EPPT M2

INTRODUCTION TO RELATIVITY

EPPT M2

INTRODUCTION TO RELATIVITY

K Young, Physics Department, CUHKThe Chinese University of Hong Kong

CHAPTER 4

APPLICATIONS OF THE LORENTZ

TRANSFORMATION

CHAPTER 4

APPLICATIONS OF THE LORENTZ

TRANSFORMATION

ObjectivesObjectives

Length contraction Concept of simultaneity

Time dilation– Twin paradox

Transformation of velocity Adding velocities Four-velocity

Length ContractionLength Contraction

Choice of UnitsChoice of Units

In this ChapterIn this Chapter

c =1

1 2

1 221 /

v vv

v v c

ExampleExample

2/ c

ExampleExample

Measure separation between 2 ends of a rod

0t L / c

ExampleExample

2 21 /

tt

V c

2/ c

Length contractionLength contraction

Formula for contraction Concept of simultaneity Paradoxes

Length contractionLength contraction

S

x

y

VS'

x'

y'

L0

What is length L as it appears to S?

Definition of lengthDefinition of length

xA xB

0 txL when

At the same time!

( )

( )

x x t

x x t

Use of Lorentz transformationUse of Lorentz transformation

Both are correct Which is more convenient?

Rod is fixed in S', x' = L0 alwaysx = L when t = 0

/0LL

A moving rod appears contracted

00 LL

What if we use the other equation?What if we use the other equation?( )x x t

00

LL

0 0t t

Simultaneity is not absolute

NOT simultaneous in S' 2 events are simultaneous in S

(What are 2 events?)

0

( )t t x Generally

2 events which are – simultaneous in S (t = 0)– but occurring in different places (x 0)

would not be simultaneous in S' (t' 0)

0 0 0

2L0

D BA

E C

ProblemProblem

Seen by S' co-moving with train

0 0B D

L Lt t

c c

S on ground sees train moving at V = c

Event BEvent BVt L ct L

ct

Vt

0 02

0 /

1

(1 )

1

1 1

B

Lt t

c VL

c

L L

c c

Sign? 0?

Event DEvent D

1

10

c

LtD

0 1

1B

Lt

c

1

10

c

LtD

0B D

Lt t

c

Are they simultaneous?

2L0

D BA

E C

Lack of symmetry?Lack of symmetry? All observers equivalent? Symmetry S S'? L < L0???

We're equivalent I'm special

ParadoxParadox

ParadoxParadox

Hole of length L0

Rod of length L0, moving at V

Push both ends of rod at the same time

Can rod go through?

V

At rest with hole Rod contracted

Goes through Does not go through

At rest with rod Hole contracted

Observer SObserver S Observer S'Observer S'

ParadoxParadox

V Hole of length L0

Rod of length L0, moving at V

Push both ends of rod at the same time

Can rod go through?

At the same time in SAt the same time in S

At the same time in S' ?

S S'

Time DilationTime Dilation

Time dilationTime dilation

What is time t as it appears to S? t is the time separation between 2 events. Which 2 events?

S

1

2

VS'

1'

2'

0

Both are correct Which is more convenient?

Clock is fixed to S' (co-moving frame), x' = 0

( )

( )

t t x

t t x

'tt

Moving observer measures a longer time

Proper Time

Lack of symmetry?Lack of symmetry?

We are equivalent I'm special

Twin ParadoxTwin Paradox

Twin paradoxTwin paradox

Who is older? Is there symmetry? Motion (velocity) is relative

Acceleration is absolute — S' has travelled Clock shows shorter time

S

S'

ExampleExample

PQ

10 ly

0.5

According to Q, ?t

According to P,

?t Who has aged more?

ExampleExample

Who has experienced acceleration?

Who is the “moving observer”?

PQ

t t

t t t

t

Experimental proof: elementary particleExperimental proof: elementary particle

S

S /

01

2

Tt

N N

/

01

2

t T

N N

/

01

2

Tt

N

T T

/

01

2

Tt

N

T T Lifetime appears longer.

Clearly verified.

Other clocks?Other clocks?

Atomic clocks Quartz watches Biological clocks Weak decays Strong decays

Do these all "slow down" when moving?

Analyze in detail lnvoke Principle of Relativity

Discrepancy not allowed

Study laws of physics (e.g. EM) rather than phenomena

L A W S

Transformation of VelocityTransformation of Velocity

Transformation of velocityTransformation of velocity

Galilean transformation Relativistic transformation

– Using Lorentz transformation directly– Using addition of "angles"

P

Transformation of velocityTransformation of velocity

1. Galilean1. Galilean

V

Vtx

x'

x x Vt

v v V

x xv v

t t

"Addition of velocities"

Same t !!

2. Relativistic2. Relativistic

/

/ 1

x x t x t

t x t x t

Note +( )

( )

x x t

t x t

A. Using Lorentz transformationA. Using Lorentz transformation

1

v

v

/

/ 1

x x t x t

t x t x t

Vvt

xv

t

x

21 /

v Vv

v V c

Cannot add to more than c If v' or V << c, the reduce to Galilean

1

v

v

21 /

v Vv

v V c

2/ c1 2

1 21

v vv

v v

"0.01 + 0.01""0.01 + 0.01" "0.9 + 0.9""0.9 + 0.9"

ExampleExample 1 2 1 2: 1v v v v 1 2 1 2: 1v v v v

0.01 0.01

1 (0.01)(0.01)

0.02

1.0001

0.019998

0.9 0.9

1 (0.9)(0.9)

1.8

1.81

0.9945

B. Using addition of anglesB. Using addition of angles

S S' P

2

2

1

1

Angle

Vel

21

21

21

21

tanhtanh1

tanhtanh

)tanh(

tanh

21

21

1

Easy to do multiple additions

1 Obvious that resultant satisfies

Four VelocityFour Velocity

Four velocityFour velocity

Velocity transforms in a complicated nonlinear manner

1

v Vv

v V

V framev, v' particle

Displacement is 4-vectorDisplacement is 4-vectort

xx

y

z

Simple case: 0y z

tx

x

4-vector transforms as

t tL

x x

cosh sinh

sinh coshL

Velocity does not transform simply

because we divide by , andt

is not an invariant,t

t t

transforms simply;x

If we divide by a constant

(e.g. 3.14), the result is still a 4-vector

Hint: Divide by a universal time

proper time

1u x

/1

/

t t

x x

called four -velocity

u

u L u

transfroms linearly

For relative motion along x:

u u

EvaluationEvaluation

t

t

x xv

t

t

/

/

tu

x v

x

y

z

vu

v

v

ObjectivesObjectives

Length contraction Concept of simultaneity

Time dilation– Twin paradox

Transformation of velocity Adding velocities Four-velocity

AcknowledgmentAcknowledgment

I thank Miss HY Shik and Mr HT Fung for design