eppt m2 introduction to relativity
DESCRIPTION
EPPT M2 INTRODUCTION TO RELATIVITY. K Young , Physics Department, CUHK The Chinese University of Hong Kong. CHAPTER 4 APPLICATIONS OF THE LORENTZ TRANSFORMATION. Objectives. Length contraction Concept of simultaneity Time dilation Twin paradox Transformation of velocity - PowerPoint PPT PresentationTRANSCRIPT
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EPPT M2
INTRODUCTION TO RELATIVITY
EPPT M2
INTRODUCTION TO RELATIVITY
K Young, Physics Department, CUHKThe Chinese University of Hong Kong
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CHAPTER 4
APPLICATIONS OF THE LORENTZ
TRANSFORMATION
CHAPTER 4
APPLICATIONS OF THE LORENTZ
TRANSFORMATION
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ObjectivesObjectives
Length contraction Concept of simultaneity
Time dilation– Twin paradox
Transformation of velocity Adding velocities Four-velocity
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Length ContractionLength Contraction
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Choice of UnitsChoice of Units
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In this ChapterIn this Chapter
c =1
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1 2
1 221 /
v vv
v v c
ExampleExample
2/ c
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ExampleExample
Measure separation between 2 ends of a rod
0t L / c
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ExampleExample
2 21 /
tt
V c
2/ c
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Length contractionLength contraction
Formula for contraction Concept of simultaneity Paradoxes
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Length contractionLength contraction
S
x
y
VS'
x'
y'
L0
What is length L as it appears to S?
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Definition of lengthDefinition of length
xA xB
0 txL when
At the same time!
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( )
( )
x x t
x x t
Use of Lorentz transformationUse of Lorentz transformation
Both are correct Which is more convenient?
Rod is fixed in S', x' = L0 alwaysx = L when t = 0
/0LL
A moving rod appears contracted
00 LL
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What if we use the other equation?What if we use the other equation?( )x x t
00
LL
0 0t t
Simultaneity is not absolute
NOT simultaneous in S' 2 events are simultaneous in S
(What are 2 events?)
0
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( )t t x Generally
2 events which are – simultaneous in S (t = 0)– but occurring in different places (x 0)
would not be simultaneous in S' (t' 0)
0 0 0
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2L0
D BA
E C
ProblemProblem
Seen by S' co-moving with train
0 0B D
L Lt t
c c
S on ground sees train moving at V = c
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Event BEvent BVt L ct L
ct
Vt
0 02
0 /
1
(1 )
1
1 1
B
Lt t
c VL
c
L L
c c
Sign? 0?
Event DEvent D
1
10
c
LtD
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0 1
1B
Lt
c
1
10
c
LtD
0B D
Lt t
c
Are they simultaneous?
2L0
D BA
E C
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Lack of symmetry?Lack of symmetry? All observers equivalent? Symmetry S S'? L < L0???
We're equivalent I'm special
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ParadoxParadox
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ParadoxParadox
Hole of length L0
Rod of length L0, moving at V
Push both ends of rod at the same time
Can rod go through?
V
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At rest with hole Rod contracted
Goes through Does not go through
At rest with rod Hole contracted
Observer SObserver S Observer S'Observer S'
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ParadoxParadox
V Hole of length L0
Rod of length L0, moving at V
Push both ends of rod at the same time
Can rod go through?
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At the same time in SAt the same time in S
At the same time in S' ?
S S'
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Time DilationTime Dilation
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Time dilationTime dilation
What is time t as it appears to S? t is the time separation between 2 events. Which 2 events?
S
1
2
VS'
1'
2'
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0
Both are correct Which is more convenient?
Clock is fixed to S' (co-moving frame), x' = 0
( )
( )
t t x
t t x
'tt
Moving observer measures a longer time
Proper Time
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Lack of symmetry?Lack of symmetry?
We are equivalent I'm special
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Twin ParadoxTwin Paradox
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Twin paradoxTwin paradox
Who is older? Is there symmetry? Motion (velocity) is relative
Acceleration is absolute — S' has travelled Clock shows shorter time
S
S'
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ExampleExample
PQ
10 ly
0.5
According to Q, ?t
According to P,
?t Who has aged more?
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ExampleExample
Who has experienced acceleration?
Who is the “moving observer”?
PQ
t t
t t t
t
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Experimental proof: elementary particleExperimental proof: elementary particle
S
S /
01
2
Tt
N N
/
01
2
t T
N N
/
01
2
Tt
N
T T
/
01
2
Tt
N
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T T Lifetime appears longer.
Clearly verified.
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Other clocks?Other clocks?
Atomic clocks Quartz watches Biological clocks Weak decays Strong decays
Do these all "slow down" when moving?
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Analyze in detail lnvoke Principle of Relativity
Discrepancy not allowed
Study laws of physics (e.g. EM) rather than phenomena
L A W S
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Transformation of VelocityTransformation of Velocity
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Transformation of velocityTransformation of velocity
Galilean transformation Relativistic transformation
– Using Lorentz transformation directly– Using addition of "angles"
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P
Transformation of velocityTransformation of velocity
1. Galilean1. Galilean
V
Vtx
x'
x x Vt
v v V
x xv v
t t
"Addition of velocities"
Same t !!
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2. Relativistic2. Relativistic
/
/ 1
x x t x t
t x t x t
Note +( )
( )
x x t
t x t
A. Using Lorentz transformationA. Using Lorentz transformation
1
v
v
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/
/ 1
x x t x t
t x t x t
Vvt
xv
t
x
21 /
v Vv
v V c
Cannot add to more than c If v' or V << c, the reduce to Galilean
1
v
v
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21 /
v Vv
v V c
2/ c1 2
1 21
v vv
v v
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"0.01 + 0.01""0.01 + 0.01" "0.9 + 0.9""0.9 + 0.9"
ExampleExample 1 2 1 2: 1v v v v 1 2 1 2: 1v v v v
0.01 0.01
1 (0.01)(0.01)
0.02
1.0001
0.019998
0.9 0.9
1 (0.9)(0.9)
1.8
1.81
0.9945
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B. Using addition of anglesB. Using addition of angles
S S' P
2
2
1
1
Angle
Vel
21
21
21
21
tanhtanh1
tanhtanh
)tanh(
tanh
21
21
1
Easy to do multiple additions
1 Obvious that resultant satisfies
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Four VelocityFour Velocity
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Four velocityFour velocity
Velocity transforms in a complicated nonlinear manner
1
v Vv
v V
V framev, v' particle
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Displacement is 4-vectorDisplacement is 4-vectort
xx
y
z
Simple case: 0y z
tx
x
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4-vector transforms as
t tL
x x
cosh sinh
sinh coshL
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Velocity does not transform simply
because we divide by , andt
is not an invariant,t
t t
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transforms simply;x
If we divide by a constant
(e.g. 3.14), the result is still a 4-vector
Hint: Divide by a universal time
proper time
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1u x
/1
/
t t
x x
called four -velocity
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u
u L u
transfroms linearly
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For relative motion along x:
u u
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EvaluationEvaluation
t
t
x xv
t
t
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/
/
tu
x v
x
y
z
vu
v
v
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ObjectivesObjectives
Length contraction Concept of simultaneity
Time dilation– Twin paradox
Transformation of velocity Adding velocities Four-velocity
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AcknowledgmentAcknowledgment
I thank Miss HY Shik and Mr HT Fung for design