Domina Eberle Spencer and Uma Y. Shama- Stellar Aberration and the Postulates on the Velocity of...

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Physics Essays volume 9, number 3, 1996

S t e l la r A b e r r a t i o n a n d t h e P o s t u l a t e s o n th e V e l o c i t y o f L i g h t

D o m i n a E b e rl e S p e n c e r a n d U m a Y . S h a m a

A b s t r a c t

This pap er presents a s im ple pr oo f of the validity o f the universal t ime postulate on theveloci ty o f light based on the experimental d ata on s tel lar aberrat ion observed by Bradley in

1728. I t is demonstrated that Eins tein's postulate on the veloci ty o f l ight predicts s tel lar

aberration correctly in coordinate systems in wh ich the star is stationary bu t fa lls to pre dict

the pheno meno n at al l in the earthbound coordinate sys tem in which i t is observed.

Ke y word s : ve loc i ty of l igh t , s t e ll a r aberra t ion , pos tu la tes on the ve loc i ty of l igh t

1 . I N T R O D U C T I O N

This paper c l a r if i e s the re l a tionship be twee n the three pos tu-

l a tes on the ve loc i ty of l igh t sugges ted by Eins te in (1905), R i t z(1908), and M oon an d Spencer (1956) . I t summ ar izes the

previous a t tempts to de te rmine which p os tu la te i s cor rec t : t he

ana lysi s of the da ta on b inary s t ars, t he Micbe l son-M or ley

exper iment , t he Miche l son-Gale exper iment , and the Sagnac

exper iment . The paper then desc r ibes the phenomen on of s t e ll a r

aber ra t ion in two coordina te sys tems and demons t ra tes tha t on ly

one o f the three pos tu lates on the ve loc i ty of l igh t i s t enable in

Eucl idea n space.

2 . T H E P O S T U L A T E S O N T H E V E L O C I T Y O F L I G H T

The f i rs t scient is t who had the fores ight to real ize that a

formal pos tu la te on the ve loc i ty of l igh t was neces sa ry was

Eins tein. In 1905 Ein s tein 1) propo sed:

Pos tu la te I : The v e loc i ty of l igh t in f ree space i s a lways a

cons tan t c i r respec tive of the m ot ion o f the source or rece iver.

As a resu l t o f th i s postu late , un iversa l t ime no lon ger had an y

meaning except in l abora tori es whose re l a tive ve loc i ty was ze ro .

Eins te in h imm lf recognized tha t Pos tu la te I w as n ot en t i re ly

satis factory. On ly two years la ter he sug gested that i t be

mo dified, 29 His revised postulate is essent ial ly equivalent to wha t

has been ca l ledO)

Pos tu la te I* : The ve loc i ty of l igh t in f ree space i s a cons tan ti r respec t ive of the ve loc i ty of source or rece iver in any ine r t i a l

coordinate sys tem.

This means tha t in any ine r t i a l coord ina te sys tem x ,y , z l ight

emit ted at t im e rI t rave ls outw ards in spheres o f radius r x,

Fi g . 1 ,

r = c ( t - r l ) , ( I )

who se cente r a lway s rema ins at the point/~(ri ), ,l (Zi) ,~r wh ere

the source was a t the ins tan t of em i~ io n r I . Th e equa t ion for any

poin t x , y , z on th i s spher ica l wave f ront a t t im e t i s

I x - ~ ( r z ) ] 2 + [ y - n ( r z ) ] 2 + [ z - ? % ) 1 2 _ _ ( r z ) 2 . ( 2 )

Solving for r t we have

r z = ( [ x - ~ ( r i ) ] : + l y - , l ( r z ) ] ~ + [ z - r ( r z ) ] ~ ) ~ a . O )

Accordins to the pos tu la te , c locks cannot be sy nchronized unless

the re is no re l a tive mot ion be tween source and rece iver . T hus

the concept of un iversal t ime becom es untenable for l abora tor i es

that are in relat ive motion.

Eins te in ' s pos tu la te was not imm edia te ly accepted . The

in tu i tive ly a tt rac tive concept of un iversa l t ime w as n ot l igh t ly

abandoned. In 1908 the Swiss physicis t Ri tz ,(4) wh o was w orkin g

at Got t ingen , was unw i l ling to accept th is pos tu la te an d propo sed

a bal lis tic pos tulate of his o wn :

Pos tu la te H : The ve loc i ty of l igh t in f ree space i s a cons tan t c

wi th respec t to the source a t the ins tan t o f emis s ion .

