Cosmological Origin of Irreversibility, Time, And Time Anisotropies

7/30/2019 Cosmological Origin of Irreversibility, Time, And Time Anisotropies

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Foundations o f Phy sics , Vol . 6 , N o. 4 , 1976

C osm ological O rigin o f Irreversibi li ty , T ime, and

Time Anisotropies . I

B e n j am i n G a l - O r 1

R e c e i v e d M a y 2 3 , 1 9 7 5

Cau sal l inks am ong the thermo dynam ic, e lectrodynamie, and cosm ological

arrows o f t ime are exp la ined w i th in the f ram ew ork o f a new theory derived

fro m New ton ian grav ita t ion or genera l relat iv is tic theory. The mas ter asym -

m etry so derived i s emp loyed to deduce the second pos tu la te o f thermodynamics

in terms of d iss ipation fun ction or entropic grow th. Discussing Olbers ' para dox

and em ploying a "laborator y-universe principle of equivalence," the theory

dem onstrates h ow the expansion o o u r isotropie universe af fects a l l irreversible

processes on earth . Gravitation and the observed expanding space becom e theindirect causes o f therm al gradients an d irreversible processes channeling

energy fr om planetary and gala ct ic cores , through colder regions, a l l the w ay

into the unsaturable s ink o f expanding in tergalactic space. The new form ula-

t ions rep lace the ax iomat ic forma l i sm s o f c lass ica l and con t inuum thermo-

dynamics . The funda me nta l ro le p layed by the expans ion (bu lk ) v i scos ity i s

s tressed . Other poss ib le in terac t ions amo ng cosmology , thermodyna mics , and

e l e ct r o d yn a mic s a r e r e v ie we d a n d a n a l y z e d f r o m a n e w v i ewp o in t.

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

T r a d i t i o n a l l y s c ie n ti st s d i s ti n g u is h a m o n g a n u m b e r o f t i m e a s y m m e t r i e s ,

n o t a b l y a m o n g t h e t h e r m o d y n a m i c , e l e c tr o m a g n e t i c , c o s m o l o g ic a l , b i o -

l o g ic a l, a n d ( m o r e r e c e n tl y ) t h e m i c r o s c o p i c a r r o w s o f t im e . T h i s s e p a r a t i o n

n o t o n l y b r e a k s u p t h e f o u n d a t i o n s o f p h y s i c s i n t o d i s j o in t e d c h a n n e l s , b u t

i t h a s c r e a t e d a s e r i o u s c r is is a t t h e b a s i s o f m o s t o f o u r t h e o r i e s . (1 -9 ) T h i s

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

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

q u e s t io n s , a s a m p l i n g o f w h i c h f o l lo w s :

1 De.partment of A eronautical Engineering, Techn ion-lsrael Institute of Technology,Haifa, Israel.

407

© 1976 Plen um Publis hing Corporation, 227 West 17th Street , Ne w York, N.Y. 10011. N'o par t of thispubli,sation 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 ofthe publisher.



408 Gal -Or

1. A r e t h e o r i g in s o f t h e r m o d y n a m i c , e le c t r o m a g n e t i c , a n d m i c r o -

s c o p i c i r re v e r si b il it ie s k n o w n ? C a n t h e y b e p h y s i c a l l y l i n k e d w i t h g r a v i t a t i o n

a n d c o s m o l o g y , a n d d e d u c e d f r o m g e n e r a l re l a t iv i t y ?

2 . I s t h e s e p a r a t io n a m o n g th e c a u se s o f t h e a f o r e m e n t i o n e d t im e

a s y m m e t r ie s j u s t if i e d ? O r a r e t h e y n o m o r e t h a n d i ff e r e nt m a n i f e st a t io n s o f

o n e a n d t h e s a m e e f f e c t ? I .e . , i n w h a t s e n se m i g h t th e s e a r r o w s b e p h y s i c a l l y

c o r r e la t e d w i t h e a c h o t h e r a n d w i t h a s in g le " m a s t e r a s y m m e t r y " ? W h a t a r e

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

poss ibi l i t ies ?

3 . Ca n i r r eve r s ib i li t y (o r t he es t ab l i shed s t a t i s ti ca l l aw o f i nc rease o f

e n t r o p y , " H - t h e o r e m s , " " m i x i n g , " " M a r k o v p r o c e s s e s , " e t c . ) b e d e d u c e d

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

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

s ib i li ty s m u g g l e d i n t o t h e v a r i o u s t h e o r ie s o f s t a t i s ti c a l m e c h a n i c s ? I s i t a n

a c c i d e n t t h a t n o o n e h a s f o u n d a p r o o f o f ir re v e rs ib i li ty b y u s in g H - t h e o r e m s ,

e t c ., a f t e r a l m o s t a c e n t u r y o f tr y i n g ? I f it t u r n s o u t ( as w e s h o w i n P a r t I I o f

t h i s w o r k ) t h a t s t a t i s t i c a l m e c h a n i c s , w i t h a l l i t s p o w e r f u l a p p l i c a t i o n s , h a s

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

a b o u t i ts c l a im t o f u n d a m e n t a l i t y ?

I n t r y i n g t o a n s w e r t h e s e q u e s ti o n s , I h a v e f o u n d i t u s e f u l t o d e m o n s t r a t ef ir st th e i n d i re c t l in k s b e t w e e n c o s m o l o g y a n d t h e r m o d y n a m i c s . T h e se l i n k s

a r e t o b e i l lu s t r a t e d b y o u r l a b o r a t o r y - u n i v e r s e p r i n c i p le o f e q u i v a l e n c e ,

w h i c h i s d e s c r ib e d b e l o w .

2 . L A B O R A T O R Y - U N I V E R S E P R I N C I P L E O F E Q U I V A L E N C E

T h e p r i n c i p l e to b e d e s c r i b e d h e r e is b a s e d o n t w o s e t s o f o b s e r v a t i o n s ,b o t h o f w h i c h h a v e b e e n t h o r o u g h l y c o n f i r m e d b y i n d e p e n d e n t m e th od s.(1 °-1 4)

T h e s e a r e :

( i) Th e un ive r se acces s ib le t o ou r obs e rva t i on s con t a ins a ve ry l a rge

n u m b e r o f s e l f- g e n e r a ti n g g a l a c ti c e n e r g y s o u r c e s , w h i c h , o n s u f f ic i e n tl y l a rg e

s ca le , a re h o m o g e n e o u s l y ( t h o u g h r a n d o m l y ) a n d i s o tr o p i c a ll y d i s t r i b u te d

( the cosmolog i ca l p r i nc ip l e ) .

( ii) Th e un ive r se acces s ib le to o u r o bse rv a t i ons i s evo lv ing (i .e ., t he

o b s e r v e d i s o t r o p i c r e c e s s i o n i n t h e s y s t e m s o f g a l a x i e s a n d s u p e r g a l a x i e s ) .T h e s e o b s e r v a t i o n s v e r if y t h e t h e o r e t ic a l p r e d ic t io n s o f N e w t o n i a n t h e o r y o f

g r a v i t a t i o n a n d o f E i n s t e i n ' s g e n e r a l r e l a t i v i t y , n a m e l y , t h a t t h e u n i v e r s e

e x p a n d s .



Co sm ologica l Orig in of Irreversibil ity , Tim e, and Tim e Anisotropies . I 409

T h e s e c o n c l u s i o n s , a n d o t h e r s t h a t f o l l o w , l a r g e l y r e l y o n r e c e n t d i s -

c o v e r ie s t h a t i n c l u d e t h e d i s c o v e r y o f c o s m i c m i c r o w a v e b a c k g r o u n d

r a d i a t i o n ( w h i c h is i s o t r o p i c t o w i t h i n l es s t h a n 0 . 1 5 ~ a n d is a r e li c

o f e a rl ie r , h o t t e r e p o c h s ) , o n t h e e x t e n d e d H u b b l e r e la t i o n , o n r a d i o s o u r c e

c o u n t s , a n d o n ve r if ie d t r a n s f o r m a t i o n s o f e n e r g y i n s t el la r c o r e s b y

n u c l e o s y n t h e s i s .

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

t h e d o m a i n o f N e w t o n i a n t h e o r y . I t is, h o w e v e r , o n l y a n a n a lo g . I m a g i n e a

l a rg e n u m b e r o f u n i f o r m l y d i s t r ib u t e d m a s s es o f a b o u t e q u a l s iz e, e ac h w i th

i ts o w n s e l f -g e n e r a t i n g e n e r g y s o u r c e , l o c a t e d i n s id e a r i g id (nonexpanding)

l a b o r a t o r y r o o m w i t h r e f le c t i n g ( i s o la t e d ) w a ll s. I t is e v i d e n t f r o m o u r

o b s e r v a t i o n s t h a t a f t e r a fi n it e t i m e ( d e p e n d i n g o n t h e s iz e o f t h e l a b o r a t o r ya n d o n d e t a il s o f t h e m a s s e s ) , a t h e r m a l e q u i l i b r i u m w i ll b e e s t a b l i s h e d

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

t e m p e r a t u r e g r a d i e n t s w i l l v a n i s h .

