Special Theory of Relativity
Frame of Reference
A coordinate system relative to which the measurements are taken is known as frame of reference. A coordinate system in which the law of inertial is valid is known as inertial frame of reference, while the accelerated frame is known as non-inertial frame of reference.
Special Theory of Relativity
Postulates of Special Theory of Relativity:
In 1905, Albert Einstein formulated his special theory of relativity in terms of two postulates:
1. Principle of Relativity
The laws of Physics have the same form in all frames of reference moving with constant velocities with respect to one another. It can also be stated as
“The laws of Physics are invariant o a transformation between all inertial frames”
2. Principle of Constancy of Speed of Light
The speed of light in free space has the same value for all observers regardless of their state of motion. It can also be stated as:
The speed of light in free space has the same value “c” in all inertial frames of references.
Relativistic Effects:
Theory
of Relativity shows the effects of relative motion on physical quantities. These
effects are observed at relativistic speed. (The speed 
is
called relativistic speed, where c is speed of light.)
Discussion:
The first postulate shows that laws of Physics are absolute and universal and are same for all inertial observers. So the laws of Physics that hold for one inertial observer can’t be violated for any other inertial observer.
To understand 2nd postulate, consider three observers A, B and C at rest in three different inertial frames.
· A flash of light emitted by observer A is observed by him to travel at speed c.
·
If
the frame of B is moving away from A at a speed of 
then
according to Galilean Kinematics, B measure the speed of flash emitted by A: 
.
·
If
the frame C is moving towards A with a speed of then
according to Galilean Transformation, C measures the value 
for
the speed of flash emitted by A.
However according to 2nd postulate all the three observers measure the same speed of flash of light. However ordinary objects don’t obey 2nd postulate e.g.,
But
velocities of waves and particles moving at speeds close to c behave in this
wave. When Einstein put forward these postulates, there was no experimental
test for the verification of these postulates. However, in 1964 a proton
accelerator produced a beam of neutral pions (
which
rapidly decay into 
:
Now are
electromagnetic waves and move with the speed of light. The speed of moving
Pions was measured equal to 
.
According to
Galileo, the emitted
in the direction of motion of Pions should have a speed equal to 
But
the measured speed of was
equal to 
.
This is consistent with 2nd postulate.
Galilean Transformation
Consider two observers in two different inertial
frames 
and
.
Frame is
at rest and is
moving with uniform velocity 
along
x-axis with respect to frame .
Suppose at 
,
the origins of two frames coincide.
Both
the observers observe the same event. The position and time of event observed
by is
denoted by 
and
position and time of the event observed by is
denoted by 
.
According to Galilean Transformation:
Fundamental Equation of Special Theory of Relativity
Consider two observers in two different inertial
frames and
.
Frame is
at rest and is
moving with uniform velocity along
x-axis with respect to frame .
Suppose at ,
the origins of two frames coincide.
Both
the observers observe the same event. The position and time of event observed
by is
denoted by and
position and time of the event observed by is
denoted by .
Consider a wave of light starts from 
and
at
with
speed .
Let the wave reaches a point P after time 
from
and
takes the time 
to
reach at P from point .
Then the distance covered by light ray from point to
point 
:
And the distance
covered by light ray from point 
to
point P:
Comparing these equations, we get:
This is the fundamental equation of special theory of relativity given by Einstein in 1905.
Galilean Transformations Doesn’t Satisfy the Fundamental Equation of Relativity
Applying
the values of 
from
Galilean Transformation in Fundamental Equation of Relativity
This is clearly
impossible until .
Hence Galilean Transformation fail to satisfy Fundamental Equation of
Relativity.
Lorentz Transformations Satisfy the Fundamental Equation of Relativity
Hence we need such transformations which satisfy Fundamental Equation of Special Theory of Relativity. Such transformations are called Lorentz Transformation. These are given below:
Derivation of Lorentz Transformation
Consider
two inertial frames of reference and
.
Frame is
at rest and is
moving with velocity in
the direction of increasing 
.
As
relative velocity is
along 
.
Moreover, and
coincides
at ,
so: 
and
.
So the fundamental equation of special theory of relativity 
becomes:
·
As
move
along with
relative velocity v with respect to ‘S’, so distance covered by with
respect to S after time will
be 
.
·
As
O also appear to move along negative with
relative velocity 
.
