Theory of relativity Albert Einstein Some important definitions

Theory of relativity Albert Einstein

Some important definitions Event: A sudden occurrence of a happening in space at a given time Observer: Anything which measure the happening Frame of reference: A system of co-ordinate to measure the event Inertial frame of reference: A frame of reference in which Newton’s law of motion are applicable. Every frame of reference either at rest or uniform motion are inertial Non-Inertial frame of reference: Newton’s law of motion are no longer valid Earth rotate around the sun and about its on axis with an angular acceleration very-2 small so for practical purpose, it can be assumed as inertial frame of reference

Galilean Transformations Position and velocity are variant under Galilean transformation Acceleration is invariant under Galilean transformation Galilean Transformations Inverse Galilean Transformations

The Michelson Morley experiment set up

Light rays motion The time difference between two ray reaching at observer is given by ∆t= lv 2/c 3 And path difference is given by =∆x= lv 2/c 2

If the whole system is rotated through 90 then maximum path difference = 2 lv 2/c 2 The condition of bright fringes is given by 2 lv 2/c 2= nλ n = (2 lv 2/c 2)/n In Michelson Morley experiment l=11 m, v = 3 X 104 m/s and wavelength of light = 5. 5 X 107 m N=0. 4 Experimentally no fringe shift was observed in experiment. The experiment concluded that there is no relative motion between earth and ether

Interpretation of negative results �Michelson view: The ether is dragged along the earth �Fitzgerald and Lorentz View: Bodies contract along the direction of motion �Einstein View: Velocity of light in space is universal constant

Einstein theory of relativity 1. All the law of physics are same in all inertial frame of reference either at rest or in uniform motion 2. The sped of light in the free space is same for all observer regardless of their state of motion Lorentz transformation Equations Based upon theory of relativity, Lorentz derive transformation equations given by Derive Lorentz transformation Equations

Proper Time And Time Dilation

Proper Length and Length Contraction � Measure positions at endpoints at same time in frame S’ and in frame S, L’=x 2’-x 1’

Special-relativistic length contraction Both 1 meter Meter sticks Frame 2 Vertical: 1 meter; horizontal: shorter than 1 meter. V close to c Frame 1 11

Special-relativistic length contraction (continued) Both observers measure lengths instantaneously. y Frame 2 V Frame 1 x 12

Velocity addition 3 -D Velocity

Variation of mass with velocity According to classical mechanics mass of a body does not change with velocity. But according to relativistic mechanics mass of a body does not remains constant but varies with velocity according to relation Derive it At c = 0, m = m 0 The body at rest have a specific mass at v = c, m 0 = 0 The rest mass of body is zero Consider Photon with rest mass zero and Having energy hυ

If u 1 and u 2 are the velocity of body relative to O’ given by While at instant of collision total mass = m 1+m 2 and velocity = v Applying law of conservation of momentum we have M 1 u 1+m 2 u 2 = (m 1+m 2)v After solving we have If body b is at rest u 2 = 0 and m = mo. The system is reduced to a single body of mass mo moving with velocity u 1 = v We have

MASS-ENERGY EQUIVALENCE RELATION Consider the example of fire produce from wood burning, the mass of ash is less than the mass of wood Why What are the source of light and heat energy in combustion process? What are the source of nuclear energy? Answer is given in Einstein theory of relativity According to mass-energy equivalence mass is converted into energy and vice versa according to relation Prove it

Examples of mass energy equivalence 1. Pair production: Pair of electron and positron are formed when gamma radiation of energy 1. 02 Mev interact with nucleus. 2. Pair annihilation: A positron and electron combined with each other and a photon of energy 1. 02 Me. V is produced. 3. Mass defect: During a nuclear reaction the mass of initial reactants is more than that of product. The difference of mass is called mass defect. Which is converted into energy during nuclear reaction.