R |
e c e p t io n o f Sp e c ia l R e la t i v it y |
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8. 225 / ST S . 042, P h y s i c s i n t h e 20t h C e n t u r y Pr of e s s or Da v id K a is e r , 2 3 S e pte mbe r 2 0 2 0
1. Ea rl y R e a c t i o n s t o R e l a t i vi t y
2 . M i n k o ws k i an d S p aceti m e
3. C a m b ri dg e W ra n g l e rs an d R e l at i v i t y
R e la t i v it y R e c a p
Ei n s t e i n w a s i n s p i r e d b y Er n s t M a c h ’ s “ p o s i t i v i s m ” t o fo c u s o n “ o b j e c ts o f p o s i ti v e e xp e ri e n c e ” — phenom ena th at c o u l d b e o b s e r v e d o r m e as u re d — and he nc e be g an by f o c u s i n g o n ki n e ma t i c s (m o ti o n o f o b j e c ts ) rath e r th an dy nam i cs (s tu d y o f fo rc e s ).
“O l y m p i a A c a d e m y ”: S o l o v i n e , H a b i c h t , E i n s t e i n , c a . 1 9 0 5
Image is in the public domain.
1. T he l a w s of ph ysi cs a r e v a l i d i n a ny f r a m e of r ef er ence mo v i n g a t a c o n s t a n t s p e e d ( in e r tia l fram e s o f re fe re n c e ).
2. T he speed of l i ght c is c o n s t a n t , in d e p e n d e n t o f t h e
mo t i o n o f t h e s o u r c e .
Fr o m t h e re l a t i v i t y o f s i m u l t a n e i t y , E in s t e in w a s le d t o consi der le n gth con tra ction and tim e d ila tio n : a ll b a s e d o n h o w (m o v i n g ) o b s e r v e rs w o u l d p e rfo r m m e as u re m e n ts , rath e r
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E a rly R e a c t io n s t o R e la t i v it y
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Second r ea ct i on: E i nst ei n ’ s 1905 pa per pr esent ed a cl ev er re - deri v at i on of r esu l t s th at h ad p re v i o u s l y b e e n fo u n d b y ph ysi ci st s l i k e H endr i k L or ent z i n t he 1890s . A f t er a l l , L or ent z ha d a l r ea dy pu bl i shed on le n gth con tra ction and d e r i v e d
th e fac to r = 1 / [ 1 – ( v / c ) 2 ] 1/ 2 .
“L o r en t z - Ei n st e i n t h e o r y ”
Sw i s s F e d e r a l P a t e n t O f f i c e , B e r n
Fe w r e c o g n i z e d t h a t L o r e n t z ’ s w o r k
assum ed an e the r , w he r e as E i nste i n had
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di sm i ssed t he et her a s “super f l uous . ”
As l a t e a s 1 9 1 3 , a l e a d i n g Br i t i s h p h y s i c i s t wr o t e t h a t t h e “ a b s t r u s e c o n c e p t i o n s ” o f Ei n s t e i n ’ s wo r k w e r e “ m o s t f o r e i g n t o o u r h a b i t s o f t h o u g h t , ” a n d “ a s y e t s c a r c e l y a n y o n e i n t h i s c o u n t r y pr of essed t o u nder st and, or at l east t o appr eci at e t hem . ”
E a rly R e a c t io n s t o R e la t i v it y
“L o r en t z - Ei n st e i n t h e o r y ”
1909: as s i s t an t p r o f e s s o r i n Z u r i c h 1911: p r o f e s s o r i n P r ag u e
1912: p r o f e s s o r i n Z u r i c h
1914: P r u s s i an Ac ad e m y o f Sc i e n c e s , B e r l i n
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Sw i s s F e d e r a l P a t e n t O f f i c e , B e r n
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Q u e s tion s ?
Min k o w s k i a n d Sp a c e t im e
ET H Zür i c h , c a . 1 9 0 0 .
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On e o f t h e f e w r e s e a r c h e r s wh o di d begi n t o pa y atte nti on to E i nste i n ’ s w or k w as He r m a n Mi n k o w s k i , o n e o f Ei n s t e i n ’ s f o r m e r m a t h e m a t i c s t e a c h e r s a t t h e ETH * i n Zür i c h .
