1978.04.26 - Stefan Marinov - The Coordinate Transformations of the Absolute Space-Time Theory

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Foundations o f Phy sics, VoL 9, Nos. 5/6, 1979

The Coordinate Transformations of the Absolute

Space-Time Theory

S t e f a n M a r i n o v ~, ~

Received April 26, 1978

In the light o f our recently per form ed experime nts, revealing the an isotropy

c~fl ight velocity in any fl a m e moving with respect to absolute space, we show

that the Lor ent z transJbrmation, where the relativity o f light velocity is given

implicitly through the relativity o f the time coordinates, mu st be treated fr om

an absolute point o f view if one seeks to preserve its adequacy to physical

reality. Then w e ptvp ose a new transformation (whic h is to be considered as a

legitimate companion of the Loren tz transformation) wherein the relativity

o f light velocity is given e xplicitly and the time coordinates are absolute.

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

Af te r the pe r fo rm ance o f our "cou p led -mi r ro r s" exper im en t m and espec ial ly

af ter i t s repet i t ion in the so-cal led in ter fe rome tr ic var iant (') no dou bts can

rem ain that the veloc i ty of light is d i rect ion dep end ent ( see a lso Refs . 3 and 4)

and tha t abso lute space is a real i ty p lact ical ly detectable in a labora tory .Thus , jus t a t th i s ve ry mo me nt w hen man k ind , a f t e r a heavy ba tt l e

las t ing several decades, has f inal ly and def in ite ly re jected the Ne wto nian

abso lu te space- t ime concep t ions and the ae ther mode l fo r l igh t p ropaga t ion ,

per f id ious exper imen t p roduces a new puzz le which canno t be exp la ined

wi th in the f r amew ork o f mod ern h igh-ve l0c ity physics.

Are we on the th resho ld o f a new dram at i c c r i s is? Hav e we to r ev ise

once more our space- t ime concep t ions? Have phys ic i s t s o f d i f f e r en t o r i en -

ta t ions and erudi t ion , phi losophers , th inkers , and f ic t ion wr i ters to g ive

b i r th t o t h o u sa n d s o f n e w p a p e rs a n d b o o k s?Ou r def in i te answ er is : N o! M an kin d has s imply to re turn to the o ld ,

1 Labora tory for Fundamental Physical Problems, Sofia, Bulgaria.Present address: 83, rue St6phanie, 1020 Bruxelles, Belgium.

445

0015-90t8]79[0600-0445503.00/0 @) t979 Plenum Publishing Corporation



446 Mar inov

na tu ra l , s imple , and c lea r Newton ian and n ine teen th cen tu ry concep t ions .

How ever , as the d iMectical laws require , cer ta in e lem ents o f the unsuccessful

r evo lu t ion a lways r emain insc r ibed on the banner s o f the coun te r r evo lu t ion

which com es to replace the former . In a ser ies of papers which are no w

in the press, we sho w which e lements o f m od ern high-veloci ty physics are

to be in t roduced in the o ld Newton ian mathemat ica l appara tus wi th the

a im of ob ta in ing a theory adequa te to phys ica l r ea l i ty , wh ich we have

called the absolute space-time theory.

In the present paper we shal l consider the problem of coordina te t rans-

form at ion s in h igh-veloci ty physics , which, for h is tor ical reasons, we shall

also call relativistic.

Before b roach in g th is p rob lem we mus t s t a te tha t in abso lu te space- t ime

theory we work only wi th three undef inable physical not ions: (a) space,

(b) t ime, and (c) energy (mat ter ) .

These three physical quant i t ies cannot be def ined a t a l l and, appeal ing

to the in tu i t ive ideas of the reader , w e can say only: (a) Space i s that which

extends. (b) Time i s that which endures . (c) Energy is that which exists.

The points in space are cal led material ( i f thei r energy is d i f ferent f rom

zero) o r immaterial ( i f thei r ene rgy is equal to zero) . The m ater ia l points

( i .e . , the lumps of energy) can move in space. The space in which the energy

of the w hole w or ld is a t res t i s ca l led absolute. The space a t t ached to a mate r i a l

system ( i .e . , a combinat ion of mater ia l points) which moves wi th respect to

absolute space is cal led relative. W hen al l points o f a mater ia l system move

with the same veloci ty in absolute space th is sy stem is cal led inertial a n d

the relat ive space at tached to i t is also cal led inert ial .

Mate r i a l po in t s o f an im por ta n t c lass , cal led pho to ns , p ropag a te a lways

with veloci ty c in absolute space. This asser t ion is hypothet ica l and represents

the axiom at ical basis of the so-cal led aether model fo r l igh t p ropaga t ion .

We cal l th is aether concept ion Newtonian, emphas iz ing in such a way tha ti t has no th ing in common wi th the wave mode l fo r l igh t p ropaga t ion . More

de ta i l s abou t our concep t ion fo r l igh t p ropaga t ion can be found in Ref . 5 .

Space in tervals can be measured by r ig id rods ( i .e . , mater ia l systems the

d i s t ances be tween whose po in t s do no t change) and t ime in te rva l s can be

measured by so-cal led l ight c locks. S ince the veloci ty of l ight in absolute

space i s a universal constant , the l ight c lock represents the most accurate ,

exact , and theoretically the m os t simp le clock.

2 . T H E L I G H T C L O C K

The l ight c lock represents a l ight sou rce and a m ir ror p laced a t a

cer ta in d is tance in f ro nt o f i t, ca l led the "a rm " of the c lock. I f th is



The Coordinate Transformations of the Absolute Spa ce-Time Theory 447

" a r m " h a s d l e n g t h u n it s , t h e n a n y p h o t o n ( o r sh o r t e n o u g h p a c k a g e o f

ph oto ns) wi ll re turn to the l ight source , being ref lected by the mir ror , af ter

= 2d/~ (1)

t ime uni ts . The t ime in terval T i s cal led the per iod of the l ight c lock.

A clock that res ts in absolute space i s ca l led an absolu te c lock a n d a

clock tha t m oves wi th a cer ta in veloci ty V is cal led a proper c lock . L e t u s

now es tab l ish the per iod o f a p ro per l igh t c lock wi th " a r m " d .

