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' x'

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

Invariance of Spacetime Interval

Brehme Spacetime Diagram Exchange Ot axis and Ot' axis

Brehme Spacetime Diagram ct' ct x' O • x

Oblique Coordinates ct O • x

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

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

World Line

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

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

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

World Line ct • E ct2 ct1 • x O x1 x2

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

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

Question: world line 與 trajectory 有何不同？

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

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

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

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

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

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

Time Dilation

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

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

Time Dilation • • • • • • • ct' ct E2 C2 A2 proper time c E1 ct x' A1

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

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

Time Dilation • • • • • • • ct' ct C2 A2 E2 ct' c proper time C1 x'

Simultaneity

World Line of Light 角平分線 ct O • x

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

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

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

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

Length Contraction

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

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

Off Synchronization

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

Lorentz Transformation

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

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

Comparison of Loedel Diagram and Brehme Diagram

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

Summary

Geometry; Invariance of Spacetime; Constancy of Speed of Light