1973.03.26 - stefan marinov - velocity of light in a moving medium according to the absolute...

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International Journal o f Theoretical Physics, Vol. 9, No . 2 (1974), pp. 139-144

Veloc ity o f L ight in a M oving M edium A ccording

to the A bsolute Sp ace-Time Theory

S TEFAN M A R I N O V

uL E lin Pelin 2 2 , Sofia 2 1 , Bulgaria

Received: 26 March 1973

A b s t r a c t

Proceeding from our absolute space-time conceptions and applying the 'hitch-hiker 'mod el (so-called by us) for the propagation of light in a m edium, w e obtain the generalformula fo r the light velocity in a m oving medium including terms o f second orde r h~v/c. This formula is identified with tha t o ne obtained by proceeding from th e Lo rentztransformation.

W h a t is l ig h t ? W h a t is t h e m e c h a n i sm o f p ro p a g a t i o n o f l i g h t ? - -

D e s p i t e t h e h i g h l e v e l t o w h i c h s c i e n c e h a s b e e n d e v e l o p e d i n t h e l a s t

c e n t u r y , t h e r e h a s n o t b e e n a f i r m a n d c l e a r an s w e r t o t h e s e q u e s t i o n s ,N o w t w o s u b s ta n t ia l ly d i f fe r en t m o d e ls o f li g h t ar e c o m m o n i n p h y s ic s

a n d , a l t h o u g h e x c l u d in g e a c h o t h e r , m a n y p h e n o m e n a a r e e x p la i n e d b y

t h e o n e m o d e l , m a n y b y t h e o t h e r a n d m a n y b y b o t h . T h e s e m o d e l s a r e :

(1 ) T h e c o r p u s c u l a r ( N e w t o n ' s ) m o d e l .

(2 ) T h e w a v e ( H u y g h e n s ' ) m o d e l .

I n o u r a b s o l u t e s p a c e - t i m e t h e o r y w e u s e o n l y t h e c o r p u s c u l a r m o d e l .

W e i n t r o d u c e t h e n o t i o n o f th e ' p e r i o d ' o f a p h o t o n ( i. e. , o f a n y l ig h t

c o r p u s c l e ) a s f o l lo w s : T h e p e r i o d T is th e t i m e f o r w h i c h a g i v e n p h o t o n

i s e m i t t e d o r a b s o r b e d , o r t h e ti m e f o r w h i c h w e c a n a s s e r t w i t h c e r t a i n t yt h a t a p h o t o n p r o p a g a t i n g w i t h v e l o c it y c i n v a c u u m ( w i t h r e sp e c t to t h e

r e f e r en c e f r a m e u s e d ) a n d c r o s s in g a g i v e n s u r f a c e h a s i n d e e d c r o s s e d t h is

s u r fa c e . T h e q u a l i t y v i n ve r s e t o t h e p e r i o d is c a l le d t h e f r e q u e n c y .

S i n c e t h e r e is a c e r ta i n t im e T d u r i n g w h i c h t h e p h o t o n i s e m i t t e d , w e c a n

i m a g i n e i t a s a n ' a r r o w ' o r a s a ' m a c h i n e - g u n b u r s t ' w i t h l e n g t h 2 = cT ,

c a l le d th e w a v e l e n g t h . N o w t h e fo l l o w i n g q u e s t i o n a ri s e s: W h e n t h e s o u r c e

m o v e s w i t h a c e r t a i n v e l o c i t y v i n t h e r e f e r e n c e f r a m e u s e d , w o u l d t h e

" a r r o w ' ( o r t h e s in g le b u l l e ts o f t h e ' b u r s t ' ) m o v e w i t h a v e l o c i t y d i f f er e n t

