UBC Theses and Dissertations

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UBC Theses and Dissertations

Some properties of a cosmological model containing anti-matter Matz, Detlef 1959

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SOME PROPERTIES OP A COSMOLOGICAL MODEL CONTAINING ANTI-MATTER b y DETLEP MATZ B.Sc, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 19^ 8 A THESIS SUBMITTED I N PARTIAL FULFILMENT OP THE REQUIREMENTS POR THE DEGREE OP MASTER OF SCIENCE i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1959 - i i -ABSTRACT The c h i e f a i m o f t h i s work i s t o i n v e s t i g a t e c o s m o l o g i c a l c o n s e q u e n c e s o f a h y p o t h e s i s p u t f o r w a r d b y M o r r i s O n and G o l d i n 1956. T h e s e a u t h o r s p o s t u l a t e the e x i s t e n c e o f e q u a l amounts o f m a t t e r and a n t i -m a t t e r i n o u r u n i v e r s e . A b a n d o n i n g t h 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 y a t t r i b u t e n e g a t i v e g r a v i t a t i o n a l m a s s . t o a n t i - n u c l e o n s . The r e s u l t i s a d r a s t i c a l t e r a -t i o n i n the f i e l d e q u a t i o n f o r the g r a v i t a t i o n a l p o t e n t i a l . I n t h e f i r s t t h r e e c h a p t e r s N e w t o n i a n C o s m o l o g y i s d e v e l o p e d f r o m b a s i c p r i n c i p l e s . The e q u a t i o n s d e s c r i b i n g a u n i v e r s e c o n s i s t i n g o f m a t t e r a r e s e t up and s o l v e d . I n c h a p t e r I V t h e h y p o t h e s i s o f M o r r i s o n and G o l d i s i n t r o d u c e d , and the r e s u l t i n g m o d e l f o r t h e u n i v e r s e i s compared w i t h m o d e l s o b t a i n e d i n c h a p t e r I I I . I t i s c o n c l u d e d t h a t w i t h i n t h e framework o f t h e model c o n s i d e r e d , the h y p o t h e s i s of M o r r i s o n and G o l d i s i n c o m p a t i b l e w i t h t h e o b s e r v a t i o n a l e v i d e n c e , b e c a u s e i t l e a d s t o an age o f the u n i v e r s e o f b e t w e e n 1.3 and 1.9$ b i l l i o n years, w h i c h i s l e s s t h a n the age d e r i v e d f r o m o t h e r g e o l o g i c a l and a s t r o p h y s i c a l d a t a . I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e partment The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 6*, C a n a d a . - i i i -TABLE: OF C;OETEI.T.S PAGE ABSTRACT I i ACKIOWIEDGEMEETS v CHAPTER I. TEgmOSOGT AND POSTULATES 1.1. The Universe I. 1.2. Cosaiiologlcal Principle 1 1 . 3 . Mateix. re)presenta.tlon>. of velocity fields' 2 I.fjL. . Special, subcase and example: 3' I I . DYSTAMICS OF THE UNIVERSE II . 1 . F i e l d equations: 5 11.2, l a c t a t i o n a l velocity field. 6. 11.3. I s o t r o p i c velocity f i e l d 8 I I I . SOLUTION, OF THE COSM0L0GICAL EQUATION' 111.1. First. Integration of th© equation 10 111.2. ; Hyperbolic ttBlv©rs© 11 111.3. E l l i p t i c universe: 11 III»l}.. Parabolic tnaivers© 12 IV/. ANTI-MATTER IV.I. Alteration, of the gravitational f i e l d equation 13 IV.2. General solution, of the cosmologlcal equation 20 IV'.3• Types of universes 22 - i v -COMPARISOH ¥ITH EXPERIMENTAL EVIDENCE V . l . Intoo duetion V .2. Exclusion, of the hyperbolic and parabolic models V .3. Comparison, of the e l l i p t i c model tilth the anti-matter' model Graph. I Gr-aph II Graphs III, IV, and V Graph VC