Acc ording to Postulate II , l igh t emit ted at t ime r n t ravels

outward in spher ica l waves o f rad ius r n , F ig . 2 ,

rn = c(t - 7n). (4)

Ritz mad e the ball is t ic assum ption that the cen ter o f thesespheres mov es a t the ve loc i ty tha t the source had a t the ins tan

of emis sion . Thus the pos i t ion of the cen te r a t time t i s

l i ( r ~ ) + (t - r T T ) ( d U d r ) ( r T T ) ,

~ l ( rn ) + ( t - 7 n ) ( d T I I d r ) ( r u ) ,

~ra ) + (t - rn)(dr

Thus , accord ing to Pos tu la te I I , t he equ a t ion for any poin t on the

47 6



Domina Eborle Spencer and Uma Y. Shama

Figure 1. Postulate I*.

spherical wave front is

Ix - ~(rn ) - ( t - r 1 1 ) ( d ~ / d z ) ( T n ) ] 2

+ [y - n(r n) - ( t - r n ) ( d n / d r ) ( r n ) l 2

+ [z - ~(rn ) - ( t - r u ) ( d ~ / d r ) ( r u ) ] 2 = (rn) 2.

Or, solving for riI,

rn =

lx2

1/2

(5)

(6)

third postulate on the velocity o f light:

According to this postulate, clocks can be synchronized if thelaboratories are moving at a constant relative velocity. Thus if

Postulate II is valid, universal time must be abandoned only for

accelerated laboratories. With Postulate II the concept of

universal time can be extended to all laboratories whose relative

velocity is a constant.

It was not until 1956 that another important postulate on the

velocity o f light w as prop osed. Moo n and Spencer ~3 asked

whether there is any postulate on the velocity of light that will

permit the concept of u n i v e r s a l t i me to be established for a ll

l a b o r a t o r i e s , e v e n t h o s e t h a t a r e a c c e l e ra t e d . T h u s was born the

Pos tulate in : In any coordinate system in which the light source

is stationary, the velocity of light in free space is a constant c.

Postulate III reduces to Postulate II w henever there is noacceleration of source or receiver. It was subsequently found

necessary 3) to modify Postulate III by inserting the wo rd

"inertial," just as in Postulate I*:

Pos tulate i n . : In an inertial coordinate system that is no

moving with respect to the source, the velocity of light in free

space is a constant c.

In an inertial coordinate system in which the source is always

at the origin, light travels at velocity c, so if light is emitted at

time rim the wav e front is a sphere with center at the origin and

radius, Fig. 3,

r m = c(t - rm). (7)

In an inertial coordinate system x ' , y ' , z ' in which the source is

stationary, the equation of the sphere is

(X, )2 + (y, )2 + (Z, )2 = ( rm)2 . (8 )

However, in an inertial coordinate system x , y , z in which the

source moves along a path ~(t),~(t),~(t),

477



Stellar Aberration and the Postulates on the Velocity of Light

Figure 2. Postulate II .

x ' = x -

y ' = y -

z ' = z -

Thus the equation of the spherical wave front, Eq. (8), becomes

[x - ~(t)] 2 + [y - 17(012 + [Z - ~ t ) ] 2 = ( r l l l ) 2 , (9 )

or, solving for rm,

r m -- (Ix - ~( t)l 2 + LY - ~( t) l 2 + [z - ['(0 ]2) 112. (10)

The light emitted from the source at time r m travels outward in

a sphere of ever-increasing radius rm w hose center always

remains at the source no matter how the source moves. Postu-

late III is the o nly postulate that permits clocks to b e

synchronized even in accelerated laboratories. (5,~ Thus it can be

called the "universal time postulate."

In 1989 Moon et al. (6) first showed that all three of these

postulates can be expressed mathematically by the same equa-

tion,

(r)R = C(t -- l"p.), (11)

if the radii (r)R are suitably defined as shown in Fig. 4. For

Postulate I*, (r)x is the distance from where the source was at

the time of emission ri to where the receiver is at time t. For

Postulate II , (r) n is the distance from wh ere the source woul d

have been at the time of reception t, if it had continued to move

at the velocity with which it was moving at the time of emission

zn, to where the receiver is at time t. For Postulate III , (r) m is

the distance from where the source is at time of reception t towhere the receiver is at time t.

3 . W H I C H P O S T U L A T E IS C O R R E C T ?

It is one thing to formulate a postulate. It is quite a different

problem to determine whether or not a postulate is consistent

with experimental evidence.

The serious consideration of any postulate other than Postu-

late I was nipped in the bud in 1913 when de Sitter~7) poi nte d out

that according to Postulate II , distant binary stars should exhibit

very strange behavior that had not been observed: infinite

Doppler shifts, multiple images, and apparent variation in

magnitude.