C o p y r i g h t © 1 9 74 P l e n u m P u b l i s h in g C o m p a n y L i m i te d . N o p a r t o f t h i s p u b l i c a ti o n m a y b e r e p r o d u c e d ,stored i n a r e t r ie v a l s y s t e m , o r t r a n s m i t t e d , in a n y f o r m o r b y a n y mea ns, e lectronic , mech anical , p h o t o -c o p y i n g , m i c r o f il m i n g , r e c o r d i n g o r o t h e r w i s e , w i t h o u t w r i tt e n p e r m i s s i o n o f P l e n u m P u b l i s h i n g C o m p a n y

L i m i t e d .1 3 9

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1.40 STEFAN MARINOV

from c 9. According to the answer given to this question there are possiblytwo different models:

(a) The 'arrow' (Ritz') model, according to which the photon moveswith a velocity representing the vector sum of v and c, while thewavelength remains constant.

(b) The 'burst' (Marinov's) model, according to which the photonmoves always with velocity c and only the wavelength (i.e., thedistances between the single bullets of the 'burst') change.

For the mechanism of propagation of light in a medium we use the'hitch-hiker' model (so-called by us). According to this model the photon

is a hitch-hiker walking with velocity c and the molecules (the atoms) ofthe medium are cars driving with velocity v (c > v). Since the walker would

be tired if he walked all the time (then his velocity will be the highestI),

he takes any ruth car on the road (we suppose that the distance between thecars are the same) and rests there a definite time (if he drove all the time

his velocity will be the lowest !). If v < c, then the mean velocity of the hitch-hiker will be cm = c /n , where 1 In is that part of the time during which, onaverage, the hitch-hiker walks and 1 - (l/n) is that part of the time which

the hitch-hiker spends in the cars.Now using this model for the propagation o f the photons in a medium,

we shall calculate their velocity when the medium moves with respect to

the observer. The factor n is called the refractive index of the medium; c isthe velocity of light in vacuum and c , , = c /n is the velocity of light in the

medium when it is at rest with respect to the observer. In the same manneras the hitch-hiker takes a rest in any ruth car, so the photon is 'absorbed'

by any ruth molecule which it meets on its way and there is a definite timeafter which the photon is again 're-emitted'.

Let us suppose first that the medium rests in the frame o f reference usedand that the light propagating with velocity c /n makes an angle 0' with thex-axis (Fig. 1). As supposed previously, any photon, on average, moves1/nth part of the time and [I - (1/n)]th par t of the time rests absorbed by the

molecules.Let us then suppose that the medium moves with velocity v along the

x-axis only during this time when the photon is absorbed by some moleculeand let us suppose that during the time between the re-emission and nextabsorption the medium is at rest. If we consider the path of the photonbetween two successive absorptions, then this path could be presented bythe broken line A B C in Fig. 1. Supposing that the time between two succes-sive absorptions is chosen for a uni t of time, i.e., that

A B B C+ - - = 1 ( 1 )

v c

we shall have

AB=(1 _ 1 ) . v , B C = c /n (2)

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VELOCITY OF LIGHT IN A MOVING MEDIUM 141

tJ

g 9

A

Figure l . - -Thepa thsofaphotonmoving inamediumwi threspec t to the res tandmoving

flames.

I f n o w w e s u p p o s e t h a t t h e m e d i u m m o v e s w i th v e l o c i ty v d u r i n g t h e

w h o l e t i m e , t h e n t h e n e x t ( r u t h ) m o l e c u l e w i l l b e c a u g h t n o t a t p o i n t C

b u t a t p o i n t D , w h e r e t h e d i s t a n c e C D i s c o v e r e d b y t h i s m o l e c u l e i n t h e

t i m e i n w h i c h t h e p h o t o n c o v e r s d i s t an c e B D , i .e . ,

C D = v / n (3)

