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

CHAPTER 3 MOVING REFERENCE FRAMES

Objectives  Galilean transformation  M-M experiment  Derive L transformation from c = const  Explicit form of L transformation  Inverse transformation

Basic subject is an event e.g. E = bullet shot from rifle E = (t, x, y, z) E = (t, x)

S V S' Vt  S' moving relative to S with velocity V along x  Basic object is an event E E

Galilean Transformation

Galilean transformation y x x' P Vt V y' x' x

Galilean transformation Velocities "add"

Michelson-Morley Experiment

L Galilean c c c  V c + V V

MM experiment  No effect found  Galilean transformation wrong

Details of MM experiment

 "Train" = Earth V ~ 3  10  4 m s -1 V/c ~ 10  4  Difference is

 There is no way to stop this "train" and compare with the case V = 0  Instead, compare rays parallel and perpendicular to direction of motion

Say  ~

How did MM measure such a small difference?

A sketch of the Michelson-Morley experiment L L V A S D C B

 Interference  ~ ?

 No effect found  Speed of light is same in all reference frames

No absolute motion NO!! YES!! cc c - Vc + V cc V V

Derivation of Lorentz Transformation

Derivation of Lorentz transformation  Basic object is an event E  Linear assumption [x'] = [L] [x]  Identify an invariant  Condition on transformation matrix

Basic object An event E V stone hitting ground atom emits photon e.g.

Linear assumption 16 coefficients Simpler notation

Consider the 4-dimensional coordinate Express as column vector

In 2-D case

Identify invariant    2 can be negative  Claim: M-M experiment 

Emit at (0, 0, 0, 0) Why proportional? S M-M  S': same argument, same c   Proportional E receive at (t, x, y, z)

 Invariance = MM  Therefore (up to a sign)  Consider reverse transformation  Independent of direction  Proportional

Invariant interval 1 D space 3 D space Minkowski

Condition on transformation matrix

Compare chapter 2

 3 conditions 

Relate to relative velocity S S'

How to remember signs?

Galilean limit

Inverse Transformation

What you should not do

1 Do not hide a genuine difference  Euclidean x w Closed Finite

 Minkowski Open Infinite t x

2Genuine i / Fake i QM Rel Impossible to keep track! But only for the "genuine" i not for the "fake" i

Choice of Units

 Use same units  Choose c = 1 x y x in m y in km 1km = c 1m

Similarly  Choose  All formulas simpler  Can multiply / divide by c n

Example Time = 3.0 m ??? Time = 3.0 m = s

Example ??? Energy = kg J = kg m 2 s -2

Actual units  time = year  distance = light year

 distance = m s s Actual units  time = ns = s = 0.3 m

Standard of length and time Optical transition  1 tick = 1 period T  1 rod = 1 wavelength  Velocity of light because defined quantity

Combining two transformations

Claim Prove it!

Inverse transformation

 Invert algebraically 

Four Vectors

Objectives  Index notation  Galilean transformation  M-M experiment  Derive L transformation from c = const  Explicit form of L transformation  Inverse transformation

Acknowledgment =I thank Miss HYShik and Mr HT Fung for design