Phys-103-Lec-3-Special-Theory-of-Relativity-2025.pptx

PHYSICS

1. 1

2. 2 The Lorentz-FitzGerald Contraction Another hypothesis proposed independently by both H. A. Lorentz and G. F. FitzGerald suggested that the length ℓ 1 , in the direction of the motion was contracted by a factor of …thus making the path lengths equal to account for the zero phase shift. This, however, was an ad hoc assumption that could not be experimentally tested.

3. 3 The Lorentz Transformations The special set of linear transformations that: preserve the constancy of the speed of light ( c ) between inertial observers; and, account for the problem of simultaneity between these observers known as the Lorentz transformation equations

4. 4 Remarks If v << c , i.e., β ≈ 0 and ≈ 1, we see these equations reduce to the familiar Galilean transformation. Space and time are now not separated. For non-imaginary transformations, the frame velocity cannot exceed c .

5. 5 Time Dilation and Length Contraction Time Dilation : Clocks in K’ run slow with respect to stationary clocks in K. Length Contraction : Lengths in K’ are contracted with respect to the same lengths stationary in K. Consequences of the Lorentz Transformation:

6. 6 Time Dilation To understand time dilation the idea of proper time must be understood: The term proper time , T , is the time difference between two events occurring at the same position in a system as measured by a clock at that position. Same location

7. 7 Not Proper Time Beginning and ending of the event occur at different positions Time Dilation

8. 8 1)  T ’ > T or the time measured between two events at different positions is greater than the time between the same events at one position: time dilation. 2) The events do not occur at the same space and time coordinates in the two system 3) System K requires 1 clock and K’ requires 2 clocks. Time Dilation

9. 9 1)  Time Dilation occurs due to the special property of light having a constant maximum speed 2) As an object approaches the speed of light, time relative to other frames of reference slows down. 3) One of the odd results of this it that anyone on the fast moving object and anyone in a different frame of reference will feel time pass at a natural rate Time Dilation

10. 10 4) However, it also appears that time slows down for the other frame of reference. Scientists have confirmed that time dilation exists by placing a very precise atomic clock on a jet plane and flying it for a period of time, while placing an identical clock in a stationary lab. Time Dilation

11. 11 Time Dilation

12. 12 The clock on the aircraft had lost a bit of time compared to the clock left in the stationary lab. Time dilation formula: Time Dilation

13. 13 t = the dilated time interval is measured by the observer that is in motion with respect to the event = this observer views the event (the fast moving object) from a different place. t = the proper time interval is measured by the observer at rest with respect to the event = this observer views the event (the fast moving object) from the same place. Time Dilation

14. 14 1. An astronaut is travelling at a constant speed of relative to the Earth toward a distant star. If this trip takes 25 years as measured by an observer on Earth, how long does it take relative to the observer on the space vehicle?   Time Dilation

15. 15 = time measured in the astronaut’s frame (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

16. 16   Time Dilation

17. 17   Time Dilation

18. 18 2. An astronaut is travelling at a constant speed of relative to the Earth through space. According to the time devices on the space vehicle, her trip lasted 0.500 years. How long did this trip last relative to an observer on Earth?   Time Dilation

19. 19 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

20. 20   Time Dilation

21. 21   Time Dilation

22. 22 3. You take a trip through space and when you return to Earth you have aged 4.0 years. Your friend who remained on Earth has aged 11.0 years. How fast was your spaceship travelling on your trip? Time Dilation

23. 23 (proper time) time measured in Earth’s frame Your speed speed of light   Time Dilation

24. 24   Time Dilation

25. 25   Time Dilation

26. 26 4. An astronaut is traveling at a constant speed of relative to Earth through space. According to timing devices aboard the space vehicle, the trip took . How long did the trip take if measured relative to Earth?   Time Dilation

27. 27 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

28. 28   Time Dilation

29. 29   Time Dilation

30. 30 5. An astronaut is travelling at a constant speed of through space relative to Earth. According to timing devices aboard the space vehicle the trip lasted years. How long did the trip take is measured relative to Earth?   Time Dilation

31. 31 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

32. 32   Time Dilation

33. 33   Time Dilation

34. 34 6. An astronaut is traveling at a constant speed of measured relative to Earth through space. If this trip takes as measured on Earth, how long does it take relative to the space vehicle?   Time Dilation

35. 35 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

36. 36   Time Dilation

37. 37   Time Dilation

38. 38 7. You are traveling through space at a constant speed of relative to Earth. If your friend who stayed back on Earth aged during the time of your trip, how many years did you age?   Time Dilation

39. 39 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

40. 40   Time Dilation

41. 41   Time Dilation

42. 42 8. Your teacher is taking your class on a field trip to a distant planet. The space bus is traveling at a constant speed of 0.96c relative to the Earth. Your teacher prepared an exam before leaving the Earth that was set to take 2.5 hours. If the students complete this exam while on the field trip, how much time passes for those that have stayed behind at the school on Earth while those on the space bus write it? Time Dilation

43. 43 (proper time) time measured in Earth’s frame speed of the astronaut speed of light   Time Dilation

44. 44   Time Dilation

45. 45   Time Dilation

46. 46 Length Contraction To understand length contraction the idea of proper length must be understood: Let an observer in each system K and K’ have a meter stick at rest in their own system such that each measure the same length at rest. The length as measured at rest is called the proper length .

47. 47 Length Contraction Length contraction is considered when an object has traveled with the velocity of light. So, relativity arrives into the picture. Therefore, one can say that length contraction happens when an object is traveling at the speed of light. This is described as the decrease in length if a body is traveling with the velocity of light linked to the observer.

48. 48 Length Contraction Formula: Where L = relativistic distance L = proper length

49. 49 Length Contraction Formula: An observer on Earth measures the length of a spacecraft travelling at speed of 0.700c to be 78m long. Determine the proper length of the spacecraft

50. 50 Length Contraction Formula: Where relativistic distance speed of spacecraft proper length  

51. 51 Length Contraction Formula: An object at rest is 5 m long, but when it drives past a stationary observer, the observer measures it to be only 4.5 m long. Determine how fast the object is moving

52. 52 Length Contraction Formula: Where proper length (at rest) relativistic distance speed  

53. 53 Length Contraction Formula: A spacecraft passes you at a speed of 0.8c. The proper length of the spacecraft is 120 m. Determine the length that you measure as it passes you.

54. 54 Length Contraction Formula: Where proper length (at rest) relativistic distance speed  

55. 55 Length Contraction Formula: Assignment 1. A car with proper length of 2.5 m moves past you at speed of v, and you measure its length to be 2.2 m. Determine the speed of the car

56. 56 Length Contraction Formula: 2. A space vehicle that is 50.0 m long is travelling at a constant speed of 0.75c past an observer standing on Earth. How long does the space vehicle appear to the observer?

57. 57 Length Contraction Formula: 3. A space vehicle is travelling at a constant speed of 0.955c parallel to a brick wall that is 115 m long. How long does the wall appear to an observer in the space vehicle?

58. 58 Length Contraction Formula: 4. A space vehicle that is 35.0 m long is traveling at a constant speed of m/s past an observer standing on Earth. How long does the space vehicle appear to the observer?