Ei n s t e i n us e d t o c ut M i n k o w s k i ’ s c l a s s e s a s a s t ud e n t — bor r o w i ng not es f r om f r i ends t o cr am bef or e t he exam s — and , not su r pr i si ng l y , M i nk o w sk i thou g ht l i ttl e of E i nste i n ’ s tal e n ts . A fte r a fri e n d o f E i n s te i n ’ s e n c o u rag e d M i n k o w s k i to re ad E i n s te i n ’ s 1 9 0 5 p ap e r , M i n k o w s k i re p l i e d : “ I re al l y wo u l d n ’ t h a v e t h o u g h t E i n s t e i n c a p a b l e o f t h a t . ”
To a n o t h e r , M i n k o w s k i re c a lle d , “ It c a m e a s a tre m e n d o u s s u r p ris e , fo r in h is s tu d e n t d a y s Ei n s t e i n h a d b e e n a l a z y d o g . . . . H e n e v e r b o t h e r e d a b o ut m a t h e m a t i c s a t a l l . ”
*Ei d g e n ö s s i s c h e T e c h n i s c h e H o c h s c h ul e : F e d e r a l P o l y t e c h n i c I n s t i t ut e
Min k o w s k i a n d Sp a c e t im e
Onc e Minkow ski did r e ad Einst ein ’ s paper, he bec ame convince d t h at Einste in had done a poo r jo b , m aking t hings unne ce ssar ily c omplic ate d, a nd missi ng t he real point. So he r e fo r mulat e d the w or k in his own w ay in 1908.
Minkowski spe c ialize d in geomet r y . He h ad e ve n w r itte n a b ook ( firs t published in 1896 ) on The G eome t r y o f Numbers , in w hic h h e had c ast t he abstrac t subjec t of numbe r theor y int o ge ome tric al te r m s .
When he turne d to Einst ei n ’ s w ork on t he e le ct r ody namic s of moving bodie s, he did so as a geome te r , rathe r t han a s a p hy sic ist who foc used on pe r fo r ming me asurements or a philosophe r int ere ste d in M ac hian p osit ivism.
W o rldlin e s
t
Us e c o o r d i n a t e s s u c h t h a t l i g h t t r a v e l s o n e u n i t of space i n eac h u ni t of t i m e , e . g . [ t ] ~ s e c o n d s and [ x ] ~ l i g h t - seconds . ( I n ef f ect , w e scal e c = 1 . )
me si t t i ng st i l l
me o u t f o r a jo g , v < c
lig h t
x
L in e s o f S im u lt a n e it y
t
t = 3
t = 2
If I s tan d e q u i d i s tan t fro m l o c ati o n s A and B and I r e c e i v e l i g ht si g nal s fr om A and B at the sam e t i m e , t hen t he l i ght si gnal s m u st ha v e been em i t t ed si m ul t aneousl y (s i n c e c = c o n s t a n t ) .
me In m y re fe re n c e fram e , lin e s of s im u lta n e ity ar e par al l el t o t he x axi s .
t = 1
A B x
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
Wo r l d l i n e s f o r t h e f r o n t , m i d d l e , a n d bac k of a mo v i n g trai n , as o b s e r v e d w i th i n th e o ri g i n al c o o rd i n ate s y s te m .
A x
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
A r i d e r i n t h e m i d d l e o f t h e ( m o v i n g ) trai n re c e i v e s l i g h t s i g n al s fro m A and B at th e s am e ti m e . S o th e e v e n ts A and B mu s t ha v e been si m ul t aneous (s i n c e c = c o n s t a n t ) .
B
A x
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
A r i d e r i n t h e m i d d l e o f t h e ( m o v i n g ) trai n re c e i v e s l i g h t s i g n al s fro m A and B at th e s am e ti m e . S o th e e v e n ts A and B mu s t ha v e been si m ul t aneous (s i n c e c = c o n s t a n t ) .
