1974.02.08 - stefan marinov - the fundamental conception of the absolute space-time theory

8/3/2019 1974.02.08 - Stefan Marinov - The Fundamental Conception of the Absolute Space-Time Theory

http://slidepdf.com/reader/full/19740208-stefan-marinov-the-fundamental-conception-of-the-absolute-space-time 1/24

In ternat ional Journal o f Theoret ical P hysics , Vol. 13, No. 3 (1975), pp. 189-212

T h e E x p e r i m e n t a l V e r i f i c a t i o n o f t h e A b s o l u t e

S p a c e - T i m e T h e o r y - I

T h e F u n d a m e n t a l C o n c e p t i o n o f th e A b s o l u t e S pa c e -T i m e T h e o r y

S TEF A N M A R I N O V

ul. Elin Pelin 22, Sofia 1421, Bulgaria

Received: 8 February 1974

Abstract

The fundamental postulate of our theo ry is the c onstancy of light velocity only withrespect to absolute space. This postulate was proved right by our recently performed'coupled-mirrors ' experime nt (Marinov, 1974). In the p resen t paper it is shown tha t th e

so-called (by us) Newtonian and Einsteinian time synchronisations lead respectively to

the Galilean and Lorentz transformations. Both types of synchronisation can be practi-caUy realised, hence b oth corresponding trans format ions describe the physical reality at

low as well as at high velocities of the material points. The concep tion tha t the Eins tein

time dilation is an absolute phen omen on and th e Lorent z length contraction a fiction is

defended.

1. Introduction

Our absolute space-t ime theory (Marinov, in preparat ion) fmds i ts experi-

mentum crucis in the 'coupled-mirrors ' exper im ent recen t ly per tbrm ed by us

(Marinov, 1974). This exper iment has undo ubte dly show n that the Einsteinprinciple o f relat ivi ty is invalid and th at the hyp oth etica l m otionless "lumi-ni ferous ether ' of the n ineteen th cen tury in which l ight propagates wi th

velo ci ty c in al l direct ions is a physical real i ty w hich w e call abso lute space.After the de velopm ent of the 'coupled-mirrors ' exper iment , theoret ical

physics has to thoro ughly revise the fundam ental space- time conc epts defend ed

by conv ent ional physical th eory , who se impor tan t basis i s the th eory of rela-t iv ity , and in man y aspects return to the o ld New tonian absolute concept ion .

© 1975 Plenum Publishing Corporation. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, microfilming, recording, or otherwise, without written

permission of the publisher.

189



19 0 STEFAN MARINOV

H o w e v e r , w e m u s t e m p h a s is e t h a t w e a ls o w o r k w i t h t h e L o r e n t z t r a n sf o r m a -

t i on an d do n o t r e j ec t i t . Hence , a l m os t a ll f o r m u l ae o f conv en t i ona l phys i ca l

t he o r y f i nd a p l ace in abs o l u t e s pace - ti m e t heo r y , t hus t he r ev i si on o f t he

m a t hem a t i ca l appa r a t u s is ve r y l im i t ed .I n t h i s pape r we s ha ll e xp ou nd ou r ba s ic s pace - t im e co nc ep t and w e s ha ll

s h o w h o w w e a r r iv e a t t h e f o l lo w i n g t w o v e r y im p o r t a n t c o n c lu s io n s :

( a ) Th e E i ns te i n t im e d i l a t ion is an abs o l u t e phe no m en on ( a s s uppo s ed by

L o r e n t z ) a n d n o t a re l at iv e p h e n o m e n o n ( a s s u p p o s e d b y E i n s t e in ) .

( b ) Th e L or en t z leng t h co n t r a c t i on i s pu r e f i c t i on , i .e . , i t is ne i t he r an abs o -

l u t e p h e n o m e n o n (a s s u p p o s e d b y L o r e n t z ) n o r a r e la t iv e p h e n o m e n o n

( as s uppos ed by E i ns t e i n ).

2 . N e w t o n i a n a n d E i n st ei n ia n T i m e S y n c h r o n i sa t io n s

Acc or d i ng t o ou r t he o r y l i gh t p r opaga t e s wi t h ve l oc i t y e a l ong a ll d i r ec ti onson l y i n abs o l u t e s pace. Th e de f i n i t i on o f abs o l u t e s pace is g i ven i n M ar i nov

( 1 9 7 2 ) . S i n c e o u r ' c o u p l e d - m i r r o r s ' e x p e r i m e n t h a s n o t y e t g i v en a r e li a bl e

quan t i t a t ive va l ue f o r the abs o l u t e ea r t h ve l oc i t y , and s ince the a s t r onom er s

a lso c an no t of f e r such a re l iab le va lue , w e sha ll assume th a t absolu te space is

t h a t i n w h i c h t h e c e n t r e o f m a s s o f o u r G a l a x y r es ts .

I t is we l l kn ow n t h a t p hys i c s is geom e t r y p l u s t im e . Hen ce any phys i c i s t

m u s t s t a r t w i t h t h e p r o b l e m o f h o w t i m e is t o b e m e a s u r e d .

W e s ha ll s uppo s e t ha t w e have a c l ock wh i ch o pe r a t e s a t t he s am e r a t e a s

' d i e Rgd e r an de r g r os s en W e l t enuh r ' , i. e . , t ha t t h i s c l ock pe r f o r m s a pe r iod i ca l

m o t i o n w i t h e x a c t l y e q u al p e r io d s . I f w e w a n t t o h a v e a t s o m e o t h e r p l ac e o f

t h e u s e d r e f er e n c e fr a m e a n o t h e r ' d a u g h t e r ' c l o c k w h i c h w o u l d s h o w t h e s a m et i m e a s ou r 'm o t h e r ' c l ock , i .e . , who s e po i n t e r s s how a t any a b s o l u t e m o m e n t

t he s am e r ead i ng on i ts c l ock - face a s t he p o i n t e r s o f the 'm o t h e r ' c l ock , t hen

we hav e tw o poss ib i l it i es of rea l iz ing th i s :

( a ) B e t w e e n t h e ' m o t h e r ' a n d ' d a u g h t e r ' c l o c k s w e p l a c e a l o n g r ig id s ha f t

wh i ch is r o t a t i ng a t a cons t an t angu l a r ve l oc i t y de t e r m i ned , s ay , by t he

' m o t h e r ' c l o c k . L e t u s h a v e t w o c o g - w h e el s o n b o t h e n d s o f t h e s h a f t

and l e t u s num ber any t w o cogs wh i ch l ie aga inst e ach o t he r on t he

oppo s i t e ends o f a g iven s ha ft ' s gene r a t i on . L e t u s now s uppos e t ha t a t

t he beg i nn i ng o f any t i m e i n t e r va l chos en a s a t i m e un i t a de f i n i t e cog

o f t h e f ir s t c o g -w h e e l c o m e s i n t o u c h w i t h t h e ' m o t h e r ' c l o c k . I f t h ep o i n t e r s o f t h e " d a u g h te r ' c l o c k s h o w t h e r e a d in g w h i c h i s 'c o m m u n i -

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

c o n t a c t w i t h t h e 'd a u g h t e r ' c l o c k , t h e n w e s a y t h a t a N e w t o n i a n t i m e

s y n c h r o n i s a t io n is m a i n t a i n e d b e t w e e n b o t h c lo c k s.

( b ) F r om t he 'm o t h e r ' c l ock we s end a l igh t s ignal a t t he beg i nn i ng o f any

t i m e i n t e r va l chos en a s a ti m e un i t . I f t he po i n t e r s o f t he ' da ugh t e r '

c l oc k s how t h e r ead i ng wh i ch t h e l i gh t signal ha s "com m u ni ca t ed ' , p lu s

t h e t i m e r / c , w h e r e r is th e d i s ta n c e b e t w e e n b o t h c l o ck s , t h e n w e s a yt h a t a n E i n st e in i a n t i m e s y n c h r o n i s a t io n is m a i n t a in e d b e t w e e n b o t h

c locks .

W h e n i n t r o d u c i n g t h e E i n s te i n ia n t i m e s y n c h r o n i s a t io n w e m a k e t h ea s s u m p t i o n t ha t l igh t p r opaga t e s w i t h a ve l oc i t y wh i ch ha s t he s am e num er i ca l



EXPERIMENTAL VERIFICATION OF ABSOLUTE SPACE-TIME THEORY--I 191

va lue a long a l l d i r ec t i ons i n a n y i n e r ti a l f r a m e o f r e fe r e n c e , i .e ., in a n y f r a m e

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

I f o u r ' m o t h e r ' a n d ' d a u g h t e r ' c l o c k s r e s t i n a b s o l u te s p a c e , th e n t h e i r

N e w t o n i a n a n d E i n s te i n ia n t i m e s y n c h r o n i s a t i o n s le a d t o t h e s a m e r e su l t, i .e .,t w o ' d a u g h t e r ' c l o c k s p l a c e d a t t h e s a m e s p a c e p o i n t a n d s y n c h r o n i s e d r e s p e c-

t i v e ly i n N e w t o n i a n a n d E i n s te i n ia n m a n n e r w i lt s h o w t h e s a m e re a d i n g o n t h e i r

c l o c k - fa c e . H o w e v e r , i f o u r c l o c k s m o v e w i t h a c e r t a i n v e l o c i t y i n a b s o l u t e

s p a ce , t h e n t w o s u c h ' d a u g h t e r ' c l o c k s w i ll s h o w d i f f e r e n t r e a d i n gs , a n d f r o m

t h is v a ri a n c e , w i t h t h e h e l p o f r a n d c , o n e c o u l d d e t e r m i n e t h e c o m p o n e n t V

o f th e u n k n o w n a b s o l u t e v e l o c i ty a l o n g t h e l in e c o n n e c t i n g t h e ' m o t h e r ' c l o c k

t o b o t h ' d a u g h t e r ' c l o c k s . T h i s i s d u e t o t h e f a c t t h a t v e l o c i t y o f l ig h t in a f r a m e

m o v i n g w i t h v e l o c i t y V in a b s o l u t e s p a c e is eq u a l t o c - V a l o n g a d ir e c t i o n

pa r a l le l t o V and t o c + V a long a d i r ec t i on an t i pa r a l l e l t o V .

