EPPT M2 INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK The Chinese University of Hong...

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