Length Contraction, Length Contraction in Relativity, What is Length Contraction?

LENGTH CONTRACTION IN RELATIVITY

When a body moves with very high speed v relative to an observer then it appears to be contracted by a factor in the direction of motion only. This concept is known as length contraction and this shows clearly that length is not absolute but relative.

Let us consider two inertial frame s and s’. s’ is moving with velocity v relative to s along +x direction. Frame s is at rest. Let a rod AB is laying along the x axis in frame s. If x1 and x2 be coordinate of ends A and B respectively then the length L0 of the rod in frame s is given by 

 Let an observer in s’ measure the length of the same rod and its x1’ and x2’ are the coordinate of end A and B of the rod at the same time then the length of the rod L will be given by

According to Inverse Lorentz Transformation we have

Hence the length of rod moving with velocity v relative to the observer is contracted by a factor  in the direction of motion. This phenomenon is also called Lorentz-Fitzgerald Contraction.

Lorentz Transformation, Lorentz Transformation Equation, What is Lorentz Transformation?

LORENTZ TRANSFORMATION, LORENTZ TRANSFORMATION EQUATION, WHAT IS LORENTZ TRANSFORMATION?

Lorentz Transformation is a set of equations which correlates space-time coordinate of an event occurring in space as observed by two observers in different frame of reference moving with constant velocity relative to each other.

To derive equation let us consider two inertial frame s and s’. s’ is moving with velocity v, relative to s along the direction of x-axis. Let us consider that at an instant the origin o and o’ of two frames. Suppose a pulse of light is generated at t=0 at the origin. It will spread into the space and consider the situation when pulse reaches at P. Observers at o and o’ measure coordinate of P as (x,y,z,t) and (x’,y’,z’,t’). When pulse is observed by an observer at s then 

As (z,z’) and (y,y’) axes are perpendicular to the direction of motion coordinate will remain unaffected.

Now y = y’ and z = z’

So that from Eq. (1) and (2), we get

Where k is independent of x and t.

If we suppose that the system s is moving relative to s’ with velocity v along +x direction. Then

Putting Eq. (4) into Eq. (5) we get

Putting x’ from Eq. (4) and t’ from Eq. (6) in Eq. (3) we get

This is an identity and hence the coefficient of various power of x and t must vanish separately. Equating the coefficient of t*t to zero, we get

Putting this value in Eq. (5) we get

Also putting in Eq. (6), we get the equation

Therefore Lorentz Transformation Equations are:

Inverse Transformation Equations are:

Michelson-Morley Experiment, Michelson Interferometer, Morley Interferometer

Michelson-Morley Experiment, Michelson Interferometer, Morley Interferometer

ETHER HYPOTHESIS

A material medium is a necessity for the propagation of waves. It was considered that light propagate through ether as the sound wave propagate through air. Ether affects all space. An interesting question is whether relative motion between the earth and ether can be detected. If such a motion can be detected, we can choose a fixed frame of reference in stationary ether. In 1887, Michelson and Morley set out to measure the relative velocity of earth with respect to the ether. The principle of the experiment lies is nothing, the shift in fringes in Michelson Interferometer due to the difference in time taken by light to travel along and opposite the direction of motion of the earth. The time taken by a beam of light to travel along the direction of motion of earth is greater than that of traveled  distance opposite to the direction of earth. Surprisingly, despite best efforts, the presence of ether could not be detected.

MICHELSON-MORLEY EXPERIMENT

The main aim of Michelson-Morley Experiment was to find evidence confirming the presence of an absolute Frame of Reference called Ether. Ether was thought of as a medium which fill all spaces and penetrates all matter. It was considered to be rigid, Invisible, Mass less and perfectly transparent elastic, non-resistive, continuous and with negligible density.

Michelson’s Interferometer consists of two plane mirrors M1 and M2 arranged perpendicular to create coherent sources by division of amplitude. The light from the sources goes to the beam splitter P and then divides itself to go to M1 and M2 separately. The reflected light mirror M1 and M2 is allowed to interfere and interference fringes are observed.

The two mirrors M1 and M2 are placed at equal distance  from the plate. It is assumed that the earth is moving through the ether with velocity along the direction of the incident light.

Due to the motion of the apparatus with the motion of earth, the time taken by ray 1 travelling to mirror M1 and ray 2 travelling to mirror M2 would not be the same.

If c is the velocity of light through ether, then using Galilean Transformation, the velocity of light along the direction of incident light is c-v for light from P to M2 and c+v for light from M2 to P also 

The whole apparatus was rotated by 90 degree and the fringe pattern was observed to measure the amount of fringe shift.