Graphs VII,, VIII, and IX. Graph X. Graph XI TABLES 24-25 26 Table I Table II APPENDICES' A.. Determination of © B„ Calculation of <A and. ft „• C. Points- for the maximum and minimum curves of both models B1BLI0GRAPHT facing- pag-e 3 facing page facing page 11 facing page 12 facing page 22 facing page 27 facing page 32 33 34-3 0 3 1 32 •35 - V -A.CKM OMLED-GEMENT'S I am Indebted to Br. P. A. Kaempffer' far sazggestlm«g this thesis, and for his helpful discussions- throughout the course of the research. I am also grateful to the Hationa! Research Council of Canada for financial assistance in. the form of a Bursary. -1-CHAPTER l t TERMINOLOGY AND POSTULATES. I . 1. The U n i v e r s e . The u n i v e r s e c o n s i s t s o f a huge c o l l e c t i o n o f g a l a x i e s w h i c h a r e s e p a r a t e d b y d i s -t a n c e s much l a r g e r t h a n t h e i r maximum d i a m e t e r s . We c a n t h e r e f o r e c h o o s e an e l e m e n t o f volume dV s u c h "that: volume o c c u p i e d b y one galaxy^c,<LV<&volume o f u n i v e r s e . I t i s t h e n r e a s o n a b l e - t o d e s c r i b e t h e u n i v e r s e i n terms o f the d e n s i t y J ^ j ^ J . j t ) , "the v e l o c i t y Of (X,*J» J i*") anl" i f d e s i r a b l e , i n terms o f t h e p r e s s u r e p^ -x:,^ , ^  5 -fcj. We s h a l l a l s o r e s t r i c t o u r s e l v e s t o E u c l i d e a n G e o m e t r y . A s s u m i n g f u r t h e r the v a l i d i t y o f the l a w s o f New-t o n i a n M e c h a n i c s , i n c l u d i n g Newton's l a w o f g r a v i t a t i o n , one o b t a i n s a d e s c r i p t i o n , o f the u n i v e r s e known as "New-t o n i a n C osmology", w h i c h was f i r s t s t u d i e d i n t h i s f o r m b y M i l n e a i d M c C r e a i n 193U-I . 2. The C o s m o l o f t i c a l P r i n c i p l e . M i n e and : M c C r e a make t h e f o l l o w i n g two b a s i c a s s u m p t i o n s : (1) C h o i c e o f t h e c o o r d i n a t e s y s t e m : As a s u i t a b l e c h o i c e o f " o b s e r v e r s " ( o r c o o r d i n a t e s y s t e m s ) we d e s c r i b e t h e u n i v e r s e e x c l u s i v e l y f r o m t h e p o i n t o f v i e w o f o b s e r v e r s swimming w i t h t h e u n i v e r s e . (2) B a s i c P o s t u l a t e : A t any g i v e n t i m e a l l o b s e r v e r s h a v e t h e same viex* o f t h e u n i v e r s e . T h i s i m p o r t a n t a s s u m p t i o n i s c a l l e d t h e C o s m o l o g i c a l P r i n c i p l e ( o r the H o m o g e n e i t y P o s t u l a t e ) . -2-As a d i r e c t c o n s e q u e n c e o f (2) -fj 4"-) t h e v e l o c i t y f i e l d s a r e r e s t r i c t e d , t o ^ ^ j f u n c t i o n s l i n e a r i n t h e c o o r d i n a t e s , . F o r w i t h t h e a i d o f t h e d i a g r a m we see t h a t what o b s e r v e r (7) s e e s a t x s h o u l d be i d e n t i c a l w i t h what o b s e r v e r @ s e e s Sitx+A. We must t h u s h a v e : 1.1) ftt+<k,t) = fCZjtJ I t a l s o f o l l o w s t h a t : 1.2) H = <*?(*ji)+'Sf(Ji't) . S i n c e x a n d b a r e a r b i t r a r y v e c t o r s , we f i n d t h a t t h e most g e n e r a l f u n c t i o n s s a t i s f y i n g t h e s e p o s t u l a t e s a r e : 1.3) f * f*> \ f= t L i ) inhere Q.;K(-t) a r e a r b i t r a r y f u n c t i o n s . F r o m now on we s h a l l u s e t h e u s u a l summation c o n v e n t i o n s and w r i t e : <Vi ^  aub) xK 3 ( t« J,*, 3 s «• I, JJ, 3 ) . Thus f o r a c o m p l e t e d e s c r i p t i o n o f t h e u n i v e r s e we jau s t d e t e r m i n e t e n unknowns,- t h e CL^ffc) s and J ? ( i j . 