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Lecture Twelve

Explore Brehme and Loedel diagrams, relativistic vista of spacetime geometry, invariance of distance, and spacetime interval. Understanding principles like constancy of light speed, time dilation, simultaneity, length contraction, and Lorentz transformation in Cartesian coordinates. Discover the differences between world lines and trajectories and grasp the concept of off-synchronization and proper time.

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Lecture Twelve

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  1. Lecture Twelve

  2. Spacetime Geometry: Brehme Diagramand Loedel Diagram

  3. Relativistic Kinematics: Relativistic Vista of Spacetime

  4. Geometry of Relativity

  5. Cartesian Coordinates y y • P (x, y) • x O x

  6. Cartesian Coordinates y' y' • P (x', y') • O x' x'

  7. Cartesian Coordinates invariance of distance y y' y' y • P (x', y') P (x, y) • x O x x' x'

  8. Invariance of Spacetime Interval

  9. Brehme Spacetime Diagram Exchange Ot axis and Ot' axis

  10. Brehme Spacetime Diagram ct' ct x' O • x

  11. Oblique Coordinates ct O • x

  12. Brehme Diagram (perpendicular components) ct • E (ct, x) ct • x O x

  13. Loedel Diagram (parallel components) ct ct • E (ct, x) • x O x

  14. World Line

  15. World Line ct ct3 ct2 • E ct1 • x x1 x2 x3 O

  16. World Line ct rest at x in  for all time t parallel to t -axis • E • • x x O

  17. World Line ct' ct rest at x' in ' for all time t' parallel to t' -axis perpendicular to x -axis •E x' • x' • x O

  18. World Line ct • E ct2 ct1 • x O x1 x2

  19. World Line of Light 角平分線 ct T • • E 3 4 ct 2 1 X O • • x x

  20. World Line of O' ct x ct • E ct ct x • x O x

  21. Question:world line 與 trajectory 有何不同？

  22. Loedel Diagram ct' ct x' • E x' ct' ct' x' • x O

  23. Loedel Diagram ct' ct x' • E x' ct' ct' • x O x'

  24. Loedel Diagram ct' ct • • E (ct, x) or E(ct', x') ct • x' ct' x' • • • x O x

  25. Principle of Constancy of Light Speed ct' ct • • E(ct, x) E ct x' • O • x x

  26. Principle of Constancy of Light Speed ct' ct • E(ct',x') E • ct' x' • x' O • x

  27. Principle of Constancy of Light Speed ct' ct • • E(ct , x) or (ct',x') ct • ct' x' • x' • O • x x

  28. Time Dilation

  29. Time Dilation ct' ct • E2 C2 • • A2 • c E1 ct x' • • A1 C1 • x' • x O

  30. Time Dilation ct' ct • • E2 B2 C2 • ct • A2 proper time • • c E1 B1 ct x' • • A1 C1 • x' same place in  ' • x O

  31. Time Dilation ct' ct • E2 C2 • • A2 proper time • c E1 ct x' • • A1 C1 • x' • x O

  32. Time Dilation ct' ct • C2 • • A2 E2 ct c • C1 x' • • E1 A1 x • • x O

  33. Time Dilation ct' ct • C2 • • A2 E2 • ct' B2 c • ct' x' proper time C1 • • E1 A1 • B1 x • • x O same place in 

  34. Time Dilation ct' ct • C2 • • A2 E2 ct' c • C1 x' proper time • • E1 A1 x • • x O

  35. Simultaneity

  36. World Line of Light 角平分線 ct O • x

  37. v ct' D • O' • C • • • • ct O B A • B • v • x' C D • O' C • • • • • • • • O • B A A D • x O v D O' C • • • • • • O B A • • v D O' C • • • • • • B O A

  38. v Events C and D ct' D • O' • C • • • • ct O B A v • • x' C D O' C • • • • • • • • • • O B A A D • x O v D O' C • simultaneous in  ' • • • • • O t'C = t'D B A tD < tC • •

  39. ct' • O' • • • • ct • -v O x' O' • • • • • • -v O • x O O' • • • • • • -v O O' • • • • • • -v O

  40. ct ct' E2 after E1 in  ' In  , E2 and E1 are simultaneous • ct2 • • • E1(x,t2) or (x',t2') E1 E2 • ct2' x' E2' • • • E2' before E1 in  x' • • • x x O In  ', E2' and E1aresimultaneous •

  41. Length Contraction

  42. Length Contraction ct' ct • L ct1 • • • simultaneous measurements at time t1 in  x' B • A • L0 (proper length) O • x world lines of A and B

  43. Length Contraction ct' ct • L • • ct'1 • simultaneous measurements at time t'1 in  ' x' world lines of A and B A B • • O • x L0 (proper length)

  44. Off Synchronization

  45. Off -Synchronization ct ct' c = L sin  = L v/c L ct' • • • ct(proper time) Time dilation : ct' =   ct Time dilation :ct =  (ct' - c ) x' • x O • L trailing clock leading clock

  46. Lorentz Transformation

  47. Lorentz Transformation ct' ct • B • E (ct, x) or E(ct', x') D • x' ct ct' • x' C C' • • • x O A x

  48. Lorentz Transformation ct' ct x • B • E (ct, x) or E(ct', x') D • x' • D' ct • C ct' x' • • x x O A

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