In h e r re fe re n c e fram e , lin e s of s im u lta n e ity
ar e par al l e l to the x ’ axi s .
x ’
B
A x
L in e s o f Sim u lt a n e it y
t t ’
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
A r i d e r i n t h e m i d d l e o f t h e ( m o v i n g ) trai n re c e i v e s l i g h t s i g n al s fro m A and B at th e s am e ti m e . S o th e e v e n ts A and B mu s t ha v e been si m ul t aneous (s i n c e c = c o n s t a n t ) .
In h e r re fe re n c e fram e , lin e s of s im u lta n e ity
ar e par al l e l to the x ’ axi s .
x ’
B
Th e t ’ axi s i s j u st the wo r l d l i n e of t he poi nt x ’ = 0 .
A x
Th e a n g l e bet w een t he x and x ’ ax e s and be tw e e n the t and t ’ ax e s i s the sam e : t a n = v / c .
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
B
Le n g t h c o n t r a c t i o n : w e m u s t m e a s u re t h e lo c a t io n s o f t h e f ro n t a n d t h e b a c k o f t h e trai n at t h e sam e t i m e .
x ’
A B ’ x
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
Le n g t h c o n t r a c t i o n : w e m u s t m e a s u re t h e lo c a t io n s o f t h e f ro n t a n d t h e b a c k o f t h e trai n at t h e sam e t i m e .
We me a s u r e t h e l e n g t h o f t h e t r a i n t o b e
L ’ = AB ’ , s in c e A and B ’ ar e si m u l tane ou s .
x ’
B
A L ’ B ’ x
L in e s o f Sim u lt a n e it y
t
ba c k of tr a in
mi d d l e of t r a i n
fr o n t o f tr a in
Le n g t h c o n t r a c t i o n : w e m u s t m e a s u re t h e lo c a t io n s o f t h e f ro n t a n d t h e b a c k o f t h e trai n at t h e sam e t i m e .
We me a s u r e t h e l e n g t h o f t h e t r a i n t o b e
L ’ = AB ’ , s in c e A and B ’ ar e si m u l tane ou s .
x ’
B
The r i d e r o n t he t r a i n me a s u r e s t h e l e n g t h of t he t r ai n t o be L = AB , s in c e A and B
L ar e si m u l tane ou s .
L ’ = L /
A L ’ B ’ x
To M i n k o w s k i , l e n g t h c o n t r a c t i o n wa s m e r e l y a g e o m e t r i c a l e f f e c t of pr oj ect i ng ont o appr opr i at e ax es .
R o t a t io n s a n d I n v a r ia n c e
y
y 1
P = ( x , y )
1 1
x 1
x
Us i n g h i s n e w space - tim e d ia g ra m s , Mi n k o ws k i n e x t d e mo n s t r a t e d t h a t t h e Lo r e n t z t r a n s f o r ma t i o n wa s n o t h i n g b u t a ge o m e t r i c e f f e c t : a r o t a t i o n i n s p a c e - ti m e .
Fi r s t c o n s i d e r r o t a t i o n s i n t h e x - y pl ane .
R o t a t io n s a n d I n v a r ia n c e
y
y ’ 1
y ’
y 1
P = ( x ’ , y ’ )
1
1
x 1
x
x ’ 1
x ’
Us i n g h i s n e w space - tim e d ia g ra m s , Mi n k o ws k i n e x t d e mo n s t r a t e d t h a t t h e Lo r e n t z t r a n s f o r ma t i o n wa s n o t h i n g b u t a ge o m e t r i c e f f e c t : a r o t a t i o n i n s p a c e - ti m e .
Fi r s t c o n s i d e r r o t a t i o n s i n t h e x - y pl ane .
R o t a t io n s a n d I n v a r ia n c e
y
y ’ 1
y ’
y 1
P = ( x ’ , y ’ )
1
1
x 1
x
x ’ 1
x ’
Us i n g h i s n e w space - tim e d ia g ra m s , Mi n k o ws k i n e x t d e mo n s t r a t e d t h a t t h e Lo r e n t z t r a n s f o r ma t i o n wa s n o t h i n g b u t a ge o m e t r i c e f f e c t : a r o t a t i o n i n s p a c e - ti m e .