I n o u r ' c o u p l e d - m i r r o r s ' e x p e r i m e n t ( M a r in o v , 1 9 7 4 ) w e h a v e re a li s ed f o r

t h e f i rs t t i m e i n t h e h i s t o r y o f p h y s ic s a c o m b i n a t i o n o f th e N e w t o n i a n a n d

E i n s t e in i a n t i m e s y n c h r o n i s a t i o n s a n d t h i s g a ve u s t h e p o s s i b i li t y o f d e t e r m i n -

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

3 . T he G aIi le an and L o r e n t z T r ans fo r m a t ions

I f w e h a v e t w o f r a m e s o f r e f e r e n c e m o v i n g w i t h r e s p e c t t o e a c h o t h e r , i .e .,

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

t h e N e w t o n i a n a n d E i n s t e i n i a n s y n c h r o n i s a t i o n s w o u l d l e a d t o t w o d i f f e r e n t

t y p e s o f 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 t h e e l e m e n t s o f m o t i o n o f a g iv e n m a t e r ia l

p o i n t w h o s e m o t i o n i s c o n s i d e r e d in b o t h f r a m e s . T h e N e w t o n i a n s y n c h r o n i s a -

t i o n l e ad s t o t h e G a l il e a n t r a n s f o r m a t i o n f o r m u l a e a n d t h e E i n s t e in i a n s y n -c h r o n i s a t i o n l e a d s t o 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 .

We s ha l l f i r s t deduce t he G a l i l e an t r ans f o r ma t ion .

L e t u s h a v e (F i g . 1 ) t w o f r a m e s K a n d K ' b e t w e e n w h i c h t h e r e i s t h e c a s e

Iy , y ' ( O ) '

y ' ( t ) j P ( O ) P ( t)

Il t J . 7 0 I / 3 0 '

t --

X' ,~ X,X '

l_

Figure 1 .- Tw o space frames moving with respect to each other, th e tran sformati onbetween being a special one.



1 9 2 S T E F A N M A R I N O V

o f a s p e ci a l tr a n s f o r m a t i o n , i .e ., a t t h e i ni ti a l z e ro m o m e n t b o t h f r a m e s h a v e

c o i n c i d e d a n d f r a m e K ' 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 p o s i ti v e d i r e ct i o n

o f th e x - a x i s o f fr a m e K ( o r f r a m e K p r o c e e d s w i t h t h e s a m e v e l o c i t y a l o n g

t h e n e g a t i v e d i r e c ti o n o f th e x ' -a x i s o f f r a m e K ' ) . F o r t h e s a k e o f si m p l i c i tyw e h a v e s h o w n a t w o - d i m e n s i o n a l c a s e in F ig . 1 .

L e t p o i n t P b e a t re s t in K ' . F o r t h e i n i t ia l z e r o m o m e n t , t o = to = 0 , t h e

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

m o m e n t t ( t o w h i c h in f ra m e K ' t h e m o m e n t t ' c o r r e s p o n d s ) t h e ra d iu s v e c t o rs

o f p o i n t P i n f r a m e s K a n d K ' a r e r e s p e c t i v e l y r a n d r ' (= r ~ ). I t i s o b v i o u s t h a t

t h e y - a n d z - c o m p o n e n t s o f r a n d r ' a r e e q u a l , i. e .,

y = y ' , z = z ' ( 3 . 1 )

O n l y t h e x - c o m p o n e n t s a r e d if f e re n t a t a n y d i f f e re n t m o m e n t . T o f i n d t h e

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 th e x - c o m p o n e n t s l e t u s a s s u m e t h a t a t t h e i n it ia lz e r o m o m e n t w e s e n d a p h o t o n f r o m t h e c o m m o n f r a m e s ' o r ig in t o th e p r o j e c ti o n

o f p o i n t P o n t h e x -a x e s . W h e n t h e N e w t o n i a n t i m e s y n c h r o n i s a t i o n i s u s e d w e

s h o u l d f i n d t h a t t h i s p h o t o n r e a c h es t h e p r o j e c t io n o f P a t th e m o m e n t s

x x t

t = t ' = ~ ( 3 . 2 )C' c-V

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

m o m e n t s

X fX tr =- -

t = c + V ' c ( 3 . 3 )

i f w e a s s u m e t h a t f r a m e K ' 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 .

I n b o t h c a s es it m u s t b e

X X r X X t

c c - V ' c + V c

r e s p e c t i v e l y , i . e . ,

( 3 . 4 )

a n d

F r o m ( 3 . 2 ) a n d ( 3 . 5 ), a s w e l l a s f r o m ( 3 . 3 ) a n d ( 3 . 5 ) , w e i m m e d i a t e l yo b t a i n

x = x ' + V . t ' ( 3 . 6 )

x ' = x - V . t ( 3 . 7 )

F o r m u l a e ( 3 . 6 ) , ( 3 . 7 ) , ( 3 . 1 ) a n d ( 3 . 5 ) r e p r e s e n t t h e s p e c i a l G a l i l e a n tr a n s -

f o r m a t i o n w h i c h i s t h e m a t h e m a t i c a l b a s is o f s o -c a l le d n o n - r el a ti v i st ic

m e c h a n i c s .

L e t u s n o w d e d u 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 t h i s p u r p o s e t h e

t = t ' ( 3 . 5 )



E X P E R I M E N T A L V E R I F I C A T I O N O F A B S O L U T E S P A C E -T IM E T H E O R Y - I 1 9 3

E i n s t e i n i a n t i m e s y n c h r o n i s a t i o n m u s t b e u s e d , a n d i n s t e a d o f f o r m u l a e ( 5 . 2 )

a n d ( 3 . 3 ) w e s h o u l d c o n s i d e r , i n b o t h c a s e s ,

X X ~

t = - , t ' = - ( 3 . 8 )c c

W e s e e t h a t w h e n t h e E i n s t e in i a n s y n c h r o n i s a t i o n i s u s e d t h e c o u p l e s o f

f o r m u l a e ( 3 . 2 ) a n d ( 3 . 3 ) m u s t b e r e p l a c e d b y t h e u n i q u e c o u p l e ( 3 .8 ) . T h u s

w e h a v e a ls o t o r e p l a c e b o t h f o r m u l a e ( 3 . 4 ) b y a u n i q u e f o r m u l a . T h i s is d o n e

b y m u l t i p l y in g f o r m u l a e ( 3 . 4 ) a n d t a k i n g t h e s q u a r e r o o t giv in g

x . J ( I - V ) J ( l + V ) (3 .9 )

F r o m h e r e , u s in g th e s e c o n d f o r m u l a ( 3 . 8 ), w e g e t

x ' + g . t 'X ~-

j ( 1(3.10)

a n d u s i n g th e f i rs t f o r m u l a ( 3 . 8 ) w e g e t

x - V . tx ' - ( 3 . i 1 )

j ( l )S u b s t i t u t i n g ( 3 . 1 0 ) i n t o ( 3 . t t ) w e o b t a i n

t ~ + X r . m / c 2

t - ( 3 . 1 2 )

a n d s u b s t i t u t i n g ( 3 . 1 1 ) i n t o ( 3 . 1 0 ) w e o b t a i n

t ' - t - x . V / c 2 ( 3 . 1 3 )

F o r m u l a e ( 3 . 1 0 ) , ( 3 . 1 1 ) , ( 3 . 1 ) , ( 3 . 1 2 ) , a n d ( 3 . 1 3 ) r e p r e s e n t t h e s p e c i a l

L o r e n t z t r a n s f o r m a t i o n w h i c h is t h e m a t h e m a t i c a l b a s i s o f s o -c a l le d r e la t iv i s ti c

m e c h a n i c s .A t t h e g iv e n d e d u c t i o n o f t h e L o r e n t z t r a n s f o r m a t i o n t h e m o m e n t s t a n d

t ' ( f o r w h i c h w e h av e o b t a i n e d t h e t r a n s f o r m a t i o n f o r m u l a e ) a r e s u c h th a t a

p h o t o n s e n t a t t h e i n i ti a l z e r o m o m e n t , t o = to = O , f r o m t h e o r ig i n s o f f r a m e s

K a n d K ' , a l o n g th e i r x - a x e s , j u s t r e a c h e s t h e p r o j e c t i o n s o f p o i n t P o n t h e

x - a x e s a t t h e m o m e n t t ( o r t ' ) .

W e sha ll now s upp ose t he m os t gene ra l c a se w h e re t and t ' a r e a rb i t r a ry . In

s u c h a c a se w e s e n d a p h o t o n f r o m t h e o r ig in o f th e m o v i n g fr a m e K ' a t s o m e

i n it ia l m o m e n t t o ~ 0 ( o r t o ~ 0 ) a n d i t r e a c h e s t h e p r o j e c t i o n o f p o i n t P o n t h e



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

x - a x e s a t t h e a r b i t r a r y m o m e n t t ( o r t ' ) . I n t h is g e n e ra l ca s e, if w e s u p p o s e t h a t

the r e s t f r ame K i s a t t ach ed to abso lu t e space , we sha l l have , on the g roun ds o f

f o r m u l a e (3 . 1 0 ) , ( 3 . 1 1 ) , ( 3 . 1 2 ) , a n d ( 3 . 1 3 ) ,

x ' + V . ( t ' - t~ )t - t o -

X p - -

x - X o - V . ( t - to)

X t . V

t ' - t o + 02

2 ( 1 )(x - X o ) . V

t - t o c 2t t i

( 3 . 1 4 )

w h e r e X o is t h e x - c o o r d in a t e o f th e o r ig in O ' o f f r a m e K ' a t t he m o m e n t t o

( o r t o ) w h e n t h e ' p h o t o n - r u n n e r ' is s e n t f r o m O ' a l on g th e x - a x e s ; t h i s p h o t o n

r e a c h es t h e p r o j e c t i o n x ( o r x ' ) o f p o i n t P a t t h e m o m e n t t ( o r t ' ) . H e n c e

Xo = V . t o (3 .15 )

I f w e f u r t h e r a s s u m e

to = to . 1 - ( 3 .16 )

t h e n f o r m u l a e ( 3 .1 4 ) r e d u c e t o f o r m u l a e ( 3 . 1 0 ) , ( 3 . 1 1 ) , ( 3 . 1 2 ) , a n d ( 3 .1 3 ) .