F i r s t we sha ll suppose tha t the "a rm " i s pe rpend icu la r to V and such

a l ight clock wil l be cal led transverse . D e n o t i n g b y T = ' a n d T / ' the t imes

in which l igh t cover s the " a r m " d o f the t r ansverse l igh t c lock " the re" and" b a c k , " w e h a v e

c~(r L') 2 = cl~ + V~ (r ~') 2,

f rom which

c2(T r") 2 = d e -~- V2 (I~ ") 2 (2 )

2d TT~_ = 7 2 ' + 7"=" = (3)

e(1 - - V21e2) 112 ( I - V21e2) lz

Th en we sha ll suppose tha t the " a r m " is pa ra ll e l t o V and such a l igh t

c lock wi l l be cal led longitudinal. Denoting by T(, ' and Tin)" he t imes in which

l igh t cover s the "a rm " d o f the long i tud ina l ligh t c lock " the r e" and "ba ck ,"

we have ( fo r the case where the so urce -mi r ro r vec to r po in t s a long the same

direct ion as the c lock 's veloci ty)

cT~,' = d + VT) I',

f rom which

c T , " = d - - V T , " (4 )

2d 7"

T,j ......T, ' + Tj ," = c~(l - - V2/c ~) = - 1 ~ V2/c 2 (5 )

Hence we have the fo l lowing conc lus ions :

( i ) T i le ra te of a proper l ight c lock is d i f ferent than the ra te of an

absolute c lock.

( ii ) The ra te o f a pr op er l ight c lock depends on the or ien ta t ion of it s

"a rm" wi th r espec t to i t s ve loc i ty .

3. T H E H I G H - V E L O C I T Y A X I O M

The h i s to r i c exper imen t o f Miche l son and Mor ley showed tha t the r a t e

o f a p ro per l igh t c lock does no t depend on the o r i en ta t ion o f i ts "a rm ."



448 Marinov

Two hypotheses have been proposed for the explanation of this experimental

fact, which contradicts the Newtonian aether conception:

(a) The Lorentz Hypothesis. Every rigid body contracts those of its

dimensions that are parallel to its velocity by the tactor (1 -- VY/cY)I/~. Hence

the "ar m" of the longitudinal light do ck becomes d~ : d(1 -- VY/cY)~/~",

while the "a rm " of the transverse clock remains d± : d, and one obtains

Ta = Tl~ = 7"/(1 - - V2/c2 ) 1/2 (6)

(b) The Einstein Hypothesis. In every inertial relative space the velocity

of light is isotropic and equal to c. Thus the period of any proper light clock

will be given by formula (1).

Einstein rejected the Newtonian aether hypothesis mor e radical ly than

w a s d e m a n d e d by the Michelsom~Morley experiment. This experiment only

showed that the "there-and-back," i.e., bidirec t ional , light velocity is iso-

tropic in any relative space, but it gives no information whether the "there"

and "back," i.e., unidirec t ional , light velocities are also isotropic.

Einstein made the radical assumption about the unidirectional light

velocity constancy, proceeding from the general principle of relativity, which

asserts that there is no physical possibility for registering the motion of an

inertial material system.

Before the performance of our "coupled-mirrors" experiment(1,2~ there

was no experiment contradicting the general (Einstein) principle of relativity

and the hypothesis for the unidirectional light velocity constancy. But

following its performance we have to revise Einstein's hypothesis, reject

the general principle of relativity, and make it clear that from the Michelson-

Morley experiment no such radical conclusion is to be drawn.

The axiomatical grounds of high-velocity (relativistic) physics is givenby the tenth axiom of our absolute space-time theory, (6) which reads:

The material points called photons move with velocity c along all

directions in absolute space and their velocity does not depend on their

history. Light clocks with equal "arms" have the same rate in any frame,

independent of the orientation of their "arms." At any point of any frame

the time unit is to be defined by the period of light clocks with equal "arms,"

independent of the velocity of the frame and the local concentration of

matter.

Now proceeding from this axiom, we shall show which coordinate trans-

formations we must have in high-velocity physics.



The Coordinate Transformat ions of the Absolute Space-Time Theory 449

4 . T H E G A L I L E A N T R A N S F O R M A T I O N

A l l t r a n s f o r m a t i o n s o f t h e s p a c e a n d t i m e c o o r d i n a t e s w h i c h w e c o n s i d e r

a r e b e t w e e n a f r a m e K a t ta c h e d t o a b s o l u t e s p a c e (c a ll e d a b s o lu t e f r a m e )

a n d a f r a m e K ' m o v i n g i n e r ti a l ly w i t h a v e l o c i ty V ( c a ll e d re la t ive f i 'am e) .

T o a v o i d t r iv i a l c o n s t a n t s w e sh a l l c o n s i d e r t h e s o - c a l le d h o m o g e n e o u s t r a n s -

f o r m a t i o n , i . e . , w e s h a l l s u p p o s e t h a t a t t h e i n i t i a l z e r o m o m e n t t h e o r i g i n s

o f b o t h f r a m e s c o i n c i d e ( s ee F ig . 1, w h e r e f o r s i m p l i c i t y ' s s a k e a t w o - d i -

m e n s i o n a l c a s e i s p r e s e n t e d ) .

L e t u s h a v e a p o i n t P w h o s e r a d i u s v e c t o r i n K i s r ( c a l l e d t h e abso lu te

radius vector) a n d w h o s e r a d i u s v e c t o r in K ' i s r ' ( c a ll e d th e relative radius

vector). T h e r a d i u s v e c t o r o f th e o r i g i n o f f r a m e K ' i n f r a m e K is R ( c a ll e d

t h e t ransient radius vector) . I t i s

R = V t = V o t 0 ( 7 )

w h e r e t is t h e t i m e r e a d o n a c l o c k a t r e s t i n f r a m e K ( a n a b s o l u t e c l o c k )

a n d V i s t h e v e l o c i t y o f f r a m e K ' m e a s u r e d o n t h is c l o c k , w h i le t o is th e

t i m e r e a d o n a c l o c k a t r e s t i n K ' ( a p r o p e r c l o c k ) a n d V 0 is th e v e l o c i t y o f

f r a m e K ' m e a s u r e d o n t h is c lo c k .