T h u s n o w t h e d i s t a n ce c o v e r e d b y t h e p h o t o n b e t w e e n t w o s uc c es si ve re -

e m i s s i o n a n d a b s o r p t i o n w i l t b e n o t B C b u t

w h e r e

j ( c 2 v 2B D = B E + E D = ~ - ~-~. in 2 ~9 + . co s ( 4 )

i s t h e a n g l e b e t w e e n t h e ' f r e e p a t h ' o f th e p h o t o n a n d t h e x - a x is w i t h r e s p e c t

t o t h e o b s e r v e r , w h i l e O ' i s t h e a n g l e b e t w e e n t h e ' f r e e p a t h ' o f t h e p h o t o n

a n d t h e x - ax i s w i t h r e s p e c t t o t h e m e d i u m , a n d

• C E v / n . v . ,

"~ - - . s in ip ~ c ' s i n 0~ = a rc sm ~ = c /n (6 )

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

s ee f u r t h e r , i t is e n o u g h t o c o n s i d e r i t w i t h a n a c c u r a c y o f fi rs t o r d e r i n

I ) / C ,

4 , = 0 ' - ( 5 )

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142 S T E F A N M A R I N O V

W i t h i n t h e s a m e a c c u r a c y o f f ir s t o r d e r i n v / c w e c a n w r i t e , h a v i n g i n

m i n d ( 5) a n d ( 6),

cos f f = cos 0 ' + v . s in2 0 ' (7)c

T h e d i s t a n c e c o v e r e d b y t h e p h o t o n b e t w e e n t w o s u c c e ss iv e a b s o r p t i o n s

w i t h r e s p e c t t o t h e o b s e r v e r w i ll b e

A D 2 = A B z + B D 2 + 2 . A B . B D . c o s O (8 )

S u b s t i t u t i n g h e r e (2 ), ( 4 ) a n d ( 7) a n d w o r k i n g w i t h a n a c c u r a c y o f s e c o n d

o r d e r i n v / c ( o b v i o u s l y i n s u c h a c a s e i t is e n o u g h t o t a k e c o s ~ k - - a n d t h u s

a l so e - - w i t h a n a c c u r a c y o f f ir st o r d e r i n v / c ) , w e o b t a i n

J c ( . ~ v . c O ' ) c , 1 v 2A D = + 2 . - - n . c o s + v z = - + v . c O S 0 n + 2 " ~ . n . s i n 2 0 ' (9)

T o o b t a i n t h e m e a n v e l o c i ty o f t h e p h o t o n w i t h r e sp e c t t o t h e o b s e rv e r

w e h a v e t o d i v i d e t h e d i s t a n c e A D b y t h e t i m e f o r w h i c h t h e b r o k e n l in e

A B D i s c o v e r e d . T h i s t i m e , t a k e n w i t h a n a c c u r a c y o f s e c o n d o r d e r i n

v / c , i s

A B B D A B ~ / ( B C z - C E 2 ) + D Et i n = - - + = - - +

/3 C V C

A B B C 1 C D 2 s in2@ C D= - - + - - . - - + ......... .c o s ~

v c 2 B C 2 " c c

v 1 v 2= 1 + ~ . c o s 0 ' + ~ . s inz 0' (10)

c . n 2 " c . n

wh ere w e have use d (1 ), (2 ) , (3) an d (7 ).T h u s f o r t h e m e a n v e l o ci ty o f th e p h o t o n i n t h e m o v i n g m e d i u m m e a s -

u r e d b y t h e o b s e r v e r a t r e s t w e g e t , w i t h a n a c c u r a c y o f s e c o n d o r d e r

i n v / c ,

. . . . + v . I - .c o s 0 'Cm t m n

( ) ( 1 ) 0,a 1 O ' 1 v2 .n 1 - ~ s i n 2 (11)c . n " 1 - ~ . co s 2 + 2 " - ' - c ' - - '

T h e f a c to r

1

= 1 - n-- (12)

i s ca l l ed the Fresne l ' s d rag coe f f i c ien t .I f w e w a n t t o i n t r o d u c e t h e a n g l e 0 b e tw e e n t h e x - a x is a n d t h e a v e r a g e