Due to rotation of apparatus,

The expected fringe shift was evaluated using the formula in Equ ..(6) and was calculated to be 0.4 but actually even though the experiment was done very precisely and number of times, no fringe shift was observed. This null result could not be explained by Michelson and Morley at that point of time.

EXPLANATION OF NEGATIVE RESULT

1.   ETHER-DRAG HYPOTHESIS: The hypothesis assumed that ether moved along with the earth while the later moved through space. Thus there was no relative motion between the earth and ether. This implied that the fringe shift goes to zero.

2. FITZGERALD-LORENTZ CONTRACTION HYPOTHESIS: Lorentz and Fitzgerald put forth the suggestion that there was contraction of bodies along the direction of their motion through the ether by a factor of 

.This hypothesis was rejected because it was purely mathematical and had no experimental evidence to support itself.

3. CONSTANCY OF THE SPEED OF LIGHT: In order to preserve the ether hypothesis it was proposed that light travel with a constant speed with respect to the source. This was in contradiction to the wave theory, thus a negative result of the Michelson-Morley experiment could not be explained through the proposed hypothesis at that time.

Frame of Reference – Inertial and Non Inertial Frame of Reference

FRAME OF REFERENCE – INERTIAL AND NON-INERTIAL FRAME OF REFERENCE

FRAME OF REFERENCE

A frame of reference is a coordinate system relative to which the position and motion of a body may be described.

“So, a frame of reference is a part of the description of the motion.”

The simplest frame of reference is the Cartesian system of coordinates, in which the position of the particle or body is specified by its three coordinates x, y, z along the three perpendicular axis.

If at any instant, the position vector drawn from the origin to the particle is given by 

The position vector should be expressed as function of time, then the velocity and acceleration of the particle can be determined as 

So in general for the location of an event in frame of reference we require its position and time of occurrence for a coordinate (x, y, z, t).

TYPES OF FRAME OF REFERENCE

Frame of Reference is of two types:

1.      INERTIAL FRAME OF REFERENCE

When bodies in a frame of reference obey Newton’s Law of Inertia and Other Law of Newtonian Mechanics, This Frame is called Inertial Frame of Reference. In this case, No external force acts on the body. It is either at rest or moving with constant velocity.

Thus in an inertial frame, if a body does not experience any external force, its acceleration a is given by

IS EARTH AN INERTIAL FRAME?

The earth is not really an inertial frame of reference because of its orbital motion about the sun and rotational motion about its own axis. In both these motion, Centripetal acceleration are present, so any frame of reference set up earth or earth itself, as is where is cannot be considered as an inertial frame of reference but in the present study we are considering speed of the order of 3 lakh km per second and earth is moving only with 30 km/s, so the effect of and revolution of earth may be ignored. Thus for all practical purposes, earth or any frame of reference set up on earth is regarded as an Inertial Frame of Reference.

2.      NON-INERTIAL FRAME OF REFERENCE

When a body is accelerated in a frame of reference, the frame of reference is called Non-Inertial Frame of Reference. The law of Inertia is not valid for an accelerated Frame of Reference.

The simplest example of a Non-Inertial Frame is provided by a rotating merry-go-round.

Theory of Relativity - Relativistic Mechanism - Einstein Theory of Relativity

THEORY OF RELATIVITY, EINSTEIN THEORY OF RELATIVITY

The theory that deals with relativity motions of bodies is called Theory of Relativity. The Theory of Relativity shows that the Classical Mechanics is not applicable to bodies moving with velocity nearer to the velocity of light. According to Classical Theory, Space and Time are absolute. The flow of time is assumed to be confirmed in all situations, the movement of time and time intervals are identical in all frames of reference and that, similarly, lengths are also identical in all frames. But these concepts are not valid at high velocities. The theory of Relativity provides many new concepts such as:

1)      The length of a moving body contracts in the direction of motion.

2)      Clock in motion slow-down.

3)      At high speed, speed and time are no more separate but merge into space-time continuous.

4)      The mass of a moving body is greater than its mass when at rest.

CONCEPT OF RELATIVITY

(A)   Let us consider a river flowing and two persons (A and B) are standing on the banks of the river facing towards each other. Now the person A will say that house is on the right side while the Person B will say that house is on the left side. But actually the position of house is fixed. This indicates that position is relative.

(B)   If someone asks whether ball is large or small? If we compare the ball with earth then we will say that ball is small but if we compare the ball with an atom, then we will say that ball is large. This indicates that size is relative.

Similarly time and motion are also relative.