Phy

sics

2D

Lec

ture

Slid

esO

ct 6

Viv

ek S

harm

aU

CSD

Phy

sics

New

Rul

es o

f Coo

rdin

ate

Tran

sfor

mat

ion

Nee

ded

•Th

e G

alile

an/N

ewto

nian

rule

s of t

rans

form

atio

n co

uld

not

hand

les f

ram

es o

f ref

s or o

bjec

ts tr

avel

ing

fast

–

V ≈

C(li

ke v

= 0

.1 c

or 0

.8c

or 1

.0c)

•Ei

nste

in’s

pos

tula

tes l

ed to

–

Des

truct

ion

of c

once

pt o

f sim

ulta

neity

( ∆

t ≠∆

t’)

–M

ovin

g cl

ocks

run

slow

er

–M

ovin

g ro

ds s

hrin

k

•Le

ts fo

rmal

ize

this

in te

rms o

f gen

eral

rule

s of c

oord

inat

e tra

nsfo

rmat

ion

: Lor

entz

Tran

sfor

mat

ion

–R

ecal

l the

Gal

ilean

tran

sfor

m

•X

’ = (x

-vt)

•T’

= T

–Th

ese

rule

s th

at w

ork

ok fo

r mul

e ca

rts n

ow m

ust b

e m

odifi

ed

for r

ocke

t shi

ps w

ith V

≈C

Dis

cove

ring

The

Cor

rect

Tra

nsfo

rmat

ion

Rul

e'

gu

ess

'

()

('

''

'

g

uess

)

xxv

Gx

xvt

xxvt

txvt

Gx

=−

→=

=+

+→

=−

Nee

d to

figu

re o

ut

func

tiona

l for

m o

f G !

G m

ust b

e di

men

sion

less

G

doe

s no

t dep

end

on x

,y,z

,tB

ut G

dep

ends

on

v/c

G is

sym

met

ricA

s v/

c→0

, G →

1R

ocke

t in

S’ (

x’,y

’,z’,t

’) fra

me

mov

ing

with

vel

ocity

v w

.r.t o

bser

ver o

n fra

me

S (x

,y,z

,t)Fl

ashb

ulb

mou

nted

on

rock

et e

mits

pul

se o

f lig

ht a

t the

inst

anto

rigin

s of

S,S

’ coi

ncid

eTh

at in

stan

t cor

resp

onds

to t

= t’

= 0

. Lig

ht tr

avel

s as

a s

pher

ical

wav

e, o

rigin

is a

t O,O

’

Do

a T

houg

ht E

xper

imen

t: R

ocke

t Mot

ion

alon

g x

axis

Spe

ed o

f lig

ht

is c

for b

oth

obse

rver

s

Exa

min

e a

poin

t P (a

t dis

tanc

e r f

rom

O a

nd r’

from

O’)

on

the

Sph

eric

al W

avef

ront

The

dist

ance

to p

oint

P fr

om O

: r =

ct

The

dist

ance

to p

oint

P fr

om O

: r’

= ct

’ C

lear

ly t

and

t’ m

ust b

e di

ffere

ntt ≠

t’

Dis

cove

ring

Lore

ntz

Tran

sfro

mat

ion

for (

x,y,

z,t)

Mot

ion

is a

long

x-x

’ axi

s, s

o y,

z u

ncha

nged

y’=y

,

z’ =

zE

xam

ine

poin

ts x

or x

’ whe

re s

pher

ical

wav

ecr

osse

s th

e ho

rizon

tal a

xes:

x =

r , x

’ =r’

22

22

22

2

'

'

( -

) ,

' (

-)

()

[]

1 o

r =

1

(/

)

('

')

(

'')

'

()

xct

Gxvt

Gt

xvt

c

vct

Gct

tc

c

xct

Gxvt

xct

G

Gc

vctvt

Gv

vtv

c

t

xxvt

γ

γ

==

⇒=

∴

⎡⎤

∴=

−+

−⎢

⎥⎣

⎦⇒

=−

==

+

== −

+

=

∴=

−

22

2

2

2

2

2

22

1si

nce

1

, (

'')

(

()

')

'

1

'1

,

'(

)

'