For the next forty years Einstein's postulate on the velocity of

light reigned unchallenged. In 1953 Moon and Spencer~s

reexamined the data on binary stars. I t was shown that Postu-

late II need not necessarily be abandoned: there are two possibil-

ities. The data on binary stars could be explained if

(1) Postulate I is employed in Euclidean space;

(2) Postulate U is employed in Riemannian space.

The analysis of the binary star data considering all three pos-

478



Domina Eborle Spencer and Urea Y. Shama

Figure 3. Postulate III*.

tulates on the velocity of l ight was undertaken by Moo n et a l p )

in 1989. This paper showed that

(1) both P ostulates I a nd III were consistent with all the data on

binary stars in Eu clidean space;

(2) Postulate II must be abandoned unless space was assumed tobe Riem~nnlan.

The famous Michelson-Morley<1~ experiment of 1887 was

~nalyzed by Mo on n) in 1993 from the point of view of each of

the three postulates on th e velocity of light. I f the experiment

was analyzed in a coordinate system attached to the laboratory,

all three postulates on the velocity o f light were show n to predict

the null fringe shift that was observed experimentally. If the

experiment was described in a coordinate system moving at a

constant velocity relative to the laboratory, the result was

entirely different. Postulate I does not predict the null fringe

shift unless the FitzGerald contraction of length, which is a

function of velocity, is introduced. Howev er, with Postulates IIand HI the mill effect is predicted in co ordinate systems that are

mov ing at a constant velocity relative to the laboratory without

any need fo r contraction o f length.

In linearly accelerated coordinate systems the situation is

different. Postulate I requires a FitzGerald-type contraction that

is a function o f both velocity an d acceleration. Postulate ILl

necessitates the introduction o f a FitzGerald-type contraction that

is a fu nction of acceleration. OMy Postulate 111predicts the null

frin ge shift with out any nee d fo r a l'~tzGerald-type contraction

even in linearly accelerated coordinate systems. Thus there are

three possible conclusions that are cott~istent with the

Michelson-Mofley experiment:

(1) Postulate I: Distance is an invariant only in stationary coor-

dinate systems but is contracted in any moving coordinate

system.(2) Postulate II: Distance is an invariant on ly in u uaccelerated

coordinate systems and is contracted in accelerated co-

ordinate systems.

(3) Postulate III: Distance is an invariant in all coordinate

systems, even those that are linearly accelerated.

The Michelson -Gale 12) experiment was analyzed by M oo n et

al. ~ in 1990. This experiment could not be explained by either

Postulate I or Postulate HI unless they were both modified to

exclude rotating coordinate systems. Thus the Michelson-Gale

experiment necessitated the introduction o f Po stulates I* ~nd

III*. Both o f these modified postulates permitted the p rediction

of precisely the fringe shift measured by M ichelson and Gale.The Sagnac(13) expe rimen t also pro duc es a me asurab le frin ge

shift. Analysis of this experiment by Mo on et al . (14) has shown

that the experimental fringe shifts are predicted by bo th Po stu-

lates 1" and III*.

4 . STELLAR ABERRATION IN THE STELLAR COORDI-

N A T E SY ST E MThe phenom enon of stellar aberration has been ably treated by

Hayden,o5) who questioned the Einstein equations, wh ich wer e

supposed to explain the well-known result. The phen om enon o

479



Stellar Aberration and the Postulates on the V elocity of Light

v ( t )n ( x ) + V y ( ~ ) ( t - ~ n ) , v(* ~n ) ~ ' - - . . .

( t '

v ( x ( t ) , y ( t ) ,

r i

z ( t )

Figure 4 . The three postula tes on the ve loci ty of l ight .

s te l la r aberra t ion w as discovered by Bradley, (103 wh o found tha t

t he t e le scope mus t b e t i pped fo rward b y an ang l e

= ( v / c ) s i n 0 ( 1 2 )

wh en the te lescope is moving a t a re la t ive ve loci ty v toward the

star and the star i s a t an angle 0 above the hor izonta l . The

maximum orb i ta l ve loc i t y o f t he Ea r th i s abou t 1 0 -4c , so t he

max imum va lue o f v t c i s approximate ly 10 -4 tad , or approx -imate ly 20.5".

In a coordinate system in which the star i s s ta t ionary the

analysis i s the same for Pos tula tes I* , I I , and I I I* , Fig . 5 . Light

t ravels f rom the star a t ve loci ty c a t angle 0 wi th the hor izonta l.

