euclid eberle moon- a postulational formulation of the michelson-morley experiment

8/3/2019 Euclid Eberle Moon- A Postulational Formulation of the Michelson-Morley Experiment

1/5

Physics Essays volume 6, numb er 4, 1

A Postulational Formulation of the Michelson-Morley Experiment

E u c l i d E b e r l e M o o n

A b s t r a c tIn th is pap er the Michelson-Mo rley exper iment i s analyzed f ro m the poin t of v iewpos tulates on the prop aga tion o f l ight: Pos tulate I (Einstein, 1905), P ostula te II (Ritand Pos tulate III (Moon and Spencer, 1956). In a reference system tha t is stat ionrespect to the laboratory, al l three pos tulates pred ict the experimentally determ ined nuThis pap er inves tiga tes the predic t ions based on each pos tu la te in coordinate sys temat a constant ve loci ty wi th respect to the labora tory and in coordinate sys tems uniformly accelerated in a straight l ine.

K ey words: veloci ty of l ight , Michelson-Morley exper iment , in terpre ta t ion oM ichelson-M orley exper iment , postula tes on the veloci ty of l ight , F i tzGerald continterpretation of the FitzGerald contraction, accelerated coordinate systems, constanity coo rdinate systems

1 . I N T R O D U C T I O NIn 1881 Michelson f i rst performed an exper imen t to m easure

the speed of the Ear th through the e ther, the postulated mediumin which l ight waves were supposed to t ravel . The exp er imentconclus ively proved that the e ther hypothes is was mis takenbecause no f r inge shi f t was observed. Al though not des igned forthis purpose, the M iche lson -M orley ex perim ent 1) is ofteninterpreted as a proof that the speed of l ight is constant in allinertial reference frames. As we shall see, this is not the only

conclus ion that can be drawn f ro m this exper iment .In th is paper the Michelson-Morley exper iment wi l l be

analyzed from th e point o f view of three postulates: that ofEinste in, ~2) wh ich will be called Postu late I; that o f R itz, (3) calledPostu late II, an d that o f M oo n an d Spe ncer, ~4~ called PostulateI II . The exper iment wi ll be examined as in terpre ted wi th each o fthe three postulates, f irst, in a referen ce frame that is stationarywith respect to the source; next , in a f rame moving a t a constantvelocity with respect to the source; and, f inally, in a referenceframe that is un iform ly accelerated in a straight line.

2 . T H E P O S T U L AT E S O N T H E V E L O C I T Y O F L I G H TBefore we a t tempt to analyze the M ichelson-M orley exper i -

ment, i t is important to visualize the three postulates on theveloci ty of l ight.

Postu la te I (Einstein, 1905): The velo city of l ight is a constantc independent of the m ot ions of source and receiver.

We can visualize this as l ight emanating from the source inspheres whose center remains where the source was a t theinstant of emission, Fig. 1.

Figure 1. Postulate I on the velocity of l ight, sugg est

Einstein in 1905.

Postu la te I I (Ritz, 1908) : According to Ri tz , the or ig in ol ight spheres m oves l inear ly wi th whatever v eloci ty the shad at the t ime of emission, Fig. 2.

Postulate HI (Mo on and Spencer, 1956) : The or ig in of thespheres a lways remains a t the source regardless of the m otthe source after emission, F ig. 3.

48


2/5

A Postulational Formu lation of the Miehelson-Morley Experiment

F igu re 2 . Pos tu l a t e I I on t he ve loc i t y o f li gh t , sugges t ed by R i t zin 1908.

F igu re 3 . Pos tu l a t e I I I on t he ve loc i t y o f l i gh t , suggesMoon and Spence r i n 1956 .

3 . I N T H E L A B O R AT O R Y R E F E R E N C E F R A M ETw o beams o f l igh t l e ave the l i gh t sou rce a t in s t an t t = 0 ,

F ig . 4 . T he beam sho wn ho r i zon t a l l y i n F ig . 4 r eaches mi r ro rM~ a t t ime t t , i s re f lec ted back to the rec e iver, and ar r ives a tt ime t3. The l eng th o f bo th a rms i s L . S ince the l i gh t sou rce i ss t a t i ona ry i n t he l abo ra to ry r e f e r ence f r ame , t he ve loc i t y o f l igh ti s c according to Pos tu la tes I , I I , and I I I . The l ight t ravel ing inthe pa th show n ve r t i c a l ly i n F ig . 4 r eaches the s econd mi r ro rM Ea t t ime h and a r r i ve s back a t t he r ece ive r a t time t4. The re fo re ,i n t he l abo ra to ry r e f e r ence f r ame ,

L = Ct l , L = c ( t a - t l ) ,(1 )

L = c t 2 , L = c ( t 4 - t2 ).