B y c o n t r a s t, a l a b o r a t o r y r o o m w i t h f le x ib l e w a ll s, c o n t i n u o u s l y m a i n -

t a ine d in a s t a t e o f su f fi c i e n t ly f a s t expansion, c a n n o t a c h i e v e e q u i l i b r i u m

( s in c e th e d e n s i t y o f r a d i a t i o n n e a r t h e e m i t t e rs d e c r e a s e s w i t h t i m e ) , t h e

r e s u l t b e i n g o n e - d i r e c t i o n a l , u n r e l a x e d , n e t e n e r g y f lu x e s f r o m e m i t t e rs t o

t h e u n s a t u r a b l e s u r r o u n d i n g s . T h i s c a u s e - a n d - e f f e ct l i n k is indirect a n d

i n v o l v e s n o a c t i o n a t a d i s t a n c e , s o t h a t i f e x p a n s i o n c e as e s, b o t h t h e f l u x e sa n d t h e g r a d i e n t s p e r s i s t, a l b e i t o n l y f o r a f in i te re l a x a t i o n t i m e d u r i n g w h i c h

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

w i t h d e c r e a s i n g n u m b e r o f s o u r c e s) . S i n c e t h e d i s t r i b u t i o n o f t h e s o u r c e s is

isotropic ( t h o u g h r a n d o m ) , a static r o o m o f a s t r o n o m i c a l s iz e w o u l d r e a c h

e q u i l i b r i u m a t a l l i ts p a r t s a t a b o u t t h e s a m e f in i te ti m e , p r o v i d e d a l l s o u r c e s

h a v e t h e s a m e m a x i m u m t e m p e r a t u r e .

S i n c e e n e r g y a l w a y s f lo w s i n o p p o s i t e d i r e c t i o n s a l o n g a l i n e o f si g h t

c o n n e c t i n g a n y t w o n e i g h b o r i n g s o u r c e s , t h e r e e x i s t s a p o i n t o n t h i s l i n e a t

w h i c h t h e n e t f l u x v a n i s h e s a n d e a c h s o u r c e i s e n c l o s e d i n a n i m a g i n a r yadiabatic envelope c o n s i s t i n g o f a n i n f in i t y o f s u c h p o i n t s . E a c h s u c h a d i a -

b a t i c cell c a n n o w b e c o n s i d e r e d a s a n e x p a n d i n g r o o m , equivalent t o t h e

o r i g in a l r o o m b u t o f s m a l le r si ze , a n d i ts e v o l u t i o n a w a y f r o m o r t o w a r d

e q u i l i b r i u m w o u l d b e a l s o equivalent t o t h a t o f i ts o r i g i n a l c o u n t e r p a r t .

E x t r a p o l a t i n g , a c c o r d i n g l y , t o a n y a s t r o n o m i c a l s c a l e , e a c h s o u r c e a s s u m e s

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

the : s t a t i s t i c a l ly a v e r a g e d cell vo lum e pe r supergalaxy .

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

bounde d ( i . e . , c lo se d ) o r i n f in i t e ( i . e . , ope n ) , s inc e f o r bo th c a se s the a ve r a gee v o l u t i o n o b s e r v e d i n s id e a t y p i c a l c el l i s e q u i v a l e n t t o t h a t o f th e " l a r g e r

l a b o r a t o r y " a n d w o u l d s u f fi ce f o r r e a c h i n g the same g e n e r a l c o n c l u s i o n

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



4 1 0 G a l - O r

h a s not b e e n static f o r a l o n g t i m e ) C h a n g i n g t h e s h a p e s o f t h e s o u rc e s ,

r e s o l v i n g t h e m i n t o b i l l io n s o f s m a l le r u n i ts , r o t a t i n g a p a r t o f o r th e e n t i r e

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

c o s m i c r a y s , e l e c t r o m a g n e t i c w a v e s , e l e c t r o m a g n e t i c d i ss i p a t i o n , e t c . - - a l l

t he se , i n t h em s e l ve s , c a n n o t c h a n g e t h e a f o r e m e n t i o n e d c o n c l u s i o n , w h i c h i s

a ls o s u p p o r t e d b y a d d i t io n a l a n d i n d e p e n d e n t o b s e r v a t i o n s , m -1 4) F o r

i n s ta n c e , t h i s ef fe c t is c o n f i r m e d b y o b s e r v a t i o n s i n si d e a n d n e a r o u r o w n

g a la x y . F u r t h e r m o r e , t h e r o t a t i o n a l m o t i o n o f n e w l y c re a t e d n e u t r o n s ta r s

is f o u n d t o b e s u b j ec t to e l e c t ro m a g n e t i c " b r a k i n g , " w i t h m e c h a n i c a l e n e r g y

d i s s ip a t e d i n t o r a d i a n t e n e r g y i n t h e c o r o n a o f t h e s ta r a n d e v e n t u a l l y l o s t i n

s p a c e n 3) ( s e e b e l o w ) .

O b s e r v e d m -x 4~ e l e c t ro m a g n e t i c e n e r g y e m i s s i o n s b y s u p e r n o v a r e m -n a n t s a l s o s u g g e s t t h a t s i m i l a r d i s s i p a t i o n e f fe c ts a r e d u e t o f lu i d a n d r a d i a t i o n

s t r e s s e s , m a g n e t i c f i e l d s , a n d t u r b u l e n t v i s c o s i t i e s . E v e n r e c o u r s e t o c u r v e d

s p a c e a n d d i s t r ib u t i o n o f t h e g a l a x ie s a c c o r d i n g t o s iz e a n d t y p e l e av e

t h e g e n e r a l r e su l t u n c h a n g e d i n s o f a r as t h e " l a b o r a t o r y " a c c es s ib l e t o o u r

o b s e r v a t i o n s e x p a n d s a n d r e m a i n s i s o t r o p i c .

R e c e n t i n v e s t i g a t i o n s (15,aG) o f a p o s s i b l e i n i t i a l c o s m i c a n i s o t r o p i c

e v o l u t io n h a v e s h o w n t h a t i t m u s t r a p i d ly v a n i s h b y d is s ip a t i o n . S o m e o f

t h e se c o m p l e x i t i e s m a y c a u s e l o c a l d i s t o r t io n s o r e v e n a n e t fl o w o f e n e r g y

f r o m a t o o s m a l l ce ll t o a n o t h e r , t h e r e b y f o r c i n g u s t o r e s o r t t o a l a r g e r c el lu n t i l i s o t r o p y h a s b e e n r e s t o r e d ?

S u m m a r i z i n g t h e d e d u c t i o n s f r o m o u r p r in c ip l e, w e a r r iv e a t th e

f o l lo w i n g a d d i t i o n a l c o n c l u s i o n s :

1 . T h e o b s e r v e d e x p a n s i o n c a u s e s s p a c e t o a c t a s a n unsaturable

t h e r m o d y n a m i c s in k .

2 . T h e origin o f ir r ev e r s ib i li ty c a n b e t r a c e d a ll t h e w a y b a c k i n t im e

a n d o u t i n to expanding space.

2 W heth er the process consists in "real i ty " o f expansion or c on tract ion is essential ly a

matter for the defini t ion of t ime (see below). What is important here is the negative

conclusion.3 Black holes, i f verif ied, prov ide on ly add it ional sinks to the already unsa turab le sink

o f exp and ing interstel lar space. Re cen t observations strong ly indicate, however, th atblack holes p lay a m ore im por tan t role in the generat ion o f ir revers ibi li ty in o ur universe

by supplying a sink which remains unsaturated even if the expansion reverses into acontract ion (see also the sect ion on the origin of t ime for a discussion of the effect of

cont ract ion on our theory) .4 G old 's s tar in a ( stat ic) b ox m odeYm has only a weak conn ect ion wi th the cosmological

ar row and m ay s t il l l ead to sy mm etr ic bou nd ary condit ions . This res tr ic tion i s obviated

in the present laboratory-cell-universe equivalence, which leads to a supergalaxy in an

expanding adiabatic envelope.



Co sm ologica l Orig in o f Irreversibi li ty , Tim e, and Tim e Anisotropies . I 411

3. The first cosmological time asymmetry ( a c c o r d i n g t o o u r s c h o o l , t h e

master asymmetry) i s t h a t a s s o c i a t e d w i t h th e e x p a n s i o n o f t h e u n iv e r s e

p r o p e r . I f , f o r i n s t a n c e , w e p h o t o g r a p h a g r o u p o f s u p e r g a l a x i e s a t i n s t a n t s

t l a n d t 2 , t h e i r d if f e r e n t m u t u a l s e p a r a t i o n R 1 a n d R ~ w o u l d t e l l u s w h e t h e r

t l > t2 o r t 2 > t l . T h i s a r r o w o f ti m e is i n d e p e n d e n t o f a n y c o s m o l o g i c a l

m o d e l ( i n c l u d i n g t h e s t e a d y - s t a t e m o d e l , w h i c h a l s o t a k e s i t i n t o a c c o u n t ) .

F o r t i m e - d e p e n d e n t m o d e l s , w e d e f i n e t h e d e c r e a s i n g d e n s i t y o f r a d i a t i o n

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

space as t he third cosmological time asymmetry. (2~)

4 . T h e s e c o n d a n d t h i r d t im e a s y m m e t r i e s a r e ( i n d e p e n d e n t l y ) c r e a t e d

a n d d o m i n a t e d b y th e m a s t e r a s y m m e t r y . T o t h e b e s t o f o u r k n o w l e d g e ,

rever se li nks a re phy s i ca l l y impo ss ib l e , i .e . , no phy s i ca l t heo ry based ono b s e r v a t io n s a n d p r e s e n t i n f o r m a t i o n c a n b e d e v is e d in w h i c h t h e o b s e r ve d

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

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

ga l a t i c sou rces t o co ld space .

