W a v e s i n t h e E t h e r

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8 .2 2 5 / S T S .0 4 2 , P h y s ic s in t h e 2 0 t h C e n t u r y P r o f e s s o r D a v id K a is e r , 1 6 S e p t e m b e r 2 0 2 0

1. W a v e s f o r M a x w e l l a n d Lo r e n t z

2. T h e M i c h e l so n - Mo r l e y E x p e r i m e n t

3. Lo r e n t z C o n t r a c t i o n

* S e e L e c tu r e No te s o n “ El e c t r o dy n am i c s f o r M ax w e l l an d L o r e n t z ” f o r m o r e de t ai l e d de r i v at i o n s an d di s c u s s i o n .

Ma x w e l l ’ s “ T r a n s v e r s e U n d u l a t i o n s ”

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In 1 8 6 5 , M a x w el l co n cl u d ed t h a t l i g h t w a s not hi ng ot he r t ha n “ t r a ns v e r s e u nd u l a t i ons ” of E an d B fi e l d s p r o p a g a t i n g i n t h e e t h e r .

Wh y ? R e c a l l t h a t D =  E an d B =  H (  is th e d ie le c tr ic c o n s ta n t,  is th e m a g n e tic pe r m e a bi l i t y ) . I n t he e t he r ,  →  0 ,  →  0 .

In a s o u rce - fr e e r e g i o n o f sp a c e ,  = J = 0 .

wa v e e q u a t i o n s !

wh e r e v is th e s p e e d o f th e tr a v e lin g w a v e . C o n s id e r m o tio n in o n e s p a tia l d im e n s io n , x → x :

ƒ  ƒ 

x / λ

t / T

wh e r e v is th e s p e e d o f th e tr a v e lin g w a v e . M a x w e ll h a d f o u n d

Wh e n h e s u b s t i t u t e d t h e b e s t - kn o w n v al ue s f o r  0 an d  0 , h e f o u n d

wh e r e c = 3 x 10 5 km / s , t h e s p e e d o f l i gh t ( kn o w n f r o m as t r o n o m i c al m e as ur e m e n t s ) . S o

he c onc l u d e d t ha t l i g ht w a s no t h i ng o t h e r t h an wa v e s o f E an d B , t r a v e l i n g t h r o u g h t h e lu m in if e r o u s e th e r a t a s p e e d

Be g i nni ng i n t he 1 8 8 0 s , t he D u t c h m a t he m a t i c a l ph y s i c i s t He n d r i k L o r e n t z tr ie d to g e n e r a liz e Ma x w e l l ’ s w o r k : w h a t i f t h e e m i t t e r o r r e c e i v e r o f l i g h t w a s mo v i n g ?

Ga l i l e a n c o o r d i n a t e t r a n s f o r ma t i o n :

An d r i e s v a n E e r t v e l t , “ D u t c h s h i p s s a i l i n g o f f a r o c k y s h o r e , ” c a . 1 6 1 0

Image is in the public domain.

Bu t t he n M a x w e l l ’ s e q u a t i ons y i e l d e d , a n d s i m i l a r l y fo r B ’.

If , t h e n t h e s o l u t i o n s

Lor e nt z w a s c onc e r ne d : t he Ea r t h is m o v in g Ed v a r d M u n c h , “ T h e S c r e a m , ” 1 8 9 3

th r o u g h th e e th e r , a n d y e t we me a s u r e l i g h t t o be ha v e l i k e s i ne s a nd c os i ne s a l l t he t i m e . . .

Image is in the public domain.

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Hi s r e s p o n s e ( p a r t 1 , 1 8 9 5 ) : i n t r o d u c e a ne w t i m e v ar i ab l e , “ l o c a l tim e , ” t ’ = t ’ ( x , t , v ) . T h e n h e c o u l d s a v e t h e f o r m o f M a x w e l l ’ s wa v e e q u a t i o n , e v e n f o r b o d i e s i n m o t i o n .

Lor e nt z a l w a y s c ons i d e r e d t hi s a “ m a t he m a t i c a l t r i c k . ” Loc a l t i m e w a s “ f i c t i t i ou s , ” a nd ge n u i n e t i m e t al w a ys re f e r r ed t o t h e et h er r es t f r am e .

Qu e s t i o n s ?

c

Li k e a l l hi s c ol l e a g u e s , Lor e nt z k ne w t ha t l i g ht pr opa g a t e d i n t he et h er . A n at u ral q u es t i o n t o as k : C o u l d o n e d et ect t h e E ar t h ’ s mo t i o n t h r o u g h t h e e t h e r , b a s e d o n t h e b e h a v i o r o f l i g h t o n E a r t h ?