Fi r s t c o n s i d e r r o t a t i o n s i n t h e x - y pl ane .
R o t a t io n s a n d I n v a r ia n c e
y
y ’ 1
y ’
y 1
P = ( x ’ , y ’ )
1
1
x 1
x
x ’ 1
x ’
Us i n g h i s n e w space - tim e d ia g ra m s , Mi n k o ws k i n e x t d e mo n s t r a t e d t h a t t h e Lo r e n t z t r a n s f o r ma t i o n wa s n o t h i n g b u t a ge o m e t r i c e f f e c t : a r o t a t i o n i n s p a c e - ti m e .
Fi r s t c o n s i d e r r o t a t i o n s i n t h e x - y pl ane .
R o t a t io n s a n d I n v a r ia n c e
Mi n k o ws k i d e mo n s t r a t e d t h a t o n e
t
t 1
P = ( t 1 , x 1 )
ca n f ol l o w t he sa m e st e ps f or tran s fo r m ati o n s i n space - tim e .
x 1 x
R o t a t io n s a n d I n v a r ia n c e
Mi n k o ws k i d e mo n s t r a t e d t h a t o n e
t
t ’
t 1
t ’
P = ( t ’ , x ’ )
1
1
1
x ’
x ’ 1
x 1
x
ca n f ol l o w t he sa m e st e ps f or tran s fo r m ati o n s i n space - tim e .
To M i n k o w s k i , t h e Lo r e n t z tra n s fo r m a tio n wa s n o t h i n g b u t a ge o m e t r i c a l r o t a t i o n i n s p a c e - ti m e .
R e c a l l : t a n = v / c
R o t a t io n s a n d In v a ria n c e
y
y ’
P
d
x
x ’
To M i n k o w s k i , t h e g e o m e t r i c a l a p p r o a c h re v e a le d a n e v e n m o re im p o r ta n t le s s o n : ev e n am i d r ot at i ons , som e quant i t y sh oul d r em ai n in v a r ia n t.
In th e E u c l i d e an p l an e , th e di st ance bet w een poi nt s does not c hang e , ev en i f one ro ta te s fro m ( x , y ) to ( x ’ , y ’ ):
R o t a t io n s a n d In v a ria n c e
t
t ’
P
s
x ’
x
Mi n k o ws k i d e mo n s t r a t e d t h a t a space - tim e in te r v a l , s , re m a in e d in v a r ia n t un de r t h e Lo r e n t z t r a n s f o r m a t i o n s :
R o t a t io n s a n d In v a ria n c e
t t ’
P
To M i n k o w s k i , space - tim e wa s t h e u l t i m a t e r e a l i t y — not space and t i m e se par at el y , and cer t ai nl y not t he Ma c h i a n - posi t i vi st descr i pt i ons of m easu r em ent s th at E i n s te i n h ad b e l ab o re d i n h i s p ap e r .
s
x ’
x
R o t a t io n s a n d In v a ria n c e
t t ’
P
To M i n k o w s k i , space - tim e wa s t h e u l t i m a t e r e a l i t y — not space and t i m e se par at el y , and cer t ai nl y not t he Ma c h i a n - posi t i vi st descr i pt i ons of m easu r em ent s th at E i n s te i n h ad b e l ab o re d i n h i s p ap e r .
“H e n c e f o r t h s p a c e b y i t s e l f , a n d t i m e b y itse lf , a r e doom e d to f a de a w a y into m e r e sha d o w s , a nd only a k ind of u nion of the tw o w ill pr e se r v e inde pe nde nc e . ” (1 9 0 8 )
s
x ’
x
Q u e s tion s ?
C a m b ridg e W ra n g le rs a n d R e la t i v it y
Wr i t t e n “ T r i p o s ” e x a m i n a t i o n a t C a m b r i d g e , mi d - 19 th cen t u r y
Image is in the public domain.
“ Im age ch ar ges ” i n el ect r ost a t i cs: sol v e f or t he f i el d l i nes of a c har g e near a g r ou nded condu ct i ng pl at e .