A c c o r d i n g t o th e t r a d it i o n a l N e w t o n i a n c o n c e p t i o n s w e s h al l h a v e

r ' : r - - Vt , r = r ' + Voto (8)

A d d i n g t h e se t w o e q u a t i o n s , w e o b t a i n ( 7). I f w e a s s u m e t h a t t h e d o c k s

a t t a c h e d t o K a n d K ' r e a d th e s a m e t im e , w e h a v e

t = t o , V = V 0 (9)

Fig. 1.

X

A homogeneous transformation betweentwo inertial frames of reference.

8 2 5 / 9 / 5 / 6 - 9



4 5 0 M a r i n o v

T h e f i r s t f o r m u l a ( 8 ) r e p r e s e n t s t h e d i r e c t a n d t h e s e c o n d f o r m u l a ( 8 )

t h e i n v e r s e h o m o g e n e o u s Galilean transformation w h e r e t h e r e l a t i o n s ( 9)

a r e t o b e t a k e n i n t o a c c o u n t .

5 . T H E L O R E N T Z T R A N S F O R M A T I O N

N o w w e s h a ll s e a rc h f o r a t r a n s f o r m a t i o n o f t h e s p a c e a n d t i m e c o o r -

d i n a t e s w h i c h w i l l l e a d t o t h e r e l a t i o n T ± = T~E b e t w e e n t h e p e r i o d s o f

t r a n s v e r s e a n d l o n g i t u d i n a l l ig h t cl o c k s , a s r e q u i r e d b y o u r t e n t h a x i o m .

L e t u s d e c o m p o s e ( F i g. 1) t h e r a d iu s v e c t o r s r a n d r ' in t o c o m p o n e n t sr ± , r ± " a n d r , , r ~ , r e s p e c t i v e l y , p e r p e n d i c u l a r a n d p a r a l l e l t o t h e d i r e c t i o n

o f p r o p a g a t io n o f K ' .

A c c o r d i n g t o th e t ra d i t io n a l N e w t o n i a n c o n c e p t i o n s w e h a v e

r ' = r ± ' ÷ r t l ' = r . + ( r i ~ - - V t ) = r - - V t (10)

W e c a n m e e t t h e r e q u i r e m e n t T ± = T , o f o u r t e n th a x i o m i f w e t a k e

t h e p a r al le l c o m p o n e n t o f t h e r e l at iv e r a d iu s v e c t o r c o n t r a c t e d b y t h e f a c t o r

(1 - - V2/c~)1/~ w h e n e x p r e s s e d b y th e c o o r d i n a t e s i n f ra m e K , i . e ., i f w e

axiomat ical ly a s s u m e a s v al id i n s te a d o f th e N e w t o n i a n r e l at io n s

r ± = r ± ' , r~, - - V t = r , ' ( 1 1 )

t h e " L o r e n t z i a n " r e l a t i o n s

r ~ = r ± ' , r ~, - - V t = r ~ E '( 1 - - V~IcZ )112 ( 1 2 )

T h i s " c o n t r a c t i o n " ( w h e n r~t - - V t is e x p re s s e d b y r ~ ' ) o r " d i l a t i o n "

( w h e n r , ' is e x p r e s s e d b y r , - - V t ) is n e i th e r a p h y s i c a l ef fe ct , a s supposedb y L o r e n t z , n o r a r e su l t o f m e a s u r e m e n t , a s obtained b y E i n st ei n . A c c o r d i n g

t o o u r t h e o r y r n ' a n d r , - - V t r e p r e s e n t t h e same l e n g t h ( d i s t a n c e ) b e t w e e n

t w o m a t e r i a l p o i n t s w h i c h c a n b e c o n n e c t e d b y a ri gi d r o d o r w h i c h c a n m o v e

w i t h r e s p e c t t o o n e a n o t h e r , a s w e l l a s b e t w e e n t w o n o n m a t e r i a l p o i n t s ,

t a k e n a t a g i v e n m o m e n t . ( N . B . A b o u t l e n g t h s o n e c a n s p e a k a t a g i v en m o -

m e n t o n l y I) T h u s r ~ ' a n d r , - - Y t a r e equal a n d w e w r i t e t h e s e c o n d r e l a t i o n

( 12 ) o n l y b e c a u s e t h e v e l o c i t y o f li g h t d o e s not h a v e a n e x a c t a e t h e r - N e w -

t o n i a n c h a r a c t e r. M a k i n g a tr a n s i t io n f r o m ( l l ) t o ( 12 ), w e in t r o d u c e a

blunt mathematical contradiction i n to t h e t r a d it i o n a l N e w t o n i a n m a t h e m a t i c a la p p a r a t u s . A s w e s h o w i n d e t a i l i n R e f . 7 , t h i s m a t h e m a t i c a l c o n t r a d i c t i o n

r e m a i n s i n t h e f o r m u l a s a n d w e m u s t s t a t e t h a t a f t e r y e a r s o f in t en s iv e

m a t h e m a t i c a l s p e c u l at io n s w e h a v e f o u n d n o w a y t o g e t r id o f it. W e a s k

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



The Coord inate Transformationsof the Absolute Space-Tim e Theory 451

t h e o r y f o r m a t h e m a t i c a l i m p e r f e c t io n . T h i s i m p e r f e c t i o n e xi st s i n N a t u r e

i t s e l f . W e m u s t r e a l i z e o n c e a n d f o r a l l t h a t l i g h t d o e s n o t h a v e a n e x a c t

a e t h e r - N e w t o n i a n c h a r a c t e r o f p r o p a g a t i o n , s in c e i ts " t h e r e - a n d - b a c k " v e l o-

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

th e aether-Newtonian c o n c e p t i o n s i t m u s t b e a n i s o t r o p i c . W e h a v e c a l l e d

t h is p e c u l i a r i t y i n t h e p r o p a g a t i o n o f li g h t th e aether-Marinov c h a r a c t e r o f

l i g h t p r o p a g a t i o n .

