1. SPECIAL THEORY OF RELATIVITY

2. Inertial frame Fig1. Frame S’ moves in the +x direction with the speed v relative to frame S.

3. Galelian Transformation From figure 1 we can obtain Galelian transformation equations as x’=x-vt (1.1) y’=y (1.2) z’=z (1.3) t’=t (1.4) Therefore, v’ x =v x -v In case of light, c’=c-v (1.5)

4. Newton’s idea and inertial reference frame The classical laws of physics were formulated by Newton in the Principia in 1687. Newton Motion of a particle has to be described relative to an inertial frame in which the particle, not subjected to external forces, will move at a constant velocity in a straight line. Two inertial frames are related in that they move in a fixed direction at a constant speed with respect to each other. Time in the frames differs by a constant and all times can be described relative to an absolute time.

5. What is special about special theory of relativity Einstein published this theory in 1905. The word special here means that we restrict ourselves to observers in uniform relative motion. This is as opposed the his General Theory of Relativity of 1916; this theory considers observers in any state of uniform motion including relative acceleration. It turns out that the general theory is also a theory of gravitation.

6. Basic postulates of Special Theory of Relativity 1. The laws of physics take the same form in all inertial frames. 2. In any inertial frame, the velocity of light c is the same whether the light is emitted by a body at rest or by a body in uniform motion.

7. Lorentz Transformation Develops a set of Transformation equations directly form the basic postulates of STR. A reasonable guess for the relationship between x and x / is x / =k(x-vt) (2.1) Where k is a factor of proportionality that does not depend upon either x or t but may be a function of v.

8. Lorentz Transformation-2 Justification of the choice (2.1) 1.It is linear in x and x /, so a single event in frame S corresponds to a single event in S /. 2.It is simple, and a simple solution to a problem should be explored first. 3.It has the possibility of reducing to eqn. 1.1, which we know to be correct in ordinary mechanics.

9. Lorentz Transformation-3 Equation of Physics must be same in both S and S’, we need only change the sign of v. Therefore, x=k(x’+vt’) (2.2) The factor k must be same in both frame of reference since there is no difference between S and S’ other than in the sign of v.There will not be any change in y,y’ and z,z’ since they are normal to the direction of v.. y’=y (2.3) z’=z (2.4)

10. Lorentz Transformation-4 The time coordinates t and t’, how ever, are not equal. We can see this by substituting the value of x’ given in 2.1 into 2.2. We obtain x=k 2 (x-vt)+kvt’ Or, (2.5)

11. Assume that at t=0, t’=0 both the frames (S and S’) has same origin and a flare is set off at the common origin. Now if observers of both the frame measures the speed with which the light spreads out. Both obsevers must find same speed c, which means x=ct (2.6) x’=ct’ (2.7) Substituting both x’ and t’ in equation (2.7) with the help of (2.1) and (2.5) we obtain Lorentz Transformation-5

12. Lorentz Transformation-6 Solving the last equation for x we obtain Equating this equation with (2.6) we get (2.8)

13. Lorentz Transformation-7

14. Inverse Lorentz Transformation

15. Length contraction A rod is lying along the x’ axis of the moving frame S’. If an observer determines the coordinates of its ends to be x’ 1 and x’ 2 then the length of the rod is L 0 = x’ 2 - x’ 1 Therefore, L 0 is the length of the rod in a frame in which the rod is at rest. Now using Lorentz transformation Where L is the length measured in S

16. Length contraction-2 Therefore, L=L 0 (1-v 2 /c 2 ) 1/2 The Length of an object in motion with respect to an observer appears to the observer to be shorter than when its at rest with respect to him, this phenomenon is known as Lorentz FitzGerald contraction.

17. Time dilation Assume a clock is placed at x’ in the moving frame S’. When an observer in S’ measures a time interval t 0 =t’ 2 -t’ 1 The observer in S, will measure this interval as A stationary clock measures a longer time interval between events occurring in a moving frame of reference than does a clock in the moving frame

18. Velocity Addition-1 Let us consider something which is moving relative to both S and S’. An observer in S measures three component of velocity to be

19. Velocity Addition-2 To an observer in S’ they are

20. Velocity Addition-3 By differentiating the Lorentz transform equations for x’, y’, z’ and t’, we obtain

21. Now we can write, This is relativistic velocity transformation equation. Its Inverse transformation equation is Velocity Addition-4

22. Velocity Addition-5 By applying the same technique we can obtain transformation for V y and V z as

23. Velocity addition-6 Example: Let V / x =c, that is, if a ray of light is emitted in the moving reference frame S’ in its direction of motion relative to S, an observer in frame S will measure the velocity

24. The relativity of mass-1 Consider an elastic collision of two identical particles A and B. Particle A has been at rest at frame S and Particle B in frame S’. Then at the same instant A is thrown in the +y direction at the speed V A. While B thrown in –y’ direction at the speed V’ B where V A = V’ B

25. Relativity of Mass-2 When the two particles collide, A rebounds in –y direction at the speed V A, while B rebounds in the +y’ direction at the speed V’ B. If the particles are thrown from positions Y apart an observer in S finds the collision occurs at y=.5Y and the one in S’ finds that it occurs at y’=.5Y. The round trip time T 0 for a as measured in frame S is therefore And it is the same for B in S’,

26. Relativity of Mass-3 Since momentum is conserved in S frame, Where m A and m B are the masses of A and B respectively. In S the speed V B is Where T is the time required for B to make its round trip as measured in S. In S’, however B’s trip requires T 0 where

27. Relativity of Mass-4 We can write V B in terms of T 0 as Inserting these equations in the equation of momentum conservation we get

28. Relativity of mass-5 Our original hypothesis was that and B are identical when at rest with respect to an observer ; the difference between m A and m B therefore means, measurement of mass depends upon the relative velocity between an observer and whatever he is observing.

29. Relativity of Mass-6 Consider a similar case when V A and V B are very small. Now an observer in S will see B approach A with velocity v and make a collision and then continue on. In S m A =m 0 (rest mass of the particle) and m B =m Therefore,

30. Relativity of Mass-7 The mass of a body moving at the speed v relative to an observer is larger than its mass when at rest relative to the observer by the factor