5 . T h e v e r y e x i st e n c e o f a u n iv e r s a l s i n k ( w h i c h r e m a i n s u n s a t u r a t e d

a s l o n g a s s p a c e k e e p s e x p a n d i n g ) i s t h e indirect c a u s e o f t h e o b s e r v e d ,

u n l i m i t e d , o n e - d i r e c t i o n a l e n e r g y l o ss e s f r o m t h e s u r f a c e s o f a ll g a l a c t ic s t a rs

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

( se e b e l o w ) o f e n e r g y - d e n s i t y a n d t e m p e r a t u r e g r a d i e n t s o r i g i n a t i n g a t t h ec o r e s o f g a l a t ic e m i t te r s ( w h e r e , p r o v i d e d t h e t e m p e r a t u r e i s s u ff ic i e n tl y h i g h ,

n u c l e a r r e a c t i o n s g e n e r a t e th e e n e r g y to b e t r a n s p o r t e d t o o u t e r r e g i o n s b y

r a d i a t i o n , c o n v e c t i o n , c o n d u c t i o n , e t c . ) , a n d e x t e n d i n g t h r o u g h t h e o u t e r

l a y e r s o f th e v a r i o u s p l a n e t s , st a rs , a n d n e b u l a e a n d t h r o u g h o u t t h e g a l a x y

i tself .

6 . T h e s e c a u s a l l in k s a r e i n d e p e n d l y s u p p o r t e d b y b o t h m a c r o c o s m i c

o b s e r v a t i o n s a n d p r e d i c t i o n s o f g e n e r a l r e la t i v it y , a s w e l l a s b y t h e t h e o r ie s o f

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

s t a b il i ty . T h e r e f o r e , i f r e l a t iv i t y i s u n a c c e p t a b l e a s t h e u n i v e r s a l f r a m e w o r k o f

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

re l a t i v is t i c ob se rv a t i ons an d N ew ton ian g rav i t a t i on . 121)

3 . T H E O L B E R S P A R A D O X

W h a t m a y t h e t e m p e r a t u r e b e i n a n o n e x p a n d i n g ( st at ic ) w o r l d o f t h e

p r e s e n t d i m e n s i o n s ?I n t r y i n g t o a n s w e r t h i s q u e s t i o n , l e t m e r e f e r f i r s t t o a n o l d p r o b l e m

o r i g i n a l l y r a i s e d b y E . H a l l e y ( 1 6 5 6- 1 74 2 ) a n d r e v i v e d b y H . W . M . O l b e r s

( 1 75 8 -1 8 4 0) , e v e n t u a l l y c o m i n g t o b e k n o w n a s O l b e rs p a r a d o x . (4) G e n e r a l l y



412 Gal-Or

s p e a k in g , t h e p a r a d o x s ta te s t h a t i f o n e t a k e s i n t o a c c o u n t t h e o b s e r v e d

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

t h e r e s u l t i n g s k y - b r i g h t n e s s ( e q u i l i b r i u m ) t e m p e r a t u r e i n a s t a t i c i n t e r -g a l a ct ic s p a ce s h o u l d b e a r o u n d 6 0 0 0 K ( w h i c h is a b o u t t h e s u r f a c e t e m p e r a -

t u r e o f th e s u n) . Y e t t h e o b s e r v e d n i g h t s k y is d a rk . T h i s c o n t r a d i c t i o n h a d

b e e n k n o w n f o r m a n y y e a r s , b e f o r e t h e s o l u t i o n w a s f o u n d i n t h e 1 9 2 0 s

t h r o u g h t he E i n s t e i n - F r i e d m a n n - L e m a i t r e - H u b b l e d i sc o v er ie s o f t h e

e x p a n d i n g u n i v e r s e.

T h e s i m p l es t m a t h e m a t i c a l f o r m u l a t i o n o f O l b e r s' p a r a d o x i s as fo l lo w s .

S u p p o s e w e h a v e a n i n f i n it e l y o l d a n d i n f in i te in e x t e n t u n i v e r s e w i t h

E u c l i d e a n g e o m e t r y , a n d w i t h a u n i f o r m d i s t r ib u t i o n o f r a d i a t in g o b j e c ts .

I n a s p h e r i c a l s h e ll o f th i c k n e s s d r , l o c a t e d a t d i s t a n c e r f r o m e a r t h , t h e r e a r e4nrrr 2 d r s u c h o b j e c t s , w h e r e n i s t h e i r n u m b e r p e r u n i t v o l u m e . T h e t o t a l

c o n t r i b u t i o n o f r a d i a t i o n f r o m a ll s u c h o b j e c ts is t h e n ( 4 m r r 2 d r ) (L / 4 7 r r 2 ),

w h e r e w e f ir st a s s u m e 5 t h a t e a c h o b j e c t h a s t h e s a m e l u m i n o s i t y L . T h e

c o n t r i b u t i o n o f ra d i a t i o n f r o m t h e e n ti r e u n i v e r se w o u l d t h e n b e i n f i n i t e .

H o w e v e r , a fi n i te r e s u l t i s o b t a i n e d i f a f in i t e c r o s s s e c t i o n A ( n o r m a l t o t h e

l in e o f s ig h t f r o m e a r t h ) is a s s u m e d f o r e a c h o b j e c t . E a c h o b j e c t w o u l d t h e n

b l o c k r a d i a t i o n f r o m a l l o b j e c t s b e y o n d i t a n d l a y i n g i n t h e s a m e s o l i d

a ng le A / r L T o t a l b l o c k a g e o f r a d i a t i o n r e su l ts w h e n t h e e n t i re s o l id a n g l e

4~r i s cov ere d , i . e . , w he n

f o R dr = 47r7rrZn(A / r 2)

S o lv ing f o r R g ive s R = 1 ~ A n , t h e d i s t a n c e w h e r e t o t a l b l o c k a g e r e s u l t s .

T h u s t h e t o t a l r a d i a t i o n B f r o m a l l g a l a c t i c s o u r c e s w i t h i n t h e d i s t a n c e R

a m o u n t s t o

~ R=I /An

B = n L d r ~ L / A~0

w h i c h is o f th e o r d e r o f t h e s u r f a ce b r i g h tn e s s o f e a c h o b j e c t (i n c o n t r a d i c t i o n

t o t h e o b s e r v e d n i g h t - s k y b r ig h t n e s s) . V a r i o u s a t t e m p t s h a v e b e e n m a d e t o

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

v a r i o u s o b s e r v a t i o n s o f e x p a n d i n g i n t e rg a l a c t ic s p a ce . F o r i n s ta n c e , t h e

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

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

b e f o r e . A n a r ti fi ci a l r e s o l u t io n c a n b e o b t a i n e d b y c h o p p i n g u p t h e u n i v e r se

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

Bondi /18)

O n t h e o t h e r h a n d , t h e r e d - s h i f t s z , w h i c h a r e s y s t e m a t i c a l l y o b s e r v e d

See, however, below.



Cosm ological Origin o f l l'reversibility, Tim e, and Tim e Anisotropies. I 41 3

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

b u t a l s o m o d i f ic a t i o n o f t h e p r e v i o u s i n t eg r a l f o r B s o t h a t

foR n LB = [1 + (Ho/e)r] 2 dr

w h e r e u s e h a s b e e n m a d e o f H u b b l e ' s l aw z = Hor/C. H e r e H 0 i s H u b b l e ' s

c o n s t a n t f o r t h e p r e s e n t e p o c h [ H 3 ~ = : ( 1 .7 7 ~ 0 .2 ) × 1 0~° yea rs ( l°q an d c

i s t h e v e l o c i t y o f li g h t. W i t h t h i s r e s u l t B c o n v e r g e s t o a v a l u e w h i c h i s

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

I n t h i s w a y o n e l in k s l o c a l t h e r m o d y n a m i c s w i t h t h e l ar g e -s c al e s t r u c t u r e

o f t h e e x p a n d i n g u n i v e r se ! B u t h o w d o e s t h e e x i s te n c e o f t h e d a r k s k y a ff e ct

a l l i r r e ve r s ib l e p r oc e s se s i n o u r o w n a t m o s p h e r e ? T h i s q u e s t i o n i s d i s c u s s e d

s e p a r a t e l y i n a l a t e r s e c t io n .

B e f o r e c lo s i n g th i s s e c t io n I m u s t , h o w e v e r , r e e x a m i n e t h e q u e s t i o n o f

m a x i m u m t e m p e r a t u r e in a s ta t ic u n i v e r s e . T h e r e a r e t w o a d d i t i o n a l f a c t o r s

h e r e :

( i) T h e o b s e r v e d s o u r c e s s h o w d i s t r i b u t i o n in t h e v a l u e s o f L .

(i i) A s t a ti c u n i v e rs e m u s t r e a c h a m a x i m u m t e m p e r a t u r e e q u a l t o t h e

c o r e ( a n d n o t t o t h e s u r f a c e !) t e m p e r a t u r e T~ o f s t ab l e a d v a n c e d s t a rs i n t h e

ga la x ie s .