1.3. M a t r i x r e p r e s e n t a t i o n o f v e l o c i t y f i e l d s . To v i s u a l i z e the v e l o c i t y f i e l d s r e p r e s e n t e d b y t h e m a t r i x fan j » ^ -*-3 c o n v e n i e n " t t o w r i t e t h e m a t r i x e l e m e n t s i n th e f o r m : - 3 -where ^ a " ' l < ^ Q " c ) - ^ t'u I s the s y m m e t r i c a l p a r t o f (a i(1<.), t h a t i s : 0. u = < * K/ , a n d i . - a«.i)= Q*CHXS t h e s k e w - s y m m e t r i c a l p a r t o f (0-,'K), t h a t i s : =— 4Kt' • L e t us t a k e the s p e c i a l c a s e of a.t- k = Then (o.t-K j d e s c r i b e s an i r r o t a t i o n a l v e l o c i t y f i e l d . P o r , t a k i n g t h e v e c t o r c u r l o f v , we h a v e : [c~tit)x = >gt = «*„ - « „ . = o , and s i m i l a r l y : Thus i n a n i r r o t a t i o n a l f i e l d we h a v e o n l y s e v e n "unknowns l e f t , s i n c e (a,fcJ i s s y m m e t r i c . I . k . S p e c i a l s u b c a s e aid example. A p a r t i c u l a r l y s i m p l e and i m p o r t a n t s u b c a s e i s t h a t o f s p h e r i c a l l y sym-m e t r i c v e l o c i t y f i e l d s , a l l o w i n g o n l y c o n t r a c t i o n o r e x p a n s i o n i n r a d i a l d i r e c t i o n , r e p r e s e n t e d b y : *<M = f&) S*K , o r : V^faix, ; At^-fit)** j ^ f i ) * * . E x a m p l e : To i l l u s t r a t e t h e C o s m o l o g i c a l P r i n c i p l e , g r a p h I and I I h a v e b e e n drawn, t h a t i s , one d e p i c t i n g an i r r o t a t i o n a l , and one a r o t a t i o n a l v e l o c i t y f i e l d f o r two d i f f e r e n t o b s e r v e r s swimming w i t h the u n i v e r s e . I n e a c h g r a p h the " v e l o c i t y f i e l d as observer(£) s e e s i t was drawn f i r s t ( t h e t y p e o f f i e l d i s i n d i c a t e d on e a c h g r a p h ) . 11" i » >. r a. | R o t a t i o n a t v e l o c i t y fie td (v.-r-klXi , * , ) . T h e n o b s e r v e r (j£)'s f i e l d was c o n s t r u c t e d as f o l l o w s . Prom t h e C o s m o l o g i c a l P r i n c i p l e i t \2- f o l l o w s {see d i a g r a m ) : £~ ^ \ T($) =<¥(*)- ^L?) . S i n c e <qr_ (-$} $) > the v e l o c i t y components f o r o b s e r v e r (Dare: where a,b a r e c o n s t a n t s ( f o r example: CL = 0.9L } J0r=-O'l i n g r a p h I ) . Thus I f o b s e r v e r (g)'/ f i e l d i s drawn I n t h i s manner, he w i l l h a v e t h e same v i e w o f the u n i v e r s e as does o b s e r v e r ® . CHAPTER I I : DYNAMICS OP THE UNIVERSE. I I . 1. F i e l d equations. We now state the funda-mental laws waicb. are assumed to hold u n i v e r s a l l y . (1) Conservation of mass. This law Is most conven-i e n t l y expressed as an equation of continuity, which i n absence of either creation or a n n i h i l a t i o n of matter reads: _^ I 3 : - 1 } H T W (2) Conservation of momentum. L e t ^ " ^ be the density of the k component of momentum. Then the equation of continuity f o r each component independently i s : 11.2) ^ - ( f ^ + A . (S^t^i) - ?o*CA _ P \ <P where^?is the g r a v i t a t i o n a l p o t e n t i a l per u n i t mass. (3) Newton's law of g r a v i t a t i o n . The law i n d i f f e r e n -t i a l form i s : 11.3) V