T h u s , i f w e w i sh to m e e t t h e r e q u i r e m e n t T c = T ~ , w e h a v e t o w r it e

i n s t e a d o f r e la t i o n ( 10 ) t h e f o l l o w i n g t r a n s f o r m a t i o n o f t h e r a d i u s v e c t o r s i n

f r a m e s K a n d K ' :

, , ri~ - - V t (13 )r ' = r I + r~ : r± --~ (1 - - V~ Ie2) i~

T h i s f o r m u l a , w r it te n i n s u c h a m a n n e r t h a t o n l y th e a b s o l u t e r a d i u s

v e c t o r r is r e p r e s e n t e d , b u t n o t i ts tr a n s v e r s e a n d l o n g i t u d i n a l c o m p o n e n t s

r ± , r ~ , h a s t h e f o r m

1 _ _ 1 ] r v V tr ' = r + l [ ( 1 - V2 /e2) /2 ( 1 - V ~ / c ~ ) ~ I V ( 14 )

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

r ' t o r . H e r e w e h a v e t w o p o s s i b i l i t i e s :

( a) T o a s s u m e t h a t a l so i n f r a m e K ' t h e v e l o c i t y o f l ig h t is i s o t ro p i c a n d

e q u a l t o e ( t h e L o r e n t z w a y ) .

( b ) T o a s s u m e t h a t t h e v e l o c i t y o f li g h t is is o t r o p i c a n d e q u a l t o e

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

T h e L o r e n t z w a y le a d s t o t r a n s f o r m a t i o n o f th e t im e c o o r d i n a t e s w h e r et h e r a d i u s v e c t o r s s h o u l d a p p e a r , i . e . , t o r e l a t i v e t i m e c o o r d i n a t e s , w h i l e

t h e M a r i n o v w a y l e ad s t o t r a n s f o r m a t i o n o f th e t i m e c o o r d i n a t e s w h e r e t h e

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

N o w w e s h a ll f o l lo w t h e f i rs t w a y a n d i n S e c t i o n 6 th e s e c o n d .

I f t h e v e l o c i t y o f li g h t in f r a m e K ' is a s s u m e d t o b e i s o t ro p i c a n d e q u a l

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

w i t h r e s p e c t t o K " ( a n d m e a s u r e d o n a c l o c k a t t a c h e d t o K ') i s e q u a l a n d

w i t h o p p o s i t e s ig n t o t h e v e l o c i t y V w i t h w h i c h f r a m e K ' m o v e s w i t h r e s p e ct

t o K ( a n d m e a s u r e d o n a c l o c k a t t a c h e d t o K ) , w e c a n w r i t e ( le t u s n o t e t h a tb o t h t h e s e a s s u m p t i o n s f o l l o w f r o m t h e p r i n c i p l e o f r e l at i v it y )

l [ i ]r = r ' + ( 1 - - V ~/c~)ln - - 1

t ~

÷ V( t - v ' ~ l d ) l n f

(15)



452 Marinov

Adding formulas (14) and (15), we obtain

[L ( 1 - V ~ l c ~ ) ' l 2 ~ - - ( 1 - V ~ l c ~ ) l l ~

1 .1 r' • V t'

= [ ( 1 - - V21c2) l i2 - - lJ ~ + ( l - - V21e2) ' i ~ (16)

If in th is form ula we subst i tu te r ' f rom (14), we obtain the t ransfo rm ation

formula fo r t ime in which . t ' i s expressed th rough t and r ,

t f = t - - r • V/ c a

( 1 - V2te2) 1/2 (17)

On the oth er ha nd , if in form ula (16) we substitute r fro m (15), we obta in

the t ran s form at ion fo rm ula tb r t ime in which t is expressed th rough t ' a nd r ',

t" + r ' . Vl c ~ (I s)t - (1 - v21d)t/~

Formulas (14) and (17) represent the direct , and formulas (15) and

(18) the inverse homogeneous Loren t z t rans f ormat i on . These fo rmulas show

tha t n ot only are the radius vectors r and r ' tw o different quanti t ies , but

a lso the t ime coordinates t and t ' a re two different quanti t ies and ar e to

be called the absolute t ime coordinate and the re la t i ve t ime coordinate.

Thus, as the t ime coordinates in the Lorentz t ransformation are re la t ive

quanti t ies , the l ight velocity constan cy in th is t ransfo rm atio n is only apparent .

In Ref. 7 [see form ulas (4.33) a nd (4.34) there] we show how , proceeding

f rom the Loren tz t rans format ion , one can ob ta in the express ions fo r the

l ight veloci ty in an y iner t ia l f ram e wh ich are adequate to physical real ity .

Hence, according to absolute space-t ime theory, the Einste in general

pr inciple of re la t iv i ty is inval id and the Lorentz t ransformation is adequateto physical real i ty only i f i t is t reated from our absolute point of v iew.

Since Einste in t reats th e l ight veloci ty cons tancy as a physical fact and

the general pr inciple o f re la t iv i ty as a law of N ature , we consider the Lo rentz

transformation in the context of specia l re la t iv i ty as inadequate to physical

reality.

Note that we consider the Gali le i l imited pr inciple of re la t iv i ty as

adequ ate to physical real i ty . This pr inciple asser ts that there is no mech anical

phys ica l phenomenon by whose he lp one can es tab l i sh the ine r t ia l mot ion

of a g iven mate r ia l sys tem. Hence fo r the mechan ica l phenomena anyinertial relative space is isotropic.

For e lectromagnetic phenomena the pr inciple of re la t iv i ty does not

hold good. Thus for the e lectromagnetic phenomena the iner t ia l re la t ive

spaces are not isotropic.