T h i s a s s u m p t i o n ( n a m e l y , t h a t t i m e g o e s a t a sl o w e r r a te a c c o r d i n g t or e l a t io n ( 3 . 1 5 ) i n a n y f r a m e m o v i n g a t v e l o c i t y V w i t h r e s p e c t t o a b s o l u t e

s p a c e is f u n d a m e n t a l i n 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 a n d i s ca ll ed t h e

a b s o l u t e t i m e d i l a ti o n . W e c a n c o n s i d er t h i s a s s u m p t i o n a s a r es u l t o f t h e

L o r e n t z t r a n s f o r m a t i o n , b e c a u s e i f w e p l a c e i n t o ( 3 . 1 2 ) x ' = 0 , w e o b t a i n

t ' = t . 1 - - ~ - ( 3 .1 7 )

T h e o p p o s i t e a s s u m p t i o n ( w h i c h w o u l d f o l l o w f r o m ( 3 . 1 3 ) i f w e i n s e r t

x = 0 ) c a n n o t b e m a d e b e c a u s e o n l y f ra m e K ' ( t o g e t h e r w i t h t h e a t t a c h e d K ' -

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

( t o g e t h e r w i t h t h e a t t a c h e d K - c l o c k r e a d in g a b s o l u t e t i m e ) m o v e s w i t h r e s p e c t

t o f r a m e K ' .H o w e v e r , w e m u s t e m p h a s i s e t h a t r e l a t i o n ( 3 . 1 7 ) i s n o t a n a b s o l u t e l o g ic a l

r e s u lt o f th e L o r e n t z t r a n s f o r m a t i o n b e c a u s e t h e e x i s t e n c e o f a b s o l u t e s p a c e is

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

s y m m e t r ic c h a r ac t e r f r o m a m a t h e m a t i c a l v i e w p o i n t . A s a m a t t e r o f f a c t , th e

s p e ci a l t h e o r y o f re l a ti v i ty , w h i c h w o r k s w i t h t h e L o r e n t z t r a n s f o r m a t i o n , d o e s

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

a n d h a s e n d e a v o u r e d ( d e s p i t e t h e r e s is t a n ce o f t h e h e a l t h y m i n d o f s e v er a l

g e n e r a ti o n s o f p h y s ic i s ts ) t o t r e a t t h e t i m e d i l a ti o n a s a re l at iv e p h e n o m e n o n .



EXPERIMENTAL VERIFI CATION OF ABSOLUTE SPACE-TIME THEORY--I 195

In Section 6 we give further motivations in favour of our absolute time

dilation dogma. Other motivations will be given in future papers which are

being prepared for printing.

We shall also need the formutae for our so-called restricted Lorentz trans-formation. The restricted transformation has the same character as the special

one, with the unique difference that the relative velocity V is no t parallel to

the x-axes of frames K and K ' but has an arbitrary direction. The formulae for

the restricted Lorentz transformation can easily be obtained if we consider the

radius vector r o f an arbitrary point P as a sum of its vector components rll and

r±, which are respectively parallel and perpendicular to V, and if we apply the

special Lorentz transformation to rll and r±. We therefore obtain

r = r ' + V 2 - . - - ~ - + - V 2 . V

~ - 1 ~ ( 3 .1 8 )

t r ' . V ~t : . + - 7 - ]

and the parallel inverse formulae if we should express r' and t ' by r and t.

In conclusion, we can say that the Galilean and Lorentz transformations

represent two d i f f e ren t mathematical implements which are used for the

description of the same physical reality, i.e., they represent two slightlydifferent images of the same object. We do not agree with the conventional

opinion that the Galilean transformat ion represents only a limited case of the

Lorentz transformation when V ~ c. We defend the assertion that the Galilean

transformat ion is also to be used when high velocity material systems are

considered. The difference between these two transformations is determined

only by the different character of synchronisation of clocks remote in space.

4. Space In tervals in No n-Re lat iv is t ic M echanics

Let us take a rod which is at rest in the used reference frame. We canmeasure its length (i.e., the space interval between both ends), with the help

of a standard length, during a specific time, the duration o f which is of no

importance.

However, if this rod moves in the used reference frame, then we cannot

proceed in such a manner. We now have to register the 'track' which the rod

would leave for a certain time interval. Af ter measuring this 't rack', which re-

presents a 'rod at rest' in the used frame, we can calculate the true length of

the moving rod if we know its velocity and the duration of the corresponding

time interval. Of course, if the time interval is insignificantly short, then it is

not necessary to make such a correction over the measured 'track'.

As an example let us measure the length of a train which moves with vel-




o c i t y v . I f w e r u n a c e r t a i n t i m e t w i t h a v e l o c i t y c ( c > v ) , p a r a l le l w i t h t h e

t r a in f r o m t h e l a s t c a r ri a g e t o t h e l o c o m o t i v e , w e s h a ll c o v e r a d i s t a n c e

r = r o + v . t ( 4 . 1 )

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

t h e s t a t i o n .

B o t h e n d s o f d i st a n ce r c a n b e m a r k e d b y u s r e la t iv e t o t h e g r o u n d ( r el a-

t iv e t o t h e r a i lw a y ) , a n d , h a v i n g t h e s e t w o s c o r e s , w e c a n m e a s u r e d i s t a n c e r

d u r i n g s p e c i f i c l o n g t i m e - i n t e r v a l .

B u t w e c a n r u n w i t h v e l o c i t y c o n t h e t o p o f t h e c a r r ia g e s ( a s w e h a v e s e e n

m a n y t i m e s i n t h e m o v i e s ) a n d , t h r o w i n g t w o s t o n e s, m a r k t h e ' t r a c k ' w i t h

r e s p e c t t o t h e r a i l w a y .

I n t h e f i rs t ca s e , i .e . , w h e n o u r v e l o c i t y c i s t a k e n w i t h r e s p e c t t o t h e g r o u n d ,

w e s h a ll h a v e

r

t = - - ( 4 . 2 )C

a n d , i n t h e s e c o n d c a s e , i .e . , w h e n o u r v e l o c i t y c is t a k e n w i t h r e s p e c t t o t h e

t r a i n , w e s h a l l h a v e

rot = - - ( 4 . 3 )

C

S u b s t i t u t i n g ( 4 . 2 ) a n d ( 4 . 3 ) i n t o ( 4 . 1 ) , w e o b t a i n , r e s p e c t i v e l y

ro ( : )r - , r = r o . 1 + ( 4 .4 )

/)

1 - - - -C

W e n o t e t h a t t h e s e t w o r e l a t i o n s d i f f e r w i t h i n s e c o n d - o r d e r t e r m s o f v / c .

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

t h a t i n t h e f i rs t ca s e c is t h e v e l o c i t y o f t h e ' r u n n e r ' w i t h r e s p e c t t o t h e g r o u n d ,a n d i n t h e s e c o n d c a s e c is t h e v e l o c i t y o f t h e ' r u n n e r ' w i t h r e s p e c t t o t h e t r a in .

I f i n t h e f i r st f o r m u l a o f ( 4 . 4 ) w e p u t c + v i n s t e a d o f c, w e s h o u l d o b t a i n t h e

s e c o n d f o r m u l a o f ( 4 . 4 ), a n d i f i n th e s e c o n d f o r m u l a w e p u t c - v i n s te a d o f

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

L e t u s n o w c o n s i d e r t h e m o s t g e n er a l ca se , w h e r e t h e v e l o c i t y o f t h e m o v i n g

r o d i s n o t p a r a ll e l t o i t s le n g t h ( s e e F i g. 2 ) . S i m i l ar r e s u l t s c o u l d b e o b t a i n e d i f

w e w a n t t o k n o w t h e d i s t a n ce b e t w e e n a p o i n t q , m o v i n g w i t h a n a r b i t r a r y

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

T h e r e a r e t w o p o s s ib i li t ie s o f m e a s u r i n g t h e l e n g t h o f t h e m o v i n g r o d P o Q o ,

o r t h e d i s ta n c e b e t w e e n t h e r e st p o i n t P o a n d t h e m o v i n g p o i n t q w h e n t h e

l a t t e r c r o s s e s t h e s p a c e p o i n t Q o :



E X P E R I M E N T A L V E R I F I C A T I O N O F A B S O L U T E S P A C E - T I M E T H E O R Y - - I 1 9 7

o r

/ t o

/

PO P x

F i g u r e 2 . - A ' p h o t o n - r u n n e r ' g o i n g ' th e r e ' o r 'b a c k ' .

( a ) e i t h e r t o s t a r t f r o m p o i n t q ( w h e n i t c ro s s es Q o ) a t t h e m o m e n t t o ,

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

t h e d i s t a n c e r o = Q o P o t o a r r iv e a t p o i n t P o a t t h e m o m e n t

t = t o + r o ( 4 . 5 )e

w h i c h w e s h a l l ca ll r e c e p t i o n m o m e n t ;

( b ) o r t o s t a rt f r o m p o i n t P o a t th e e m i s si o n m o m e n t t o an d c o v e r i ng w i t h

v e l o c i t y c t h e d i s t a n c e r = P o Q t o c a t c h p o i n t q ( w h e n i t c ro s s e s Q ) a t

t h e r ec e p ti o n m o m e n t

F

t = t o + - ( 4 . 6 )C

W e h a ve t o u s e t h e r e l a t i o n

r = ro + v . ( t - t o ) ( 4 . 7 )

I f i n t o t h i s e q u a t i o n w e f i r s t p l a c e (4 . 7 ) , w e s h o u l d o b t a i n , u s i n g F i g. 2 ,

r = r o . 1 + 2 . - - . c o s 0 o + ( 4 .8 )C

)= ro . 1 - f i - . s in s 0 + - - 'c cos 0 (4 . 9 )

w h e r e 0 o i s t h e a n gl e b e t w e e n v a n d r o ( th e v e c t o r c o n n e c t i n g p o i n t P o w i t h

p o i n t q a t t h e e m i s s i o n m o m e n t ) a n d i s c a l l e d t h e e m i s s i o n a n g l e , whi le 0 i s the




ang le be t w een v and r ( t he vec t o r conn ec t i ng po i n t Po w i t h po i n t q a t t he r ecep-

t i on m om en t ) and i s ca l led the r e c e p t i on ang le .

