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

Chapter 1 INTRODUCTION

Questions of interest in relativity  Behavior of particles at high speeds   Energy / momentum of particles at high speeds; their interactions  Twin paradox; length contraction  Black holes  Cosmology; expansions of universe

Common Theme  How does the same phenomenon appear to different observers?  How is the same phenomenon described in different coordinate systems?

Example v

Objectives  Physics independent of coordinates  Rotation of coordinates  Principle of relativity  Experimental basis  Applications

Physics Independent of Coordinates

Physics independent of coordinates  Physics is absolute  Coordinates are arbitrary  Physics independent of coordinates

Coordinate Transformations

 Rotation leads to vectors x y x' y'

Moving coordinates leads to Special Relativity V

General transformation leads to General Relativity

Rotation of coordinates  Linear relationship  Vectors and matrices  Rotation matrix

3D notation bold Cartesian

Coordinates are relative Study coordinate transformations x' y' L  x y L End point = r

 r y x y' r '' x' 

Properties of rotation matrices Addition theorem for sin, cos

Addition theorem

Principle of Relativity

Physical law: different observers  Variables covariant  Equation invariant  Depends on linear transformation

Physical law: different observers a F

Principle 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 basis  SR: Michelson-Morley experiment: The speed of light is the same for all observers  GR: All objects fall at the same acceleration in a gravitational field  Both known to great precision  Thought to be exact

Order of magnitude of effect  Particle moving at speed v  Speed of light c  Dimensionless ratio v c 

Order of magnitude of effect   Sign of does not matter

Another expression

Order of magnitude of effect  Gravity important in GR

Example What is clock error (seconds/day) due to  speed  height 3 km 1000 km/hr

Applications

 Relativistic kinematics and dynamics — collisions  Mass-energy equivalence  Relation between E & M  Theory of gravity

Applications  Astrophysics  Cosmology  Global Positioning System (GPS)  Constraining other laws of physics

Relativistic kinematics & dynamics Only need to do this once and for all SS'

Mass-energy equivalence From relativistic kinematics & dynamics, new concept of E, P, m Important for nuclear physics & high energy physics

High 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 Theory  When E > E 0 =mc 2, particles can be created / destroyed  Theoretical description requires relativistic quantum field theory

S B Electricity Magnetism q v S' q

Gravity =If a o = g, cannot tell apart =If we understand transformation to an accelerating frame, then we understand gravity?? SS' aoao g

BUT

Astrophysics — gravity important

Black hole — heuristic derivation M m R Escape?

Black hole — heuristic derivation Escape? M m R Max speed = c

Black hole — heuristic derivation Escape M m R Cannot Escape

Black hole — heuristic derivation M m R

Mixture of Newtonian + relativistic Not really legitimate OK for order-of -magnitude estimate

Global Positioning System (GPS)

GPS Accuracy ~ 10 m

GPS

Cosmology =Depends on gravity =In detail: Einstein's theory

Constraining other laws of physics =Laws must be invariant =Limited possibilities

Objectives =Physics independent of coordinates =Rotation of coordinates =Principle of relativity =Experimental basis =Applications

Acknowledgment =I thank Miss HY Shik and Mr HT Fung for design