Mu c h l i k e Mi n k o ws k i , o t h e r r e s e a r c h e r s wh o e n c o u n t e r e d Ei n s t e i n ’ s 1 9 0 5 p a p e r r e a d i t f r o m w i t h i n t h e i r o w n “r ef er ence f r a m es . ” F or C a m br i dg e W r a ngl er s , E i nst ei n ’ s wo r k p r o v i d e d s o m e e x c i t i n g t e c h n i q u e s f o r ( f u r t h e r ) expl or i ng t he beha vi or of cl a sses of di f f er ent i a l eq u a t i ons and the i r sol u ti ons .
field lin es m u s t in ter s ect th e p la te a t r igh t a n gles
On e c a n f i n d t h e e x a c t s o l u t i o n by r e p la c i n g th e co n d u ctin g p la te with a n “im a g e c h a r g e” p la ced a n eq u a l d is ta n ce b e lo w t h e p l a t e .
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C a m b ridg e W ra n g le rs a n d R e la t i v it y
Me t h o d o f “ i n v e r s i o n ” : ma p a
K
B
A
O
di f f i cul t pr obl em i nt o a si m pl er one .
An y p o i n t ( s u c h a s A ) i n s i d e th e ci r cl e of r a di u s K ma y b e ma p p e d t o an “ i n v e r se poi nt” B by r e q u i r i n g
OA x OB = K 2
C a m b ridg e W ra n g le rs a n d R e la t i v it y
Me t h o d o f “ i n v e r s i o n ” : ma p a
K
B
A
O
di f f i cul t pr obl em i nt o a si m pl er one .
An y p o i n t ( s u c h a s A ) i n s i d e th e ci r cl e of r a di u s K ma y b e ma p p e d t o an “ i n v e r se poi nt” B by r e q u i r i n g
OA x OB = K 2
If th e p o i n t A mo v e s a l o n g t h e cu r v e pq , t h e n t h e in v e rs e p o in t B mo v e s a l o n g t h e c u r v e p ’ q ’ .
Su c h a t r a nsf or m a t i on pr eser ves ang l es , b u t n o t le n g t h s : “ c o n fo r m a l t ra n s fo r m a t io n . ”
C a m b ridg e W ra n g le rs a n d R e la t i v it y
If w e w an t to s o l v e fo r th e fi e l d
lin e s a n d e q u ip o t e n t ia l s u rf a c e s d u e to a c h arg e at l o c ati o n O near a
gr o u n d e d c o n d u c t i n g s p h e r e cent er ed a t B , w e m a y u s e in v e rs io n
O B to tre at a s i m p l e r p ro b l e m :
C a m b ridg e W ra n g le rs a n d R e la t i v it y
If w e w an t to s o l v e fo r th e fi e l d
O A B
lin e s a n d e q u ip o t e n t ia l s u rf a c e s d u e to a c h arg e at l o c ati o n O near a
gr o u n d e d c o n d u c t i n g s p h e r e cent er ed a t B , w e m a y u s e in v e rs io n to tre at a s i m p l e r p ro b l e m :
Co n s i de r a c h a r g e d c o n duc t i n g spher e cent er ed at A . D ra w t h e fie ld lin e s a n d eq u i pot ent i a l su r f a ces fo r th e s i m p l e p ro b l e m .
C a m b ridg e W ra n g le rs a n d R e la t i v it y
If w e w an t to s o l v e fo r th e fi e l d
K
O A B
lin e s a n d e q u ip o t e n t ia l s u rf a c e s d u e to a c h arg e at l o c ati o n O near a
gr o u n d e d c o n d u c t i n g s p h e r e cent er ed a t B , w e m a y u s e in v e rs io n to tre at a s i m p l e r p ro b l e m :
Co n s i de r a c h a r g e d c o n duc t i n g spher e cent er ed at A . D ra w t h e fie ld lin e s a n d eq u i pot ent i a l su r f a ces fo r th e s i m p l e p ro b l e m . E n c l o s e A in a spher e of r adi u s K .