The Coordinate Transformations of the Absolute Spa ce-T ime Theory 453

Ho wev er , as Minkow ski has show n, i f we conside r a 4-space in which

the three space coord inates in any iner t ia l f rame are uni f ied wi th the cor-

re spond ing t ime coo rd ina t e mu l t ip l i ed by c (and by the imag ina ry un i t ) ,

t hen th is 4 -space tu rns ou t t o be i so t rop ic and hom ogeneou s . As the Ga l i l ean

t rans fo rm at ions ma ke a g roup in the 3 -space , so the Lor en tz t rans fo rma t ions

make a group in the 4-space . This i s a grea t mathemat ica l advantage and the

fou r -d im ens iona l ma them at i ca l appa ra tus de ve loped by Minkowsk i has been

an e no rm ous he lp in t he inves t igat ion o f h igh -ve loc ity physi cal pheno men a .

In ou r abso lu te sp ace - t ime theo ry we wo rk in t ens ive ly w i th the fou r -

d imens iona l ma thema t i ca l fo rma l i sm o f Minkowsk i , keep ing a lways in mind

that the four th d imension i s no t a t ime ax is , bu t a length ax is a long which

the t ime coor d inate s are mu l t ip l ied by the ve loci ty of l igh t, a nd here the

appa ren t abso lu t eness o f the l igh t ve loc ity is a lways c onnec ted wi th the

re l a tiv i ty o f the t ime co o rd ina t es . As a ma t t e r o f fac t , the t ime c oo rd ina t es

a re abso lu t e and l igh t ve loc ity re la t ive , a s i n t he M ar in ov t rans f o rm at ion

and as we have show n by the he lp o f numero us exper imen t sJ 8~

We mus t no te and emphas i ze tha t i f expe r imen t s a re set up where on ly

e l ec t romagne t i c phenomena a re i nvo lved , t hen the p r inc ip l e o f re l a t iv i ty

apparen t ly ho lds because o f the m u tua l cance l la t ion o f t he abso lu t e e f fec ts

tha t appea r . Th i s p r inc ip le b reaks dow n on ly fo r expe r imen t s where combin edelec t rom agnet ic and mec hanica l p hen om ena are involved , as is the case with

our "cou p led -m i r ro r s" exper ime n t a,2/ and Br i scoe ' s u l t ra son ic "c oup led -

t ran sm i t ters " exper im ent . ("~

6 . T H E M A R I N O V T R A N S F O R M A T I O N

To ob ta in a trans fo rm at ion o f t he space and t ime coo rd ina t es adequa te

to phys i ca l rea l i t y we sha l l p roceed f rom our t en th ax iom, no t ing tha t now

we shall no t take in to ac cou nt the inf luence of the gravi ta t ing masses on

the ra te of the ligh t c locks , which prob lem was consid ered in Ref . 6 .

Thus acco rd ing to t he t en th ax iom:

(a ) L igh t c locks w i th equa l "a rms" have the same ra t e , i ndependen t o f

t h e o r i e n t a t io n o f t h e ir " a rm s . "

(b) In any frame the t ime uni t i s to be def ined by the per iod of l igh t

c locks w i th equa l "a r m s ," i ndepende n t o f t he ve loc ity o f t he f rame .

As we have shown in Sect ion 2 , the f i rs t asser t ion dras t ica l ly cont rad ic tsthe t rad i t i ona l Newton ian concep t ions . The second asse r t i on rep resen t s no

such d ras t i c con t rad ic t ion , because in t he f ram ewo rk o f t he t rad i t iona l

Newton ian space - t ime concep t ions one can a l so de f ine the t ime un i t i n any

ine r t i a l ~ rame by the pe r iod o f l i gh t c locks w i th equa l "a rms ." However ,



454 Marinov

in the t rad i t iona l Newton ian f ramework , the inconven ience ex is t s tha t one

has fu r the r to de f ine tha t the "a rms" o f the l igh t c locks mus t a lways make

the same ang le wi th the ve loc i ty o f the ine r t ia l f rame used- - fo r example ,

the i r "a rms" mus t be pe rpend icu la r to th i s ve loc i ty . In such a manner the

abso lu te t ime d i la t ion phen om eno n wi lt be in t roduced a l so in to the t rad i t iona l

Newtonian theory . Thus , a t f i rs t g lance , i t seems that the second asser t ion

has n ot such a " na tu ra l" character as the f irs t one a nd represents only" a

s t ipula t ion . H owever , i t tu rns ou t tha t n o t on ly do the pe r iods o f ligh t

c locks become g rea te r when they move wi th g rea te r ve loc i ty in abso lu te

space (we repeat , a phenomenon which exis ts a lso in the t radi t ional New-

tonian theory) , but so do the per iods of many other physica l processes ( the

per iods of a tomic c locks , the mean l ives of decaying e lementary par t ic les) .

So far there is no exper imenta l evidence permit t ing one to asser t tha t the

per iod of any sys tem (say , the per iod of a spr ing c lock, the pulse of a man)

becomes greater with the increase of i ts absolu te veloci ty . This pro blem needs

addi t ional theore t ica l and exper imenta l invest iga t ion . At any ra te , we th ink

the s ta tement about the t ime di la t ion is to be considered not as a s t ipula t ion

bu t a s an ax iomat ica l a s se r t ion a l ien to the t rad i t iona l Newton ian theory .

Le t us f ind f ir st how the Ga l i lean t rans fo rm at ion fo rm ulas a re to be

wri t ten i f one assumes tha t in an y iner t ia l f rame the t ime un i t is to be def ined

by the per iod of l ight c locks with equal "arms," supposing for def in i teness

tha t the "a rms" o f the l igh t c locks mus t be a lways pe rpend icu la r to the

absolute veloci t ies of the f rames.

The pe r iod o f an abso lu te l igh t c lock wi th "a rm" equa l to d i s g iven

by f o rm ula (1). The pe r iod T O o f a p roper l ight c lock wi th the same "a rm "

and whic h moves a t ve loci ty V (we repeat , V is assum ed to be perp endicular

to the " ar m ") wil l be g iven by form ula (3) , where we have to wri te 7"= = To.

I f (a t an appropr ia te cho ice o f d ) we choose T as a t ime un i t in f rame K

(called an absolute second) a n d To as a t im e un i t in f ra me K ' (ca lled a propersecond), then i t i s c lear tha t when between two events , t absolu te seconds

and to proper seconds have e lapsed, the re la t ion between the m wil l be

to/t = T/To = (1 - - V2/c2)1/2 (19)

where T and To are measu red in the same t ime uni ts (absolute or proper) .