I f n o w i n t o e q u a t i o n ( 4 . 7 ) w e p l a ce ( 4 . 6 ) , w e s h o u l d o b t a i n , u s in g F i g. 2 ,

o r

r or ? ( v 2 0 ) v1 - ~ - . s i n 2 0 - - -c ' C ° S 0 °

(4 . 10)

r o

r= / [ v 2 ... ( 4 . 1 1 )

1 - 2 . - - . c o s O +C

The d i s t ance ro can be ca l led t he e m i s s i o n d i s t a n c e and t he ' t r ack ' d i s t ancer can be ca l led t he r e c e p t i on d i s t anc e .

L e t u s n o w f i n d th e r e l a t i o n b e t w e e n r o a n d r w h e n t h e ' r u n n e r ' c o v e r s,

w i t h v e l o c i t y c , s o m e m i d d l e d i s t a n c e r m , s t a r ti ng a t t he emi s s ion mo m en t t o

f r o m Q m ( o r f r o m P o ) a n d a r r iv i ng a t t h e r e c e p t i o n m o m e n t

t = t o + r m ( 4 . 1 2 )

C

a t Po (o r a t Q m ) .

We can now w r i t e ( see F i g . 2 )

r_m ; r ' c ° s O - r o . c o s Oo

C V( 4 . 1 3 )

When t he ' r unn e r ' cove r s d is t ance ro be t w een t he emi s s i on and o bse rva t i on

m o m e n t s , i t is

r o = r . c o s 0 - to.C OS 0 o(4 . 14)

C V

f r o m w h e r e

r = ro.V'c/ + cos 0o (4 . 15 )cos 0

and w h en t he ' r un ne r ' cov e r s d i s t ance r be t w ee n t he emi s s ion and r ecep t i on

moment s , i t i s

r r . c o s 0 - r 0 . c o s 0 o- ( 4 . 1 6 )

C v

f r o m w h e r e

cos 0o

r = rO . co s 0 - v /c ( 4 . 1 7 )



E X P E R I M E N T A L V E R I F I C A T I O N O F A B S O L U T E S P A C E - T I M E T H E O R Y - I 1 9 9

R e l a t i o n s ( 4 . t 5 ) a n d ( 4 . 1 7 ) d i f f e r w i t h i n s e c o n d - o r d e r t e r m s v / c a n d c a n b e

c a l l e d n o n - r e la t i v is t i c r e l a t i o n s b e t w e e n r o , r , 0 o , 0 , v , a n d c b e c a u s e , u n t i l n o w ,

w e ha v e w o r k e d o n l y w i th t h e N e w t o n i a n c o n c e p t .

L e t u s n o w f i n d a r e l a t i o n b e t w e e n r o , r , 0 o , 0 , v , a n d c w h i c h w o u l d c o r r e -s p o n d t o c o n d i t i o n ( 4 . 1 3 ). F o r t h i s r e a so n w e h a v e t o d e f in e a n e x p r e s s io n f o r

rm t h r o u g h r o , r , 0 o , 0 , v , a n d c , s u b s t a n t i a l l y d i f f e r e n t f r o m ( 4 . 1 3 ) , a n d ,

p u t t i n g t h i s e x p r e s s i o n i n t o ( 4 . 1 3 ) , o b t a i n a s u i t a b l e r e l a t i o n b e t w e e n r o , r , 0 o ,

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

s u cc e ss , an d w e f o u n d t h e f o l l o w i n g t o b e t h e m o s t r e a s o n a b le p a t h : L e t u s

m u l t i p l y f o r m u l a e ( 4 . 1 5 ) a n d ( 4 . 1 7 ) a n d l e t u s ta k e t h e s q u a r e r o o t : w e o b t a i n

r ro .N / \ c o s O _ v / c . c - ~ s O ] ( 4 . 1 8 )

W e c a n n o w c a ll ( 4 . t 8 ) t h e r e la t iv i s ti c e x p r e s s i o n b e t w e e n r a n d r o , b e c a u s et h e w a y i n w h i c h w e p a s s f r o m b o t h n o n - re l at i vi s ti c f o r m u l a e ( 4 . 1 5 ) a n d ( 4 . 1 7 )

t o t h e u n i q u e f o r m u l a ( 4 .1 8 ) is s im i l ar t o h o w w e p a s s ed f r o m b o t h n o n -

r e la t iv i s ti c f o r m u l a e ( 3 . 4 ) t o t h e u n i q u e r e l at i vi s ti c f o r m u l a ( 3 . 9 ). H o w e v e r , t h e

m a t h e m a t i c a l e s s e n ce o f r e l a t i o n ( 4 . 1 8 ) i s n o w v e r y t r a n s p a r e n t a n d c l e a r f r o m

a n o n - re l a ti v is t ic p o i n t o f v i e w , b e c a u s e i t i s o b v i o u s t h a t r e l a t i o n ( 4 . 1 8 )

c o r r e s p o n d s t o t h e c a se w h e r e t h e ' r u n n e r ' c o v e r s s o m e m i d d l e d i s ta n c e r m

b e t w e e n t h e e m i s s i o n a n d r e c e p t i o n m o m e n t s .

F r o m F i g. 2 w e h a ve

r o s i n 0( 4 . 1 9 )

r s in 0o

a n d f r o m ( 4 . 1 8 ) a n d ( 4 . 1 9 ) w e o b t a i n t h e f o l l o w i n g r e l at i on s b e t w e e n t h e

a n g l e s 0 o a n d 0

c o s O o + v / c c o s 0 - v / cc o s 0 = , c o s O o = ( 4 . 2 0 )

I + - . c o s Oo i - - . c o s OC C

F r o m f o r m u l a e ( 4 . 1 9 ) a n d ( 4 . 2 0 ) w e f i n d

2 ( t 9+ - - . c o s 0 o -

¢

. 0f r o m w h e r e

1 + c . c o s Oo

r = r 0 . .~ . . . . .

_ 2_ . co s 0C

( 4 . 2 2 )




F r o m ( 4 . 2 1 ) w e g e t

v . v = t -¢-i" (4.23)

S i nce i t i s app rox i m a t e l y

cos Oo = co s O m + a , cos 0 = cos Om - a ( 4 . 2 4 )

w here a is a pos i ti ve o r nega t ive qu an t i t y and O m is t he ang le be t w een v and

r m ( t h e v e c t o r c o n n e c t i n g p o i n t P o w i t h p o i n t q a t t h e m i d d l e m o m e n t t m be-

t w ee n t he emi s s i on and r ecep t i on mom ent s ) , w h i ch is ca l led t he m i d d l e a n g l e ,

t h e n w e c a n w r it e ( 4 .2 2 ) a p p r o x i m a t e l y i n t h e f o r m

/ / 1 + ~ - .c o s 0 m ~

r = r ° " / I v...... I ( 4 . 2 5 )

~/ \I--7.COS0m

We mu s t emphas i se t ha t f o rm ul ae (4 .21 ) a r e i d e n t i c a l , w hi l e f o rmul ae (4 . 8 )

a n d ( 4 . 9 ), o n t h e o n e h a n d , a n d f o r m u l a e ( 4 . 1 0 ) a n d ( 4 . 1 1 ) , o n t h e o t h e r h a n d ,

are d i f f e r e n t . So fo r t he l ong i t ud i na l ca se 00 = 0 = 0 , in s t ead o f the t w o fo r -

m u l a e ( 4 . 4 ) , o b t a i n e d w h e n p r o c e e d in g , r es p e c t iv e l y f r o m f o r m u l a e ( 4 . 1 0 ) a n d

( 4 . 1 1 ) a n d f o r m u l a e (4 . 8 ) a n d ( 4 . 9 ) , w e o b t a i n t h e u n i q u e f o r m u l a

/[l+v/c~r = r o . d ~ l - ' 7 ~ ] ( 4 . 2 6 )

w h e n p r o c e e d i n g f r o m f o r m u l a e (4 . 2 1 ) .

H ow ever , i t is i m po r t an t t o n o t e t ha t f o r t he t r ansve r se case 0o = rr /2

f o r m u l a e (4 . 2 1 ) , ( 4 . 1 0 ) a n d ( 4 . 8 ) - t h e l a st w i t h in a n a c c u r a c y o f s e c o n d o rd e r

in v / c - g i v e the same resul t :

r 0r - (4 .2 7)

Y 0 )and fo r t he t r ansve r se case 0 - - r r/ 2 f o rmu l ae (4 . 21 ) , ( 4 . 9 ) a nd (4 . 11 ) - - t he l a s t

w i t h i n an ac curacy o f s econd o rd e r in v / c - a ga i n g ive t he s ame r e su lt :

Thu s t he d i f f e r ence be t w een t he non- r e l a ti v i st i c f o rmu l ae (4 . 8 ) , ( 4 . 9 ) ,

( 4 . 10 ) and (4 . 11 ) a nd t he r e la t iv i s ti c f o rmu l ae (4 . 2 t ) i s no t so d r a s ti c .