Howeve r , i f t he t e l e scope i s i n mot ion a t ve loc i t y v i n t he

horizonta l d i rec tion, the te lescope mus t be or iented a t angle (0 -

A0). From the l aw o f s i nes ,

vAt /sin A0 = cAt /sin (0 - A0) , (13)

where ~t i s the t ime requi red for l ight to t raverse the length of

the te lescope. Since A0 is smal l compared to 0 and v << c ,

AO ~ , ( v l c ) s in0 , , V s in0 (14)1 + ( v / c ) cos 0 c

jus t a s was found by B rad l ey .

This i s essent ia l ly the explanat ion of s te l la r aberra t ion given

by Bergmann. (17) Th us w e can c onclu de that, since in a stellar

coordinate system a l l three postula tes on the ve loci ty of l ight

predic t tha t the ve loci ty of l ight i s c , a l l three are in agreement

wi th Bradley 's exper imenta l resul t .

5 . S T E L L A R A B E R R A T I O N I N A N E A R T H B O U N D

C O O R D I N A T E S Y S T E M

But ast ronomical measurements are ac tual ly made in e a r t h -

b o u n d labora tor ies . I t i s necessary to be able to expla in them

both in coordinate systems in wh ich the star i s s ta t ionary and in

coordinate systems in which the te lescope i s s ta t ionary.

As shown in F ig . 6 , i n an ea r thbound coord ina t e sys t em Pos-

tula te I* predic ts an ent i re ly di f ferent resul t f rom Postula tes I I

and II I* . The star i s a t angle 0 abov e the hor izon a t t ime ofemission r . In th is coordinate system the star i s t ravel ing to the

lef t a t ve loci ty v . According to Postula te I* , l ight t ravels f rom

where the star was a t t ime r to where the te lescope i s a t t ime t

a long t he d is t ance r I shown in F ig . 6 . The subsequen t mot ion o f

the star does n ot a ffec t the d istance r I . T herefore , P o s t u l a t e I *

p r e d i c t s e r r o n e o u s l y t h a t t h e r e i s n o s t e l l a r a b e r r a t i o n i n t h e

e a r t h b o u n d c o o r d i n at e s y s t e m .

Since the pa th of the star i s approximate ly l inear o ver the t ime

interval to be co nsidered, P ostula tes I I and II I* cannot b e

dist inguished in this exper iment . According to both postula tes,

the distance r n = r m is measu red f ro m whe re the star i s a t t ime

t to where the te lescope i s a t t ime t as shown in Fig. 6 . Theangle A0 betw een the di rec t ion r I and the di rec t ion of r~ and rm

of Fig. 4 i s the ansle o f s te l la r aberra t ion. To pro ve this , we

have o nly to use e lementary t r igonom etry. Since r n = r m =

c ( t - r n) = c ( t - r il l) and the star has mo ved a distance v( t -

rn) = v( t - rm) betw een t ime rn = rm and l ime t , by the law

of sines,

v ( t - r ) = c ( t - r ) (15

sin A0 sin (0 - A0) '"

4 80



D o m i n a E b o r le S p o n c er a n d U r e a Y . S h a m s

Star

C

T e l e s c o p e

V A t

O-AO

0

Figu re 5. Stel lar aberrat ion in a coord inate sys tem in wh ich the s tar is s ta tionary and the te lescope is in m otion , Postulates I*, II ,

and I I I* .

But s ince A0 is a very smal l angle and s ince A0 << 0 and

v < < c ,

AO ~ v /cs inO. (16)

Th us Po stulates 11 and III* both pred ict exactly the stellar

aberrat ion discovered by Bradley in 1728. But Postulate l ' fa i l s

to pre dic t any stellar aberration in an earthbound coordinate

system.

6 . C O N C L U S I O N S

Since a cont rad ic tion has been demon s t ra ted , i f t he phenom

enon o f s t e l la r aber ra t ion i s ana lyzed in s t e ll a r and ea r thboun

coordinate sys tems, Postulate I* must be rejected. Postulate I

mus t be re j ec ted because o f the b inary s t a r da ta unless we

assume that space is Riemannian rather than Eucl idean.

