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Spacetime Geometry: Brehme Diagram and Loedel Diagram
Relativistic Kinematics: Relativistic Vista of Spacetime
Geometry of Relativity
Cartesian Coordinates P O x y (x, y) x y
Cartesian Coordinates P O x'x' y'y' (x', y') x'x' y'y'
Cartesian Coordinates P O x'x' y'y' (x', y') x'x' y'y' y x y x invariance of distance (x, y)P
Invariance of Spacetime Interval
Brehme Spacetime Diagram Exchange Ot axis and Ot' axis
Brehme Spacetime Diagram O ct x x'x' ct'
Oblique Coordinates O ct x
Brehme Diagram (perpendicular components) E x x ct O (ct, x)
Loedel Diagram (parallel components) E x x ct O (ct, x)
E O x ct x1x1 ct 1 x2x2 ct 2 x3x3 ct 3
World Line E O x ct x rest at x in for all time t parallel to t -axis
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
World Line E O x ct x1x1 x2x2 ct 2 ct 1
World Line of Light E Ox ct ct x X T 1 2 3 4 角平分線
World Line of O' E O ct x x x x
Question: world line 與 trajectory 有何不同？
Loedel Diagram E O ct x x'x' ct' x'x' x'x'
Loedel Diagram E O ct x x'x' ct' x'x' x'x'
Loedel Diagram O ct x x'x' ct' E(ct, x) ct' x'x' ct x or E(ct', x')
Principle of Constancy of Light Speed O ct x x'x' ct' E(ct, x) x ct E
Principle of Constancy of Light Speed O ct x x'x' ct' E(ct', x')E x'x' ct'
Principle of Constancy of Light Speed O ct x x'x' ct' E(ct, x) or (ct', x') x'x' ct' x ct
O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2
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
Time Dilation O ct x x'x' ct' cc x'x' E1E1 E2E2 ct A1A1 A2A2 C1C1 C2C2 proper time
Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ctct
Time Dilation O ct x x'x' ct' cc B1B1 B2B2 A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' same place in proper time
Time Dilation O ct x x'x' ct' cc A1A1 A2A2 x E1E1 E2E2 C1C1 C2C2 ct' proper time
World Line of Light Ox ct 角平分線
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
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
O'O' O O'O' O'O' O'O' O O O x ct ct' x'x' O -v
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 '
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
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
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'
O ct x x'x' ct' E(ct, x) ct' x'x' ct x or E(ct', x') A B C D C'C'
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
Comparison of Loedel Diagram and Brehme Diagram
Loedel Diagram Parallel Component Contravariant Component Brehme Diagram Perpendicular Component Covariant Component
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