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

e x h a u s t e d m o s t o f t h e i r h y d r o g e n , h e l i u m , e t c . a n d t h e i r co r e c o n ta i n s t h e

m o s t s t a b l e e l e m e n t - - i r o n . (4) I r o n , h o w e v e r , c a n n o t r e a c h 1 0 ~ ° K , s i n ce i t

m u s t u n d e r g o e ndo the rmic d e c o m p o s i t i o n b a c k t o h e l i u m ( w h i c h g e n e r a t e s

e n e r g y a t l o w e r t e m p e r a t u r e , v i z. 1 0 7 -1 0 8 K ) . (4) T h u s i n s t a b le a d v a n c e d

s t a r s T ~ c a n n o t e x c e e d 1 0~° K . I f w e n o w p i c t u r e t h a t t h e u n i v e r s e " s t o p s

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

a t w h i c h a ll s p a c e w o u l d b e a t t h e u n i f o r m m a x i m u m t e m p e r a t u r e T~ ( a n d

n o t a t 6 0 0 0 K a s is c u r r e n t l y a s s u m e d ) .N a t u r a l l y o n e is te m p t e d t o a s k w h a t h a p p e n s i f t h e u n i v e rs e s ta r ts t o

c o n t r a c t ? A p p a r e n t l y c o n t r a c t i o n m e a n s b l u e - s h i f e d r a d i a t i o n w i t h n e g a t i v e

z v a l u e s t h a t w o u l d c a u s e B to i n c r e a s e b e y o n d t h e s u r f a c e b r ig h t n e s s o f a n y

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

s t a r t t o converge f r o m ho t t e r spac e o n t o colder galac t ic masses . I r r e ve r s ib i l i t y

w o u l d t h e n reappear , w i t h a ll i t s s p e c if ic t i m e a n i s o t r o p i e s . W o u l d t h e s e

i r r e v e r s i b i l i t i e s a n d t i m e a r r o w s b e p o i n t i n g i n oppos i te d irec t ion t o t h a t w e

o b s e rv e t o d a y ? W o u l d t h e c o n v e r g en c e o f r ad i a t io n f r o m s p ac e o n t o m a t t e r

c a u s e o n e t o o b s e r v e a d v a n c e d p o t e n t i a ls i n s t e a d o f t h e c u r r e n t l y o b s e r v e dr e t a r d e d p o t e n t i a l s ? H o w e v e r , th e se f u n d a m e n t a l q u e s t io n s o f t h e r m o d y n a -

m i c s a n d e l e c t r o d y n a m i c s ar e l i n k e d to t h e v e r y c o n c e p t o f t im e a n d i ts

o r i g i n , t o p i c s r e e x a m i n e d i n l a t e r s e c t i o n s .



414 Gal-Or

4 . D E R I V A T IO N O F T H E M A S T E R T I M E A S Y M M E T R Y F R O M

G R A V I T A T IO N T H E O R I E S

I n t h i s s e c t io n I d e r i v e t h e m a s t e r t i m e a s y m m e t r y f r o m s im p l i fi ed

( r e v e r s i b l e ) N e w t o n i a n o r g e n e r a l r e l a t i v i s t i c m e c h a n i c s . T h e r e s u l t i n g

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

w i t h t h e r e v e r s ib l e e n e r g y e q u a t i o n s o a s t o p r o d u c e e q u i v a l e n t i n e q u l a it i e s

o f t h e " s e c o n d l a w , " e t c .

C o n s i d e r a v e r y l ar g e g a s c lo u d , w h i c h , l i k e t h e o b s e r v e d u n i v e r s e a t

l a r ge , i s i so t r op ic . (4~ O ne c a n th i nk o f t he b i l l i ons o f ga l a x ie s e i the r a s t he

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

g a l a c t i c ( a t o m i c o r i o n i z e d ) g a s , c o n d e n s a t i o n s t h a t a c t a s t r a c e r s f o r t h e

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

v i c in i ty . I c o n s i d e r f i rs t t h e N e w t o n i a n t h e o r y , w h i c h n o t o n l y is m a t h e -

m a t i c a l l y si m p l e r, b u t a l s o l e ad s t o m a n y i m p o r t a n t r e su l ts t h a t a r e e s se n t ia l ly

t h e s a m e a s i n g e n e ra l re l a ti v i ty . F o r t h i s ca s e N e w t o n i a n d y n a m i c s g iv es

d 2 R 4 7 r Gp ( t~ ) d ZR a n d d R

d t~ - = 3 R 2 ' d t- d i - # 0 ( I )

w h e r e R i s a t i m e - d e p e n d e n t s c a l e f a c t o r ( w h o s e c h a n g e i n t i m e m a y b e

t h o u g h t o f as r e p r e s e n t i n g th e c h a n g e i n ti m e o f t h e d i s t a n ce b e t w e e n a n y

t w o s u p e rg a l ax i e s) , G is th e N e w t o n i a n c o n s t a n t o f g ra v i t at i o n , a n d p ( G ) is th e

d e n s i t y o f t h e m e d i u m a t a s t a n d a r d t i m e t~ w h e n R ( t s ) i s a r b i t r a r i l y f ixe d a s

e q u a l t o u n i t y . C o n s e r v a t i o n o f m a t t e r t h e n g i ve s

p ( t ) R ~ = p ( t 3 (2)

E q u a t i o n (1 ) a d m i t s n o s t a t ic s o l u t i o n s ( w h i c h r e q u i r e t h e f i rs t a n d s e c o n d

t i m e d e r i v a t i v e s o f R t o v a n i s h ) . C o n s e q u e n t l y , E q . ( 1) d e m o n s t r a t e s t h a t

e v e n N e w t o n i a n t h e o r y d i c t a t e s a s y s t e m a t i c c o n t r a c t i o n o r e x p a n s i o n o n a

t a r g e s c al e o v e r w h i c h t h e u n i v e r s e is a p p r o x i m a t e l y u n i f o r m . T h e r e f o r e ,

t h is o t h e r w i s e r e v e rs i b le e q u a t i o n is e m b e d d e d w i t h a n i m p o r t a n t t i m e a s y m -

m e t r y , n a m e l y c o m p u l s o r y c o n t r a c t i o n o r e x p a n s io n .

I n t e g r a t i o n o f E q . ( 1 ) l e a d s t o

d R ~ 2 _ 8 z r G p ( t ~ )

--dF-J 3 Rk (3)

w h e r e k is a c o n s t a n t o f i n t e g r a t i o n .N e w t o n i a n d y n a m i c s , h o w e v e r , c a n n o t a c c o u n t f o r m a n y e f f e c t s , s u c h

a s t h e o b s e r v e d b e h a v i o r o f li g h t i n s t r o n g g r a v i t a ti o n a l f ie ld s , c u r v e d s p a c e -

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




a g a s c l o u d , n o n a d d i t i v i t y o f s e p a r a t e e f fe c ts , e tc . F o r t h e p r e s e n t a n a l y s i s

E i n s t e i n ' s o r i g i n a l fi e ld e q u a t i o n s r e d u c e t o t h e s im p l e s e t

d R ] 2d t ] = ~ - G p R 2 - - lc (4 )

2 d ZR 1 ( d R y 2 G p kR d t 2 - ? ~ \ d t ! = - - 8 ~ c2 R 2 (5 )

S u b j e c t t o E q . ( 2 ), o n e i m m e d i a t e l y n o t e s t h a t t h e r e l a ti v is t ic e q u a t i o n (4 ) is

i d e n t ic a l to t h e N e w t o n i a n e q u a t i o n (3 ). H e n c e d e s p i te a l l t h e f u n d a m e n t a l

d i ff e r en c e s b e t w e e n g e n e r a l r e la t i v it y a n d N e w t o n i a n t h e o r y , t h e s c al e

f a c t o r R s a ti sf ie s t h e s a m e e q u a t i o n i n b o t h t h e o r i e s p r o v i d e d t h e e f fe c t o f

p r e s s u r e [c f. E q . (5 )] i s n e g l e ct e d . T h i s i s t h e w e l l - k n o w n M i l n e - M c C r e a

t h e o r e m . C o n s e q u e n t l y t h e r e q u i r e m e n t o f n o n s t a t i c so lu t i ons l eads t o t he

s a m e c l a s s if i c a ti o n o f m o d e l s a n d w o r l d e v o l u t i o n s in b o t h t h e o r i e s . I n t h e

s e c o n d c a s e t h e p h y s i c a l m e a n i n g o f k i s d i f f e r e n t , i . e . , i n s t e a d o f s p e a k i n g

a b o u t b o u n d ( k > 0 ) a n d u n b o u n d ( k < 0 ) c lo u d s , w e t h i n k o f " c l o s e d " a n d

o p e n space , o r s p h e r i c a l a n d h y p e r b o l i c space , o r os c i l l a t i ng a n d e v e r - e x p a n d i n g

s p a c e . T h e c a s e k = 0 is t h e n c h a r a c t e r i z e d b y E u c l i d e a n t h r e e - d i m e n s i o n a l

s p a c e . T h e r e e x i s t, o f c o u rs e , o t h e r i m p o r t a n t m o d e l s i n r e la t iv i s ti c c o s m o l o g y ,

i n p a r t ic u l a r t h o s e r e l a t e d t o t h e e a r l y e v o lu t i o n in w h i c h r a d i a t i o n d o m i n a t e d( a n d i ts p r e s s u r e c o u l d n o t b e n e g l e c t e d ).