So the distance covered by with
respect to after
time is
equal to 
.
These two requirements can be satisfied by putting:
and
In equation (1)
and (2), if we know values of ‘
’
and ‘
’,
then we can find the relations between 
and
,
which satisfy equation (A).
To find ‘’
and ‘’,
we put the value of 
from
equation (1) in equation (2), we get:
Putting the
values of and
from
equations (1) and (3) in equation (A):
This
relation must hold for all values of and
.
So the coefficients of 
,
and
must
be zero separately. So we get three equations:
From equation (6), we have:
Now consider equation (5):
Putting values
of ‘’
and ‘’
in equation (1) and (2), we get:
Where 
is
called Lorentz’s factor.
Now putting 
in
equation (3), we get:
Conclusion:
The set of transformation equations
is called
Lorentz’s Transformation. When 
i.e.,
,
the Lorentz Transformations transform into Galilean transformation. Thus the
Galilean Transformations is special case of Lorentz Transformation.
Inverse Lorentz Transformation
The Lorentz Transformation equations are as follows:
From equation (4):
Putting the
value of in
equation (1):
Now consider equation (1):
Putting value of
in
equation (4), we get:
Hence the Inverse Lorentz Transformations are:
Important Note:
We can obtain
the inverse Lorentz Transformations just by interchanging primed and unprimed
coordinates and replacing by
.
Transformation of Velocities
The
equations of Lorentz Transformations can be used to get a relation between
velocity 
of
a particle measured by an observer in frame
and velocity 
of
the same particle measured by an observer in frame
who is moving with velocity with
respect to .
Suppose
according to 
,
particle moves from 
to
and
according to 
,
the particle moves from 
to
.
The -component
of velocity measured
by will
be:
By Lorentz Transformations, we have:
Also,
Putting values in equation (1):
The 
-component
of velocity measured
by will
be:
As 
Putting values in equation (2), we get:
Similarly,
It should be
noted that 
and
even
though 
and
.
This is another difference between Lorentz and Galilean transformations.
Inverse Velocity Transformations
We
can obtain inverse velocity transformation from equations of velocity
transformation simply by changing by
replacing
primed coordinates with unprimed coordinates and vice versa. So inverse
velocity transformations are:
Lorentz Velocity Transformations under Non-Relativistic Limit
Under
non-relativistic limit (i.e., for 
,
we put 
)
the equations of velocity transformations takes the form:
This set of equation is called Galilean Velocity Transformation. So under non-relativistic limits, the Lorentz Velocity Transformation change into Galilean Velocity Transformation.
The Lorentz Velocity Transformation and Einstein’s 2nd Postulate
We can derive the result of Einstein’s 2nd postulate from Lorentz Velocity Transformations. According to Einstein’s 2nd postulate, the speed of light is constant for all observers. So speed c measured by an observer must also be measured to be c by any other observer.
Suppose
the two observers observe a common event of passage of light beam along x-axis
in frame and
.
According to observer in ,
the velocity of light beam along x-axis 
and
.
So, according to Lorentz Velocity Transformation, the velocity measured by :
And
Therefore,
velocities measured by 
are
.
So the observer also
measures the same speed. Hence the speed of light is same for all observers.
CONSEQUENCE OF SPECIAL THEORY OF RELATIVITY
Relativity of Time
Consider
two frames of references and
.
is
at rest and is
moving with uniform velocity with
respect to .Suppose
an event occurs at one and same place ‘x’ in frame .
·
The
duration of event measured by the observer in frame is
.
·
The
duration of same event measured by the observer in frame is
.
By using Lorentz Transformation:
Now
Because
the event occurs at the same place, therefore: 
.
Equation (1) takes the form:
Since
So,
Hence
the observer in frame will
conclude that the clock in frame is
slowed down i.e., time is dilated.
Relativity of Length (Length Contraction)
Consider a rod
lying at rest along x-axis in stationary frame S. Let the coordinates of its
ends in this frame be 
and
.
Then, length 
of
the rod is called proper length and described as:
Let the length
of the rod seen in moving frame moving
with velocity be
.
Let the coordinates of ends of the rod in 
frame
of reference (FOR) are 
and
.
Then the length of
the rod observed with respect to 
will
be:
It should be noted that the measurements are made simultaneously in both frames.
By Lorentz Transformation, we have
-
Speed of Proton 
Momentum 
As 
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