'[1

xxvt

xx

vtvt

xxvt

xxvt

vt

xx

tx

xt

tv

vvv

xt

tv x

v

v

v

c

v

tv

ct

γ γγ

γγ

γ

γγ

γγ

γγ

γ

γ

γ

γγγ

=+

⇒=

−+

∴

⎡⎤

⎡⎤

∴=

−+

=−

+⎢

⎥⎢

⎥⎣

⎦⎣

⎛⎞

⎛⎞

−=

−⎜

⎟⎜

⎟⎝

⎠

⎦

⎡⎤

⎛⎞

∴=

+⎝

−⎢

=−

−+

=

⎥⎜

⎟⎝

⎠⎣

⎦

⎛⎞

⇒=

−⎜ ⎝

⎠

+⎟ ⎠

21

vxt

cγ

⎡⎡

⎤⎛

⎞−

⎜⎟

⎢⎥

⎝

⎤−

=⎢

⎠⎥

⎣⎣

⎢⎦

⎦⎥

Lore

ntz

Tran

sfor

mat

ion

Bet

wee

n R

ef F

ram

es

2

' ' ''(

)y

yz

zv

ttx

t c

xv x

γ γ=

⎛⎞

=−

⎜⎟

⎝

=−

⎠

=Lore

ntz

Tran

sfor

mat

ion

2

' '

'') '

'

(

vtx

xvt

tc

y zx

y z

γ γ=+

⎛⎞

=+

⎜⎟

⎝

=

⎠

=

Inve

rse

Lore

ntz

Tran

sfor

mat

ion

As

v→0

, Gal

ilean

Tra

nsfo

rmat

ion

is re

cove

red,

as

per r

equi

rem

ent

Not

ice

: SPA

CE

and

TIM

E C

oord

inat

es m

ixed

up

!!!

Lore

ntz

Tran

sfor

m fo

r Pai

r of E

vent

s

Can

und

erst

and

Sim

ulta

neity

, Len

gth

cont

ract

ion

& T

ime

dila

tion

form

ulae

from

this

Tim

e di

latio

n: B

ulb

in S

fram

e tu

rned

on

at t

1&

off

at t 2

: W

hat ∆

t’di

d S

’ mea

sure

?tw

o ev

ents

occ

ur a

t sam

e pl

ace

in S

fram

e =>

∆x

= 0

∆t’

= γ

∆t

(∆t=

pro

per t

ime)

S

x

S’

X’

Leng

th C

ontra

ctio

n: R

uler

mea

sure

d in

S b

etw

een

x 1&

x2

: W

hat ∆

x’di

d S

’ mea

sure

?tw

o en

ds m

easu

red

at s

ame

time

in S

’ fra

me

=> ∆

t’ =

0 ∆

x=

γ(∆

x’ +

0 )

=> ∆

x’ =

∆x

/ γ(∆

x=

prop

er le

ngth

)

x 1x 2

rule

r

Lore

ntz

Vel

ocity

Tra

nsfo

rmat

ion

Rul

e '

''

21

x''

''

21

x'

2

x'

2

x'

2'

In S

' fra

me,

u

,

u,

u1

For v

<<

c, u

(G

ali

divi

de b

y dt

'

'

lean

Tra

ns. R

esto

r

()

(

ed)

)

x

x

xxx

dxtt

dt

dxvdt

vdt

dxc

vdt

dtdx

dxv

dxc

uu

u

d

vv

t

cvγ

γ

−=

=−

−=

− −==

=

=−

−

−

−

SS

’v

u

S an

d S’

are

mea

surin

g

ant’s

spe

ed u

alo

ng x

, y, z

ax

es

2

'

2

'

2

divi

de b

y dt

on

(1)

Ther

e is

a c

hang

e in

vel

ocity

in th

e di

rect

ion

to S

-S' m

otio

n

',

' '(

H

)

'

S

!

R

()

x

yyy

uudy

dy dy

vdt

dtdx

dyc

u

uv

dydt

dxc

v cγ

γ

γ= =

= ⊥

−

−

=

=

−

Vel

ocity

Tra

nsfo

rmat

ion

Per

pend

icul

ar to

S-S

’ mot

ion

'

2

Sim

ilarly

Z

com

pone

nt o

fA

nt' s

vel

ocity

tra

nsfo

rms

(1)

as

zz

x

uu

v cu

γ=

−

Inve

rse

Lore

ntz

Vel

ocity

Tra

nsfo

rmat

ion

'x

'

' '

2 ' 2 ' 2

Inve

rse

Vel

ocity

Tra

nsfo

rm:

(1

u

)

11 ()

yy

z

x

z

x

x

x

uv

vu uu

v c u vc

u

c

u u

γ γ

=+

=++

=+

As

usua

l, re

plac

e v

⇒-v

Doe

s Lo

rent

zTr

ansf

orm

“wor

k” ?

