Albert Einstein was one of the greatest physicists of his time. He is best known in physics for
his famous theory of relativity which was formulated by him in 1905 in his article "On the
electrodynamics of moving bodies". He was also awarded Nobel Prize in Physics in 1921 for
his explanation of photoelectric effect.
Before we really deal with Special Theory of Relativity, it is necessary for us to understand
what do we actually mean by motion?
Motion:
When we say that something is moving, what we mean is that its position
relative to something else is changing. A passenger moves relative to an airplane; the
airplane moves relative to the earth; the earth moves relative to the sun; the sun moves
relative to the galaxy of stars (The Milky Way) and so on.
Non-relativistic motion and a Brief Review of Newton's Laws of Motion
Non-relativistic Motion: If a material particle performs motion with a velocity much smaller
than the velocity of light then the motion of the material particle is known as non-relativistic
motion. Such motion of the particle is described by the classical laws of motion (e.g.
Newton’s Laws of motion or the laws of Maxwell’s electromagnetic theory).
Let's review certain basic concepts of motion, namely Newton's first two Laws of
Motion, which are presumably as basic and fundamental as any natural law can be:
(i) The Law of Inertia: A body which has no force acting on it will move with
uniform motion (that is, with constant speed and direction).
(ii) The Force Law: If a force acts on a body, it will not move uniformly, but will
be accelerated in the direction of the force at a rate proportional to the force, and
inversely proportional to its inertia, or mass.
Now, these two laws appear to be very simple and obvious, perfectly reasonable and
correct. So much so, that if we see an object which is moving uniformly, we presume that
it must not have any force (or at least, any net force) acting on it; whereas if we see an
object which is accelerating, we presume it must have some force acting on it, in the
direction of its acceleration. 4 Relativistic Motion and Special Theory of Relativity:
Relativistic Motion: Motion of any material particle with a velocity comparable with the
velocity of light is known as relativistic motion. Such a motion is described by the laws of
Special Theory of Relativity.
Frame of Reference:
A frame of reference is simply that portion of the world around us, which we use to
measure the motion of moving bodies. In simple words, a frame of reference is nothing
but a coordinate system to which an observer is attached and with respect to which the
observations can be taken.
For all practical purposes, the world around us appears to be at rest, and insofar as that
statement is true, then any motion we measure relative to our surroundings is correctly
observed, and if a motion appears uniform, it must truly be uniform, and if the motion
appears non-uniform, then it must truly be non-uniform.. If so, then in any frame of reference
that is moving absolutely uniformly relative to the everyday world, the Law of Inertia will
still be correct. Depending on this fact the frames of reference are divided in to two
categories, namely Inertial and non-inertial frames of reference.
:Inertial frame of reference .
All frames of reference, in which the Law of Inertia is correct, are called inertial frames. In
such frames the material particle or object will not change its state of motion until some
unbalanced force stats acting on it.
Non-Inertial Reference Frames
Frames of reference, in which the Law of Inertia is not correct, are called non-inertiaframes.
But how in the world could we possibly find ourselves in such a situation that the Law of
Inertia would appear to be wrong? To see this, consider the case uniformly moving car. Any
measurement or observation of a moving object has an error, relative to the everyday world,
but an this error is as constant as the motion of the car, so that uniform motion still appeared
uniform, and non-uniform motion still appeared non-uniform. But what if the motion of the car is not constant? Then the error introduced into measurements of the moving object would
be changing with time, which would make a constant motion look non-uniform, and
therefore accelerated, which would suggest that a force is acting on the moving object, which
is obviously incorrect, if it is really moving with constant motion. In other words, if our
frame of reference has a non-uniform, or accelerated motion, then the Law of
Inertia will appear to be wrong, and you must be in a non-inertial frame of
reference. So, although all frames of reference which are moving uniformly relative to an
inertial reference frame are also inertial reference frames, all frames of reference which are
moving non-uniformly (are accelerated) relative to an inertial reference frame are noninertial reference frames.
In Special Theory of Relativity we deal only with Inertial frames of reference.
Event:
An event (physical phenomenon) is described/represented by a point in the frame of
reference. This point is given by the space and time coordinates which indicate the place and
time where the event has taken place.
Space and Time:
As mentioned earlier, the Special Theory of Relativity is nothing but the measurement of any
physical phenomenon in space and time. For this it is absolutely necessary to know certain
properties of space and time. According to the Newtonian concept the space is considered
to be homogeneous and isotropic ( by homogeneity we mean that all points of the space
are equivalent to each other and by isotropy we mean that all the directions in space are
equivalent which means it does not matter at which point or in which direction a
physical phenomenon takes place the observations will be sa`me) and time is absolute
i.e. time of occurrence of any physical phenomenon for different observers in different
parts of the world will be same.
Einstein was the first person who modified the laws of classical mechanics by modifying the
Newtonian concepts of space and time. According to Einstein the space is considered to be
homogeneous and isotropic (by homogeneity we mean that all points of the space are
equivalent to each other and by isotropy we mean that all the directions in space are
equivalent which means it does not matter at which point or in which direction a physical phenomenon takes place the observations will be same) but time is relative i.e.
time of occurrence of any physical phenomenon for different observers in different
parts of the world will be different.
POSTULATES OF SPECIAL RELATIVITY:-
On the basis of this modified concept of space and time, Einstein formulated two postulates
which form the basis of Special Theory of Relativity. These postulates are
(i) The laws of physics are the same for all inertial frames of reference.
(The laws by which the states of physical systems undergo change are not affected,
whether these changes of state are referred to the one or the other of two systems of
coordinates in uniform translatory motion).