Thus the only postulate on the velocity of l ight that is in

agreement with all the experiments hitherto analyzed on a

comparative basis is Postulate III*. This postulate is consis ten

481



Stellar Aberration and the Postulates o n the Ve locity of Ligh t

P o s t u l a t e I * P o s t u l a t e s II a n d I I I*

S t a r S t a rv _ v ( t - ~ , , , )

131

C

C

rn = r~H

r I = C( t - T i ) - - C ( t - z i I )

0 / J / J \ 0

t - t

F i g u r e 6 . S t e l l a r a b e r r a t i o n i n a c o o r d i n a t e s y s t e m i n w h i c h t h e s t a r i s m o v i n g a n d t h e t e l e s c o p e i s s t a t i o n a r y .

w i t h t h e b i n a r y s t a r d a t a e v e n i n E u c l i d e a n s p a c e . I t p r e d i c t s t h e

n u l l re s u lt o f th e M i c b e l s o n - M o f l e y e x p e r im e n t w i t h o u t th e

n e c e s s i t y f o r a F i t z G e r a l d c o n t r a c t i o n . I t a l s o p r e d i c t s t h e c o r r e c t

f r i n g e s h i f t f o r b o t h t h e M i c h e l s o n - G a l e a n d t h e S a g n a c e x p e r i -

m e n t s . A n d i t p r e d i c t s t h e s t e l la r a b e r r a t i o n o b s e r v e d b y B r a d l e y .

O n t h e b a s i s o f th e e x p e r i m e n t a l r e s u l t s h i t h e r to a n a l y z e d w e

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

a n i n v a f i a n t , s p a c e i s E u c l i d e a n , a n d t h e a n g l e b e t w e e n r I a n d r l a

s h o w n i n F i g . 4 i s t h e a n g l e o f a b e r r a t i o n t h a t a s t r o n o m e r s h a v e

b e e n o b s e r v i n g f o r n e a r ly t h r e e c e n t u r ie s . U n l e s s f u r t h e r a n a l y s i s

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

c o n c l u d e t h a t the velocity o f light is not a constant in al l

coordinate systems but is a constant in any inertial coordinate

system in which the source of light is stationary.

R e c e i v e d 2 4 A p r i l 1 9 9 5 .

4 82



Domina Eberle Spencer and U rea Y. Shama

R 6 s u m 6Cet art ic le pr~sente une preuve sim ple de la val idi t~ du po stula t d 'universe l temps s ur la

v i tesse de la lumi~re ~ part i r d es donn~es exp~rimentales sur l 'aber rat ion ste l laire observ~e

pa r Bradley en 1728. I I est ds que le pos tulat d 'Einste in sur la v i tesse de la lumi~re

donne une pred ic t i on adequa te dans un sy s t~ne de co ord onn ~ dans l aqueUe l '~ to i le e s t

s tat ionnaire mais la pr~ l ic t ion est compl~tement erron~e lorsqu'observ~ pa r ra ppor t r un

syst~Vne de coordonn ~es li~ au r~f~ rentiel terrestre.

R ef erences

1. A. Einstein, Ann. Phys. 17, 891 (1905).

2 . Idem, Jahrbnch Radioaktivitat IV , 42 2; V, 98 (Berichti-

gunsen, 1907).

3. P. M oon, D.E . Spencer, and E.E. M oon, Phys. Essays 3,

431 (1990).

4. W. Ritz, Ann. Chim. Phys. 13, 145 (1908).

5. P. M oon and D.E. Spencer, Phi los . Sci . 23, 216 (1956).

6. P. Moon, D.E. Spencer, and E.E. Moon, Phys. Essays 2,

268 (1989).7. W . de Sitter, Phys. Z. 14, 1267 (1913).

8. P. M oona nd D.E. Spencer, J . Op t . Soc. Am. 43, 635 (1953).

9. P. Moon, D.E. Spencer, and E.E. Moon, Phys. Essays 2,

275 (1989).

D o m i n a E b e r l e Sp en ce r

University of Connecticut

Storrs , C onnecticut 06268 U .S.A.

U m a Y . S h a m a

Bridgewater State College

Bridgewater, Massachusetts 02324 U.S.A.

10. A.A. Michelson and E.H. Morley, Am. J . Sci . 34, 333

( 1 8 8 7 ) .

I I. E . E . M o o n , P h y s . E s s a y s 6 , 4 8 7 ( 1 9 9 3 ).

12. A.A . M ichelson, Astrophys. I. 6 1, 137 (1925); A.A .

Michelson and H.G . G ale, ibid. , 140.

13. G. Sagnac, C. R. Acad. Sci. 1 57, 708, 141 0 (1913).

14. P. M oon, D.E . Spencer, and U.Y . S h~m a, Phys. Essays 4,

249 (1991).

15. H.C . H ayden, "Stellar Aber ration," to be published;

"Stellar Aberration and the Street-Lamp Pa rado x," G alilean

Electrodyn., to be published.

16. J. Bradley, philos. Trans. 35, 637 (1728).

17. P.G. Bermnann, Introduct ion to the T heory o f Relat iv ity

(Prentice-Hall, NY, 1942).

48 3

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