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Faster-Than-Light Paradoxes in Special Relativity

Explore the principles of special relativity, Lorentz transformation, time dilation, and paradoxes. Understand relativistic rockets and resolve complex scenarios. Dive into the Twin Paradox and more.

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Faster-Than-Light Paradoxes in Special Relativity

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  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/SR/TwinParadox/twin_vase.html

  39. Twin Paradox - Resolution Infinite Acceleration http://math.ucr.edu/home/baez/physics/Relativity/SR/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

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