Under th is s t ipula t ion we shal l obta in f rom (7) and (19)

V Vo (20)V° -- (1 -- V2/cU)1/~' V :: (1 -- Vo2/cU)1/2

Formulas (8), to which we attach the relations (19) and (20), represent

the d i rec t and inverse homogeneous relativistic Galilean transformation.

In these form ulas , V is the veloci ty of f rame K ' w ith respect to ab solute

space ( i .e . , to f rame K) measured in absolute seconds (ca l led the absolute



The Coordinate Transformations of the Absolute Sp ace-Tim e Theory 4 55

transient velocity), V0 i s t he sam e ve loc i ty m e asure d in p rop e r seconds

(ca l led the proper transient veloc i ty) , and c i s the ve loc i ty of l ight a long the

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

t he " a r m " o f t h e p r o p e r c l o c k m e a s u r e d in p r o p e r s e co n ds .

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

would com e to the r e su l t t ha t a t r ansve r se and a long i tud ina l l i gh t c lock

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

c o o r d i n a t e s a d e q u a t e t o p h y s i c a l r e al it y , a t th e a s s u m p t i o n o f th e t i m e

d i l a ti o n d o g m a , w o u l d b e g i v en b y t h e r e la t iv i st ic G a l i l e a n t r a n s f o r m a t i o n .

H o w e v e r , t h e t r a d i t i o n a l N e w t o n i a n c o n c e p t i o n s l e a d t o t h e c o n c l u s i o n t h a t

a t r ansve r se and a long i tud ina l l i gh t c lock have d i f fe r en t r a t e s ( see Sec tion 2 ) .

O n t h e o t h e r h a n d , e x p e r i m e n t s ( t h e h i s t o r i c M i e h e l s o n - M o r l e y e x p e r i m e n t

was the f ir s t) hav e show n tha t t he r a t e s o f a t r ansve r se and a long i tud ina l

l igh t c lock a r e equa l . We have a s sum ed th i s em pi r i ca l f ac t a s an ax iom a t i ca l

asser t ion, without trying to explain why Nature is made in such a manner.

T h e i n t r o d u c t i o n o f t h is a x i o m a t i c a l ( e m p i r i c a l ) a s s e r t i o n i n t o t h e G a l i l ea n

t r a n s f o r m a t i o n l e a d s t o t h e M a r i n o v t r a n s f o r m a t i o n .

T h i s is to b e d o n e i n t h e f o l l o w i n g m a n n e r : E x a c t l y i n t h e s a m e w a y

as in Sec t ion 5 , we com e to the co nc lus ion tha t i f we wish to m ee t t he r e -

q u i r e m e n t o f o u r t e n t h a x i o m a b o u t t h e i n d e p e n d e n c e o f t h e li g h t c l o c k ' si a t e o n t h e o r i e n t a t io n o f t h e c lo c k ' s " a r m , " t h e t r a n s f o r m a t i o n b e t w e e n

the r ad ius vec to r s r and r ' i s t o be wr i t t en in the fo rm (14) . To ob ta in the

i n v e r s e t r a n s f o r m a t i o n w e p r o c e e d f r o m t h e f o r m u l a [ s e e ( 1 2 ) ]

r = r j_ + r~ = r j_' + r ,' ( 1 - - V2/c2) 1/2 + V t (21)

Thi s f o rm ula , wr i t t en in such a m anne r tha t on ly the r e l a t ive r ad ius

v e c t o r r ' i s r e p r e s e n t e d , b u t n o t i t s p e r p e n d i c u l a r a n d p a r a l l e l c o m p o n e n t s

r± ' a nd r ~ , ha s the fo rm

r = r ' 4- {[(1 - - V2/c2)1/2 -- 1]r ' • V /V 2 + t}V (22)

From form ulas ( 14) and (22) , i n a m anne r s im i l a r t o tha t used in Sec t ion

5 , we ob ta in fo rm ula (19) .

Le t us com bine fo rm ulas ( 14) and (19) , and then fo rm ulas ( 22) and

(19), expre ss ing in bo th l a s t f o rm ula s V th ro ug h V 0 accord ing to the second

re la t ion (20) :

r l = r - ~ - V (23)~L(i- V 2 / c 2 ) 1 /2 J _ ( 1 - P~/c2)1/2,

to = t(1 - - V~/c2)1/2

l[ ' -1 ] r''v° Ir = r' + (1 + Vo2/C~)1/2 Vo----~ 4- to V0 (24)

t = t0(1 + Vo2/Ce)X/2



456 Marinov

Formulas (23) represent the d i rect , and formulas (24) the inverse ,

h o m o g e n e o u s Marinov t rans format ion .

Let us now ob ta in the Mar inov t r ansfo rmat ion fo rmulas fo r ve loc i t i e s .

Wri t ing in the f i r s t form ulas (23) and (24) dr , dt, dr', dto instead of r , t , r ' , to,

d iv id ing them by dt, and in t roduc ing the no ta t ions v = dr~dr, v' = dr'/dt,

we ob ta in

v ' = v - r ( 1 - - V2/c2)1/~ -- 1 V~ ( t - - V2/c2) 1/~ V (25)

+ 1 - - c 2 ] - - 1 - ~ / - g - + 1 V (2 6)

The veloci t ies v and v ' are measured in absolute t ime. Thus v must be

cal led the absolute absolute veloci ty (as a ru le , the f i r s t adject ive "absolute"

wi l l be omit ted) and v ' the absolute rela t ive veloci ty (as a rule, the adjective

"absolute" wi l l be omit ted) . For th is reason we have wr i t ten in (26) the

abso lu te t r ans ien t ve loc i ty V and no t the p r ope r t r ans ien t ve loc i ty V0.

Formula (25) r ep resen t s the d i r ec t and fo rmula (26) the inver se Mar inov

t r ansfo rmat io n fo r ve loc i ti e s wr i t ten in absolute t ime.

