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EPPT M2 INTRODUCTION TO RELATIVITY
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Transcript of EPPT M2 INTRODUCTION TO RELATIVITY
EPPT M2
INTRODUCTION TO RELATIVITY
EPPT M2
INTRODUCTION TO RELATIVITY
K Young, Physics Department, CUHKThe Chinese University of Hong Kong
CHAPTER 4
APPLICATIONS OF THE LORENTZ
TRANSFORMATION
CHAPTER 4
APPLICATIONS OF THE LORENTZ
TRANSFORMATION
ObjectivesObjectives
Length contraction Concept of simultaneity
Time dilation– Twin paradox
Transformation of velocity Adding velocities Four-velocity
Length ContractionLength Contraction
Choice of UnitsChoice of Units
In this ChapterIn this Chapter
c =1
1 2
1 221 /
v vv
v v c
ExampleExample
2/ c
ExampleExample
Measure separation between 2 ends of a rod
0t L / c
ExampleExample
2 21 /
tt
V c
2/ c
Length contractionLength contraction
Formula for contraction Concept of simultaneity Paradoxes
Length contractionLength contraction
S
x
y
VS'
x'
y'
L0
What is length L as it appears to S?
Definition of lengthDefinition of length
xA xB
0 txL when
At the same time!
( )
( )
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
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
( )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
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
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
0 1
1B
Lt
c
1
10
c
LtD
0B D
Lt t
c
Are they simultaneous?
2L0
D BA
E C
Lack of symmetry?Lack of symmetry? All observers equivalent? Symmetry S S'? L < L0???
We're equivalent I'm special
ParadoxParadox
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
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'
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?
At the same time in SAt the same time in S
At the same time in S' ?
S S'
Time DilationTime Dilation
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'
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
Lack of symmetry?Lack of symmetry?
We are equivalent I'm special
Twin ParadoxTwin Paradox
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'
ExampleExample
PQ
10 ly
0.5
According to Q, ?t
According to P,
?t Who has aged more?
ExampleExample
Who has experienced acceleration?
Who is the “moving observer”?
PQ
t t
t t t
t
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
T T Lifetime appears longer.
Clearly verified.
Other clocks?Other clocks?
Atomic clocks Quartz watches Biological clocks Weak decays Strong decays
Do these all "slow down" when moving?
Analyze in detail lnvoke Principle of Relativity
Discrepancy not allowed
Study laws of physics (e.g. EM) rather than phenomena
L A W S
Transformation of VelocityTransformation of Velocity
Transformation of velocityTransformation of velocity
Galilean transformation Relativistic transformation
– Using Lorentz transformation directly– Using addition of "angles"
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 !!
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
/
/ 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
21 /
v Vv
v V c
2/ c1 2
1 21
v vv
v v
"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
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
Four VelocityFour Velocity
Four velocityFour velocity
Velocity transforms in a complicated nonlinear manner
1
v Vv
v V
V framev, v' particle
Displacement is 4-vectorDisplacement is 4-vectort
xx
y
z
Simple case: 0y z
tx
x
4-vector transforms as
t tL
x x
cosh sinh
sinh coshL
Velocity does not transform simply
because we divide by , andt
is not an invariant,t
t t
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
1u x
/1
/
t t
x x
called four -velocity
u
u L u
transfroms linearly
For relative motion along x:
u u
EvaluationEvaluation
t
t
x xv
t
t
/
/
tu
x v
x
y
z
vu
v
v
ObjectivesObjectives
Length contraction Concept of simultaneity
Time dilation– Twin paradox
Transformation of velocity Adding velocities Four-velocity
AcknowledgmentAcknowledgment
I thank Miss HY Shik and Mr HT Fung for design