X f = 4TT Yf t I f there i s any amount of creation of matter per uni t time and per u n i t volume, Q, the equation should be: I I . l a ) 31 ^ 9_ ( P^-j where the p o s i t i v e sign i s for creation and the negative sign for a n n i h i l a t i o n of matter. The quantity Q may be either a constant or a function of time t. I t i s p a r t i c u l a r l y important i n the Steady-State Theory pro-posed by Bondi and Gold i n 19l|-8 ' ^ ) . p Q r reasons of s i m p l i c i t y , these authors assume that the creation Q i s constant i n space and time. Q, Is of such order that the mean density of matter of our expanding universe remains constant. I t follows that the amount of matter created i n a volume element of space-tine i s proportional to this volume element. The' factor of p r o p o r t i o n a l i t y i s : 3 X(Hubble's constant) X (mean density of the universe) X \0'H3 <j/yw- CA^-Ate'1 . Any further discussion of this case, however, would take us beryond the range of this work. Gra r W I : I r r o t a t i o n a t veEoclty f iefd (v,=.2z, , V - . l x J . -6-where %^ oLynt-can- t 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 . W r i t i n g o u t e q u a t i o n IT.2)),we h a v e : * « • <?*,• J a « ; J ) X k By u s e o f I I . l ) ) I t s i m p l i f i e s t o : U . 4 ) «_* ^ - - - ^ . E q u a t i o n s I I . l ) , I I . 3 ) , and I I . k ) a r e t h e f u n d a m e n t a l e q u a t i o n s d e t e r m i n i n g t h e d y n a m i c s o f t h e u n i v e r s e . These-e q u a t i o n s a r e n o n - l i n e a r and I n g e n e r a l c a n n o t be s o l v e d . W i t h t h e h e l p o f t h e C o s m o l o g i c a l P r i n c i p l e , however, t h e y c a n be s i m p l i f i e d t o t h e e x t e n t o f p e r m i t t i n g s o l u t i o n b y e l e m e n t a r y means. F o r , as a d i r e c t r e s u l t o f t h e C o s m o l o g i c a l P r i n c i p l e , we h a v e v e l o c i t y f i e l d s l i n e a r i n t h e c o o r d i n a t e s . T h i s f a c t , and some f u r t h e r s i m p l i -f y i n g a s s u m p t i o n s , w i l l a l l o w us t o i n t e g r a t e t h e e q u a t i o n s . I t i s I n s t r u c t i v e t o c o n s i d e r f i r s t some simple-s p e c i a l c a s e s . I T . 2 . I r r o t a t i o n a l v e l o c i t y f i e l d . H e r e we h a v e no r o t a t i o n , t h a t i s a i l t = . S u b s t i t u t i n g - <xw(t)^ j j? = fftj f w e g e t f r o m I I . l ) 1 : i i . 5 ) f ( « „ + f t , ) i + ^ j i = o and f r o m I I . k ) : i i . 6 ; + a. , , - aie m4 = - 7^ -7-D i f f e r e n t i a t i n g I I . 6 ) , we g e t : T h e r e a r e s i m i l a r e x p r e s s i o n s f o r k - * j 3 . A d d i n g t h e s e , and u s i n g I I . 3 ) • we h a v e r 11.7) a j l S L a * 3 . „ T h i s l a s t e q u a t i o n e n a b l e s u s t o p r o v e a t h e o r e m o f Neumann and Von S e e l i g e r w h i c h s t a t e s t h a t the u n i v e r s e c a n n o t be s t a t i c . P r o o f : E q u a t i o n II.5) c a n be w r i t t e n : % = - (a„ t ^ A t f « n J > a n d a f t e r d i f -f e r e n t i a t i o n : a"„ +• fljij, +*\ 3 ] ss. "^1^^ 3 o r w i t h t h e h e l p o f I I . 