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V E L O C I T Y O F L I G H T I N A M O V I N G M E D I U M 143

v e l o c i ty o f t h e p h o t o n w h i c h i s m e a s u r e d b y t h e o b s e r v e r a t r e st , w e s h a ll

h a v e

0 = 0 ' - f l (13)

w h e r e

f i = ~ ~ = v ' s i nO v . n . s i n O ( 1 4 )c /n C

i s t h e d i f f e r e n c e b e t w e e n a n g l e s 0 ' a n d 0 w h i c h i s s m a l l a n d i t i s e n o u g h t o

c o n s i d e r i t o n l y w i th a n a c c u r a c y o f fi rs t o r d e r i n v / c .

W i t h in t h e s a m e a c c u r a c y o f f ir s t o r d e r i n v / c w e c a n w r i te , h a v i n g i n

m i n d ( 1 3 ) a n d ( 1 4 ) ,

0 _ v . nco s O ' = co s . s i n z 0 (15 )c

S u b s t i tu t i n g t h i s i n t o ( 1 1 ), w e f i n d

= - . COS 0c , , n

v2 ( 1 ) t V 2

c . n " 1 - ~ . c o s 2 0 - ~ . c

F o r 0 = 0 ' = 0 f o r m u l a e s ( 1 1 ) a n d ( 1 6 ) g i v e

( 1 - ( 1 1c . n

C ~ = n + V . 1 1 v 2 1

. n . ( 1 - ~ ) . si n 20 ( 16 )

( 1 7 )

F o r 0 = ~ /2 , 0 ' = Q c / 2 ) + [ ( v . n ) / c ] f o r m u l a e ( 1 1) a n d ( 1 6) g iv e

e l y 2 ( 1 )c , , = . n . 1 - ( 18 )

n 2 " c - ~

E x a c t l y th e s a m e r e s u l ts c a n b e o b t a i n e d w h e n p r o c e e d i n g f r o m t h e

L o r e n t z t r a n s f o r m a t i o n f o r m u l a e f o r v e l o c i ty w h i c h r u n ( se e , f o r e x a m p l e ,

M o l l e r ( 1 9 5 5 ) )

v ; + v v ; . 1 - V / c

v x = " V ' v r = v ~ . V ( 1 9 )1 + v x - - 2 -- " 1 +

c 2 c 2

w h e r e v ~, v£ a r e t h e v e l o c i t y c o m p o n e n t s o f a m a t e r i a l p o i n t i n t h e m o v i n g

f r a m e o f r e f er e n c e a n d v :¢ , v y a r e t h e v e l o c i ty c o m p o n e n t s o f t h e s a m e

p o i n t i n t h e r e st fr a m e , s u p p o s i n g t h a t t h e m o v i n g f r a m e p r o c e e d s w i t h

v e l o c i t y V a l o n g t h e x - a x i s o f t h e r e s t f r a m e a n d t h e i r a x e s a r e r e s p e c t i v e l yp a r a l l e l .

P u t t i n g i n (1 9 )

, C , C

v ~ = - . c o s 0 ' , v y = - . s i n 0 ' , V = v ( 2 0 )1l n

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1 4 4 S T E FA N M A R I N O V

a n d w o r k i n g w i t h a n a c c u r a c y o f s ec o n d o r d e r i n v/c , w e o b t a i n f o rV'(v~ + vy) exac t ly fo r m ula (11).

F o r t h e c o m p o n e n t s o f t h e v e l o c i ty th e i d e n t i ty is o n l y w i t h i n t h e f ir s t

o r d e r i n v/c. I n d e e d , i f w e u s e i n th e e q u a t i o n s

e,,. co s 0 = vx, e,,. sin 0 = vy (21)

fo rm ula e (11), (15), (19) an d (20), the n we see th a t on ly the t e rm s o f ze ro

a n d f ir s t o r d e r i n v /c a r e i d e n t i c a l o n b o t h s id e s o f t h e s e e q u a t i o n s .

