EPPT M2
INTRODUCTION TO RELATIVITY
EPPT M2
INTRODUCTION TO RELATIVITY
K Young, Physics Department, CUHKThe Chinese University of Hong Kong
Chapter 1
INTRODUCTIONChapter 1
INTRODUCTION
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
Common ThemeCommon Theme
How does the same phenomenon appear to different observers?
How is the same phenomenon described in different coordinate systems?
ExampleExample
v
ObjectivesObjectives
Physics independent of coordinates Rotation of coordinates Principle of relativity Experimental basis Applications
Physics Independentof Coordinates
Physics Independentof Coordinates
Physics independentof coordinates
Physics independentof coordinates
Physics is absolute Coordinates are
arbitrary
Physics independent of coordinates
Coordinate TransformationsCoordinate Transformations
Rotation leads to vectorsRotation leads to vectors
x
y
x'y'
Moving coordinates leads to Special Relativity
Moving coordinates leads to Special Relativity
V
General transformation leads to General Relativity
General transformation leads to General Relativity
Rotation of coordinatesRotation of coordinates
Linear relationship Vectors and matrices Rotation matrix
3D notation3D notation
x
y
z
r x
y
z
p
p
p
p
r, p bold
, , x y zr Cartesian
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
cos
sin
x r
y r
r
y
x
y'
r
'x'
cos
cos( )
x r
r
sin
sin( )
y r
r
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
Properties of rotation matricesProperties of rotation matrices
cos sin( )
sin cosR
( ) ( ) ( )R R R
Addition theorem for sin, cos
( ) ( )TR R
cos sin cos sin
sin cos sin cos
cos sin
sin cos
Addition theoremAddition theorem
Principle of RelativityPrinciple of Relativity
Physical law: different observersPhysical law: different observers
Variables covariant Equation invariant Depends on linear transformation
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
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?
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
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
Order of magnitude of effectOrder of magnitude of effect
Sign of does not matter
Another expressionAnother expression2 2
21 2 2
v mv
c mc
KE2
rest energy
Order of magnitude of effectOrder of magnitude of effect
Gravity important in GR
2
PE
rest energy
ExampleExample
What is clock error (seconds/day) due to
speed
height
3 km
1000 km/hr
ApplicationsApplications
ApplicationsApplications
Relativistic kinematics and dynamics — collisions
Mass-energy equivalence Relation between E & M Theory of gravity
ApplicationsApplications
Astrophysics Cosmology Global Positioning System (GPS) Constraining other laws of physics
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
Mass-energy equivalenceMass-energy equivalence
From relativistic kinematics & dynamics, new concept of E, P, m
Important for nuclear physics & high energy physics
20E mc
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
Quantum Field TheoryQuantum Field Theory
When E > E0 =mc2, particles can be created / destroyed
Theoretical description requires relativistic quantum field theory
SB
Electricity MagnetismElectricity Magnetism
qv
S'
q
E
B
Charge
Moving charge
GravityGravity
If ao = g, cannot tell apart
If we understand transformation to an accelerating frame, then we understand gravity??
S S'ao
g
BUT
Astrophysics — gravity importantAstrophysics — gravity important
2 2 2~
U GM
Mc Rc
R
GMU
2
~
2~
Rc
GM
Black hole — heuristic derivationBlack hole — heuristic derivation
KE PE
R
GMmmv 2
2
1
M
m
R
Escape?
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
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
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
Black hole — heuristic derivationBlack hole — heuristic derivation
Mixture of Newtonian + relativisticNot really legitimateOK for order-of -magnitude estimate
2
0
2GMR
cR
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
GPSGPS
Accuracy ~ 10 m
GPSGPS
2 2 2~ ~
GM gR
Rc c
9287 10~1031010~
421 ~ 10 s
~30 km
day
CosmologyCosmology
Depends on gravity In detail: Einstein's theory
Constraining other laws of physicsConstraining other laws of physics
Laws must be invariant Limited possibilities
ObjectivesObjectives
Physics independent of coordinates Rotation of coordinates Principle of relativity Experimental basis Applications
AcknowledgmentAcknowledgment
I thank Miss HY Shik and Mr HT Fung for design