So lv ing fo r t he fou r va lues o f t ,

t~ = L / c , t3 = L / c + t~ ,(2 )

t 2 = L / c , t 4 = L / C + h .

Subst i tu t ing for t t and t2 in the express ions for t3 and t4 ,

= 2 L / c = t , . (3)

The re fo re , t he two beam s o f l i gh t r e tu rn t o t he r ece ive rs imu l t aneous ly. Thus , i f ana lyzed i n t he l abo ra to ry r e f e r enceframe, Pos tu la tes I , I I , and I I I a l l predic t tha t there wi l l be nof r inge sh i f t , i n acco rdance w i th t he expe r im en t .

4 . I N A C O N S TA N T V E L O C I T Y R E F E R E N C E F R A M EN o w a n a l y z e t h e M i c h e l s o n - M o r l e y e x p e r i m e n t i n a c o -

o rd ina t e sy s t em in wh ich t he l abo ra to ry i s mo v ing a t a cons t an tve loc i t y v in t he d i r ec t ion o f one o f t he a rms . The l i gh t beam

f (

I_

/

~ - 0 L

M , t ,

t

F i g u r e 4 . T h e M i c h e l s o n - M o r l e y e x p e r i m e n t d e s c r i b e dl abo ra to ry r e f e r ence f r ame .

tha t t rave l s i n t he d i r ec ti on o f m o t ion i s show n in F ig .5 ( a ) .T hl igh t i s emi t t ed a t t ime t = 0 f rom the sou rce , wh ich t rve loc i t y v f rom l e f t t o r i gh t. T he l i gh t con t inues t o t r aver igh t un t il i t h i ts t he m i r ro r M l a t t ime t l .

4 8 8


3/5

E u c l i d E b e r l e M

v * ,

V ~

t , ,

t _ q M ,L

L t M 'M '

Figu re 5 ( a ). T he l i gh t beam tha t t r ave l s i n the a r m p e rp e nd i c u l a rt o t he d i r ec ti on o f mo t ion i n a cons t an t v e loc i t y r e f e re nc e f r am e ,acco rd ing t o Pos tu l a t e I .

S ince the exper imenta l appara tus i s t rave l ing a t a cons tan tve loc i t y i n t h i s coo rd ina t e sy s t em, t he mi r ro r h a s mov e d adis tance v t~ f rom i t s or ig ina l pos i t ion . Hence the to ta l d i s tancethe l igh t must t rave l to reach M~ is L + v t~ . The re f lec ted beamwil l t rave l a shor te r d is tanceL - v ( t 3 - f i ) .

Accord ing t o Pos tu l a t e I , bo th i nc iden t and r e f l e c t ed l i gh tbeams t r ave l a t ve loc i t y c , so

L = v t , = c t ~ , L - v ( t 3 - t ~ ) = c ( t 3 - t , ) . (4I)

Solv in g for t l and t3 accord ing to Po s tu la te I ,

L 1t I =

c 1 - v / c '

2L 1t 3 = m

c 1 - v 2 / c 2

( 5 1 )

The l i gh t t r ave l i ng i n t he a rm pe rpend i cu l a r t o v i s d e s c r i bedin F ig . 5 (b ) . T he pa r t o f the sphe r i ca l wave em i t t e d at t = 0wh ich r eaches M2 a t time t 2 t r ave ls a t an ang l e d o w nw a rd an dthen upward a t an ang l e t o r e ach t he r ece ive r a t t i me t 4 . T hel eng th s o f t he pa th s t r ave l ed a r e

[ L ~+ ( v t 2 ) 2 ] ' ' ~

an d

( L ~ + [ v ( t 4 - t g l ~ ) " ~ .

Acco rd i ng t o Pos tu l a t e I , s i nce l i gh t a l so t r ave l s a t v e lo c i t y ca long t he se pa th s ,

[ L 2 + ( v t 9 2 ] " 2 = c t 2 ,

( L ~ + I v ( t 4 - t 9 1 2 ) ' : 2 =c ( t , - t g .

(61)

Solv ing these equat ions for tz and t4 ,

" 'X .- \

. / M , M

Fi gu re 5 (b ) . Th e l i g h t bea m th a t t r a ve l s i n t he d i r ecm o t io n i n a con s t an t v e lo c i t y r e f e r e n ce f r ame , a cco rdPos tu l a t e I .

L 1t 2 = _

C (1 - V 2 1 C 2 ) 1 / z '

2 L 1t 4 = - -

c ( 1 - v ~ l c 2 ) t ' 2 "

( 7 i

Thus the t ime t3 i s no t equal to t4 , and th is ana lys is prf r inge sh i f t .

