RELATIVITY

TOPICS TO BE DISCUSSED LIMITATIONS OF NEWTON’S LAWS OF MOTION EINSTEIN SPECIAL THEORY OF RELATIVITY SOME IMPORTANT DEFINITIONS FRAME OF REFERENCE INERTIAL FRAME OF REFERENCE NON INERTIAL FRAME OF REFERENCE GALILEAN TRANSFORMATION

LIMITATIONS OF NEWTON’S LAW OF MOTION In classical physics,space time and mass are regarded as absolute .This means that i) length of an object is independent of condition under which it is measured such as position and motion of object or the observer. ii) The time interval between two events has the same value for all observers. iii) The mass of a body is constant and is independent of its motion relative to an observer. Newtonian mechanics fails to explain the motion of particles which are moving with high speeds close to velocity of light.

EINSTEIN’S THEORY OF RELATIVITY All the limitations of Newtonian mechanics led Einstein to modify the concept of space, time and mass. He introduced the new modified laws in order to get consistent results in all situations. According to Einstein, absolute space,time and mass have no meaning .Space and time are relative.

SOME IMPORTANT DEFINITIONS PARTICLE:- A particle is a very small quantity of matter having practically no linear dimensions,but only a position. Measure of the quantity of the matter is mass of the particle. EVENT:-An event is defined as a simple or complicated occurrence taking place in space at a given time .For example collision between two cars on the road, lightning in the sky etc. OBSERVER:-An observer is a person or an equipment meant to observe or measure events.

FRAME OF REFERENCE A frame of reference is a system of co-ordinate axes which specify the position of a particle or an event in two or three dimensional space. The simplest and the most commonly used frame of reference is the Cartesian system of co- ordinates with observer at the origin. It is not essential that the position of observer should coincide with that of origin .However, it is convenient to do so.

INERTIAL FRAME OF REFERENCE Inertial Frame of reference, is that frame in which Newton’s laws of motion hold good , i.e. it moves with a constant velocity when there is no net force acting on it. Thus an inertial frame of reference is a non- accelerating frame of reference.

Let, S be an inertial frame of reference with co-ordinate axes OX,OY and OZ and the origin O. S’ is another frame of reference with origin O’ and co- ordinate axes OX’,OY’ and OZ’.

Let the two frames have their origin coincident at a certain time ,say ,t=0. After time t ,S’ moves to a position shown in fig. so that displacement of origin O’ is R=vt Let r and r’ represent position vectors of a particle P with reference to frames S and S’ respectively. then r=R+r’ Or r’=r-R=r-vt dr’/dt=dr/dt-v And d2r’/dt2=d2r/dt2 (the velocity v being constant) d2r/dt 2 and d2r/dt2 are the accelerations of the particle in frames of reference S and S’ respectively. Thus acceleration of the particle as measured in two frames of reference is always same i.e.,if the particle is at rest in inertial frame S, it would appear to be at rest in another inertial frame S’ also. Thus inertial frame of reference are non-accelerating frames.

NON INERTIAL REFERENCE FRAME When a frame of reference is accelerated relative to an inertial frame,the form of basic physical laws such as Newton’s 2nd law , becomes completely different. Such relatively accelerated frames of reference are known as non inertial frames of reference.

Consider a non inertial frame of reference S’ moving with an acceleration ao relative to any stationary frame s.then all the particles which are stationary w.r.t. frame S ,have acceleration –ao in frame S’ . Now if a particle of mass m moves with acceleration ai in frame S ,the apparent acceleration of the particle as observed in S’=ai-ao Hence the apparent force acting on the particle in frame S’ is given by F=ma=m(ai-ao)=mai-mao F=mai+Fo When mai=0 , F=Fo The force Fo=-mao is called fictitious or pseudo force. Thus a force which doesn’t really exist but appears only due to relative acceleration of the frame of reference is called fictitious force. So a non inertial frame is either a frame having uniform linear acceleration or a uniformly rotating frame.

GALILEAN TRANSFORMATION Galilean transformation is a set of equations, which represents how the co-ordinates of an event, which occurs in space at any time,in inertial frame of reference are related to the co-ordinates of same event in another frame of reference, moving with constant velocity relative to the former frame.

DERIVATION OF GALILEAN TRANSFORMATION EQUATIONS S is an inertial frame of reference with origin at O. Let it be at rest. S’ is another inertial frame of reference with origin O’. It is moving with constant velocity v along the positive direction of X or X’ axes . Let time be measured from the instant when origin O’ just coincides with origin O. Let an event occur in space at a point P at any instant. The co-ordinates of P observed by an observer in S are x,y,z,t . Similarly co-ordinates of same event as observed by an observer in S’ are x’,y’,z’,t’.

In time t , S’ covers the distance OO’ =vt along positive direction of x - axes i.e. O’A=OA-OO’ But O’A=x’,OA=x and OO’=vt So, x’=x-vt As there is no relative motion of S’ along Y and Z axes therefore y’=y and z’=z But time is absolute in classical relativity so t’=t Hence Galilean transformation are given by x’=x-vt y’=y z’=z t’=t

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