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Lecture Twelve

Lecture Twelve. Spacetime Geometry: Brehme Diagram and Loedel Diagram. Relativistic Kinematics: Relativistic Vista of Spacetime. Geometry of Relativity. Cartesian Coordinates. y. y. • P. ( x, y ). •. x. O. x. Cartesian Coordinates. y '. y '. • P. ( x ' , y ' ). •. O. x '.

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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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