I t is c le a r t h a t t h e p r o b l e m , w h i c h f o r m u l a e c o r re s p o n d b e t t e r t o r e a l i t y -

t he non- r e l a t iv i s t ic o r t he r e l a t i v i s t i c - ca nn o t be posed . T hese s l igh t l y d i f f e r en t

f o r m u l a e c o r r e s p o n d t o s li g h tl y d i f f e re n t c o n d i t io n s u n d e r w h i c h t h e ' r u n n e r '




c o v e rs w i t h v e l o c i t y c t h e d i s t a n c e b e t w e e n p o i n t s P 0 a n d q , and they a l l

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

H o w e v e r , w h e n t h e ' r u n n e r ' i s a p h o t o n , t h e n , a s n a t u r e s h o w s ( t h i s c a n b e

s e en in t h e M i c h e l so n - M o r l e y e x p e r i m e n t a n d i n t h e l o n g i tu d i n a l D o p p l e re f f e c t e x p e r i m e n t s ) , t h e r e la t iv i s ti c f o r m u l a e ( 4 . 2 1 ) c o r r e s p o n d t o r e a li t y a n d

t h e n o n - r e la t iv i s ti c f o r m u l a e d o n o t . T h u s w e h a v e t o a s s u m e t h a t d u r i n g t h e

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

wi th ve lo c i ty c . I n ou r op in ion , t h i s conc lus ion , a s a m a t t e r o f f ac t , is a r e su l t

o f t h e a b s o l u t e t i m e d i l a ti o n d o g m a .

L e t u s n o w s u p p o s e t h a t t h e r e f e r e n c e f r a m e i n F i g. 2 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 a n d t h u s p o i n t q m o v e s w i t h v e l o c i t y v in a b s o l u t e s p a c e . I f w e d e n o t e

b y Co t h e v e l o c i t y o f t h e ' p h o t o n - r u n n e r ' w i t h r e s p e c t to p o i n t q , t h e n i n s t e a d

o f f o r m u l a ( 4 .5 ) w e h a v e t o w r i t e t h e f o l l o w i n g o n e

t = t o + r o ( 4 . 2 9 )

g o

and no w fo rm ulas (4 .6 ) and (4 .29 ) wi l l be va l id together . In t h i s ca se we can

i m m e d i a t e l y o b t a i n f r o m f o r m u l a s ( 4 . 2 1 ) , ( 4 . 6 ) , a n d ( 4 . 2 9 )

C o = C .

1 - 1 - - v . c o s 0c

1 + - " cos 00 1 - ( 4 .30 )c

Those are the formulas for the ve loc i t y o f l igh t in a mov ing f ram e o f r efer -

enee in relativis tic mechanics, a c c o r d i n g t o 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 .

T h e s a m e f o rm u l a s c a n b e a ls o o b t a in 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 . L e t u s s u p p o s e t h a t f r a m e K i n F ig . 1 i s a t r e s t in a b s o l u t e

s p ac e a n d f r a m e K ' is t h e m o v i n g o n e . I f t h e ' p h o t o n - r u n n e r ' i s s e n t f r o m t h e

co inc id ing o r ig ins o f K and K ' a t t he i n i t ia l ze ro mo m en t , t o = t o = 0 , and i f

i t c a t c h e s p o i n t P r e s p e c t i v e l y a t t h e m o m e n t s t a n d t ' , w e s h o u l d h a v e ( s ee

(3 .8 ) )

r r '

- = c t ' - e ( 4 . 3 1 )t

i .e ., t h e v e l o c i t y o f l i g h t in b o t h f r a m e s h a s t h e s a m e n u m e r i c a l v al u e w h e n

t i m e i n t h e s e f r a m e s is n o t t h e s a m e b u t is t o b e t r a n s f o r m e d a c c o r d i n g t o 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 s . H o w e v e r if w e s h o u l d m e a s u r e th e v e l o c i t y

o f l ig h t i n b o t h f r a m e s i n t h e same abso lu te t im e , w e h a v e t o w r i te

r r '

- = c, - Co (4 .32 )

t t




S u b s t i t u t i n g t f r o m t h e s e c o n d f o r m u l a ( 3 . 1 8 ) i n t o t h e s e c o n d f o r m u l a

( 4 . 3 2 ) , w e g e t , k e e p i n g i n m i n d t h e s e c o n d r e l a t i o n ( 4 . 3 1 ) ,

Co - 7 - T " ~ ! . ~ = c . v

1 + - - • cos O '1 + . t , • c 2 c

( 4 . 3 3 )

O b v i o u s l y , w e c a n n o t e x p r e s s t t h r o u g h T ' a nd -~ r, b u t 7 ' t h r o u g h F a n d t

a c c o r d i n g to t h e f o r m u l a i nv e r s e t o t h e f i r s t fo r m u l a ( 3 . 1 8 ) a n d u s e t h e f i r s t

r e l a t io n ( 4 . 3 1 ) . T h e c a l c u l a ti o n i n th i s c a s e i s ' m o r e c o m p l i c a t e d a n d w e o b t a i n

7"~ V1 - - - 1 - - - - c o s O

r ' X / (~ ' 2 )_ r . t . c 2 c

( 4 . 3 4 )

5 . S p a c e I n t e r v a l s in R e l a t i v i s t i c M e c h a n i c s

I n t h is s e c ti o n w e p r o c e e 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 a n d

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

t w o p o i n t s m o v i n g w i t h v e l o c i t y V t o g e t h e r , o r o n e w i t h r e s p e c t t o t h e o t h e r .

W h e n r e f e r r in g t o a ' r o d ' , w e w i l l a l so h a v e in m i n d t h e s e c o n d c a s e .

L e t u s c o n s id e r t h e p r o b l e m a b o u t t h e l e n g t h o f a ' r o d ' , u s in g f ir s t t h e

E i n s t e in i a n t i m e s y n c h r o n i s a t i o n . H e n c e t h e m e a s u r e m e n t o f t h e l e n g th o f a

' r o d ' is t o b e p e r f o r m e d b y s e n d i n g a l ig h t s ig n a l f r o m o n e o f it s e n d s t o t h e

o t h e r . T h e n t h e d i s ta n c e b e t w e e n b o t h s c o re s l e f t in t h e r e s t f l a m e K is to b e

m e a s u r e d a n d t h e r e a l l e n g t h ( d i s ta n c e ) , i f w e k n o w t h e r e l a t i o n V / c , ca l cu l a t ed .

L e t u s h av e a ' r o d ' w h i c h h a s a n a r b i t r a r y p o s i t i o n i n f l a m e K ' ( u s e F ig . 1 )

a n d l e t u s f i n d it s 't r a c k ' i n f ra m e K . W e p r o c e e d f r o m f o r m u l a e ( 3 . 1 0 ) a n d

( 3 . 1 ) a n d b u i l d t h e d i f f e re n c e s x 2 - x t , y 2 - Y l , Z2 - z l , w h e r e ( x l , Y l , z l ) a n d(X2 , Y2 , z 2 ) a r e t h e c o o r d i n a t e s o f t h e s c o r e s l e f t i n f r a m e K w h e n a li g h t si gn a l

is se n t a t th e m o m e n t t l ( t 'l ) f r o m o n e e n d o f t h e ' r o d ' t o i ts o t h e r e n d , w h i c h

is c o v e r e d b y t h e s ig n a l a t t h e m o m e n t t 2 ( t 2 ) . L e t us squa re t hese d i f f e r ences

a n d a d d , r e s p e c t i v e l y , t h e i r le f t a n d r i g h t s id e s. T a k i n g t h e s q u a r e r o o t f r o m

t h e e q u a t i o n o b t a i n e d , a n d s u b s t i t u t i n g t h e r e

y!r 1 t 2 P r r 0

t 2 - t ' l = - - . V / [ ( x ' 2 - x O + 0 , 2 - y ' l 1 2 + ( z 2 - z ' 0 2 1 = - = (5.1 /C C C

w e o b t a i n t h e f o l lo w i n g r e l a t io n ( c f . ( 4 . 2 1 ) )



E X P E R I M E N T A L V E R I F I C A T I O N O F A B S O L U T E S P A C E - T I M E T H E O R Y - - I 2 0 3

V n o . V1 + - - . c o s Oo 1 + - -

¢ ¢

r = r 0 = r 0

. j ( 1 - c _ ~ ) . j ( 1 _ c ~ ) ( 5 .2 ,

w h e r e w e h av e u s e d th e n o t a t i o n

x ~ - x______~~ 0'r ' = c o s = c o s 0 o ( 5 .3 )

a n d n ' = n o i s t h e u n i t v e c t o r p o i n t i n g f r o m t h e i n i ti a l to t h e f i n al e n d o f th e

' r o d ' i n f ra m e K ' .

I n t h e s a m e w a y , p r o c e e d i n g f r o m f o r m u l a ( 3 . 1 1 ) a n d t h e i n v er s e f o r m u l a e

( 3 . I ) a n d u s i n g t h e c o n d i t i o n

1 - - ( 5 . 4 )t2 - t l = - - . X / [ ( x 2 - - x l ) 2 + @ 2 - Y l ) 2 + (z 2 - z 0 2 1 = rc c

w e o b t a i n t h e f o l l o w i n g r e l a t io n ( c f . ( 4 . 2 1 ) )

V n . V1 - - - - . c o s 0 1 -

¢ ¢- r . ( 5 . 5 )r ° = r 'J ( 1 - - ~ ) ~ J ( 1 - _ ~ )

w h e r e w e h av e u s e d th e n o t a t i o n

x 2 - - x 1

r- - = c o s 0 ( 5 . 6 )

a n d n i s t h e u n i t v e c t o r p o i n t i n g f r o m t h e i n i t ia l t o th e f in a l e n d o f t h e ' t r a c k '

o f o u r ' r o d ' i n f r a m e K .

R e l a t i o n s ( 5 . 2 ) a n d ( 5 . 5 ) c a n a l s o b e o b t a i n e d , p r o c e e d i n g f r o m t h e f o r m u l a e

f o r t h e r e s t r i c te d L o r e n t z t r a n s f o r m a t i o n .