To sq u a re t h i s p r ed i c t ion w i t h t h e n u l l r e su l t ob t a inedmen ta l l y, F i t z G e ra l d ~ ) p rop o se d i n 1 8 89 t ha t t he l eng tha rm i n t he d i r ec t ion o f mo t i on ac tu a l l y con t r ac t ed . I f t hein the two d i rec t ions a re ca l ledLr~,,uo~ an d L p e r l x . n d i c u l a r ,and t hequat ions for t3 and t4 a re equated , then

L i , . ,, u ~ t= L p e r p e r ~ l i e u la r ( 1 - V 2 / C 2 ) 1 / 2 ,

w hi c h i s t h e f am ou s F i t z G e ra l d co n t r a c t i on .F o r m o t i o n a t a c ons t an t v e lo c i t y i n a s tr a i gh t l i ne , Po

I I a n d I II a r e i den t i ca l . Ho w e v e r, t he d e sc r i p t i on o f t hem e n t i s d i f f e r en t , a s s ho w n i n F ig . 6 ( a ) , f o r t he beam t ri n t h e d i r ec t i on o f m o t i o n . Th e cen t e r o f the emana t i nsp he r e s i s a l ways c o i nc i den t w i th t he so u rce . So t he l igha d i s t a nce L on i t s way ou t an d on i t s way back . The re

L = c f i , L = c ( t 3 - t O . (4ISolving for t I and t3 ,

t ~ = L / c , t 3 = 2 L / c . ( 5 i i , I

T he s a me t h in g hap p e ns wh e n w e l o ok a t t he pe rpenpa t h , F ig . 6 (b ) . T he sph e r i c a l w a ves a r e a lways cen t e r edsource , and the l igh t t rave ls a d is tance L as i t goes to Mi t re turns f rom Mz. Thus

L = c t 2 , L = c ( t 4 - t 2 ) (6II , I

48


4/5

A Postulational Formulation of the Michelson-M orley E xperiment

(~0

J

VT ,

v( -t,l

t,k

L

--O

L [MIM,

F i g u r e 6 ( a ) . T h e l i g h t b e a m t h a t t r a v e l s i n t h e d i r e c t i o n o fm o t i o n i n a c o n s t a n t v e l o c i t y r e f e r e n c e f r a m e , a c c o r d i n g t oP o s t u l a t e s I I a n d I I I .

vOcU

L L

M , K

F i g u r e ( 6 b ) . T h e l i g h t b e a m t h a t t r a v e l s in t h e a r m p e r p e nt o t h e d i r e c ti o n o f m o t i o n i n a c o n s t a n t v e l o c i t y r e f e r e n c ea c c o r d i n g t o P o s t u l a t e s I I a n d I I I .

a n d

t 2 = L / c , 1 4 = 2 L / c . (7II , III)

Th erefo re , acco rd ing to Pos tu la te s I I and I I I , 13 = t4, and then u l l r e s u l t o f th e M i c h e l s o n - M o r l e y e x p e r i m e n t i s e x p l a i n e dw i t h o u t a n y n e e d o f a c o r r e c t i o n f a c t o r.

5 . I N A U N I F O R M LY A C C E L E R AT E D R E F E R E N C EF R A M E

S i m i l a r r e s u l t s c a n b e o b t a i n e d f o r a r e f e r e n c e f r a m e t h a t i su n i f o r m l y a c c e l e r a t e d i n t h e d i r e c t i o n o f o n e o f t h e a r m s o f t h eM i c h e l s o n - M o r l e y e x p e r im e n t . T h e d i s t an c e s tr a v e le d n o w d e -p e n d o n t h e a c c e l e r a t i o n a a n d t h e i n i t i a l v e l o c i t y v. F o rP o s t u l a t e I ,

L + (v t , + V 2 a t e ) = c t~ ,

L - [(vts + I h a ~ ) - ( v t I + i h a t ~ ) l = c ( t 3 -tt).

T h e n

(80

tl = L ! [1 + F ( a , v , L , c ) ] ,c I v / c

t s = 2L 1 [1 +a ( a , v , L , c ) ] .c 1 - - V 2 ] C 2

(9I)

F o r t h e o t h e r a n n ,

( U + [ v ( t4 -

[ U + ( v t~ ) 2 1 " 2 =c t ~ ,

t 2 ) + I /2 a ( t ] - t ~ ) 1 2 ) ' '2 =c ( t , - 1 9 .