F o r m u l a (1 7) w a s p r o v e d e x p e r i m e n t a ll y w i t h i n a n a c c u r a c y o f fi rs t

o r d e r i n v ie f i rs t b y F i z e a u (1 85 1). A n e x p e r i m e n t a l p r o o f o f t h i s f o r m u l a

w i t h i n a n a c c u r a c y o f s e c o n d o r d e r i n v /c i s s ti ll n o t m a d e a n d a t t h e p r e s e n t

s t a t e o f te c h n i q u e s u c h a n e x p e r i m e n t i s t o b e c o n s i d e r e d o n l y a s a c h a l l e n g e

t o t h e e x p e r im e n t o r s .

T h i s e x p e r i m e n t , s k e t c h e d b r i e fl y , w i l l a p p e a r a s f o l l o w s : L e t u s u s e t h eM i c h e l s o n i n t e r f e r o m e t e r a n d l e t u s p u t a l i q u i d w i t h r e f r a c t i v e i n d e x n i n

o n e o f it s a r m s w h o s e l e n g t h is L . W e s h o u l d o b s e r v e a c e r t a i n i n te r f e r e n c e

p i c tu r e . L e t u s t h e n s e t t h e l i q u i d i n m o t i o n w i t h v e l o c i ty v a l o n g t h e

a r m L . N o w i f w e u s e f o r m u l a ( 1 7) , w e s h o u l d e a s i l y o b t a i n , w h e n t h e l i q u i d

i s i n m o t i o n , t h e l ig h t b e a m p r o c e e d i n g a l o n g a r m L , t h e r e a n d b a c k , a n d

r e t u r n i n g t o t h e s e m i - t r a n s p a r en t m i r r o r o f t h e i n te r f e r o m e t e r w i t h a t i m e

d e l a y

L L 2 . L 2 . L . v 2t = ~ + c T ~ c / n = c -g -- .n .( n z - l ) (22)

H o w e v e r , e v e n b e f o r e p e r f o r m i n g t h i s e x p e r i m e n t , w e c a n m a k e t h e

f o l l o w i n g c o n c l u s i o n : S i n c e t h e L o r e n t z t r a n s f o r m a t i o n f o r m u l a e h a v e

s h o w n t h e i r v a l i d i t y i n m a n y d i f fe r e n t e x p e r im e n t s , t h e n t h e i d e n t i t y o f

t h e r e s u l t s o b t a i n e d , o n t h e o n e h a n d , p r o c e e d i n g f r o m o u r a b s o l u t e

s p a c e- t im e c o n c e p t i o n s a n d f r o m t h e ' h i t c h - h i k e r ' m o d e l f o r t h e p r o p a g a -

t i o n o f l i g h t i n a m e d i u m a n d , o n t h e o t h e r h a n d , f r o m t h e L o r e n t z t r a n s-

f o r m a t i o n f o r m u l a e , is v e r y s tr o n g s u p p o r t f o r

( a) o u r a b s o l u t e s p a c e - t im e t h e o r y , w h i c h d e f e n d s t h e a s s e r t i o n t h a t t h e

n o n - r e l a t i v i s t i c a n d r e l a t i v i s t i c m a t h e m a t i c a l a p p a r a t u s ( i . e . , t h e

G a l i l e a n a n d L o r e n t z t r a n s f o r m a t i o n s ) a r e n o t c o n t r a d i c t o r y ( a t

l e a s t w i t h i n a n a c c u r a c y o f s e c o n d o r d e r i n v/c), t h u s a n a b s o l u t e

space - t ime does ex i s t , and( b) o u r ' h i tc h - h i k e r ' m o d e l f o r t h e p r o p a g a t i o n o f l ig h t in m a t e ri a l

m e d i a .

References

Fizeau, H. (1851). Co m pte rendu hebdomadairedes s(anees d e l'Acaddm ie des sciences,33 ,349.

M~ ller, C. (1955). The Theory of R elativity, Section 21. Clarendon Press, Oxford.