1. SPECIAL THEORY OF RELATIVITY<br />BACKGRO UND.<br />The mentioned mechanics is a special case of relativity/relativiatic mechanics.Newtonian mechanicsis applicable in the range of velocity given by vc=0<br />Where v=velocity of the body<br /> C=velocity of light<br />Newtonian mechanicsfails in high speed region where vc=1.This difficulty was overcome by Einsteins special theory of relativity which corrects and predicts the result of mechanical experiments.<br />Macroscopic worldd is our world,where as microscopicworld is the world of particals.The basic role of makin the fundamental space as time measurement in relativistic mechanics is played by light which is an electromagnetic phenomenon.<br />GALILEAN TRANSFORMATION EQUATION<br />When the transformation of co ordinates is made from one initial frame to another initail frame of reference,it is called Galilean Transformation.<br />Suppse we want to specify a physical event occuring in space then it can be specified by (x,y,z,t).The first three co ordinates represent the position of the point where the event took place wrt three co ordinates axis whereas T represents the time at which the event took place.The same event can be localised with respect to another frame of reference(X’,y’,z’,t’).<br />In this case we will consider our frame of reference to be initial frame of reference in whihch Newtons laws of motion are valid,such a frame of reference is an unaccelerated system or it is the one moving with uniform velocity and the body in it experience zero force.<br />Consider two initial frames of reference S and S’ with their respective axis parallel to each other with the origin at 0 and 0’.The frame of reference S’ is moving with with a uniform velocity v relative to S along its x-axis.Assuming the physical event to be occuring at point P which can be located by S frame S as (x,y,z,t) and by frame S’ as (x’,y’,z’,t’).<br />Here we want to find out the relation between two set of co ordinates,classical approach will allow us to consider space and time intervals to be absolute,that means they are same for all intervals observed.<br />If the origin O and O’ of two frames of reference coincide at t=0,then the Galilean Transformation equation are <br />X’=x-vt<br />Y’=y<br />Z’=z<br />T’=t<br />Similarly the inverse Galilean transformation equation are given as <br />X=x’-vt<br />Y=y’<br />Z=z’<br />T=t’<br />It follows from transformation equation that,1)The time interval between two occurance of two events at point Pand Q should be same wrt two frames of reference S and S’ ie.<br /> t’p-t’q =tp-tq<br />2)The space interval between two points Pand Q measured at a given instant of time is same wrt frame of referance S and S’<br /> x’p=xp-vtp<br /> x’q=xq-vtq<br />(x’p-x’q)=(xp-xq)-v(tp-tq)<br />But tp=tq<br /> x’p-x’q=xp-xq.<br />Thus according to galilean transformation time intervals and space intervals are absolute ie they are same for all intervals observed irrespective of the relative velocity v of the frame of reference.Adding to classical concep that mass of a body is constant and is independent of velocity of body.<br />We conclude that classical mechanics and galilean transformation imply that length mass and time are all independent of the relative motion of the observers.<br />NEWTONIAN RELATIVITY<br />Starting from eq.x’=x-vt , we can get the time derative and hence the velocity transformation eq.<br />dx'dt=dxdt-vt<br />Acc to galilean transformation<br /> t=t’<br />ddt=ddt'<br />dx'dt=dx'dt'<br />dx'dt'=dxdt-v<br />U’x=ux-v<br />From the eq the transformation eq for acceleration is given by,<br />du'xdt=ddt(ux-v)<br />du'xdt'=ddt(ux-v)<br />Or a’x=ax <br />Also a’y=ay <br />General formula,a’=a<br /> <br /> Adding to the idea of anconvience of mass,then, F’=ma’<br /> F=ma<br /> F=F’ <br />The Newtons laws of motion and eq of motion of a partical should be exactly the same ain all initial frames.<br />Since conservation laws of classical mechanics arre all consequences of Newtons laws,it follows that “the laws os mechanics are the same in all initial frames”.In other words,in all initial frames of reference,Newtons laws and conservation principles are invarient.The two important consequences of the above result are,<br />1)No mechanical experiment carried out entirely in one initial frame can tell the observer what the motion of that frame wrt any other initial frame<br />2)There is no way that all of dermining the absolute velocity of an initial frame of reference from any mechanical experiment.<br />Since the laws of mechanics are same in all initial frames,no one initial frame is prefered over any other.<br />All initial frames are equivalent as fas mechanics is concerned,we can only speak of relative velocity of one frame wrt another frame.This is called Newtonian Relativity Quantities which remain unchanged driving transformation are called Invariance of Transformation.<br />ELECRO-MAGNETISM & NEWTORIAN RELATIVITY<br />Though the laws of mechanics are invarent under galilean transformation,the laws of electro-dynamincs are not.The speed of light is not invarent under galilean transformation.The velocity of light in a hypothetical medium either <br />C=1MoEo and is equal to 2.997825x108<br />Taking the medium as an initial frame S and observer in another frame S’,movin with a constant velocity v,relative to S,measures the speed of light C±v depending upon the direction of relative motion.This is as per the galilean transformation.Since the quantity approaches in the Maxwell’s equation of electro dynamics according to galilean transformation,electromagnetic effec may not be the same for different initial observers which means Maxwell’s eq are not invarent under galilean transformation<br />We thus conclude that either,<br />a)A relativity principle exist only for mechanics but not for electrodynamics in which case,either frame can be treated as an absolute initial frame.If this is the case we must be able to detect the presence of either experimentally<br />OR<br />b)A relativity principle exist both for mechanics and electrodynamics,and Newtons laws of mechanics are incorrect.In that case we must be able to show experimentally the incorrect in Maxwell’s formula .In this case of galilean transformation holds good.<br />OR<br />c)A relativity principle exist both for mechanics and electrodynamics,and Newtons law of mechanics are incorrect.In this case,we must be able to prove experimentally the derivation form.Newtons laws of mechanics,If this is true the galilean transformation laws are not correct laws,and we will have to formulate transformation laws,which will help the Maxwell’s laws invarent.Since Newtonian mechanics breaks dows in a high speed region uc->1,we finally conclude that the third alternative is the correct one.This alternative leads to Einstein’s relativity.<br />