Two

rock

ets

trav

el in

oppo

site

dire

ctio

ns

An

obse

rver

on

eart

h (S

) m

easu

res

spee

ds =

0.7

5cA

nd 0

.85c

for A

& B

re

spec

tivel

y

Wha

t doe

s A

mea

sure

as

B’s

spe

ed?

Plac

e an

imag

inar

y S’

fram

e on

Roc

ket A

⇒v

= 0.

75c

rela

tive

to E

arth

Obs

erve

rS

Con

sist

ent w

ith S

peci

al T

heor

y of

Rel

ativ

ity

Exa

mpl

e of

Inve

rse

velo

city

Tra

nsfo

rm

Bik

er m

oves

with

spe

ed =

0.8

cpa

st s

tatio

nary

obs

erve

r

Thro

ws

a ba

ll fo

rwar

d w

ith

spee

d =

0.7c

Wha

t doe

s st

atio

nary

ob

serv

er s

ee a

s ve

loci

ty

of b

all ?

Pla

ce S

’ fra

me

on b

iker

Bik

er s

ees

ball

spee

d

u X’=

0.7c

Spee

d of

bal

l rel

ativ

e to

st

atio

nary

obs

erve

r u X

?

Can

you

be

seen

to b

e bo

rn b

efor

e yo

ur m

othe

r?A

fram

e of

Ref

whe

re s

eque

nce

of e

vent

s is

RE

VE

RS

ED

?!!

SS’

11

''

11

(,

)

(,

)

xt

xt

u

22

''

22

(,

)

(,

)

xt

xt

I take off f

rom SD

I arrive in SF

''

21

2 '

For w

hat v

alue

of v

can

'

0

vx

ttt

tct

γ⎡

∆⎤

⎛⎞

∆=

−=

∆−

⎜⎟

⎢⎥

⎝⎠

⎣⎦

∆<

I Can

t ‘be

see

n to

arr

ive

in S

F be

fore

I ta

ke o

ff fr

om S

D

SS’

11

''

11

(,

)

(,

)

xt

xt

u

22

''

22

(,

)

(,

)

xt

xt

'

22

2

' 21

2

'

'

For w

hat v

alue

of v

0

v

can

: N

ot a

l low

e

u1

< '0

dc

vx

tt

cc

vc

u

vx

vct

c

vx

ttt

tct

γ⎡

∆⎤

⎛⎞

∆=

−=

∆−

⎜⎟

⎢⎥

⎝⎠

⎣⎦

∆<

∆∆

∆=

∆⇒

<

⇒>

⇒

<⇒

∆

>

Rel

ativ

istic

Mom

entu

m a

nd R

evis

ed N

ewto

n’s

Law

s N

eed

to g

ener

aliz

e th

e la

ws

of M

echa

nics

& N

ewto

n to

con

firm

to L

oren

tzTr

ansf

orm

and

the

Spe

cial

The

ory

of R

elat

ivity

: Exa

mpl

e : pmu

=

12

Bef

ore v 1

’=0

v 2’

21

Afte

r V

’

S’

S

12

Bef

ore

vv

21

Afte

r V

=0

P =

mv

–mv

= 0

P =

0

''

'1

2

''

12

12

21

12

22

2 2

'

2

''

befo

reaf

ter2

0,

, '

2

11

1

1

1

2

2

',

p

pafter

before

mv

pmv

m

vv

vv

vV

vv

vV

vvv

vvV

vv c

p

vv

cc

mV

mv

cc

−−

−−

==

==

=

−=

+

=−

−−

−+

=

≠

+=

=−

()

()

()

2

3/2

22

2

22

2

2

3/2

2

2

2 3/2

1(

/)

12

()(

1

(/

) : R

elat

ivis

t

)2

1(

/)

1(

/)

1

ic

1(

)

( /

/)

mu

pmu

uc

ddud

usedt

dtdu

mmu

udu

Fc

dtuc

uc

mc

mu

mu

duF

dtc

u

dpd

mu

Fdt

dtu

duc

cdt

mF

c

u

γ=

=−

=

⎡⎤

−−

⎢⎥

=+

×⎢

⎥−

−⎣

⎦⎡

⎤−

+⎢

⎥=

⎢⎥

−

⎛⎞

⎜⎟

==

⎜⎟

−⎝

⎠

⎡⎣⎦

⎤⎢

⎥=

⎢⎥

−⎣

⎦ 3/2

2

Forc

e

Sinc

e A

ccel

erat

ion

a =

Not

e: A

s /

1, a

0 !!!