( lOI)

Solv in g f or t2 and t4,

t2 = 2L [1 + f ( a , v , L , c ) ]c ( 1 - v 2 / c 2 )

2 L [ 1 + g ( a , v , L , c ) ]t 4 = c ( 1 - v a / c 2 ) 1 /2

(111)

Since t3 does no t equa l t4, a F i tzG era ld- type cont racr e q u i r e d t o p r o d u c e t h e e x p e r i m e n t a l n u l l e f f e c t . T h e n

L ~ n e ' = ( 1 - v 2 / c 2 ) '/2 1 + g ( a , v , L , c ) ( 1 2 IL r ~ i ~ i + G ( a , v , L , c ) '

a n d i n a n a c c e l e r a t e d r e f e r e n c e f r a m e w i t h P o s t u l a t e c o n t r a c t i o n d e p e n d s o n b o t h v e l o c i t y a n d a c c e l e r a t i o n .

Wi t h P o s t u l a t e I I t h e e q u a t i o n s d o n o t c o n t a i n t h e ve loc i ty v, Ins tead ,

L + 1/2 a ~ = c t l ,

L - 1/2 a ( ~ - ~ = c ( t s - t O ,

(8I I

an d

[L2 + 1/2( a t 2 z ) 2 l m= c h ,

(L2 + [1A a( ~ - ~)12)1:2 = c ( t 4 - - t 2 ) ,(10I I

= (2L/c) [1 +e ( a , c , L ) ] , (9I I

t4 = (2L/c )[1 +f ( a , c , L ) l . ( l l I I

S ince t3 does no t equa l t4 , a F i tzGe ra ld- ty pe con t racr e q u i r e d t o p r o d u c e t h e e x p e r i m e n t a l n u l l e f f e c t . H e r e ,

L m , n el = 1 + f ( a , L , c ) (12IL r ~ e a ~ 1 + e ( a , L , c )

W i t h P o s t u l a t e I I t h e c o n t r a c t i o n i s i n d e p e n d e n t o f t hv e l o c i t y, b u t d e p e n d s o n t h e a c c e l e r a t i o n ( i n a n a c c er e f e r e n c e f r a m e ) .

49 0


5/5

Euclid Eberle Moon

With Postulate III the null effect is readily p redicted even inan accelerated frame. Here,

L = ctl, L = c(t3 - t~) (8III)

and

tj = L/c , t3 = 2L/c. (9III).

Also ,

SO

L = c t 2 , L - - c ( t 4 - t 2 ), (10III)

h = L/c, t4 = 2L/c . (11III)

Consequently, ta = t4, and no contraction is needed to predictthe null fringe shift.

6 . C O N C L U S I O N SIt was found that the null result of the Michelson-Mo

experiment can be predicted withou t f i le necessity of introduthe FitzGerald contraction if Postulate III is employed. result is valid for the laboratory reference frame, for referframes m oving at a constant veloc ity relative to the laboraand for accelerated reference frames.

Postulate II requires a contraction to predict the null eonly in accelerated reference frames. Postulate I requircontraction that is a function o f both velocity and acceleratiaccelerated reference frames, the famous F itzGerald contracwhich is a function of velocity in reference frames movinconstant velocity and predicts the null effect without a co ntion in laboratory reference frames.

If Occ am's razor - - that the simplest solution is most true -- is applied to the three, Postulate III is the clear wiOf course, any theory must survive many applicationOccam's razor in many experiments to become the best exiapproximation of the truth.

Received 9 January 1992.

R 6 s u m 6

Dans ce t expos~ l ' exp~r ience de M iche lson-Mor ley es t ana lys~e se lon le po in t de rt ro i s pos tu la t s s ur l a p ropaga t ion de la lumidre . Les pos tu la t s son t l es su ivants : Pos tuEins te in (1905); P os tu la t II , Ri tz (1908) ; Po s tu la t I I I, M oon and Spen cer (1956) . Ds y st d m e d e r ~ d r e n c e s t a ti o n n a ir e p a r r a p p o r t a u l a b o r a t oi r e t o u s l e s t r o is p o s tu l a t s p r ~l ' e ffe t nu l expdr imenta lement dd te rminL Cet expos~ fa i t u ne inves t iga t ion sur les p rddbas~es sur chaque pos tu la t dans des sys tdmes de r (drence qu i se d~placent ~ vconsta nte pa r rapp ort au laboratoire ou avec accdldrat ion rect i l igne constante.

R e f e r e n c e s

1. A.A . M ichelson and E.H . M orley, Am . J. Sci. 34, 3330 8 8 7 ) .

2. A Einstein, Ann. Phys. 17, 891 (1905).3. W. Ritz, Ann. Chim. Phys. 13, 145 (1908).4. P. M oon and D.E. Spencer, Philos. Sci. 23, 216 (1956).5. G.F. FitzGerald, Letter to the Editor, Science XIII, 390

(1889).

E u c l i d E b e r l e M o o n

Massachusetts Insti tute of TechnologyCambridge , Massachusetts 02139 U.S.A .

4

euclid eberle moon- a postulational formulation of the michelson-morley experiment

Documents