I n d e e d , p r o c e e d i n g f r o m t h e f i r s t f o r m u l a ( 3 . 1 8 ) , l e t u s b u i l d t h e d i f f e r e n c e

r2 - r~ , w h e r e r l a n d r 2 a r e th e r a d iu s v e c t o r s o f t h e s c o r e s l e f t in f r a m e K w h e na l ig h t s ig n a l is s e n t a t t h e m o m e n t t l ( t ' l) f r o m o n e e n d o f o u r ' r o d ' t o i ts

o t h e r e n d w h i c h is c o v e r e d b y t h e s i gn a l a t t h e m o m e n t t 2 ( t ~ ). S q u a r i n g b o t h

s id e s o f t h e e q u a t i o n o b t a i n e d , t a k i n g t h e s q u a r e r o o t , us i ng th e n o t a t i o n s

r2 - r l = r . n , r ~ - - r ' l = r ' . n ' = r o . n o ( 5 . 7 )

a n d i n t r o d u c i n g t h e c o n d i t i o n s

r r r r g Ot2 - t l = - , t2 - t ' l . . . . ( 5 . 8 )

c c c

w e o b t a i n f o r m u l a (5 . 2 ) .



2 0 4 S T E FA N M A R I N O V

I n a s i m i la r w a y , p r o c e e d i n g f r o m t h e i n v e rs e f o r m u l a t o ( 3 . 1 8 ) , w e c a n

o b t a i n f o r m u l a ( 5 . 5 ) .

W e s h al l n o w w r i t e f o r m u l a e ( 5 . 2 ) a n d ( 5 . 5 ) i n d i f fe r e n t n o t a t i o n s .

L e t u s c o n s i d e r a g a i n th e p o i n t q p r o c e e d i n g w i t h a n a r b i t r a ry v e l o c i t y v int h e r e s t f r a m e o f r e f e r e n c e K ( s e e F ig . 3 a n d c o m p a r e i t w i t h F i g . 2 ). W h e n q

c r o s s e s t h e s p a c e p o i n t Q ' a l i g h t s i gn a l ( a ' p h o t o n - r u n n e r ' ) i s s e n t t o w a r d s

p o i n t P , w h i c h w e s h a ll c al l t h e r e f e r e n c e p o i n t . T h i s li g h t s ig n a l, c o v e r i n g

d i s t a n c e r ' , r e a c h e s P a t t h e m o m e n t t , c a l l e d t h e o b s er v at io n m o m e n t , w h e n

q c r o s s e s p o i n t Q . A t t h is v e r y m o m e n t a l i g h t s i gn a l i s s e n t f r o m P w h i c h ,

c o v e r i n g d i s t a n c e r " , c a t c h e s q w h e n i t c r o s s e s p o i n t Q " .

T h e m o m e n tr F

t ' = t - - - ( 5 . 9 )C

y

Q o "

sst

p 7 ~

Fig u r e 3 . - A ' p h o to n - r u n n e r ' g o in g ' t h e r e ' an d ' b ack ' .

a t w h i c h a l i g h t s ig n a l is s e n t f r o m q w h e n i t c ro s s e s p o i n t Q ' i s c a l l e d t h e

a d va n ce d m o m e n t .

T h e m o m e n t

r tr

t " = t + - - ( 5 . 1 0 )c

a t w h i c h a l i g h t s i g na l s e n t f r o m P r e a c h e s q w h e n i t c ro s s e s p o i n t Q " , is c a l le d

t h e r e t a r d e d m o m e n t .

S l i gh t ly d i f f e r e n t v al u e s f o r t h e a d v a n c e d a n d r e t a r d e d m o m e n t s s h o u l d b e

o b t a i n e d i f i n f o r m u l a e ( 5 . 9 ) a n d ( 5 . 1 O ) w e w r i t e r o i n s t e a d o f r ' a n d r " .

W e c a l l r ' , r " , a n d r o , r e s p e c t i v e l y , t h e a d v a n c e d , r e t a r d e d , a n d o b s e r v a t i o n

d i s tances .W h e n c o m p a r i n g F i g. 3 w i t h F i g . 2 w e m u s t t a k e i n t o a c c o u n t t h a t t h e

t r i a n g l e Q o P o Q c o r r e s p o n d s e i t h e r t o t h e t r i a n g l e Q ' P Q o r t o t h e t r i a n g l e



E X PE R IME N T A L V E R I FI C A T I O N O F A B SO L U T E SPA C E -T I ME T H E O R Y - I 2 0 5

Q P Q " . a k e a l s o i n t o a c c o u n t t h a t i n F i g . 2 t h e r a d i u s v e c t o r s r o a n d r (i .e .,

t h e u n i t v e c t o rs n o a n d n ) p o i n t f r o m t h e r e st p o i n t P t o t h e m o v i n g p o i n t q ,

w h i l e in F i g . 3 t h e u n i t v e c t o r s n ' , n , a n d n " p o i n t f r o m t h e m o v i n g p o i n t q t o

t h e r e st p o i n t P . W e a l so s ee i m m e d i a t e l y t h a t i f t h e e m i s s io n m o m e n t i s t h ea d v a n c e d m o m e n t , t h e n t h e r e c e p t i o n m o m e n t i s t h e o b s e rv a t i o n m o m e n t , a n d

i f t h e e m i ss io n m o m e n t is t h e o b s e r v at i on m o m e n t , t h e n t h e r ec e p t io n m o m e n t

is t h e r e t a r d e d m o m e n t .

T h u s , w r i t i n g i n f o r m u l a ( 5 . 2 )

r = ro , ro = r ' ,

a n d w r i t in g i n f o r m u l a ( 5 . 5 )

r 0 = ro , r = r' ,

w e o b t a i n , r e s p e c t i v e l y ,

n o = - n ' , 0 o = 0 ' , V = v ( 5 . 1 1 )

11 I t

n = - n , 0 = 0 , V = v ( 5 . 1 2 )

n ~ V V

1 - 1 - - - . c o s 0 't C -r ~ C

r ° = r " I ( 1 - v~ _~ ) " ] ( i _ v ~ )

n " . v v1 + - - 1 + - - . c o s 0 "

. C -- r " C

r ° = r " I ( 1 - v ~ ) " I ( I - v~_~)

( 5 . 1 3 )

L e t u s n o w c o n s i d e r t h e p r o b l e m a b o u t t h e l e n g t h o f a ' r o d ' u s in g N e w t o n i a nti m e y n c h r o n is a tio n .e n c e t h e l e n g t h o f a g iv e n ' r o d ' m o v i n g i n f r a m e K w i t h

v e l o c i t y v i s t o b e e s t a b l i sh e d , r e g i st e r in g t h e s c o r e s w h i c h b o t h i ts e n d s l e a v e

i n f ra m e K a t a gi v en a b s o l u t e m o m e n t .

W e m u s t e m p h a s i s e t h a t a c c o r d i n g t o o u r a b s o l u te s p a c e- t im e c o n c e p t i o n ,

a t a gi ve n m o m e n t t h e " r o d ' h a s t h e s a m e e n g th n a n y f r a m e o f r e f e re n c e .

H o w e v e r , i f w e u s e N e w t o n i a n t i m e s y n c h r o n i s a t i o n i n th e L o r e n t z t r a n s fo r m a -

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

T h e ' m o m e n t a r y ' l e n g th o f a ' r o d ' w i ll b e c a ll e d th e d i s ta n c e e t w e e n b o t h

i t s e n d s a n d w i l l b e d e n o t e d

r = V l [ ( x 2 - - x l ) 2 + @ 2 - - Y l ) 2 + ( z 2 - - z t ) 21 ( 5 . 1 4 )

w h e r e ( x l , Y l , z O a n d ( x 2 , Y 2 , z 2 ) a re t h e c o o r d i n a t e s o f b o t h e n d s o f t h e ' r o d '

w h i c h a r e re g i st e re d i n f r a m e K a t t h e s a m e m o m e n t

t l = t 2 = t ( 5 . t 5 )

I f w e c o m p a r e ( 5 . 1 4 ) w i t h ( 5 .4 ) , w e m u s t t a k e i n t o a c c o u n t t h a t i n ( 5 . 4 ) r

is t h e ' t r a c k ' d i s t a n c e l e f t b y t h e ' r o d ' i n f ra m e K d u r i n g a d e f in i t e t i m e i n t e rv a l

t 2 - t l , w h i l e r in ( 5 . 1 4 ) i s a ' m o m e n t a r y t r a c k ' l e f t b y t h e ' r o d ' i n K a t a g i v en

i n s t a n t .




L e t u s n o w w r i te t h e l e n g t h o f th i s ' r o d ' i n f r a m e K ' . S u p p o s i n g t h a t t h e

' r o d ' r e s t s i n K ' , w e s h a ll h a v e

r ' = r 0 = ~ / [ ( x ~ - x ' l) 2 + ( y ~ - y ' x ) 2 + ( z ~ - z ' , ) 2 ] ( 5 . 1 6 )

w h e r e ( x a , Y l , z a ) a n d ( x 2 , Y 2 , z : ) a r e th e c o o r d i n a t e s o f b o t h e n d s o f t h e r o d

w h i c h a r e r e g t s te r e d i n f r a m e K . S i n c e t h e r o d r e s ts m K , t h e n t h e s e c o -

o r d in a t e s c o n c e r n e v e r y m o m e n t t h er e .

U s i n g fo r m u l a ( 3 . 1 1 ) a n d t h e i n v e r se f o r m u l a e ( 3 . 1 ) i n t o ( 5 . 1 6 ) , a n d

r e m e m b e r i n g c o n d i t i o n ( 5 . 1 5 ) , w e f i n d

1r o - j o - - ~ j ' J { ( x ~ - x ' ) Z + O - ~ ) ' [ ( Y 2 - Y ' ) ~ + ( z z - z 0 2 ]( 5 . 1 7 )

I f w e u s e h e r e n o t a t i o n s ( 5 . 6 ) a n d ( 5 . 1 4 ) w e g e t

1 - . i n 2 0

r 0 = r. '

w h e r e 0 i s t h e a n g l e b e t w e e n t h e l i n e a lo n g w h i c h t h e ' r o d ' li e s a n d i t s v e l o c i t y .