I n g e n e r a l , e x c l u d i n g u n r e a l i s t i c a l t e r n a t i v e s , s u c h a s s t a t i c v a c u u m

f i e l d s , n e g a t i v e m a s s e s , a n d s i n g l e - b o d y m o d e l s , o n e c a n c o n c l u d e t h a t t h e

o r i g i n a l f ie l d e q u a t i o n s o f g e n e r a l r e l a t i v i ty ( w h i c h a r e r e v e rs ib l e u n d e r t i m e

rever sa l ) a d m i t o n l y t i m e - e v o l v i n g ( n o n s t a t i c ) s o l u t i o n s ! C o n s e q u e n t l y , t h e

o b s e r v e d w o r l d e x p a n s i o n h a s i t s or ig in e x p l a in e d w i t h i n t h e d o m a i n o f b o t h

N e w t o n i a n a n d r e l a t i v i s t i c d y n a m i c s . I a m c o n c e r n e d h e r e w i t h a n e w

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

[cf. Eqs. (1), (3), and (4)]dR/dr ~ 0 (6)

w h i c h , i n a c c o r d w i t h o b s e r v a t i o n s , m e a n s

d R / d t > 0 ( i .e . , ex pan s ion ) (7)

f o r p a s t a n d p r e s e n t e p oc h s . E v a l u a t i o n o f th e e x a c t v a lu e o f th i s ra t e o f

e x p a n s i o n f r o m a n y o f t h e c o s m o l o g i c a l m o d e l s is e n t ir e l y i m m a t e r i a l h er e .

O n l y t h e i n e q u a l i t y si g n i s o f i m p o r t a n c e , a s w i ll b e s h o w n l a t er . F o r t h a t

pu rpose I wr i t e Eq . (7 ) a sd v / d t > 0 ( expans ion ) (8)

w h e r e v = l / p .



416 Gal-Or

5. O R I G IN O F I R R E V E R S I B I L IT Y I N T H E N E W A S T R O P H Y S I C A L

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

I n t h i s s e c t ion 1 deduce t h e p o s i ti v e n a t u r e o f th e " s e c o n d l a w , " a s w e ll

a s t h e p o s i t i v e n a t u r e o f i n t e r n a l d i s s i p a ti o n , b u l k a n d s h e a r v i s c o s it ie s ( o f

m a t t e r a n d / o r r ad i a ti o n ) , a n d o f t h e C l a u s i u s - D u h e m i n eq u a li ty , f r o m

reversible c o n s e r v a ti o n e q u a t io n s o f e n e rg y a n d m a s s i n c o n j u n c t i o n w i th

t h e m a s t e r a s y m m e t r y [ E q . ( 8 )] . T h u s t h e ( r ev e r s ib l e ) f i rs t l a w o f t h e r m o -

d y n a m i c s

d u = d q ' - - d w ' ( 9 )

t a k e s t h e f o r m

w h e r e

du dq dvdt - - d t P ~ + qg + b (10)

d q / d t = - - ( 1 / p ) v . a q

b = ( l /p ) Z J ,~" F k , k = 1 , 2 , . . . , nk

P = v-z ~ L Pk] e = l

G = L Pk~k/P

d / d t ~_ ( ~ / ~ t ) + ~ . v

L = p ~ ( ~ - ~ ) , k = 1 , 2 , . . . , n

-~ = T - p i

~" = '~r ( i .e . , Tj i = Ti~)

k=l

(11)

(12)

( 1 3 )

(14)

( 1 5 )

(16)

(17)

( 1 8 )

(19)

(20)

S o f a r a l l t h e s e q u a n t i t i e s a r e str ie t ly reversible! He r e u i s t he spe c i f i c ( pe r

u n i t m a s s ) i n t e r n a l e n e r g y ; q is t h e " h e a t " a d d e d p e r u n i t m a s s , a n d ~1 i s t h e

h e a t f l u x d u e t o c o n d u c t i o n ; p is t h e s c a la r h y d r o s t a t i c p r e ss u r e o f t h e t o t a l

p r e s su r e t e n so r ~ ' , a s de f in e d in ( 18 ), wh e r e =v i s t he v i s c ou s s t r e s s t e n so r a nd

i s t h e u n i t m a t r i x w i t h e l e m e n t s 8 ~B ( t h e K r o n e c k e r d e l t a ) ; p is t h e t o t a lm a t e r i a l d e n s i t y d e f i n e d b y ( 14 ), w h e r e v i s t h e s p e c i fi c v o l u m e , a n d p~ i s t h e

m a s s d e n s i t y ( p er u n i t v o l u m e ) o f c h e m i c a l c o m p o n e n t k i n a m i x t u r e o f n

c h e m i c a l c o m p o n e n t s ( k = 1, 2 , . , n ); ~ is th e c e n t e r - o f- m a s s ( b a r y c e n t r ic )



Co sm ologica l Orig in of Irreversibil ity , Tim e, and Tim e Anisotroples . I 417

ve lo c i ty ve c to r , a n d v~¢ is t he ve lo c i ty o f c o m p on e n t k . q5 i s de f ine d b y ( 12 ) ( the

ob se r v e d f a c t t ha t q ) >/- 0 i s t o be d e d u c e d l a t e r 6 ) ; Jq i s t he to t a l he a t f l ux

d e f i n e d b y ( 2 0 ) a n d i n c l u d e s e n t h a l p y t r a n s f e r r e d b y t h e d i f f u s i o n f l u x e s

J k [ d e f in e d b y ( 1 7) ], hl~ t h u s b e i n g t h e s p e c if ic e n t h a l p y o f c o m p o n e n t k ;

f i n al ly , b is t h e ( s c a l a r ) " b o d y h e a t i n g " a s s o c i a t e d w i t h e x t e r n a l f o r c e s F k a n d

radia t ion . (4)

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

S e c t i o n 1, a n i r r e v e r s ib l e t h e o r e m b a s e d o n t h e r e v e r s i b l e e n e r g y e q u a t i o n ( 10 )

a n d t h e v e r i fi e d m a s t e r a s y m m e t r y [ E q . ( 8) ].

W e f i rs t c o n s i d e r s e v e r a l s i m p l e , w e l l -d e f i n e d s y s t e m s c o m p o s e d o f r e a l

f lu i ds w h i c h a r e ca p a b l e o f t r a n s f e r r in g m o m e n t u m b y v i s co u s m e c h a n i s m s .

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

o b t a i n a b l e , f o r i n s t a n c e , w i t h T o l m a n ' s p e r f e c t ( in v i s ci d ) f l u id a n d r a d i a t i o n

( R e f . 19, pp . 323 , 328 ). W e a l so a l low r a d ia t ive s t r es se s , wh ic h m a y be

r e s o l v e d i n t o a n i s o t r o p i c p r e s s u r e PR a n d v i s c o u s s t re s s es •R a c c o r d i n g t o

¥ = ;co q- YR (21 )

P = P a 4 - P R (22)

S i n c e i s o t r o p i c e x p a n s i o n i s th e s a m e e v e r y w h e r e (i .e ., p i s a f u n c t i o n o f ti m e

a l o n e ) , t h e r e e x is t s n o n e t t r a n s f e r o f e n e r g y f r o m o n e l o c a l e l e m e n t ( o r c e ll )t o a n o t h e r . T h e w h o l e p r o c e s s is t h e r e f o r e a d i a b a t i c , i . e. , J q - 0 , J~ = 0 ,

b = 0 , and d u / d t = 0 t h r o u g h o u t t h e m e d i u m (a s e v id e n t ly w e e x p e c t f r o m

a n y s i m p l e a na l y si s o f l o c a l c o n s e r v a t i o n o f e n er g y ). T h e e n e r g y e q u a t i o n

( 1 0 ) t h e n r e d u c e s t o

~, d v ~, 1 (~ a . ÷ ~R.) : v 5 = . 0 (23)- ( P G ~ + PR ) ~ / - - p

T h u s , e m p l o y i n g t h e m a s t e r a s y m m e t r y ( 8), w e i m m e d i a t e l y o b t a i n t h e fi rs t

i m p o r t a n t p h y s i c a l i n e q u a l i t y , n a m e l y

q~a := _ ( t / p ) ( ¥ a a q_ 7~n~. : V 6 > 0 (2 4 )

w h i c h s ig n if ie s t h a t t h e v e r y p r o c e s s o f i s o t r o p i c ( i. e. , a d i a b a t i c ) e x p a n s i o n

c a u s e s t h e m e c h a n i c a l e n e r g y p ~' d v ~ /d t t o b e c o n v e r t e d i r r e v er s ib l y in t o

i n t e r n a l e n e r g y ( s o f a r a s t h e e x p a n s i o n p r e v a i ls ) , v T h i s f i rs t i n e q u a l i t y is i n

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

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

6 Th e operation sym bol in E q. (12) denotes a do uble inner product of second-order tensors,e.g., A : B -~ tr(ABr), yielding the scalar quantity without any emb edded asym m etry!

7 If o ur new definition o f time (see Part II) is adopted, this result remains unchanged in anisotropic contraction. With conventional definition of time the opposite irreversibleprocess take s place during co ntraction.



418 Gal -Or

i n c o r p o r a t e t h i s i n e q u a l i t y a l l t h e w a y t o t h e " s e c o n d l a w " a n d t h r o u g h o u t

t h e e n t ir e s u p e r s t r u c t u r e o f t h e r m o d y n a m i c s .

