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# Theory of Special Relativity

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1. Theory of Special Relativity

2. Theory of Special Relativity Einstein is recognised for: • encompassing all of the known results • restating the laws of relativity and the required modifications for the laws of Physics to be consistent

3. Theory of Special Relativity Correspondence Principle

4. Correspondence Principle • Any new theory proposed must agree with an older theory in terms of correct predictions which the theory gave. • In this case the theory of special relativity must agree with classical mechanics at low speeds.

5. Correspondence Principle • This requirement guided Einstein in developing the transformations which are the foundation of Special Relativity. • It is a good idea to verify that these equations reduce to those from classical physics.

6. Einstein’s Postulates (1905) The postulates of special relativity are: • The laws of Physics are the same in all inertial reference frames. • The speed of light in a vacuum is c in all inertial reference frames.

7. Einstein’s Postulates (1905) • Thus there is no ether (or rather no need for an absolute frame) as there is no preferred reference frame. • The Galilean transform was replaced by a Lorentz transformation.

8. Consequences of Special Relativity • Several phenomena are predicted as a result of these formulations which conflict with our everyday experiences. • Specifically an effect on length and time.

9. Consequences of Special Relativity For example: • there is no absolute length • or absolute time • events at different locations occurring simultaneously in one frame are not simultaneous in another frame.

10. Definitions Proper frame: the reference frame in which the observed body is at rest. Proper length: the length in the frame where the object is at rest wrt the observer. Proper time: the time interval recorded by a clock attached to the observed body.

11. Consequences of Special Relativity • We will be looking at three particular phenomena: time dilation, length contraction and simultaneity.

12. Consequences of Special Relativity Time Dilation

13. V S′ S Time Dilation • To illustrate how time is altered consider the case where there is a mirror attached to the ceiling of a car moving with a velocity v.

14. D Time Dilation • An observer in the car flashes a light, the path of which is shown below. • The diagram shows the path as seen by an observer in the car.

15. D Time Dilation • Time taken for the ray to be reflected to observer is

16. D Time Dilation • The time measured by this observer is the proper time since only one clock is needed to measure the time. • The length is the proper length since the observer is at rest wrt the mirror.

17. Do A B Time Dilation • An observer outside the car will see a moving mirror. • As the car travels a distance , the light moves a distance of .

18. Do A B Time Dilation • The time taken for the light to be reflected back is • Two clocks measured the time, one at A and the other at B

19. Do A B Time Dilation • The relationship between and must be found. • First using Pythagoras we determine an expression for containing :

20. Time Dilation Simplifying we get However Therefore (Time dilation)

21. Time Dilation This can be written as (Lorentz Factor) the quantity is often represented by Therefore

22. Features of Time Dilation • The time measured in the stationary frame is longer than that measured in the moving frame. • Moving clocks run slower!! This effect is called time dilation.

23. Experimental Evidence of Time Dilation Decaying Muons

24. Decaying Muons • Muons are particles produced by cosmic radiation • have a life of . • If they move at a speed of the max. distance a muon can travel is .

25. Decaying Muons • Muons are particles produced by cosmic radiation • have a life of . • If they move at a speed of the max. distance a muon can travel is . • But muons traveled .

26. Decaying Muons • How can the muons travel ?

27. Decaying Muons • How can the muons travel ? Answer • Time Dilation

28. Decaying Muons • Consider the two reference frames – the earth frame and the muon reference frame. • The time measured by an observer is the proper time since only one clock (attached to the muons) is required. • However an observer in the earth frames needs to clocks!!

29. The Earth Frame • Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. • by a factor

30. The Earth Frame • Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. • by a factor • decay time in the earth frame,

31. The Earth Frame • Because of time dilation the lifetime of a muon is longer as measured by an observer in the earth frame. • by a factor • decay time in the earth frame, • Hence available time

32. The Muon Frame • In the moving frame the muons see a shorter length to be travelled . • Achievable in their lifetime.

33. Consequences of Special Relativity Length Contraction

34. Length Contraction • length in a moving frame of reference will appear contracted in the direction of the motion. • NB: The proper length is the length measured in which the object is at rest wrt the frame of reference.

35. v E Length Contraction Consider a spaceship moving with a velocity between two stars. Two observers, one on earth at rest with respect to the stars and the other on the ship measure the distance.

36. Earth frame Length Contraction • Since at rest wrt the stars he measures a proper length . • From his frame the time for the trip is

37. v Ship frame Length Contraction • For him the star moving towards him a velocity . • Therefore measures a shorter distance. • He measures a time and distance

38. v Ship frame Length Contraction • Substituting for t0 we get (Length contraction)

39. Features of Length Contraction • A moving observer measures a contracted length in the direction of motion. • There is no contraction perpendicular to the motion.

40. Consequences of Special Relativity Simultaneity

41. Simultaneity • Unlike our everyday experiences, two events which are simultaneous in one frame are in general not simultaneous in another.

42. v O′ O A B Simultaneity The following thought experiment highlights this. • A train is moving with a velocity to the right when its ends are struck by lightning. • Two observers at and both points midway between the two ends record the occurrence via light reaching them.

43. Stationary Frame • The light from the strikes at A and B reach the observer at the same time. • Since the distance travelled by the light is the same, he concludes that the events were simultaneous.

44. v O′ O A B Train Frame • By the time the light reaches , the observer has moved (and hence his point of reference) tA tB t′B < t′A

45. Train Frame • Since the speed of light is constant, light from B′ reaches the observer first. • Hence he concludes that the events are not simultaneous. • The lightning strikes the right side first.