐2 2𝑐 2
2𝑙 1 𝑣 2
Δt = [2 𝑐 2 ]
𝑐
𝑙𝑣 2
Δt = ----------(3)
𝑐3
The corresponding path difference will be
𝑙𝑣 2
δ = c Δt = c 𝑐 3
𝑙𝑣 2
δ= ----------(4)
𝑐2
If v is made zero, then δ would become zero.
We know that if path difference between two interfering waves changes by λ, then a shift of one fringe
appears. Thus, if N fringes are shifted then
𝛿 𝑙𝑣 2
ΔN = 𝜆 = 𝑐 2𝜆 .
In actual experiment, the apparatus was placed on a block of stone floated on mercury, was rotated by
2𝑙𝑣 2
90°, to introduce a path difference of same amount in opposite direction. Hence a shift of was
𝑐 2𝜆
expected.
To get the observable shift, experimental values are:
l = 11meter, v (velocity of earth) = 3x104 m/sec, λ = 5.5x10-7 m and c = 3x108 m/sec,
the expected shift is
2
2 X 11 X (3 X 104 )
ΔN = (3 X 108)2 X 5.5 X 10−7 = 0.4
The physicists were surprised to see that there was no shift in the fringes. The experiment was
repeated various times and at various places but no shift was observed. Thus, the motion of earth
through the ether could not be experimentally detected.
Ques: How were the negative results of Michelson- Morley experiment interpreted?
Ans: Explaination of negative result: Many explanations were given to explain negative result. Few of
them are:
1. Ether-Drag Hypothesis: According to this hypothesis, the moving earth drags ether with it. So
there is no relative motion between earth and ether. Hence no fringe shift is obtained.
2. Fitzgerald-Lorentz Contraction Hypothesis: Fitzgerald and Lorentz gives an hypothesis that all
material bodies moving through ether are contracted in direction of motion by a
𝑣2
factor√(1 − 𝑐 2 ). This contraction equalizes time t1 and t2 and no shift was observed.
3. Light Velocity Hypothesis (or Constancy of velocity of light): According to Einstein, velocity of
light is invariant in every inertial frame of reference. In other words velocity of light is constant
in free space.
4
