Lecture Twelve. Spacetime Geometry: Brehme Diagram and Loedel Diagram.
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Transcript of Lecture Twelve. Spacetime Geometry: Brehme Diagram and Loedel Diagram.
Lecture Twelve
Spacetime Geometry:
Brehme Diagramand
Loedel Diagram
Relativistic Kinematics: Relativistic Vista of Spaceti
me
Geometry of
Relativity
Cartesian Coordinates
• P
• O x
y
(x, y)
x
y
Cartesian Coordinates
• P
• O
x'
y'
(x', y')
x'
y'
Cartesian Coordinates
• P
• O
x'
y'
(x', y')
x'
y'
y
x
y
x
invariance of distance
(x, y)
P
Invariance of Spacetime Interval
Brehme Spacetime Diagram
OtxOtOOx
Exchange Ot axis and Ot' axis
Brehme Spacetime Diagram
O •
ct
x
x'
ct'
Oblique Coordinates
O •
ct
x
Brehme Diagram (perpendicular components)
• E
xx
ct
ct
O•
(ct, x)
Loedel Diagram (parallel components)
• E
xx
ct
ct
• O
(ct, x)
World Line
World Line
• E
• O x
ct
x1
ct1
x2
ct2
x3
ct3
)(tfx
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'
rest at x' in ' for all time t'
x'
ct'
parallel to t' -axis perpendicular to x -axis
World Line
• E
• O x
ct
x1 x2
ct2
ct1
12
12
:itywith veloc
motion Uniform
tt
xxv
World Line of Light
• E
• O x
ct
•
•
ct
xX
T
12
34
角平分線21 內錯角31 內錯角42
為等腰角形OXE
ctx
為等腰角形 OTE為菱形 OTEX
角平分線
World Line of O'
• E
O•
ct
xx
ct
ct ct
x
x
. torelative
of velocity theis
sin/
.1cos
sin
22
O
O
ctxv
cv
c
v
ct
x
Question:
world line 與 trajectory 有何不同？
Loedel Diagram
• E
O•
ct
x
x'
ct'
x'
ct'
ct'
x' sintcx
Loedel Diagram
• E
O•
ct
x
x'
ct'
x'
ct'
ct'
x'sintcx
Loedel Diagram
O•
ct
x
x'
ct'
• E (ct, x)
ct'
x'
ct
x
or E(ct', x')•
•
•
•
cvsin21cos
21tan
Principle of Constancy of Light Speed
O •
ct
x
x'
ct'
• E(ct, x)
x•
•ct E
Principle of Constancy of Light Speed
O •
ct
x
x'
ct'
• E(ct', x')E
• x'
•ct'
Principle of Constancy of Light Speed
O •
ct
x
x'
ct'
• E(ct , x) or (ct', x')
• x'
•ct'
•x
•ct
ct
x
t
x
tcxctx
,
Time Dilation
Time Dilation
cosctc
O•
ct
x
x'
ct'
c
•x'
•
• E1
E2
ct
•
•
•
•
A1
A2
C1
C2221cos cv
cvsin
21cos
21tan
Time Dilation
O•
ct
x
x'
ct'
c
•x'
•
• E1
E2
ct
ct
•
•
•
•
•
•
A1
A2
B1
B2
C1
C2
same place in '
proper time
cvsin
21cos
21tan
Time Dilation
cosctc 221cos cv
O•
ct
x
x'
ct'
c
•x'
•
• E1
E2
ct
•
•
•
•
A1
A2
C1
C2
221 cvt
proper time
cvsin
21cos
21tan
Time Dilation
cosctc 221cos cv
O•
ct
x
x'
ct'
c
•
•
A1
A2
•
•
• x
E1
E2
•
•
C1
C2
ct 221 cvt
cvsin
21cos
21tan
Time Dilation
O•
ct
x
x'
ct'
ct'c
•
•
•
•
B1
B2
A1
A2
•
•
• x
E1
E2
•
•
C1
C2
ct'
same place in
proper time
cvsin
21cos
21tan
Time Dilation
costcc 221cos cv
O•
ct
x
x'
ct'
c
•
•
A1
A2
•
•
• x
E1
E2
•
•
C1
C2
ct' 221 cvt
proper time
cvsin
21cos
21tan
Simultaneity
World Line of Light
• O x
ctctx 角平分線
••O'O
•
•
•
O'
O'
O'
•
•
•
O
O
O
x
ctct'
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
x
ctct'
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
tD < tC
Events C and D
••O'O
•
•
•
O'
O'
O'
•
•
•
O
O
O
x
ctct'
x'
O•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
-v
-v
-v
-v
x
ct ct'
x'
O•
• E2 •
•
•
• E2'
E1(x,t2) or (x',t2')
In ', E2' and E1 are simultaneous
•
• •
x'x
ct2'
ct2
In , E2 and E1
are simultaneous
•
•
E1
E2' before E1 in
E2 after E1 in '
•
Length Contraction
Length Contraction
O •
ct
x
x'
ct'
•
•
•A
B
world lines of A and B
•ct1 • •simultaneous
measurements at time t1 in
L0 (proper length)
L
220 1 cvLL
cvsin
21cos
21tan
Length Contraction
O •
ct
x
x'
ct'
•
• •A B
world lines of A and B•ct'1 •
•
simultaneous measurements at time t'1 in '
L0 (proper length)
L
220 1 cvLL
cvsin
21cos
21tan
Off Synchronization
Off -Synchronization
O•
ct
x
x'
ct'
• •
•
•
Time dilation : ct = (ct' - c )
Time dilation : ct' = ct
c = L sin
= L v/c
L
L
leading clock trailing clock
ct (proper
time)
ct'
Lorentz Transformation
Lorentz Transformation
O•
ct
x
x'
ct'
• E (ct, x)
ct'x'
ct
x
or E(ct', x')•
•
•
•
A
B
C
D
•C'
cvsin
21cos
21tan
CA OA COcosOCcos x
cossin ctxx
21
ctx
sin sin EA OA ctx
Lorentz Transformation
O•
ct
x
x'
ct'
• E (ct, x)
ct'x'
ct
x
or E(ct', x')•
•
•
•
A
B
C
D
cvsin
21cos
21tan
•D'
DB OBDOcosODcos tc
cossin xcttc
21
xct
sin sin EB OB xct
x
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
