# Lecture Twelve.

## Presentation on theme: "Lecture Twelve."— Presentation transcript:

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