Faster-Than-Light Paradoxes in Special Relativity

1
Faster-Than-Light Paradoxesin Special Relativity

• Brice N. Cassenti

2
Faster-Than-Light Paradoxesin 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 ParadoxBarn View

• Front door opens

17
Pole Vaulter ParadoxBarn View

• Front door opens
• Forward end of pole enters barn

18
Pole Vaulter ParadoxBarn View

• Front door opens
• Forward end of pole enters barn
• Back end of pole enters barn

19
Pole Vaulter ParadoxBarn View

• Front door opens
• Forward end of pole enters barn
• Back end of pole enters barn
• Front door closes

20
Pole Vaulter ParadoxBarn 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 ParadoxBarn 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 ParadoxBarn 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 ParadoxBarn 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 ParadoxBarn 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 ParadoxPole Vaulter View

• Front doors opens

26
Pole Vaulter ParadoxPole Vaulter View

• Front doors opens
• Forward end of pole enters barn

27
Pole Vaulter ParadoxPole Vaulter View

• Front doors opens
• Forward end of pole enters barn
• Back door opens

28
Pole Vaulter ParadoxPole Vaulter View

• Front doors opens
• Forward end of pole enters barn
• Back door opens
• Front end of pole leaves barn

29
Pole Vaulter ParadoxPole 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 ParadoxPole 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 ParadoxPole 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 ParadoxPole 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 ParadoxPole 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 ParadoxResolution

• 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 ParadoxResolution

• 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
http//math.ucr.edu/home/baez/physics/Relativity/S
R/TwinParadox/twin_vase.html
39
Twin Paradox - Resolution Infinite Acceleration
http//math.ucr.edu/home/baez/physics/Relativity/S
R/TwinParadox/twin_vase.html
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 ofFaster-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
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
  sinhqgt1, dx/dtgtc.
• 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.