eppt m2 introduction to relativity k young, physics department, cuhk the chinese university of hong...
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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 1
INTRODUCTIONChapter 1
INTRODUCTION
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Questions of interest in relativityQuestions of interest in relativity
Behavior of particles at high speeds 2
0E mc Energy / momentum of particles at high
speeds; their interactions Twin paradox; length contraction Black holes Cosmology; expansions of universe
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Common ThemeCommon Theme
How does the same phenomenon appear to different observers?
How is the same phenomenon described in different coordinate systems?
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ExampleExample
v
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ObjectivesObjectives
Physics independent of coordinates Rotation of coordinates Principle of relativity Experimental basis Applications
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Physics Independentof Coordinates
Physics Independentof Coordinates
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Physics independentof coordinates
Physics independentof coordinates
Physics is absolute Coordinates are
arbitrary
Physics independent of coordinates
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Coordinate TransformationsCoordinate Transformations
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Rotation leads to vectorsRotation leads to vectors
x
y
x'y'
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Moving coordinates leads to Special Relativity
Moving coordinates leads to Special Relativity
V
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General transformation leads to General Relativity
General transformation leads to General Relativity
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Rotation of coordinatesRotation of coordinates
Linear relationship Vectors and matrices Rotation matrix
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3D notation3D notation
x
y
z
r x
y
z
p
p
p
p
r, p bold
, , x y zr Cartesian
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Coordinates are relativeCoordinates are relative
cos
sin 0
x L x L
y L y
Study coordinate transformations
x'
y'
L
x
y
L
End point = r
linearly relatedx x
y y
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cos
sin
x r
y r
r
y
x
y'
r
'x'
cos
cos( )
x r
r
sin
sin( )
y r
r
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cos sin
sin cos
( )
x x
y
R
y
r
r
cos sinx y
cosc sios s n nir r
cos( )x r
sin( )y r
coss siin c s nor r
( sin ) cosx y
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Properties of rotation matricesProperties of rotation matrices
cos sin( )
sin cosR
( ) ( ) ( )R R R
Addition theorem for sin, cos
( ) ( )TR R
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cos sin cos sin
sin cos sin cos
cos sin
sin cos
Addition theoremAddition theorem
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Principle of RelativityPrinciple of Relativity
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Physical law: different observersPhysical law: different observers
Variables covariant Equation invariant Depends on linear transformation
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Physical law: different observersPhysical law: different observers
( ) ( ) m R RF aa
F
mF a
m F a
x xF ma y yF ma
x xF ma y yF ma
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Principle of relativityPrinciple of relativity
All valid laws of physics should take the same form in different coordinates systems invariance
All terms in valid equation must transform in the same way covariance
How do they transform?
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Experimental basisExperimental basis
SR: Michelson-Morley experiment: The speed of light is the same for all
observersGR: All objects fall at the same acceleration in
a gravitational fieldBoth known to great precisionThought to be exact
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Order of magnitude of effectOrder of magnitude of effect
Particle moving at speed v Speed of light c Dimensionless ratio
vc
8 13.0 10 msc
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Order of magnitude of effectOrder of magnitude of effect
Sign of does not matter
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Another expressionAnother expression2 2
21 2 2
v mv
c mc
KE2
rest energy
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Order of magnitude of effectOrder of magnitude of effect
Gravity important in GR
2
PE
rest energy
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ExampleExample
What is clock error (seconds/day) due to
speed
height
3 km
1000 km/hr
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ApplicationsApplications
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ApplicationsApplications
Relativistic kinematics and dynamics — collisions
Mass-energy equivalence Relation between E & M Theory of gravity
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ApplicationsApplications
Astrophysics Cosmology Global Positioning System (GPS) Constraining other laws of physics
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Relativistic kinematics & dynamicsRelativistic kinematics & dynamics
SS
SS
laws Newtonian Apply
Only need to do this once and for all
S
cv 9.0
S'
1sm3 v
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Mass-energy equivalenceMass-energy equivalence
From relativistic kinematics & dynamics, new concept of E, P, m
Important for nuclear physics & high energy physics
20E mc
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High energy physicsHigh energy physics
What is matter made of ? How do the constituents interact ?
To study experimentally Accelerate to high energy/speed Let them collide To probe short distance
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Quantum Field TheoryQuantum Field Theory
When E > E0 =mc2, particles can be created / destroyed
Theoretical description requires relativistic quantum field theory
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SB
Electricity MagnetismElectricity Magnetism
qv
S'
q
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E
B
Charge
Moving charge
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GravityGravity
If ao = g, cannot tell apart
If we understand transformation to an accelerating frame, then we understand gravity??
S S'ao
g
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BUT
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Astrophysics — gravity importantAstrophysics — gravity important
2 2 2~
U GM
Mc Rc
R
GMU
2
~
2~
Rc
GM
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Black hole — heuristic derivationBlack hole — heuristic derivation
KE PE
R
GMmmv 2
2
1
M
m
R
Escape?
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Black hole — heuristic derivationBlack hole — heuristic derivation
21
2
GMmmv
R Escape?
M
m
RMax speed = c
2
PE1 /
2 rest energy
GMm R
mc
21
2
GMmmc
R
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Black hole — heuristic derivationBlack hole — heuristic derivation
2
PE 1
rest energy 2 Escape
M
m
R
2
PE 1
rest energy 2 Cannot
Escape
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Black hole — heuristic derivationBlack hole — heuristic derivation
2
PE 1
rest energy 2
M
m
R
2
/ 1
2
GMm R
mc
02
2GMR R
c
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Black hole — heuristic derivationBlack hole — heuristic derivation
Mixture of Newtonian + relativisticNot really legitimateOK for order-of -magnitude estimate
2
0
2GMR
cR
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Global Positioning System (GPS)
Global Positioning System (GPS)
1
1012
11
19
3
2
155
4
6
20
14
13
7
8
9
21 16
18
17
observer
r
satellitev
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GPSGPS
Accuracy ~ 10 m
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GPSGPS
2 2 2~ ~
GM gR
Rc c
9287 10~1031010~
421 ~ 10 s
~30 km
day
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CosmologyCosmology
Depends on gravity In detail: Einstein's theory
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Constraining other laws of physicsConstraining other laws of physics
Laws must be invariant Limited possibilities
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ObjectivesObjectives
Physics independent of coordinates Rotation of coordinates Principle of relativity Experimental basis Applications
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AcknowledgmentAcknowledgment
I thank Miss HY Shik and Mr HT Fung for design