W h e n e v e r q ~ c a n b e n e g l e c te d in c o m p a r i s o n w i t h b a s im i l a r p r o c e d u r e

l e a ds t o b > 0 . O the r w i se t he ge n e r a l c a se g ive s q~ + b ~ > 0 .

T h e s e c o n c l u s i o n s , a s I s h a l l d e m o n s t r a t e l a t e r , are equivalent to the

second l aw o f t hermodynam ics .

6 . T H E O R I G I N O F D I S S I P A T I O N I N N E W T O N I A N F L U I D S

A N e w t o n i a n f lu i d i s d e f i n e d b y t h e l i n e a r r e l a t i o n

= [ 2 /z f ) - ? ( ~ - - ~ F ) ( t r D ) ] ~ - - 2 F D + ( § F - - K ) V . ~ ( 2 5)w h e r e

b = ½[V ~ + (V~ ) T] (26)

is t h e ( s y m m e t r i c ) r a t e - o f - d e f o r m a t i o n t e n s o r ( al so c a l l e d th e s t r e tc h i n g

t e n s o r ) . I n s o f a r a s t h e p o s i t i v e n a t u r e o f K a n d ix , t h e b u l k a n d s h e a r v i s c o s i -

t ie s , is n o t s t a t e d , ~ is " r e v e r s i b l e . " S u b s t i t u t i n g E q . (2 5 ) i n ( 1 2) a n d c o n -

s i d e ri n g E q . ( 24 ) i n c o n j u n c t i o n w i t h t h e f a c t t h a t u n d e r i s o t r o p i c e x p a n s i o n

o n l y t h e r a d i a l v e l o c i t y v r a n d i ts d e r i v a ti v e s m u s t b e t a k e n i n t o a c c o u n t , w e

o b t a i n

~ a =2_~/x I / c o r t e z K 1 CO z

w h i c h m u s t b e c o m b i n e d w i t h t h e c o n t in u i t y e q u a t i o n ( a ss u m i n g n o n u c l e a r

r e a c t i o n s f o r t h i s m e d i u m )

cop 1 8 eco-i + 7 ~ ~ r (r v J = 0 (28)

w h e r e p i s a f u n c t i o n o f t i m e a l o n e . S o l v i n g E q . ( 2 8) , w e g e t

_ _ - - P (29)

w h i c h , o n s u b s t i t u t i o n i n ( 2 7 ) , y i e l d s

1 ( 8 p ] ~ ~: > 0 (3 0 )~ = ~ \ COt !

E q u a t i o n ( 3 0 ) i s a v e r y i n t e r e s t i n g r e s u l t , s h o w i n g t h a t t he expans ion i t s e l fd ic ta tes the pos i t i ve n ature o f the bu lk v i scos i ty . T h e t h i r d i n e q u a li ty o f o u r

t h e o r y i s t h u s e x p r e s s ed b y

K > 0 (31)



Co sm ologica l Orig in of Irreversibil ity , Tim e, and Tim e Anisotropies . I 41 9

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

m o l o g i c a l e x p a n s i o n . S i g n i fi c a n tl y , t h e s h e a r v i s c o s i t y i s n o l o n g e r p r e s e n t i n

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

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

W he n v~. i s ex pre ssed

e q u a t i o n b e c o m e s

by H ub b l e ' s r e l a t io n v~. = H o r , t h e c o n t i n u i t y

d v ~ _ 1 V . v ~ = 1 _ 1 _ ~ _ ( r 2 v , ) _ 3 H o (32)d t p p r 2 ? r p

C o m b i n i n g ( 32 ) w i t h ( 30 ) a n d ( 23 ), w e o b t a i n a li n k b e t w e e n c o s m o l o g y a n d

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

_ p p ~ ( d v ~ / d t ) p ~

~: ( l / p 2 ) ( d p / d t ) ~ - - 3/ /o > 0 (33)

A l t e r n a t i v e l y , t h i s l i n k i s e x p r e s s e d a s

q )a = 9 K H o 2 / p = 3 p ~ H o / p > 0 (34)

T a k ing H g 1 = 1 .77 × 101° ye a r s , (1°) T = 2 .7 K a n d p = 10 - a ° - 10 -27 g / c m ~,

o n e f i n d s f o r i d e a l i n t e r s t e ll a r g a s a b u l k v i s c o s i t y b e t w e e n a b o u t 1 0 - 5 - 1 0 -~ P .

U n f o r t u n a t e l y , n o e x p e r i m e n t a l d a t a o n • i n s p a c e a r e n o w a v a i la b l e. (la)

B y e m p l o y i n g e q u a t i o n s o f s ta t e , (3 4 ) i n d i c a t e s t h a t t h e r a t e o f d i s s ip a -

t i o n d e c r ea s e s w i t h d e c r e a s i n g t e m p e r a t u r e s ; i n p a r t i c u l a r , a s T te n d s t o

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

T h e e n e r g y e q u a t i o n m a y , h o w e v e r , n o t b e sa ti sf ie d f o r l o w - d e n s i t y

m o n a t o m i c g a s e s ( s u c h a s h e l i u m , f o r w h i c h K --+ 0 ) , u n l e s s w e p o s t u l a t e

i s o t ro p i c , t u r b u l e n t e x p a n s i o n o r f o r m a t i o n o f l o c al , u n i f o r m l y d i s t ri b u t e d ,

s e l f= g r a v i ta t in g c o n d e n s a t i o n s o f m a t t e r . F o r b o t h c a s e s , E q . ( 2 4) le a d s t o

p~ ba = p # q ~ > 0

w h e r e b y a f o u r t h i n e q u a l i t y i s o b t a i n e d

/x = k~ a + / x R > 0

(35)

(36)

a n d ¢ ~ i s t h e d i ss i p a t io n f u n c t i o n , w h o s e m a t h e m a t i c a l c h a r a c t e r i z a t i o n i s

s u c h t h a t ~ , ~ > 0 . S i m i l a r r e s u l ts a r e o b t a i n e d w i t h n o n - N e w t o n i a n f lu id s/T M

7. D I S S I P A T I O N I N N O N E X P A N D I N G F L U I D S

T u r n i n g n o w t o t h e m o r e g e n e r a l c a s es i n v o l v i n g n o n a d i a b a t i c , n o n -

e x p a n d in g , o r n o n u n i f o r m s y st em s w i t h l o c a l v o l u m e (vL) a nd ve loc i ty ( ~L)

825/6/4-4



4 2 0 G a l - O r

d e v i a t i o n s f r o m t h e s t a n d a r d u n i f o r m e x p a n s i o n r a t e dv~/d t t r e a t e d b e f o r e , s

I a p p l y t h e f i rs t l a w o f t h e r m o d y n a m i c s [ E q . ( 10 )] t o a l oca l o b s e r v e r , co-

m o v i n g w i t h t h e l o c a l m e d i u m ' s c e n t e r - o f -m a s s v e l o c it y . S u c h a n o b s e r v e r m a y

d e c o m p o s e t h e g e n e r a l m e c h a n i c a l t e r m i n t o p~ dv~/d t a n d p L d v r / d t w h e n

t h e l as t te r m r e p r es e n t s l o c a l v o l u m e d e v i a ti o n s f r o m u n i f o r m e x p a n s i o n .

T h i s l a st m a y b e p o s i t i v e , n e g a ti v e , o r z e r o , a c c o r d i n g a s th e l oca l s y s t e m

e x p a n d s o r c on t r ac t s w i t h r e s p e c t t o th e s t a n d a r d . F o r t h e se s y s t e m s th e

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

(/3 = q ~- q - ~ r q _ G K H o v . ~ L = q ~ + ~ L + 2 p , V . ~.L ( 3 7 )

a n d E q . ( 1 0 ) i s r e w r i t t e n a s

S = --p ~ dv~ ~ q~z rcr du 1 dv L~ - q - q - q - 6 ~ H o v • = ~ - @ ~ V • j q q - p L " - ~ b

( 3 8 )

w h e r e b y 3 i s d e f i n e d .

F o r i n c o m p r e s s i b l e f l u i d s v • te l = 0 . T h u s , u s i n g (2 3 ), E q . (3 8 ) b e c o m e s

= q)L = du 1_ V pL dvLd S + " L + - y i - - h > t O

P

( 3 9 )

w h e r e , f o r N e w t o n i a n f l u id s , w e h a v e K > 0 , / ~ > 0 , i .e .,

~V i ( ~ I ) ~ V k , 2 a v i ~ ~ V i ~ O ? ) i= ~ ~x ~ ~YL-x~ + 0x-Z - ~ ~i,~ ~ i + ~ ~ i'0 ~x ~ >~ 0 (4 0)

T h e p h y s i c a l m e a n i n g o f S is re l a te d t o o u r l o c a l o b s e r v a t i o n s . S i n ce H 0 is

e x c e e d i n g l y s m a l l , o n l y q ~c is o f i n te r e s t i n l o c a l s t u d i e s , w h i l e t h e h i d d e n 9

e f f ec t o f q~ a i s o f no p r ac t i c a l v a l ue i n l oca l t he r m o dy na m i c s . I n o u r c a s e ,

h o w e v e r , (b a h a s s e r v e d t o i n t r o d u c e t h e c a u s a l o r i g i n o f i r re v e r s i b i li ty v i a

i ts d e d u c e d p o s i t iv e n a t u r e a n d t h a t o f K , / x , / ~ c , / z R , e tc . W h e n S = 0 , i t is

o n l y t h e l o c a l l y o b s e r v e d d i s s i p a t io n t h a t v a n i s h e s , w h i l e q )a c a n n o t b e m a d e

t o v a n i s h s o fa r as t h e o b s e r v e d m a s t e r a s y m m e t r y p r e v a il s.