!Its

har

der t

o ac

cele

rate

whe

n yo

,

F

u ge

t clo

se to

spee

d of

l

a=1

(/

h

)

ig

m

tuc

du dt

uc

⎡⎤

−⎣

⎦⇒

→→

Rel

ativ

istic

M

omen

tum

Forc

e A

nd

Acc

eler

atio

n

Rea

son

why

you

can

t qui

tege

t up

to th

e sp

eed

of li

ght !

PEP

PEP --

II a

ccel

erat

or sc

hem

atic

and

tunn

el v

iew

II a

ccel

erat

or sc

hem

atic

and

tunn

el v

iew

A L

inea

r Par

ticle

Acc

eler

ator

3/2

2eE

a=1

(/

)m

uc

⎡⎤

−⎣

⎦

Acc

eler

atin

g E

lect

rons

Thr

u R

F C

aviti

es

Fitti

ng a

5m

pol

e in

a 4

m b

arnh

ouse

2

farm

boy

sees

pol

e c

ontra

ctio

n fa

ctor

1(3

Stud

ent w

ith p

ole

runs

/5)

4/5

says

pol

e ju

st fi

ts i

with

v=(

3/5)

n th

e ba

rn fu

lly!

c

cc

−=

2D S

tude

ntfa

rmbo

y

2

Stud

ent s

ees b

arn

con

tract

ion

fact

or

1(3

/5)

4/5

says

bar

n is

onl

y 3.

2m lo

ng

Stud

, to

ent w

ith p

ole

runs

o sh

ort

to c

onta

in

ent

ire 5

m p

ole

!

with

v=(

3/5)

cc

c

−=

Farm

boy

says

“You

can

do

it”

Stu

dent

say

s “D

ude,

you

are

nut

s”

V =

(3/5

)c

Is th

ere

a co

ntra

dict

ion

? Is

Rel

ativ

ity w

rong

?

Hom

ewor

k: Y

ou fi

gure

out

who

is ri

ght,

if an

y an

d w

hy.

Hin

t: Th

ink

in te

rms

of o

bser

ving

thre

e e

vent

s

Fitti

ng a

5m

pol

e in

a 4

m b

arnh

ouse

?

'

'2

0

' 0

0

0L =

pro

per l

engt

h of

pol

e in

S'

Even

t A :

arriv

al o

f rig

ht e

nd o

f pol

eat

= le

ngth

of b

arn

in S

< L

left

end

of b

arn:

(t =

0, t'

=0) i

s ref

eren

ce

In

fram

e

L

=L1

S: le

ng(

/th

of p

ol)

The

te

imes

in

l

vc

−

' 2

20

B

''

0C

22

't

1(

/)

1(

/)

'1

t1

(/

)

two

fram

es a

re re

late

d:

Tim

e ga

p in

S' b

y w

hich

eve

nts B

and

C

fail

to b

e si

mul

t

1(

aneo

u

/

s

)

BC

BC

ll

vc

tvc

vv

Lt

lv

vvc

vc

==

−=

−

==

⇒

=−

−

2D S

tude

ntfa

rmbo

y

V =

(3/5

)c

Ans

wer

: S

imul

tane

ity!

Farm

boy

sees

two

even

ts a

s si

mul

tane

ous

2D s

tude

nt c

an n

ot a

gree

Fitti

ng o

f the

pol

e in

bar

n is

rela

tive

!

A: A

rriv

al o

f rig

ht e

nd o

f pol

e at

left

end

of b

arn

B: A

rriv

al o

f lef

t end

of p

ole

at le

ft en

d of

bar

n C

: Arr

ival

of r

ight

end

of p

ole

at ri

ght e

nd o

f bar

nS

= Ba

rn fr

ame,

S' =

stud

ent f

Let

ram

e

Farm

boy

Vs

2D S

tude

nt

Pol

e an

d ba

rn a

re in

rela

tive

mot

ion

u su

ch th

at

lore

ntz

cont

ract

ed le

ngth

of p

ole

= P

rope

r len

gth

of b

arn

In re

st fr

ame

of p

ole,

E

vent

B p

rece

des

C