Th e e f f e c t s h o u l d b e l i k e a b i c y c l i s t f e e l i n g a h e a d - wi n d wh e n mo v i n g t h r o u g h t h e a t mo s p h e r e .

© source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/

Be g i nni ng i n t he e a r l y 1 8 8 0 s , t he A m e r i c a n e x pe r i m e nt a l ph y s i c i s t Al b e r t Mi c h e l s o n so u g h t t o me a s u r e s u c h a n e f f e c t . H e d e s i g n e d a n d b u i l t a n e w t y p e o f d e v i c e — an in ter f er o m eter — to tr y to m e a s u r e e f f e c ts o f th e E a r th ’ s m o tio n th r o u g h th e e th e r o n th e p r o p a g a tio n o f lig h t.

Mi c h e l s o n ’ s w o r k w a s s o m e o f t h e f i r s t w o r k b y p h y s i c i s t s i n t h e U S t h a t l e a d i n g p h y s i c i s t s i n Eu r o p e c o n s i d e r e d w o r t h p a y i n g a t t e n t i o n t o . H e b e c a m e t h e f i r s t U S - ba s e d ph y s i c i s t t o w i n th e N o b e l P r iz e (1 9 0 7 ).

Mi c h e l s o n ’ s i n s i g h t w a s t o u s e t h e in ter f er en ce o f li g h t as a v er y s en s i t i v e t es t .

An a l o g y : t w o s w i m m e r s i n a r i v e r . Ea c h m u s t be g i n a t A , s w i m a d i s t a n c e L an d r et ur n t o A . Th e y c a n o n l y s w i m a t s p e e d c wi t h r e s p e c t t o

th e w a te r . M e a n w h ile , th e r i v e r f lo w s w ith a cu r ren t o f s p eed v .

Sw i m m e r 1 :

Mi c h e l s o n ’ s i n s i g h t w a s t o u s e t h e in ter f er en ce o f li g h t as a v er y s en s i t i v e t es t .

An a l o g y : t w o s w i m m e r s i n a r i v e r . Ea c h m u s t be g i n a t A , s w i m a d i s t a n c e L an d r et ur n t o A . Th e y c a n o n l y s w i m a t s p e e d c wi t h r e s p e c t t o

th e w a te r . M e a n w h ile , th e r i v e r f lo w s w ith a cu r ren t o f s p eed v .

Sw i m m e r 2 :

γ Wh o w i n s ?

De f i n e (N o te  ≥ 1 f o r 0 ≤ v ≤ c .)

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 v / c

Wh e n t h e r e i s a c u r r e n t i n t h e r i v e r ( v ≠ 0 ), S w im m e r 1 ta k e s lo n g e r to c o m p le te h e r la p th a n do e s Sw i m m e r 2 .

“s e c on d or de r ef f ect”

“I n t e rf e r o m e t e r”

If t h e l i g h t w a v es t h a t t ra v el p a t h 1 t a k e lo n g e r to c o m p le te a la p th a n th e lig h t w a v e s th a t tr a v e l p a th 2 , th e n th e w a v e s w ill a r r i v e at t h e s cr een ou t of p ha s e : c r e s t s f r o m o n e wa v e w i l l n o l o n g e r l i n e u p w i t h c r e s t s fr o m t h e o t h e r . In t e rf e r e n c e !

Image is in the public domain.

1887 e x p e r i m e n t : L = 1 1 m e t e r s ; e n t i r e a p p a r a t u s f l o a t i n g i n a v a t o f m e r c u r y t o d a m p e n vi b r at i o n s . ( Do n ’ t t r y t h i s a t h o m e ! )

no o n

ev e n i n g

Se n s i t i v e t o ( v / c ) 2 ~1 0 - 8 !

An d y e t . . . “ n u l l r e s u l t ” :

Fr o m M i ch e l s o n a n d M o r l e y ( 1 8 8 7 )

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D o t t e d c u r v e s s h o w on e - e i g h th t h e e x p e c t e d m ag n i t u de o f t h e e f f e c t ; s o l i d l i n e s s h o w t h e m e as u r e d s h i f t i n i n t e r f e r e n c e f r i n g e s .

“I t s e e m s f ai r t o c o n c l ud e f r o m t h e fi g u r e t h a t i f t h e r e i s a n y d i sp l a c e m e n t du e t o t h e r e l a t i v e m o t i o n o f t h e e a r t h an d t h e l um i n i f er o us et h er , t h i s can n o t be m u c h g r e a t e r t ha n 0 . 0 1 of t he di s t a n c e b e t w e e n t h e f r i n g e s . [ . . . ] T h e act ual d i s p l acem en t w as cer t ai n l y l es s th a n th e tw e n tie th p a r t [o f th e ex p ect ed v al u e] an d p ro b ab l y l es s t h an th e f o r tie th p a r t. ” (p p . 3 4 0 - 341)

Qu e s t i o n s ?

