Phy

sics

2D

Lec

ture

Slid

esO

ct 7

Viv

ek S

harm

aU

CSD

Phy

sics

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

' '

') '

'

(y

yz

vtx

xv

tct x

z γγ ⎛⎞

=+

⎜⎟

⎝

==+

⎠

=

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

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

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

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

y

x

S

x’

S’y’

0.75

c-0

.85c

AB

O

O’

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

dtdt

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

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

?

Hol

lyw

ood

Yar

ns !

Term

inat

or :

Can

you

be

seen

to b

e bo

rn b

efor

e yo

ur m

othe

r?

A fr

ame

of R

ef w

here

seq

uenc

e of

eve

nts

is R

EV

ER

SE

D?!

!

SS’

11

''

11

(,

)

(,

)

xt

xt

u

22

''

22

(,

)

(,

)

xt

xt

I take off f

rom SD

I arrive in SF

''

21

2 '

Rev

ersi

ng se

quen

ce o

f eve

n'

0ts

vx

tt

tc

t

tγ

⎡∆

⎤⎛

⎞∆

=−

=∆

−⎜

⎟⎢

⎥⎝

⎠⎣

⎦⇒

∆<

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

theo

ry o

f rel

ativ

ity: E

xam

ple

: 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

vv

cc

cc

pmV

mv

−−

−−

==

==

=

−=

+

=−

−−

−+

=

≠

+=

=−

Watching an Inelastic Collision between two putty balls

Def

initi

on (w

ithou

t pro

of) o

f Rel

ativ

istic

Mom

entu

m

21

(/

)mu

pmu

uc

γ=

=−

With

the

new

def

initi

on re

lativ

istic

m

omen

tum

is c

onse

rved

in a

ll fra

mes

of

refe

renc

es: D

o th

e ex

erci

se

New

Con

cept

sR

est m

ass

= m

ass

of o

bjec

t mea

sure

dIn

a fr

ame

of re

f. w

here

obj

ect i

s at

rest

2

is v

eloc

ity o

f the

obj

ect

NO

T o

f a re

fere

n

11

(/

)

!ce

fram

e u

uc

γ=

−

Nat

ure

of R

elat

ivis

tic M

omen

tum

21

(/

)mu

pmu

uc

γ=

=−

With

the

new

def

initi

on o

f R

elat

ivis

tic m

omen

tum

M

omen

tum

is c

onse

rved

in

all f

ram

es o

f ref

eren

ces

mu