Modern PhysicsModern Physics(PC301)(PC301)

Class #4 Class #4

Moore - Chapter R5 – Proper timeMoore - Chapter R5 – Proper time

Chapter R6 – Coordinate Chapter R6 – Coordinate TransformationTransformation

Don’t forget: Hand in homework Tomorrow by 10 am (8 am for webassign) and Sim2 on Friday

1) Length ContractionTrip to Alpha Centuri revisited.

2) Drawing Two-Observer Spacetime Diagramstime contractionlength contractionLorentz transformation equationsReview with some homework problemsRevisit length contraction with the diagram

3) Revisit length contraction and simultaneity with the conveyor belt painters and the barn/pole problem.

Homework QuestionsHomework Questions

Problem Set #2Problem Set #2

Due Wed by 8 am Due Wed by 8 am (Tomorrow!)(Tomorrow!)

Also Sim 2 due on FridayAlso Sim 2 due on Friday

First test in two weeksFirst test in two weeks

The Twin ParadoxThe Twin Paradox

"If we placed a living organism in a box…one could arrange that the organism, after an arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had long since given way to new generations." (Einsteins original statement of paradox - 1911)

Twin ParadoxTwin Paradox

x

t

A

C

B

Alpha Centauri Worldline

Earth Worldline

4.3y

13y

s=?

DISCUSSION

Proper Time - Shortest Possible Proper Time - Shortest Possible Time Time

st

A

BI

NI

ABAB sdtvB

A

1)1(2

12For inertial (I) Clock ->

For non-inertial (NI) clock -> ABAB sdtv

B

A

1)1(2

12

Different Views: Read ThemDifferent Views: Read ThemThis is FUNThis is FUN

Tipler: pages 50-53Ohanian: pages 57-58Epstein: 85-86Feynman (6 ideas): 77-79

Binomial ExpansionBinomial Expansion

....!3

)2)(1(

!2

)1(

!11)1(

32

xnnnxnnnx

x n

....!2

)(21

21

!1

)(21

1)1(

222

2

12

vv

v

48

3

621)1(

6422

12 vvvv

1v2

1)1(2

2

12 vv

EvidenceEvidence

Hafele & Keating (1971)

Flying to AustraliaFlying to Australia

630 m/hr- 282m/s

10360 miles=16,672,803.84m

2.6*10-8st=16,4hrs

Computer Clock Precision is about 1ns

tv )1( 2

EXTENSION

NIST ATOMIC CLOCKSNIST ATOMIC CLOCKS

(1949) NIST-1Accurate to one part in

1,0*10-11s

(1960) NIST-2 (1963) NIST-3 (1968) NIST-4

(1972) NIST-5(1975) NIST-6

accurate to one part in2,5*10-13 s

(1993) NIST-7accurate to one part in

5*10-15 s

(1999) NIST-F1accurate to one part in

1,7*10-15 s

 LeptonsThe leptons are perhaps the simplest of the elementary particles. They appear to be pointlike and seem to be truly elementary. Thus far there has been no plausible suggestion they are formed from some more fundamental particles. There are only six leptons ( displayed in Table 14.3) , plus their six antiparticles. We have already discussed the electron and muon. Each of the charged particles has an associated neutrino, named after its charged partner (for example, muon neutrino). The electron and all the neutrinos are stable. The muon decays into an electron, and the tau can decay into an electron, a muon, or even hadrons(which is most probable).  From p480-481 of Thornton and Rex

http://www.youtube.com/watch?v=T3iryBLZCOQ

Good reading for next two Good reading for next two weeksweeks

Read and summarize (type please) Chapter 1.

Read chapter 2 for greater understanding.

Located on P drive or borrow the book from me.

Essay be Isaac AsimovEssay be Isaac Asimov

Speed of LightSpeed of Light

Length DefinitionLength Definition

"In an inertial frame, an objects length is defined as the distance between two simultaneous events that occur at its ends."

Frame Dependent Quantity

Visualizing Length ContractionVisualizing Length Contraction

Two Observer Spacetime Two Observer Spacetime Diagram: TimeDiagram: Time

x

t

O

t’

t=t’

21

'

tt

'tt

tv )1( 2Slope=1/

Two Observer Spacetime Diagram: Two Observer Spacetime Diagram: LengthLength

O x

t t’

Length Contraction

x’

Slope=

Two Observer Spacetime Diagram: Two Observer Spacetime Diagram: Hyperbola RelationshipHyperbola Relationship

O x

t t’

x’

From Moore p.108

Lorentz TransformationsLorentz Transformations

O x

t t’

x’

QtQ

xQ

tQ’

xQ’

P

tPQ

xOP

tPQ=t’OQ

xOP=x’OQ

)'(' QQQ txx

)'(' QQQ xtt

)''( xtt )''( txx

)(' xtt )(' txx

Inverse Lorentz Transformations

Normal Lorentz Transformations

)(' xtt )(' txx

Generalized Normal Lorentz Transformations

The Barn and Pole Paradox: The Barn and Pole Paradox: Home FrameHome Frame

Pole Rest Length (L0) = 10ns

Home Frame:

Pole moving at = 3/5 -> L=8nsBarn Length (L0) = 8ns

An instant in time when the pole is entirely in barn with doors shut.

Seems to Make Sense

The Barn and Pole Paradox:The Barn and Pole Paradox:Other FrameOther Frame

Other Frame:

Barn moving at = -3/5 -> L=6.4nsPole Length (L0) = 10ns

How can the pole 10ns long fit into a 6.4ns barn?

The runner with the pole does not observe that the pole is enclosed in the barn.

Front of pole reaches end of barn @ -6nsEnd of pole reaches front of barn @ 0ns when pole has already left

The Barn and Pole Paradox The Barn and Pole Paradox ResolutionResolution