Lo r e n t z Co n t r a c t i o n

Lor e nt z f ol l o w e d M i c he l s on ’ s w or k c l os e l y . T o a c c ou nt f or t he n u l l r e s u l t , Lor e nt z a r g u e d t ha t ph y s i c i s t s ha d ne g l e c t e d a ph y s i c al c o nt r ac t i o n of t he a ppa r a t u s a l ong i t s d i r e c t i on of m ot i on th r o u g h th e e th e r .

v

Th e m o l e c u l e s i n t h e a r m o f t h e i n t e r f e r o m e t e r w o u l d b e

sq u ee z ed du e t o t h e i r m o t i o n t h r o u g h a p h y s i c a l , r e s i s t i v e me d i u m, m u c h l i k e a b e a c h b a l l d r a g g e d u n d e r w a t e r .

Th a t w o u l d s h o r t e n t h e le n g t h of t he a r m t ha t w a s he a d i ng di r e c t l y i n t o t h e “ e t h e r w i n d” : le n g t h c o n t r a c t i o n !

In t h a t ca s e , S w i m m er 1 w o u l d h a v e le s s d i s t a n c e t o c o v e r fo r he r l a p , a nd he r l a p - tim e w o u ld b e c o m e

Th e n  t = 0 : t h e r a c e sh o u l d be a t i e , a f t e r a l l . N o w ond e r M i c he l s on a nd M or l e y m e a s u r e d no sh i ft i n t h e i n t e rfe re n c e fri n g e s .

Lo r e n t z T r a n s f o r m a t i o n

Lor e nt z c om bi ne d hi s t w o r e s pons e s t o t he e l e c t r od y na m i c s of m o v i ng bod i e s — “l o c al t i m e ” an d l en gt h co n t r act i o n — to d e r i v e a ne w se t o f c o o rd i n a t e t ra n sfo r m a t i o n s fo r b o d i e s m o v i n g wi t h r e s p e c t t o t h e e t h e r ( re p l a c i n g th e G a lile a n tr a n s f o r m a tio n ):

γ Ga l i l e a n t r a n s f o r ma t i o n Lo r e n t z t r a n s f o r m a t i o n

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 v / c

No t e t h e f a c t o r s o f : i m p e r c e p t i b l e c h a n g e f o r v << c .

Wa v e E q u a t i o n R e s t o r e d

Wh e n L o r e n t z u s e d h i s n e w c o o r d i n a t e t r a n s f o r m a t i o n t o r e c o n s i d e r M a x w e l l ’ s w a v e eq u at i o n f o r E an d B fi e l d s , w h e n e i t h e r t h e e m i t t e r o r r e c e i v e r w e r e m o v i n g w i t h r e sp e c t t o th e e th e r , h e f o u n d th e o r ig in a l f o r m r e s to r e d :

an d l i k ew i s e f o r B ’ . E v e n o n a m o v i n g E a r t h , l i g h t s h o u l d t r a v e l a t s p e e d c an d i t s E an d B

fi e l d s sh o u l d o sc i l l a t e l i k e si n e s a n d c o si n e s .

Lo r e n t z S u m m a r y

Lor e nt z a d d r e s s e d t w o pu z z l e s a bou t t he e l e c t r od y na m i c s of m o v i ng bod i e s : ma t h e ma t i c a l

(“ lo c a l tim e ” ) a n d ex p er im en ta l (n u ll r e s u lts f r o m th e M ic h e ls o n - Mo r l e y e x p e r i m e n t ) .

v

He a r g u e d t h a t s i n c e t h e e t h e r w a s a p h y s i c a l , e l a s t i c m e d i u m , i t wo u l d e x e r t a fo r c e on obj e c t s m o v i ng t hr ou g h i t . T ha t f or c e wo u l d y i e l d a ph y s i c al c o nt r ac t i o n of obj e c t s ’ l e ng t hs a l ong t he i r di r e c t i o n s o f m o t i o n t h r o u g h t h e e t h e r .

γ

Th e e f f e c t s s c a l e d w i t h  , a n d h e n c e w o u l d r e m a i n sm a l l fo r v << c .

(A n u n k n o w n p a ten t cl er k in S w itz er l a n d h a d o th er id ea s . . . ) 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 v / c

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