8 Including, of course , nonex panding sys tems in our labora tory (which are to be consideredas deviat ions f rom a uni form expansion ra te taken as the s tandard-- for more deta i l s

on this s tandard see Par t I I ) .9 O ur th eor y is distinct fro m various new at temp ts to restore str ict causal ity in qu an tum

physics by reviving earlier pro po sals called hidden-variable theories. N eith er is it in

disagreement with them, nor with the so-cal led Mach principle.




8 . C O N N E C T I O N S W I T H C L A S S I C A L A N D C O N T I N U U M

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

In classical equilibrium thermodynamics, the existence of entropy s

defined by d s = ( d q ) r e v / T is postulated. Here T is defined by means of the

equilibrium function s = s(u , v , ck .. . . ) , where c k - - pk / p . Consequently,

d s = ~ d u + - @ v - d r @ 8 c - ~ dcT~@ " " (41)X k~l X

and the thermodynamic temperature, pressure, and chemical potentials are,

respectively, defined by

[ a s ] -1 " T ( 8 s ] . ( 8 s ) (42)

Here the symbol ( )~ denotes that all other quantities are kept constant for

each partial differentiation; Eq. (41) now takes the familiar (Gibbs) form

1P dv dc~: + (43)s = - ~ d u @ - - ~ t ~ . ..

k

Reverting to the i s o t r op i c processes described previously (in which no

net diffusion or chemical reactions take place, i.e., dce == 0), and employingEq. (8), we obtain

d u ~ = T d S ~ a d v ~d-~ dt -- p W = = 0 (44)

T ds~~ - > 0 (45)

Thus, for a gas and/or radiation obeying the state equation p = p R T -? A T e ,

where A and B are constants, the expansion causes the entropy to increase.

For a pure material undergoing a local macroscopic process, (43)becomes

d u _ T d S _ p L d r y (46)d t d t d t

which, by substituting in (39), gives the familiar Planck inequality

3 = T d S 1d t q - p V ' ~ q - b > ~0 (47)

which is the cornerstone of modern continuum thermodynamics. For an

adiabatic process in a gravitational field (47) reduces to [ T ( ds / d t) ] ~ > ~ O .

Thus, for systems involving heat transfer,

d s /d t >~ - - V . L / o T (48)



4 2 2 G a l - O r

E x p r e s s i o n (4 8) is i n f a c t th e f a m i l i a r " s e c o n d l a w " o f c la s si c al t h e r m o -

d y n a m i c s

d s ~ d ' Q / T (49)

w h i c h i n o u r th e o r y is deduced f r o m t h e m a s t e r i n e q u a l i t y (8 ), T h i s c o m p l e t e s

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

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

t h e o t h e r h a n d , w i t h t h e t h e o r i e s o f g r a v i t a t io n a n d m o d e r n c o s m o l o g y . (211

9 . E L E C T R O M A G N E T I C I R R E V E R S I B I L I T Y A N D T H E M A S T E R

T I M E A S Y M M E T R Y

N o u n i f ie d t h e o r y o f t h e r m o d y n a m i c a n d e l e c t r o m a g n e t i c ir re v e rs ib i li ti e s

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

p r o s p e c t o f o u r s c h o o l a c h i e v i n g a p a r t i a l u n i f ic a t i o n v i a a second pr incip le o f

ear th -cosmolog i ca l l inks .

W h y d o e s a n a c c e l e r a te d e l ec t ri c c h a r g e lose r a d i a n t e n e r g y to sur-

rounding space? W h y i s i t n e v e r s e e n t o receive i n c o m i n g r a d i a t i o n f i ' om

outer space? The c lass ica l equat ions o f e lec t rodynam ies are fund am enta l ly

symmetr ic under t ime reversal , s o t h a t e a c h s o l u t i o n o f t h e s e e q u a t i o n s f o rt i m e t h a s a s y m m e t r i c c o u n t e r p a r t f o r - - t . T h e s e a r e k n o w n a s re tarded

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

r e sp e c ti v e ly . M a t h e m a t i c a l l y s p e a k in g , both solut ions are val id. I n p r a c t i c e ,

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

T h e f o r m a l p r e t e x t f o r s u c h a n a p r i o r i p r o c e d u r e i s i n " c a u s a l i t y , " b u t t o

s t a t e t h a t t h e c a u s e m u s t p r e c e d e t h e e f f e c t is essent ial ly equivalent to an a

p r i o r i t i m e a s y m m e t r y , w h i c h i s i n t u r n equivalent to an a pr ior i im pos i t ion o f

e lect rodynamic i rrevers ib i l i t y! E v e n i n q u a n t u m e l e c t r o d y n a m i c s , t h e r a t e s

o f i n d u c e d u p w a r d a n d d o w n w a r d t r a n s it io n s m u s t b e s y m m e t r ic ; fo r m a l ly ,t h e a c t u a l e m i s s io n o f ra d i a t i o n is a t t r i b u t e d t o " ' spontaneous" d o w n w a r d

t r a n s i t i o n s .

T h e t i m e s y m m e t r y o f M a x w e l l 's e q u a t io n s a ll ow s o n e t o o b t a i n t h e

f a m i l i a r re tarded so lu tions

( /~(t -- ([ r -- r 1 !/C), rl )r ) d3rl ( 5 0 )

! I r r l i

I j ( t - - ( ( r - - r l i / c ) , h )A~dv(t, t )

. l r -r? i d rl (5 1)

w h e r e t h e e l e c t r i c c h a r g e d e n s i t y fi a n d t h e c u r r e n t d e n s i t y j a re s u b j e c t t o

M a x w e l l ' s e q u a t i o n s .




B y r e v e r s i n g t h e s i g n o f t , o n e o b t a i n s t h e a d v a n c e d s o l u t i o n s 1°

~ , ~ ( t , r) = f f 3( t + l ! r - - r ~ I/e), r 0_- ~.! d~r~ (52)

Aaav(t , r) = f j( t + (I r -- r 1 !/c), r l )s !r - - r~ - i d~r~ (53)

I n E qs . ( 50 ) a n d ( 51) q~ a n d A a t t im e t a r e du e to d i s tu r b a nc e s a r i s ing

earlier f r o m t h e s o u r c e , w h i l e i n E q s . ( 5 2 ) a n d ( 5 3 ) t h e y a r e d u e t o s o u r c e

d i s t u r b a n c e s a t la ter t i m e s . I n t h e f o r m e r c a s e t h e s o l u t i o n r e p r e s e n t s loss

o f r a d i a t i o n e n e r g y f r o m s o u r c e t o s p a ce , w h i le i n t h e l a t e r ca s e it g a i n s

inc,o m i n g r a d i a t i o n f r o m s p ac e . I n b o t h c a se s t h e e le c t ri c a n d m a g n e t i c fi e ldi n t e n s it i es f a l l o f f a s r -1 f o r l a rg e r , c a u s i n g t h e f lu x o f r a d i a t i o n t o d r o p o f f

a s r -'2 . T h a t r e s u l t w a s a c t u a l l y u s e d i n t h e p r e v i o u s f o r m u l a t i o n o f t h e

O l b e r s p a r a d o x .

A m o r e g e n e r a l s o l u t io n c a n b e o b t a i n e d b y c o m b i n i n g ( 5 0 ) -( 5 3 ) in t h e

f o r m

A = c~Aret + (1 -- c~)A~av (55)

w h e r e 0 ~< c~ ~< 1 a n d i n w h i c h t h e r e i s b o t h o u t g o i n g a n d i n c o m i n g r a d i a -

t i o n . I n a g r e e m e n t w i t h o b s e r v a t io n s , w e i m p o s e t h e c o n d i t i o n ~ = 1 t o

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

p u r e e l e c t r o d y n a m i c s . I t c a n n o w b e e x p l a i n e d w i th i n t h e f ra m e w o r k o f t h e

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

A r g u m e n t s t h a t l i n k e l e c t r o m a g n e t i c i r re v e r s i b i l it y t o t h e u n i v e r s e a t

l ar g e w e r e r e p o r t e d b y W h e e l e r a n d F e y n m a n . (~ T h e a d v a n t a g e o f t h e ir

a p p r o a c h i s t h a t t h e v a l u e c~ = 1 d o e s n o t h a v e t o b e a s s u m e d , b u t is d e r i v e d

f r o m t he reaction o f the universe. H o w e v e r , W h e e l e r a n d F e y n m a n a c h i e v e dt h e i r r e su l t b y i m p o s i n g a s y m m e t r i c i n it ia l c o n d i t i o n s o n t h e s y s t e m o f t h e

t y p e th a t l e a d t o t h e t h e r m o d y n a m i c a r r o w o f ti m e a n d c a us a li ty . T h e s e

q u e s t io n s w e r e e x a m i n e d b y H o g a r t h , H o y l e a n d N a r l i k a r , a n d N a r l i k a r , Ca)

w h o p o i n t e d o u t t h a t u n l i k e t h e s ta t i c u n i v e r s e a n a l y z e d b y W h e e l e r a n d

F e y n m a n , t h e a c t u a l exp a n d i n g u n i v e r s e i s n o t t i m e s y m m e t r i c . U s i n g s u c h

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

a g r ee s w i th t h e o b s e r v e d r e ta r d e d s o l u t i o n s - - a t o p i c w h i c h i s o u t s i d e t h e

s c o p e o f th i s p a p e r .