Writ ing in the f i r s t formulas (23) and (24) dr , dr, dr', dto i n s t ead o f r ,

t , r ' , to, dividing them by dto, and in t roduc ing the no ta t ions v0 = dr~dr o

fo r the proper abso lu t e ve loc i t y and v0' = dr' /dto fo r the proper rela t ive

veloci ty , we can ob ta in the Mar inov t r ansfo rmat ion fo r ve loc i t i e s wr i t t en

in proper t ime .

One also can wr i te the t ransformat ion formulas for veloci t ies in which

the re la t ive veloci ty i s expressed in prop er t ime a nd the abs olute v eloci ty

in absolute t ime. This wi l l be the Mar inov t ransformat ion for veloci t ies

wr i t ten in m i xed t i m e .

Now we sha l l wr i t e the t r ansfo rmat ion fo rmulas fo r the ve loc i t i e s '

magn i tudes . Den o t ing the ang le be tween v and V by 0 and the ang le be tweenv ' and V by 0 ' , we can wr i te formulas (25) and (26) in the fo l lowing form,

a f t e r hav ing squared them:

(v,)2 = v2[1 -- V2(sin 20 )/c ~] -- 2vV cos 0 + V ~1 -- V2/c ~ (27)

U2 = (V ') 2 [1 - - g2 (c os 20 t ) / c 2] @ 2v ' V(c os 0 t ) (1 - - V2/c2) t/2 }~ V 2 (28 )

I f we sup pose v = c and i f we wr i te v' ~ c', where c ' i s the re la t ive

l ight veloci ty me asure d in a bsolu te t ime, i .e ., the abs olute re la t ive l ight

ve loc ity , then these two equa t ions ( the second a f t e r a so lu t ion o f a quadra t i c

equat ion wi th respect to v ' ) g ive

1 - V(cos O)/e (1 - V~/c2)~/~ (29)

c' = c - 0 -- V2/c~) 1/~ = c 1 q- V(cos O')/c



The Coordinate Transformations of the Absolute Space-Time Theory 4 5 7

I f we d eno te b y Co' the pr op er re la t ive light ve locity , then i ts con nect ion

wi th the absolu te absolu te l ight ve loci ty c wi l l be

, 1 -- V(cos O)/c c

c o = c 1 -- V2/ e 2 -- 1 + V(cos O')/c (30)

and i t s connec t ion wi th the p roper abso lu t e l i gh t ve loc i ty

c

co........

(1 --- V2/c2) t/~ (31)

wi l l be the same as tha t g iven by formula (29) .

No te tha t t he ve loc i t i e s w i th re spec t t o t he mov ing f rame K ' a re ca l l ed

re la t ive , whi le the c locks a t tached to K ' a re ca l l ed p roper . On the o the r

hand, the ve loci t ies wi th respect to the res t f rame K are ca l led absolu te

and the c locks a t t ached to K a re a l so cal l ed abso lu te . T o have in the second

case a te rm inologica l d i fference s imi lar to th a t in the f i rs t case we have c on-

s ide red ca ll ing the abso lu t e c lock and abso lu t e t ime "un ive rsa l . " However ,

f ina l ly we decided to use a s ingle word , even tho ugh th is migh t som et imes

lead to misunders t and ing , because o f t he confus ion in u s ing too m any d i f -

ferent te rms.

We designate the re la t ive quant i t ies by superscr ip ts (pr imes) and the

proper quant i t ies by subscr ip ts (zeros) . For th is reason , in the Lorcntz

t ransformat ion (where t ime i s re la t ive) , we designate the re la t ive t ime

coord ina t es by supersc r ip t s (p r imes) and in the Mar inov t rans fo rmat ion

(where t ime i s absolu te) we designate the proper t ime coordinates by sub-

scripts (zeros).

The d i s t ances a re a lways abso lu t e . However , t he ae the r -Mar inov

charac t e r o f l i gh t p rop aga t ion l eads to the in t roduc t ion o f t he no t ion "p r op er

d i s t ance . " The p rob lem abou t t he e t e rna l con t rad ic t ion be tween p roperdistances and distances is considered in detai l in Ref. 7 . Here we must again

repea t t ha t t he abso lu t e and p roper t ime in t e rva l s a re p h y s i c a l l y d i f f e r e n t

q u a n t i t i e s , whi le the d i fference between pro per d is tances an d d is tances is

o n l y a c o n t r a d i c t o r y m a t h e m a t i c a l r e s u l t which appea rs because o f the

ae the r - Ma r inov c ha rac t e r o f l i gh t p rop aga t ion engendered by the b id irec -

t ional l igh t ve loci ty i so t r opy in any iner t ia l f rame.

7. G R O U P P R O P E R T I E S O F T H E M A R I N O V T R A N S F O R M A T I O N

Afte r a due exam ina t ion o f t he Mar inov t rans fo rm at ions , i t can easi ly

be e s tab l i shed tha t t h ey fo rm a g roup . As the ma them at i ca l ana lys is i n t he

genera l case i s too cumbersome, we shal l suppose , for s impl ic i ty 's sake , tha t



458 Marinov

the ve loci ties of the d i fferent f ram es a nd the i r x axes are para l le l to the x

axis of the res t (abs olu te) f ram e. Since in th is s imple case the y an d z coor-

d ina t es a re sub jec t ed to an iden t i ca l t rans fo rmat ion , we sha l l i gnore them.

From fo rmulas (23 ) we ob ta in the fo l lowing d i rec t t rans fo rmat ion

be tween the coord ina t es (x , t ) i n the abso lu t e f ram e K and the coord ina t es

(x2 , tz ) in a p rop er f ram e/<2 mo ving wi th a ve loci ty V2 (V~ <> 0) a lo ng th e

posi t ive d i rec t ion of the x ax is :

x2 -- ( x - V2t)/(1 - - V~2/c2) ~/2, t2 = t(1 -- V2~/c2)1/2 (32)

T h e i n ve r se t r a n s fo rma t i o n b e tw e e n th e c o o rd i n a te s (x l , q ) i n a p ro p e r

fra me K~ m ov ing wi th ve loci ty V~ (V~ <> 0) a long the posi t ive d i rec t ionof the x ax i s o f the re s t f ram e K a nd the coord ina t es (x, t ) i n K , acco rd ing

to formulas (24) [see also formulas (20)], is

( vl ~ ~'/~ vl tl t~x = x l 1 -- --C ~ ] -k (1 -- V~2/c2)1/2 ' t = (1 - - V12/c2) 1/2 (33)

where the ve locit ies V1 and V2 are m easu red in a bsolu te t ime.