W e c a l l r o t h e proper distance o f t h e m o v i n g ' r o d ' .

F o r 0 = r r / 2 w e o b t a i n

r o = r ( 5 . 1 9 )

a n d f o r 0 = 0 w e o b t a i n

rr 0 : j ( ,A c c o r d i n g t o o u r a b s o l u t e c o n c e p t i o n t h e d if f e r e n c e b e t w e e n t h e d i s ta n c e

r a n d t h e p r o p e r d i s t a n c e r o i s not a re s u l t o f s o m e p h y s i c a l l e n g t h c o n t r a c t i o n

( c o m m o n l y c a l l e d th e L o r e n t z c o n t r a c t i o n ) . T h i s i s a r e s u lt o f th e i n t e r f e r e n c e

o f t h e t w o s l i g ht ly d i f fe r e n t m a t h e m a t i c a l a p p a r a t u s - t h e n o n - r el a ti v is t ic a n d

t h e r e l a t i v i s t i c .

I n d e e d , u s i n g F i g. 3 a n d p e r f o r m i n g a p u r e l y n o n - r e l a t iv i s t i c c a l c u l a t i o n , w e

s h a l l h a v e

r c . V . c o s 0 = . s i n 0 ( 5 .2 1 )

B u t a c c o r d i n g t o t h e l a w o f s in e s it is

r ~ r

- - - ( 5 . 2 2 )

s in (T r - 0 ) s in 0 '



EXPERIMENTAL VERIFI CATION OF ABSOLUTE SPACE-TIME TH EO RY -I 207

s o t h a t w e c a n w r i t e ( 5 .2 1 ) i n t h e f o r m

v 0~1 - - - - . c o sC

' ( 5 . 2 3 )= F .

T h i s fo r m u l a , a n d t h e f i r s t f o r m u l a ( 5 . 1 3 ) l e a d i m m e d i a t e l y t o r e l a t i o n

(5 .18 ) .

W e m u s t e m p h a s i s e t h a t w h e n w r i ti n g e q u a t i o n ( 5 . 2 1 ) w e h a v e a s s u m e d

t h a t t h e ' p h o t o n - r u n n e r ' c o v e r s t h e e m i s s i o n ( i. e ., t h e a d v a n c e d ) d i s ta n c e . I f

w e s h o u ld a s s u m e t h a t t h e ' p h o t o n - r u n n e r ' c o v e r s t h e r e c e p t i o n ( i .e . , ob s e rv a -

t i o n ) d i s t a n ce , a n o t h e r r e l a t i o n b e t w e e n r o a n d r c a n b e o b t a i n e d w h i c h w i ll

l ead to a ' l eng th d i l a t i on ' .

H e n c e w e m u s t t o o k t o t h e d i s ta n c e r a s a ' n o n - r e la t i v i s ti c ' o b s e r v a t i o n

d i s t ance and to t he p rop e r d i s t ance r0 as a ' r e l a t i v i s t i c ' obse rv a t ion d i s t ance .

T h e s e t w o d i s t a n c e s a r e c o n n e c t e d b y r e l a t io n ( 5 . 1 8 ) . T h i s p e r m a n e n t c o n t r a -

d i c t io n b e t w e e n t h e d i s ta n c e r a n d t h e p r o p e r d i s t a n c e ro a p p e a r s , n o t a s a

r e s u lt o f s o m e p e c u l i a r p r o p e r t y o f s p ac e a n d t i m e , w h i c h t h e t h e o r y o f re la -

t i v it y h a s t ri e d t o i n t r o d u c e i n t o p h y s i c s , d e s p it e t h e r e s is t a n ce o f th e h e a l t h y

h u m a n m i n d , b u t a s a r e su l t o f t h e f a c t t h a t i n n o n -r e l at iv i s ti c m e c h a n i c s w e

a s su m e t h a t b e t w e e n t h e m o m e n t s o f e m i s s io n an d r e c e p t i o n t h e ' p h o t o n -

runne r ' cove r s e i t he r t he emis s ion o r t he r ecep t ion d i s t ances , wh i l e i n r e l a -

t iv i st ic m e c h a n i c s w e a s s u m e t h a t t h e ' p h o t o n - r u n n e r ' c o v e r s t h e m i d d l e

d i s t ance . W e sha l l r epe a t ( s ee t he en d o f Sec t ion 4 ) , i n t he bas i s o f t h i s con t r a -d i c t i on l i e s t he abso lu t e t ime d i l a t i on .

6. Time Intervals

Let u s have (F ig . 4 ) tw o so -ca l l ed l i gh t c locks , one o f wh ich ( c lock A) i s a t

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

( c l o c k B ) p e r f o r m i n g a ro t a t io n a l m o t i o n i n s u c h a m a n n e r t h a t i t s ' a r m ' a lw a y s

r e m a i n s p e r p e n d i c u l a r t o t h e l in e a r v e l o c i t y o f r o t a t io n .

I f c l o c k s A a n d B h a v e th e s a m e ' a r m s ' t h e y w i ll g o e x a c t l y a t t h e s a m e

r a t e w h e n b e i n g a t r e s t, i .e ., t w o p h o t o n p a c k a g e s le f t t o g e t h e r , s a y , f r o m t h e i r

l e f t m i r r o r s , w i ll r e a c h t h e m i r r o r s a t t h e s a m e t i m e .

H o w e v e r , if c lo c k B p e r f o r m s t h e a b o v e - m e n t i o n e d r o t a t i o n a l m o t i o n , i t s

p h o t o n p a c k a g e w i ll a l w a y s a rr iv e , w i t h a s p e c i f ic t i m e d e l a y , l a t e r t h a n t h e

c o r r es p o n d i n g p h o t o n p a c k a g e i n c l o c k A . I n d e e d , t h e p h o t o n p a c k a g e s h a v e

t o c o v e r t h e d i s ta n c e 2 . r o b e t w e e n t w o r e f l e c t io n s i n c lo c k A a n d t h e d i s ta n c e

2. r 02 . r -

i n c l o c k B , w h e r e ro i s t h e l e n g t h o f t h e l i g h t c l o c k ' s ' a r m ' .

( 6 . 1 )



2 0 8 STEFAN MARINOV

T h i s f o r m u l a c a n b e o b t a i n e d f r o m t h e f i r st f o r m u l a o f (4 . 2 1 ) - - o r f r o m

form ulae (4 .8 ) an d (4 .1 0 ) - - a t t he con d i t i on 0o = ~ r/2, a s we l l a s f ro m the se cond

f o rm u l a o f ( 4 . 2 1 ) - o r f r o m f o r m u la e (4 . 9 ) a nd ( 4 . 1 1 ) - a t t h e c o n d it io n

c o s 0 = v /c ( see Fig . 4) .T h u s i f w e c h o o s e t h e u n i t o f t im e a s t h e t i m e b e t w e e n t w o s u c c es s iv e

r e f le c t i o n s o f a p h o t o n p a c k a g e i n a l ig h t c l o c k w i t h a g iv e n ' a r m ' ro , th e n t h e

l igh t c lock A wi l l have

2 . r ono = (6 .2 )

c

a b s o l u t e s e c o n d s i n a u n i t o f t im e .

T h e l i g h t c l o c k B w i ll a ls o h a v e n o a b s o l u t e s e c o n d s i n a u n i t o f t i m e w h e n

b e i n g a t r e s t a n d

2 . r 2 . t o

n = e - e . J ( 1 _ Ve~) ( 6 . 3 )

a b s o l u t e s e c o n d s in a u n i t o f t i m e w h e n b e i n g in m o t i o n .

F r o m ( 6 . 2 ) a n d ( 6 . 3 ) w e d r a w t h e c o n c l u s i o n t h a t c l o c k B g o e s a t a sl o w e r

ra t e and i f , f o r a ce r t a in ab so lu t e t im e in t e rva l, s ay fo r one r evo lu t ion o f c lock

B , t h e r e a d i n g o f c l o c k A i s t A - t im e - u n i t s , t h e n t h e r e a d in g o f c l o c k B w i ll b e

j r 1o = t . - ( 6 .4 )

B- t im e-un i t s , s i nce i t i s t o / t = n o / n . We cal l t t im e in t erv a l a n d t o p r o p e r t i m ein terva l .

T h i s d e d u c t i o n o f t h e t i m e d i l a ti o n h a s a n e n t i r e l y n o n - re l a ti v i st ic c h a r a c t e r .

I t is cl e a r f r o m F ig . 4 t h a t c l o c k B m o v e s w i t h r e s p e c t t o a b s o l u t e s p a c e a n d

c l o c k A is a t r e s t. I n t h e o p p o s i t e c a s e w e h a v e t o a s s u m e t h a t t h e w h o l e w o r l d

r o t a t e s a b o u t c l o c k B ; o b v i o u s l y , th i s i s n o n s e n s e .

I n F ig . 4 t h e m o t i o n o f c l o c k B i s n o n - i n e rt i al d u r i n g t h e w h o l e p e r i o d o f

s e p a r a t i o n f r o m c l o c k A . W e s h al l n o w s h o w t h a t w e w i ll a ls o o b t a i n t h e s a m ee f f e c t o f t im e d i l a t i o n w h e n t h e m o t i o n o f c l o c k B i s i n e rt ia l d u r i n g t h e p r e -

d o m i n a n t p a r t o f t h e s e p a r a t i o n ti m e .

Indee d , l e t u s have (F ig . 5 ) a l i gh t c lock A , w h ich i s a t r e s t i n abso lu t e space ,

a n d a n i d e n t i c a l li g h t c l o c k B w h i c h p a s s e s n e a r i t ( a t p o i n t b ) w i t h v e l o c i t y v.

U n t i l t h e p o i n t b ' t h e l i g h t c l o c k B m o v e s i n e r ti a l ly w i t h t h e s a m e v e l o c i t y v .