A c c o r d i n g t o t h e absorber theory o f W h e e le r a n d F e y n m a n , t h es y m m e t r y , u n d e r t i m e r e v e r s a l , i s c o n f i n e d t o i n t e r a c t i o n s b e t w e e n p a i r s o f

lo See , how ever, Ref. 20.



4 2 4 G a l - O r

p a r t i cl e s , w h i l e i n a l a r g e - s c al e ( a s y m m e t r i c ) e n s e m b l e o f c h a r g e d p a r t i c l es ,

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

a s y m m e t r y in e l e c t ro d y n a m i c s . I n o r d e r t o c r e a t e t h e n e c e s s a ry a sy m m e t r y ,

t h e u n i v e r s e m u s t a c t a s a perfect absorber o f e l e c t ro m a g n e t i c d i s t u r b a n c e s

g e n e r a t e d w i t h in i t. B u t h o w c a n t h e w o r l d a c t l ik e a p e r f e c t a b s o r b e r ?

W h e e l e r a n d F e y n m a n r e st ri c te d th e i r t h e o r y t o a static u n i v e r s e . T h e r e f o r e ,

t h e i r a r g u m e n t s a r e n o t c o m p l e t e o r c o n v i n c i n g . B is only in the nonstatic,

unsaturable expa nding space th at a per fect absorbtion becom es feasible,

thereby giving rise to eIectrodynamic irreversibility.

H e r e , a g a i n , t h e m a s t e r a s y m m e t r y s t an d s a t t h e c e n t e r o f t h e p i c t u r e

a n d c a n b e d i r e c t ly c o m b i n e d ( in a n u m b e r o f d if f er e n t m a t h e m a t i c a l f o r -

m u l a t i o n s (G)) w i t h t h e r e v e r s i b le e l e c t r o m a g n e t i c e q u a t i o n s , s o a s t o deduce

t h e o b s e r v e d e l e c t r o d y n a m i c i rr e v er s ib i li ty . In a f u t u r e d e v e l o p m e n t t h e

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

d i v e rg e n c e o f t h e w o r l d l i ne s a c c o r d i n g t o t h e o b s e r v e d " r e d - s h i f t e d " m a s t e r

a s y m m e t r y , o r f r o m g e n e r a l r e l a t i v i t y . A n o t h e r p o s s i b i l i t y i s v i a t h e m a s t e r

a s y m m e t r y t h a t c a u s e s in t e r st e ll a r s p a ce t o c h a n g e t o w a r d v e r y l o w d e n s i t y

o f r a d i a t i o n a n d p a r t i cl e s, n a m e l y : m a s t e r a s y m m e t r y - , u n s a t u r a b l e s p a c e

( p a r ti c l e s c l o s e t o g r o u n d s t a te ) --+ s p a c e a c t i n g a s " p e r f e c t a b s o r b e r " - +

e l e c t r o d y n a m i c i r r e v e r s i b i l i t y .

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

p a r ti c le s , w h i c h i n t u r n u n d e r g o u p w a r d t r a n s it i o n s i n a s t a ti c w o r l d . Y e t a t

t h e p r e v a i l i n g t e m p e r a t u r e o f 2 .7 K t h e p a r t ic l e s a r e m a i n t a i n e d c lo s e to

t h e i r g r o u n d s t a te b y t h e e x p a n s i o n o f i n t e rg a l a c ti c s p ac e . T h u s , d o w n w a r d

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

a b s o r b e d .

D e e p l y e m b e d d e d i n t h is c h a i n o f c a u s a l li n k s a r e q u e s t io n s r e l a t e d t o

t h e v e r y n a tu r e o f ti m e a n d t o t h e i n t e r ac t i o n b et w e e n t h e r m o d y n a m i c s ( o f

m a t t e r ) a n d e l e c t ro m a g n e t i c r a d i a ti o n . T h e p o s s i b l e i n te r a c t io n s a r e g i v e n in

T a b l e I .

Table I . Possible Interactions Am ong Cosmology, Thermodynam ics, and Electrodynamics

M aster asymm etry --+ thermody nam ic asym metry -~ electrom agnetic asym metry (a)

M aster asym metry --+ electrom agnetic asym metry ~ thermody nam ic asym metry (b)

M aster asymm etry

/ \electromagnetic asym metry +-----+ thermody nam ic asym metry (c)i n t e r

i



Cosmologica l Orig in of Irreversibi l i ty , Time, and Time Anisotropies . I 425

I n m e c h a n i s m ( c ) t h e e l e c t r o m a g n e t i c a n d t h e r m o d y n a m i c a s y m m e t r i e s

i nc lu d e e m b e d d e d p h o t o n a n d m a t e r i a l a s y m m e t r ie s , w h i c h a r e o b s e r v a ti o n a l

m a n i f e s t a t io n s o f o n e a n d t h e s a m e b a s ic p h e n o m e n o n o f i rr e v e rs i b il it y ,

d i f fe r e n ti a te d o n l y b y o u r e p i s t e m o l o g y . M e c h a n i s m s ( a) a n d ( b) m a y a l so b e

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

t h e t h e r m o d y n a m i c o n e s i d e d n e s s o f m a t t e r i n i n t e r g a la c t i c sp a c e . ( I n a

c o m p l e t e ly t ra n s p a r e n t w o r l d , f r ee o f a g g re g a t io n s o r n o n u n i f o r m i t y d u e

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

" r e d - s h i f t e d " a s y m m e t r y is u n o b s e r v a b l e; in f a c t, gravitation i s b e h i n d o u r

a b i l i t y t o o b s e r v e r e d - s h i f ts a n d i r r e v e rs i b i li t ie s t h r o u g h t h e m o t i o n o f s e lf -

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

t h a t is " i m p r i n t e d " w i th a s y m m e t r y t h a t in t u r n , g e n e r a t e s t h e o b s e r v e d

t h e r m o d y n a m i c ir r e v e rs i b il i ty w h e n e v e r i n te r a c t io n s b e t w e e n r a d i a t i o n a n d

m a t t e r d o t a k e p l a c e [ m e c h a n i s m ( b ) ] .

I f a m a c r o s c o p i c s y s t e m c o u l d b e e n t ir e ly m a d e o f s u p e r c o n d u c t i n g

m a t t e r , w o u l d t h a t a l l o w u s t o e s ta b l i sh p r o p e r a s y m m e t r i c li n k s b e t w e e n

t h e r m o d y n a m i c s a n d r a d i at io n ? U n f o r t u n a t e l y w e d o n o t y e t k n o w . S o f a r,

w e h a v e n o m e a n s o f a s c e rt a i n i n g w h i c h o f t h e t h r e e in t e r a c t io n s is t h e p r o p e r

o n e . W e m u s t , t h e r e f o r e , a s s u m e t h a t a l l o f t h e m a r e f ea sib le .(2 1~

R E F E R E N C E S

1. B. Gal-Or, Science 176, 11 (1972).2. B. Gal-Or, Nature 230, 12 (1971).3. B. Gal-Or, Ann. N.Y. Acad. Sci. 196(A 6), 305 (1972).4. B. Gal-Or, ed., Modern Developments in Thermodynamies (Wiley, New York, 1974).5. J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945).6. J. V. Narlikar, in Modern Developments in Thermodynamies, ed. by B. G al-Or (Wiley,

New York, 1974).7. Y. Ne'eman, in Modern Developments in Thermodynamics, ed. by B. Gal-Or (Wiley-,

New York 1974).8. A. Aharony, in Modern Developments in Thermodynamics, ed. by B. Gal-Or (Wiley,

New York, 1974).9. T. Gold, ed., The Nature of Time (Cornell Univ. P ress, Ithaca, N ew Y ork, 1967).

10. A. Sandage, Quart. J. Radio-Astr. Soe. 13, 282 (1972).11. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).12. L. Motz, Astrophysics and Stellar Structure (Waltham, Massachusetts, 1970).13. Ya. B. Zeldovich and I. D. Nobikov, Relativistic Astrophysics (University of Chicago

Press, Chicago, Illinois, 1971).14. E. Novotny, Introduction to Stellar Atmospheres and Interiors (Oxford University

Press, Oxford, 1973).15. C. W. Misner, Phys. Rev. 186 , 1328 (1969).16. Y 'a. B. Zeldovich, JETP Lett. 12, 307 (1970).17. T. Gold, in Recent Developments in General Relativity (Pergamon Press, New York,

1962).



4 2 6 G a l - O r

18 . H. Bondi , Cosmology (Cambridge Univers i ty Press , Cambridge , 1961) .

19 . R . L . To lman , Relativity Thermodynamics and Cosmology (Oxford Press , Oxford ,

1933).

20 . T . Gold , in Modern Developments in Thermodynamics, e d . b y B . G a l -O r (Wi le y , Ne wYork , 1974) .

21 . B . Ga l-Or , Crisis in Western Science and Philosophy, Part 11: Crossroads in Physics( N e a m a n I n s t i tu t e o f A d v a n c e d S t ud i es a t t h e T e c h n i a n - l s r a el I n s t it u t e o f T e c h n o lo g y ,

Ha ifa , 1976) .

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Cosmological Origin of Irreversibility, Time, And Time Anisotropies