Subst i tu t ing formulas (33) in to formulas (32) , we can express the

coord ina t es i n f rame K2 th rough the coord ina t es i n f rame K~:

1 - - V12/c 2 1/~ V 1 - - V~

x2 = x~ ( 1 - V2 :lc ~ ) + t~ (t - V~lc ~)~/2 ( 1 - v ~ / c 2 ) ~ / 2

(34)(_1 - - V22/e 2 ~/2

1 V~2lc 2 )2 tl

These fo rmulas a re abso lu t e ly symmet r i c w i th re spect t o t he coord ina t es

i n b o t h f r a me s . N o w w e s ha ll p ro v e t h a t t h e s e t r a n s fo rm a t i o n s fo rm a g ro u p .

A se t of t ra nsf orm at io ns T~2, T,~3, T84 . .. . form a gr oup i f i t has thefo l lowing p roper t i e s :

1. Transitive property: T h e p ro d u c t o f t w o t r a n s fo rma t i o n s o f th e s e t

is equ iva len t t o a me m ber o f t he se t, t he p rodu c t

T1a- TI2T23 (35)

being defined as performing T~2 and T2~ successively.

If formulas (34) give a transformation T12, a transformation T28 will

have the same form in which the number I is replaced by 2 and the number2 by 3. Substituting formulas (34) for the transformation T12 into the

corresponding formulas for the transformation T2a, we obtain a trans-

formation 2K~3 which has the same form as (34) in which the number 2 is

replaced by 3.



The Coordinate Transformations o f the Absolute Space--Time Theory 459

Thus the t rans i t ive p roper ty i s p roved . Le t u s men t ion he re tha t the

t rans i t ive p roper ty fo r the Loren tz and Ga l i lean t rans fo rmat ions can be

proved on ly i f one takes in to accoun t the co r respond ing t rans fo rmat ion fo r

ve loc it ie s. The t rans i tive p roper ty fo r the Ma r inov t rans fo rma t ion i s p roved

directly, i . e . , wi thou t t ak ing in to accoun t the t rans fo rmat ion fo r ve loc i t i e s .

2. Identity property: The se t inc ludes one " iden t i ty" t r ans fo rmat ion

T~ whose p roduc t wi th an y o the r m emb er o f the se t l eaves the la t te r un -

changed . Thus

Tl~T22 = TIxT12 = T12 (36)

The identity form of the transformation (34) occurs for V~ --= Vo.3. Reciprocalproperty: Every mem ber of the se t has a uniqu e rec iprocal

(or inverse) which is also a m em ber of the set . Thus the inverse o f Tt2 is T21,

where T~I is a m emb er of the se t and

TI~T21 = Tll (37)

The rec iprocal of the t ran sfor ma tion (34) can be ob ta ined by wri t ing

the number 2 ins tead of 1 and v ice versa .

4. Associative property: I f three succeeding t ransformations are per-fo rmed , then

T12(T23Tz ,) = ( T12T23) Tsa (38)

The associa t ive proper ty can eas i ly be proved.

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

In the l ight o f the exper imen ta l demon s t ra t ion o f the an iso t ropy o f the

unid irectio nal l igh t velo city pe rfo rm ed recently by us, ~1-~) we ha ve show n

how the Loren tz t r ans fo rmat ion i s to be t rea ted f rom an abso lu te po in t o f

view, if one seeks to preserve i ts adequacy to physical reali ty. The seeming

cons tancy o f l igh t ve loc i ty appear ing in the Loren tz t r ans fo rmat ion i s due

to the in t roduc t ion o f space coord ina tes in the t rans fo rmat ion fo rmulas fo r

t ime . In the L oren tz t r ans fo rm at ion the space as we ll as the t ime coord ina tes

are re la t ive quant i t ies , whi le l ight ve loci ty is an absolu te quant i ty . We pro-

pose the Mar inov t rans fo rmat ion where the t ime coord ina tes a re abso lu tequant i t ies and l ight ve loci ty is a re la t ive quant i ty and we asser t tha t th is

t ransformation is adequate to physica l rea l i ty .

In Ref . 7 we have shown that the Eins te in t ime synchroniza t ion leads

to the Loren tz t r ans fo rmat ion , wh i le the Newton ian t ime synchron iza t ion



460 Marinov

leads to the Galilean transformation. The second assertion is true only

within an accuracy of first order in V/c. Within an accuracy of second order

in V/c the Newtonian time synchronization leads to the Marinov trans-

format ion, and this is due to the aether-Marinov character of light propa-

gation.

We have found difficulties in considering the Marinov transformation

in a 4-space. In our opinion the Lorentz transformation is much more con-

venient and productive, so that the Marinov transformation only helps us

when treating the Lorentz tlansformation from an absolute point of view,

i.e., adequately to physical reality. We hope that future investigations will

throw more light on the essence of the Lorentz and Marinov transformations,

which are simply two companions.

R E F E R E N C E S

I. S. Marinov, Czech..1. Phys. B24, 965 (1974).2. S. Marinov, Int. J. Paraphys. 11, 26 (t977).3. S. Marinov, Phys. Lett. 54A, t9 (1975).4. S. Marinov, Found. Phys. 8, 137 (1978).

5. S. Marinov, lnt. J. Theor. Phys. 9, 139 (1974).6. S. Marinov, Found. Phys. 6, 57t (1976).7. S. Marinov, Int. J. Theor. Phys. 13, 189 (1975).8. S. Marinov, Eppursi muove Centre Beige de Documentation Scientifique Bruxelles

(1977).9. J. A. Briscoe, British patent, London, No. 15089/58-884830, Application date 12 May

1958.

Documents

1978.04.26 - Stefan Marinov - The Coordinate Transformations of the Absolute Space-Time Theory