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Lecture Twelve. Spacetime Geometry: Brehme Diagram and Loedel Diagram.

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Presentation on theme: "Lecture Twelve. Spacetime Geometry: Brehme Diagram and Loedel Diagram."— Presentation transcript:

1 Lecture Twelve

2 Spacetime Geometry: Brehme Diagram and Loedel Diagram

3 Relativistic Kinematics: Relativistic Vista of Spacetime

4 Geometry of Relativity

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

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

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

8 Invariance of Spacetime Interval

9 Brehme Spacetime Diagram Exchange Ot axis and Ot' axis

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

11 Oblique Coordinates O ct x

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

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

14 World Line

15 E O x ct x1x1 ct 1 x2x2 ct 2 x3x3 ct 3

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

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

18 World Line E O x ct x1x1 x2x2 ct 2 ct 1

19 World Line of Light E Ox ct ct x X T 角平分線

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

21 Question: world line 與 trajectory 有何不同?

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

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

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

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

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

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

28 Time Dilation

29 O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2

30 Time Dilation O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 B1B1 B2B2 C1C1 C2C2 same place in  ' proper time

31 Time Dilation O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2 proper time

32 Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ctct

33 Time Dilation O ct x x'x' ct' cc B1B1 B2B2 A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' same place in  proper time

34 Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' proper time

35 Simultaneity

36

37

38 World Line of Light Ox ct 角平分線

39 O'O' O O'O' O'O' O'O' O O O x ct ct' x'x' O v v v v A A A A A B B B B C C C C D D D D C D B

40 O'O' O O'O' O'O' O O x ct ct' x'x' O v v v A A A A B B B C C C D D D C D simultaneous in  ' t' C = t' D t D < t C Events C and D

41 O'O' O O'O' O'O' O'O' O O O x ct ct' x'x' O -v

42 x ct ct' x'x' O E2E2 E2'E2' E 1 (x,t 2 ) or (x',t 2 ') In  ', E 2 ' and E 1 are simultaneous x'x' x ct 2 ' ct 2 In , E 2 and E 1 are simultaneous E1E1 E 2 ' before E 1 in  E 2 after E 1 in  '

43 Length Contraction

44 O ct x x'x' ct' A B world lines of A and B ct 1 simultaneous measurements at time t 1 in  L 0 (proper length) L

45 Length Contraction O ct x x'x' ct' A B world lines of A and B ct' 1 simultaneous measurements at time t' 1 in  ' L 0 (proper length) L

46 Off Synchronization

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

48 Lorentz Transformation

49 O ct x x'x' ct' E(ct, x) ct' x'x' ct x or E(ct', x') A B C D C'C'

50 Lorentz Transformation O ct x x'x' ct' E(ct, x) ct' x'x' ct x or E(ct', x') A B C D D'D' x

51 Comparison of Loedel Diagram and Brehme Diagram

52 Loedel Diagram Parallel Component Contravariant Component Brehme Diagram Perpendicular Component Covariant Component

53 Summary

54 Geometry; Invariance of Spacetime; Constancy of Speed of Light


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