F r o m p o i n t b ' t o p o i n t b " i t s v e l o c i t y r e d u c e s to z e r o , a n d f r o m p o i n t b " t o

po in t b ' i t s ve loc i ty i nc reases aga in t o v how ever op po s i t e ly d i r ec t ed . C lock B

t h e n b e g i n s t o m o v e i n e r t i a ll y a n d , w i t h t h i s v e l o c i t y v , a g a in p a s se s n e a r

c l o c k A .N o w a s s u m i n g t h a t t h e t i m e o f n o n -i n e r ti a l m o t i o n i s i n s ig n i f ic a n t ly s h o r t

w i t h r e s p e c t t o t h e t i m e o f it s i n e rt ia l m o t i o n , w e c a n o b t a i n , i n a p u r e l y n o n -




II

//

>

.<

Q

O

o =

3

"6

Y~

O

b~

,4

Y.




A B

I

iIIb

v

b' b"

Figure 5.-Two light clocks, the second of which performs a 'there and back' motion.

relativistic way, the relation (6.4) between the time reading to of clock B and

the time reading t of clock A for the whole time of separation.

Here again clock B is in motion; in the opposite case we have to assume

that when the mutual velocity of clocks A and B change, then clock B would

not change its velocity with respect to the whole world but the whole world

would have to change its velocity with respect to clock B; again this is nonsense.

Until now we have supposed that the 'arm' of the moving light clock B is

always perpendicular to its velocity. Now we shall show that the same result

could be obtained if we assume that the 'arm' o f clock B is parallel to its

velocity.

Indeed, in such a case the photon packages have to cover the distance 2. ro

between two successive reflections on the same mirror in clock A and thedistance

/3 /3

1 + - - 1 - - -c c 2. ro

2 . r= r o . -r o (6.5)

in clock B. This result can be obtained f rom formulae (4.21) under the condi-

tions 0o = 0 = 0 and 0o = 0 = rr.The non-relativistic relations (4.8), (4.9), (4.10), and (4.11) lead to two

formula with different terms of second order in v/c, whose geometrical mean

gives the result (6.5).

Thus we have shown that any light clock moving arbitrarily with respect to

absolute space goes at a slower rate than an identical light clock which rests

in absolute space; the relation between their readings for a definite absolute

interval of time is given by formula (6.4).

We can generalise this conclusion and assume that the time of any clock

(i.e., of any material system) which moves with respect to absolute space

advances with a slower rate than absolute time. We suppose that this close

connection between the light clock and any other clock (i.e., any other



E X P E R I M E N T A L V E R I F I C A T I O N O F A B S O L U T E S P A C E -T I M E T H E O R Y - I 2 1 1

per i od i c p roces s ) is due t o t he em pi r ica l f ac t t ha t t he ve l oc i t y o f li gh t is a

un i ve rsa l con s t an t w h i ch g ives t he num er i ca l t ie be t w een space and t i me .

We r ecord he r e t ha t t he r e su l t s ob t a i ne d i n t h is s ec t i on i mm edi a t e l y g ive

t h e e x p l a n a t i o n o f t h e h i s t o ri c al M i c h e ls o n - M o r l e y e x p e r i m e n t .I ndeed , i f t he l eng t hs o f the m ut ua l l y pe rpend i cu l a r ' a rms ' i n t he Mi che l son

i n t e r f e r o m e t e r a re r o a n d R o , t h e n t h e a b s o l u t e t i m e i n te r v al s sp e n t b y t w o

ph o t o n packages t o cove r t hese ' a rms ' t he r e an d back w i ll be

2 . ro 2 . R o

Th e co r r e spon d i ng p ro pe r t i m e i n te rva ls , i .e . , those w hi ch w il l be r ead on a

c l o c k a t t a c h e d t o t h e i n t e r f e r o m e t e r , w il l b e ( s e e (6 . 4 ) )

2 . r o 2 . R oA t o = , A To = (6 . 7 )

C C

Fo r t he i r d i f f e r ence , w h i ch calls f o r t h an e ven t ua l sh i f t i n t he i n t e r f e r e nce

f ri ng e s w h e n r o t a t i n g t h e i n t e r f e r o m e t e r w i th r e s p e c t t o t h e a b s o l u te v e l o c i t y

v o f t h e i n t e r f e r o m e t e r o r w h e n c h a n g in g t h e v e l o c i t y v , w e o b t a i n

2A t o - A To = - - . ( r o - R o) = cons t . ( 6 . 8 )

C

H e n c e , n o t o n l y 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 ( w h e r e ro = R o ) , b u t a ls ot h e K e n n e d y - T h o r n d i k e e x p e r i m e n t ( w h e r e ro ~ R o ) m u s t g iv e z e r o r e su l ts , a s

w as p r ac t i ca l l y obse rved .

7. S ome Results

W i t h t h e h e l p o f f o r m u l a e ( 5 . 1 3 ) w e c a n i m m e d i a t e ly o b t a i n e x p r e ss io n s f o r

t he so - cal led L i ~na rd -Wi eche r t po t en t i a l s .

I n d e e d , a c c o r d i n g t o o u r a b s o l u t e c o n c e p t i o n , t h e e l e c t ro m a g n e t i c 4 - p o t en -

t ia l o f a po i n t cha rge q a t a r e f e r en ce po i n t d i s t an t r ( s ee fo rm ul a (5 . t 4 ) ) f r om

it is

+->+-> q Vo

C " g o

t - 5

( 7 . 1 )

( 7 . 2 )

is t he p rop e r 4 -ve l oc i t y o f t he cha rge , ~ is it s ve l oc i t y a t a g i ven m om en t o f

o b s e r v a ti o n a n d r o (s e e f o r m u l a ( 5 . 1 7 ) ) is t h e p r o p e r m o m e n t o f o b s e r v a t io n

and ro ( s ee fo rm ul a (5 . 17 ) ) i s t he p rop e r d i s t ance be t w ee n cha rge and r e f e r enc e




p o i n t a t t h is v e r y m o m e n t ; i is t h e i m a g i n a r y u n i t ; w i t h t h e s ig n ~ w e d e n o t e a

4 - v e c t o r , w i t h t h e s ig n ~ i t s s p a c e p a r t a n d w i t h t h e s i g n - i ts t i m e p a r t .

S u b s t i t u t i n g f o r m u l a e ( 5 . 1 3 ) i n t o ( 7 . 1 ) , w e o b t a i n t h e e l e c t r o m a g n e t i c 4 -

p o t e n t i a l o f th e c h a r g e q i n t h e f o r m o f L i ~ n a r d - W i e c h e r t+-~

= q . v ~ = q v ( 7 . 3 )

C t ~/r ~ C . n . V

r . 1 - r . 1+

w h e r e v = (v , i . c ) i s t h e 4 - v e l o c i t y o f t h e c h a r g e , r ' is th e a d v a n c e d d i s t a n c e

a n d r" i s t h e r e t a r d e d d i s t a n c e .

I n 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 e d o n o t i n t r o d u c e d r a s ti c d i f fe r e n c e s

b e t w e e n e l e c t r i c i ty a n d g r a v i ta t io n . A l l f o r m u l a e w i t h w h i c h w e w o r k a r e

i d e n t i c a l i f t h e e l e c t r i c c h a r g e s a re r e p l a c e d b y m a s s e s a n d t h e i n v e r s e e l e c t r i c

c o n s t a n t ( w h i c h i n t h e s y s t e m C G S i s e q u a l t o u n i t y ) b y t h e g r a v i t at i o n al c o n -s t a n t t a k e n w i t h a n e g a t i v e s i g n .

H e n c e o n t h e b a s i s o f ( 7 . 1 ) w e o b t a i n t h a t t h e g r a v i ta t io n a l p o t e n t i a l o f a

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

d i s t a n t r f r o m i t, is t o b e p r e s e n t e d i n t h e f o r m

k 2 . m

r o . 1 -

w h e r e k 2 i s t h e g r a v i t a t i o n a l c o n s t a n t a n d r o th e p r o p e r d i s t a n c e .

T h e g r a v i ta t i o n al e n e r g y o f m a s s m a n d a m a s s M w h i c h r e st s a t th e r e f e r e n c e

p o i n t w i l t b e U = M . ¢ . U s i n g t h is f o r m f o r t h e g r a v i t a t i o n a l e n e r g y w e o b t a i n

( in P a r t I V o f o u r m a n u s c r i p t ( M a r i n o v , in p r e p a r a t i o n ) d e d i c a t e d t o g r a v it a-

t i o n ) :

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

h a l f o f t h e r e s u l t g i v e n b y g e n e r a l r e l a t iv i t y .

( b ) F o r t h e a n g u l a r d e f l e c t i o n o f a l i g h t b e a m p a s s i n g n e a r t h e s u n a r e s u lt

w h i c h r e p r e s e n t s h a l f o f t h e r e s u l t g iv e n b y g e n e r a l r e la t i v i ty .

( c ) F o r t h e g r a v i t a t io n a l f r e q u e n c y s h i f t ( t h e s o - c a l l e d ' r e d s h i f t ' ) a r e s u l t

w h i c h i s th e s a m e a s t h a t g i v en b y g e n e r a l r e l a t iv i t y .

I n o u r o p i n i o n t h e e x p e r i m e n t a l c h e c k o f t h e f i r s t t w o r e s u l t s i s n o t s u f f i ci -e n t l y r e l i a b le , s o i t i s i m p o s s i b l e t o d e c i d e w h o s e p r e d i c t i o n s b e s t c o r r e s p o n d

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

c o n s i d e r o n l y t h e ' c o u p l e d - m i r r o r s ' e x p e r i m e n t ( M a r i n o v , 1 9 7 5 ).

Re f e r ences

M arinov , S. (1972) . Physics Letters, 4 1 A , 4 3 3 .Mar inov , S. (1975) . Czechoslovak Journa l of P hysics, B24, 965 .M ar inov , S. Classica l physics , in p repa ra t ion .

1974.02.08 - stefan marinov - the fundamental conception of the absolute space-time theory

Documents