(ii) The speed of light is independent of the motion of its source and in the
free has the same value in all inertial frames of reference. (Light is always
propagated in empty space with a definite velocity c that is independent of the state of
motion of the emitting body).
(Need of space time transformations):
Postulate (1) of STR tells that the laws of Nature must be independent of the choice of
inertial frames. This means that the equations expressing these laws must retain their
structure in different inertial frames. But then these equations in different frames must be
connected by some transformations laws regarding Space and time coordinates. So there
must be some space-time transformations with respect to which these equations must retain
their structure. There are two main
such transformations, namely (i). Galilean transformations and (ii). Lorentz
transformations.
GALILEAN TRANSFORMATION:
The Galilean transformation is used to transform between the coordinates of two reference
frames which differ only by constant relative motion within the constructs of Newtonian
physics. This is the passive transformation point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light due to physics
described by Einstein's theory of relativity
Basic Assumptions regarding Space-Time representation:
In formulating these transformations
1.the space was considered homogeneous and isotropic and
2.the time was considered to be absolute.
Consider two frames S and S’, S being at rest and S’ moving with a constant velocity with
respect to S. Let two observers O and O’ observing the event at any point P from S and S’
simultaneously.
x' = x − vt
y' = y
z' = z
t' = t
Validity of Galilean Transformations:
When Galilean Transformations were tested in accordance with the postulate 1, it turned
out that they are consistent with the laws of Newtonian Mechanics but fail to satisfy the
postulate 1 for the laws of electromagnetic theory in the sense that Maxwell’s equations
and the wave equation could not retain their structure in different inertial frames under
Galilean Transformations. This gave rise to following possibilities:
Possibilities:
1. Galilean Transformations and the laws of Newtonian Mechanics are correct but the
laws of electromagnetic theory should be modified.
2. Galilean Transformations and the laws of Newtonian Mechanics are correct. The laws
of electromagnetic theory are correct only in a preferred inertial frame called Ether.
3. Galilean Transformations are not the most general space time transformations. There
exist more general (than Galilean Transformations) space time transformations. This
means that the laws of Newtonian Mechanics should be modified.
The first possibility was rejected then and there because the laws of electromagnetic
theory are the experimental facts so can not be wrong.
Concept of Ether Hypothesis:
The ether (also spelled aether) was a concept in physics made obsolete in 1905 by Einstein's
theory of special relativity. The ether concept became especially predominant in the 19th
century by the work of Young and Fresnel who revived Huygens' wave theory of light. They
replaced Newton's light corpuscles by waves propagating through the ether. In order to
explain stellar aberration, first observed in the 1720s and then shown to be caused by the
velocity of the earth relative to the velocity of Newton's light corpuscles, Young (1804)
assumed ether to be in a state of absolute rest. Maxwell showed in the 1860s that light waves
are electromagnetic waves transverse (perpendicular) to the direction of the propagation of the waves. Following Young and Fresnel, Maxwell assumed that electromagnetic waves are
vibrations of the ether. In the nineteenth century made a false statement between light waves
and sound waves or any other purely mechanical disturbances. For the propagation of sound
waves a material medium (e.g. air) is necessary. If we say that the speed of sound in air is
332m/sec., it means that this is the speed which is measured with respect to reference system
fixed in air. Therefore, these physicists postulated the existence of hypothetical medium for
transmission of light and called it ether. It was supposed to fill all the space. To explain the
very high velocity of light, the density of the ether was supposed to be vanishingly small
while its elastic modulii were supposed to be quite large. They assumed that there is a fixed
frame of reference of ether in which light travels with velocity c (3×10 m/sec) in all
directions. Since the earth is moving at speed of v = 3×10 m/sec. around the sun in its orbit,
the supporters of ether theory reasoned that there must be times of the year when the earth
has a velocity of at least 3×10 m/sec with respect to the ether. In fact for the propagation of
light waves and analogy between these waves and mechanical waves is incorrect.
Thus we see that Michleson-Morley experiment discards the idea of any privileged frame of
reference or ether and suggests that velocity of light is constant in vacuum in all inertial
frames. Later fact is the root of the relativistic discussion of physical laws.
Michelson-Morley Experiment:
The apparatus used by Michelson and Morley is known as Interferometer since it depends
upon the principle of interference of light. Light from a monochromatic source S is parallel
with the lens L and is divided into portions by the half silver plate A. One portion of the
beam travels to the mirror M1 and reflected back and other portion of the beam travels
towards M2 and is reflected back. The two reflected beams interfere and the interference
fringes are viewed through the telescope.
FThe apparatus is arranged to move along the direction of earth orbit round the sun and the
speed of movement of the apparatus is equal to v. While the apparatus is floated on
mercury and can be adjusted such that it is always along the direction of the earth
orbit round the sun. The earth considered to be stationary.
This expected shift of 0.404 of a fringe could be measured easily by the experiment
arrangement. The experiment was repeated many times, but no such displacement was
observed. So the concept of Ether medium was totally discarded.
Einstein has taken up the third possibility, namely the possibility of existence of more
general than Galilean transformations which means the possible modification of the laws of
Newtonian Mechanics. Indeed such transformations exist now and are known as Lorentz
Transformations.
17 Lorentz Transformation:-
Basic Assumptions:
1. Space is homogeneous, isotropic and time is relative and they are treated on same
footing
2. speed of light is constant in all inertial frames
3. principle of superposition holds for these transformations which means that the
transformations must be linear
4. the correspondence principle holds for these transformations which means that in the
limiting case of non-relativistic motion these transformations should reduce to
Galilean transformations.
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