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Lecture Twelve

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Spacetime Geometry: Brehme Diagram and Loedel Diagram

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Relativistic Kinematics: Relativistic Vista of Spacetime

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Geometry of Relativity

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

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

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

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Invariance of Spacetime Interval

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Brehme Spacetime Diagram Exchange Ot axis and Ot' axis

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

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Oblique Coordinates O ct x

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

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

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World Line

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

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

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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

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

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

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

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Question: world line 與 trajectory 有何不同？

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

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

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

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

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

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

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Time Dilation

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O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2

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Time Dilation O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 B1B1 B2B2 C1C1 C2C2 same place in ' proper time

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Time Dilation O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2 proper time

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Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ctct

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Time Dilation O ct x x'x' ct' cc B1B1 B2B2 A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' same place in proper time

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Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' proper time

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Simultaneity

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World Line of Light Ox ct 角平分線

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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

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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

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

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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 '

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Length Contraction

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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

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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

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Off Synchronization

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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'

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Lorentz Transformation

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

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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

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Comparison of Loedel Diagram and Brehme Diagram

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

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Summary

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Geometry; Invariance of Spacetime; Constancy of Speed of Light

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