C a m b ridg e W ra n g le rs a n d R e la t i v it y
If w e w an t to s o l v e fo r th e fi e l d
K
O A B
lin e s a n d e q u ip o t e n t ia l s u rf a c e s d u e to a c h arg e at l o c ati o n O near a
gr o u n d e d c o n d u c t i n g s p h e r e cent er ed a t B , w e m a y u s e in v e rs io n to tre at a s i m p l e r p ro b l e m :
Co n s i de r a c h a r g e d c o n duc t i n g spher e cent er ed at A . D ra w t h e fie ld lin e s a n d eq u i pot ent i a l su r f a ces fo r th e s i m p l e p ro b l e m . E n c l o s e A in a spher e of r adi u s K . M a p t h e c u r v e s to th e o ri g i n al s y s te m i n v o l v i n g B .
( An d d o a l l t h i s o n a t i me d e x a m t h a t det er m i ned y our graduat i on rank ! )
C a m b ridg e W ra n g le rs a n d R e la t i v it y
Th e m e t h o d o f i n v e r s i o n a n d c o n f o r m a l t r a n s f o r m a t i o n s h a d b e c o m e ro u t i n e fo r C am b ri d g e Wr a n g l e r s b y 1 9 0 0 . T h e s e t e c h n i q u e s a p p l i e d t o el ect r ost at i cs , t h a t is , s it u a t io n s in w h ic h t h e el ect r i c pot ent i a l di d not v a r y o v er t i m e:
Eb e n e e z e r C u n n i n g h a m and Ha r r y B a t e m a n ai m e d to g e ne r al i ze those te c hni q u e s to el ect r odynam i cs .
To t h e m , E i n s t e i n ’ s p a p e r o f f e r e d i d e a s a b o u t h o w t o i d e n t i f y a l l t r a n s f o r m a t i o n s ( v ) th at wo u l d l e a v e t h e e q u a t i o n
in v a ria n t . T h e n t h e y c o u ld u s e t h e in v e r s io n te c h n i q u e to p ro d u c e new so l ut i o ns .
Th e y s o u g h t t o p r o v e t h e y h a d f o u n d t h e mo s t g e n e r a l c l a s s of t r ansf or m at i ons ( v ).
Fo r t h e Wr a n g l e r s , th a t ’ s wh a t r e l a t i v i t y w a s a l l a b o u t — not t he et her , not M ac hi an posi t i vi sm , not M i nk o w ski ’ s space - ti m e . . .
R e c e p t io n o f R e la t i v it y : Su m m a r y
Re s e a r c h e r s d i d n o t j u s t r e a d E i n s t e i n ’ s p a p e r a n d b e c o m e dev ot ed E i nst ei ni a ns . Mo s t ph ysi ci st s i gnor ed t he paper com pl et el y f or 5 - 10 yea r s; ot her s a ssum ed i t w a s m er el y a n el a bor a t i on of L or ent z ’ s pr i or w or k.
Th o s e f e w w h o di d pa y at t ent i on di d so f r om w i t hi n t hei r o w n cont ext s . D i f f er ent a spect s of E i nst ei n ’ s pa per seem ed m or e or le s s re le v a n t t o t h e m .
Mi n k o ws k i r e - in t e r p re t e d E in s t e in ’ s p a p e r in t e r m s o f space - tim e g e o m e tr y . C u n n in g h a m a n d Ba t e m a n r e - in t e r p re t e d t h e p a p e r in t e r m s o f pr oper t i es of di f f er ent i al equat i ons and i n v e r si on pr oper t i es of t hei r sol u t i ons .
No n e of t hese r eader s seem ed t o car e m u c h abou t w as m ost i m por t ant t o E i nst ei n: t hat th e e th e r w as “ m e re l y s u p e rf l u o u s , ” an d th at, fo l l o w i n g E r n s t M ac h , o n e s h o u l d s tar t w i th ki n e ma t i c s in s t e a d o f dy nam i cs . A s w e ’v e s e e n b e fo re : sam e equat i ons , di f f er ent m eani ngs!
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STS.042J / 8.225J Einstein, Oppenheimer, Feynman: Physics in the 20th Century
Fall 20 2 0
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