1. Faster-Than-Light Paradoxes in Special Relativity
Brice N. Cassenti

2. Faster-Than-Light Paradoxes in Special Relativity
Principle of Special Relativity
Paradoxes in Special Relativity
Relativistic Rockets
Conclusions

3. Principle of Special Relativity
Postulates
Lorentz Transformation
Lorentz Contraction
Time Dilation

4. Postulates of the Special Theory of Relativity
The laws of physics are the same in all inertial coordinate systems
The speed of light is the same in all inertial coordinate systems

5. Inertial Reference Systems

6. Lorentz Transformation Derivation
Assume transformation is linear
Assume speed of light is the same in each frame

7. Relative Motion
(x,ct) v
(x’,ct’)

8. Lorentz Transformation
Inverse transformation
Use of hyperbolic functions

9. Coordinate Systems ct ‘ ct x ‘ x

10. Lorentz Contraction
Length is measured in a frame by noting end point locations at a given time.
When observing a rod in a moving frame, with its length in the direction of motion the frame, the length is shorter by:
The result holds in either frame.

11. Time Dilation
Time is measured in a frame by watching a clock fixed in the moving frame.
When observing a clock in a moving frame the clock runs slower by:
The result holds in either frame.

12. Paradoxes in Special Relativity
Pole Vaulter
Twin Paradox
Faster-than-Light Travel
Instant Messaging

13. Pole Vaulter & Barn
15 ft
10 ft

14. Pole Vaulter & Barn v
7.5 ft
10 ft

15. Pole Vaulter & Barn v
15 ft
5 ft

16. Pole Vaulter Paradox Barn View
Front door opens

17. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn

18. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn

19. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes

20. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole entirely inside barn

21. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole entirely inside barn
Back door opens

22. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole entirely inside barn
Back door opens
Front end of pole leaves barn

23. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole entirely inside barn
Back door opens
Front end of pole leaves barn
Back end of pole leaves barn

24. Pole Vaulter Paradox Barn View
Front door opens
Forward end of pole enters barn
Back end of pole enters barn
Front door closes
Pole entirely inside barn
Back door opens
Front end of pole leaves barn
Back end of pole leaves barn

25. Pole Vaulter Paradox Pole Vaulter View
Front doors opens

26. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn

27. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens

28. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn

29. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Both ends of pole never in barn simultaneously

30. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Both ends of pole never in barn simultaneously
Back end of pole enters barn

31. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Both ends of pole never in barn simultaneously
Back end of pole enters barn
Front door closes

32. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Both ends of pole never in barn simultaneously
Back end of pole enters barn
Front door closes
Back end of pole leaves barn

33. Pole Vaulter Paradox Pole Vaulter View
Front doors opens
Forward end of pole enters barn
Back door opens
Front end of pole leaves barn
Both ends of pole never in barn simultaneously
Back end of pole enters barn
Front door closes
Back end of pole leaves barn

34. Pole Vaulter Paradox Resolution
Use Lorentz transformation
In barn view, time interval from back door closing to front door opening is less than time light needs to cover distance
In pole vaulter view, time from front of pole to leave and back to enter is less than time light needs to cover distance along pole
Lorentz transformation correctly predicts both views correctly from both the barn and pole vaulter frames

35. Pole Vaulter Paradox - Conclusion
Faster-than-light signals would allow contradictions in observations

36. Twin Paradox
One of two identical twins leaves at a speed such that each twin sees the other clock running at half the rate of theirs
The traveling twin reverses speed and returns
Does each think the other has aged half as much?

37. Twin Paradox Resolution
In order to set out turn around and stop at the return the traveling twin accelerates.
The traveling twin feels the acceleration.
Hence, the traveling twin is not in an inertial frame.
The stationary twin ages twice as much as the traveling twin.

38. Twin Paradox - Resolution Constant Acceleration

39. Twin Paradox - Resolution Infinite Acceleration

40. Twin Paradox - Conclusion
Accelerating reference frames need to be treated with a more
General Theory of Relativity

41. Faster-Than-Light Travel

42. Faster-Than-Light Travel

43. Faster-Than-Light Travel

44. Faster-Than-Light Travel

45. Faster-Than-Light Travel

46. Faster-Than-Light Travel

47. Faster-Than-Light Travel

48. Faster-than-Light Travel – Conclusion
Faster-than-Light motion reverses travel through time – a time machine

49. The Mathematics of Faster-than-Light Travel
Recall
Setting
Then If
Then
Let
And Or

50. Instant Messaging

51. Instant Messaging

52. Instant Messaging

53. Instant Messaging

54. Instant Messaging

55. Instant Messaging

56. Instant Messaging

57. Instant Messaging

58. Instant Messaging – Conclusion
Instant communications is a nonlinear process (exponential) and does not satisfy the postulates of Quantum Mechanics.
The number of messages must collapse to a single message.
Faster-than-light travel may result in the collapse of the wave function

59. Relativistic Rockets Relativistic Energy Constant Acceleration Rocket
Photon Rocket

60. Relativistic Energy

61. Relativistic Energy

62. Relativistic Energy

63. Relativistic Energy

64. Relativistic Energy

65. Relativistic Energy

66. & constant in all reference frames
Relativistic Energy
There is a relativistic rest mass energy
For is imaginary
Particles are tachyons
Tachyons have never been observed
& constant in all reference frames ,

67. Relativistic Accelerating Rocket

68. Relativistic Accelerating Rocket

69. Relativistic Accelerating Rocket

70. Relativistic Accelerating Rocket
In rocket frame

71. Relativistic Accelerating Rocket
In rocket frame

72. Relativistic Accelerating Rocket
In rocket frame

73. Relativistic Accelerating Rocket
In rocket frame

74. Constant Acceleration Rocket

75. Constant Acceleration Results
In the reference frame of the rocket when sinhq>1, dx/dt>c.
Accelerating at one gravity a crew could circumnavigate the universe within their working lifetime.

76. Photon Rocket

77. Photon Rocket

78. Photon Rocket

79. Photon Rocket

80. Photon Rocket

81. Photon Rocket

82. Photon Rocket Results
Velocity parameter replaces velocity in the ‘rocket equation’

83. Photon Rocket Results
Velocity parameter replaces velocity in the ‘rocket equation’
Circumnavigating the universe at one gravity requires an enormous mass ratio

84. Photon Rocket Results
Velocity parameter replaces velocity in the ‘rocket equation’
Circumnavigating the universe at one gravity requires an enormous mass ratio
But the mass required is not larger than the mass of the universe

85. Conclusions
Faster-than-light paradoxes in the Special Theory of Relativity are:
More than curiosities
They can provide a better understanding of space and time
They can add insight into other paradoxes
Collapse of the wave function
They can lead to new physical theories
And may allow unlimited access to the universe.