7 ) : Ir-8' I *<> , w i t h e q u a l i t y o n l y i f t h e r e i s no v e l o c i t y , ondp-o. Prom 11.8) i t c a n r e a d i l y be s e e n t h a t i f j ^ o , a n d f o r as l o n g as t h e r e i s ' a n y n o n - v a n i s h i n g v e l o c i t y f i e l d a t a l l , J> c a n n o t be c o n s t a n t , w h i c h c o m p l e t e s th e p r o o f . I t must be p o i n t e d o u t , however, t h a t t h i s t h e o r e m i s n o t g e n e r a l l y t r u e i n c a s e o f r o t a t i o n a l m o t i o n , b e c a u s e i f C J H ±0, t h e n t h e e q u a t i o n s w o u l d r e a d as f o l l o w s . E q u a t i o n s II.6) s t i l l h o l d s , b u t II.7) s h o u l d now r e a d : S i m i l a r l y I I . 8 ) s h o u l d r e a d : - 8 -The e q u a t i o n I I . 7 ) c a n b e i n t e g r a t e d m o s t e a s i l y I n a s p e c i a l s u b c a s e . II.3• I s o t r o p i c , v e l o c i t y f i e l d . H e r e : ci < ( <, = ^ o r : <3„ — - °ii3 =^  -j? • E q u a t i o n . II.5) t h e n becomes: I I . 9 ) f + 3 = o and II.6) i s : 1 1 . 1 0 , W ' / V - g - . By d i f f e r e n t i a t i n g , and w i t h t h e h e l p o f I I . 7 ) we o b t a i n : 11.11) j +f*= - «f*J> . Thus i n . u s i n g I I . 9 ) and 11.11) we c a n f i n d f and t h e n b y i n t e g r a t i n g 11.10) f l n d ^ . L e t u s now I n t r o d u c e a new d i m e n s f o n l e s s v a r i a b l e R ( t ) p i s u c h t h a t : ^ • We d e t e r m i n e t h e meaning o f R.(t) as f o l l o w s . Upon i n t e g r a t i n g : w. _ d*< , r % . J _ „ . A x t(t) « V t j R W ,where I s a c o n s t a n t o f I n t e g r a t i o n o b t a i n e d b y t h e a r b i t r a r y c o n d i t i o n t h a t R("t^)=l. Prom t h e s e c o n s i d e r a t i o n s . i t f o l l o w s t h a t R ( t ) I s a u n i v e r s a l : . d i a n e n s i o h l e s s ( t i m e 'dependent .oralyj s c a l e , f a c t o r , t h a t I s , I t shows how much t h e u n i v e r s e h a s expanded o r c o n t r a c t e d . ^ A n o t h e r way o f s e e i n g t h e m e a n i n g o f R ( t ) i s t h e f o l l o w i n g way: s u p p o s e t h a t f o r t h e m o t i o n o f a p a r t i c l e . w e w r i t e I n p o l a r c o o r d i n a t e s : 11.12) 3^= P " - = U « • Upon r e p l a c i n g f ( t ) b y | and i n t e g r a t i n g 1 1 . 1 2 ) : c frit), where c i s a c o n s t a n t p f I n t e g r a t i o n . I f we now c h o o s e t h e c o n d i t i o n t h a t a t some t i m e ta, I, and , t h e n c - ^ . . T h e r e f o r e ^ M j T . w h e r e ^ . I s t h e p o s i t i o n v e c t o r ot t h e p a r t i c l e a t -tr = -t. , S u b s t i t u t i n g R i n o u r e q u a t i o n s ^ we t h e n h a v e f o r I I . 9 ) : # f t l f A / ^ - O , I n t e g r a t i n g : ^ J - - * ^ * ) , o r : T h e r e f o r e : %HJL!L *=. tf = co*«s/a^o . T h i s e x p r e s s e s t h e c o n s e r v a t i o n o f mass. I n f a c t i f we a p p l y t h e c o n d i t i o n Ufa)** I r e p r e s e n t s t h e mass c o n t a i n e d i n a u n i t volume a t t h e time"rs=£>. ft _ WW. S i m i l a r l y f o r t h e e q u a t i o n II.11): a 3 • C o m b i n a t i o n w i t h t h e e q u a t i o n above g i v e s : 11.13) ti - - ^A- • a*-The s o l u t i o n o f t h i s e q u a t i o n w i l l g i v e u s R as a f u n c t i o n o f t . - 1 0 -CHAPTER I I I ; SOLUTION OP THE COSMOLOGICAL EQUATION. I I I . l . F i r s t i n t e g r a t i o n o f the e q u a t i o n . To s o l v e t h e c o s m o l o g i c a l e q u a t i o n , l e t )&M= ?, m u l t i p l y • * & a. t h r o u g h b y (I : A ^ - ~ % and i n t e g r a t e : I I I . l ) ( > ) = I * j where a I s a c o n s t a n t o f i n t e g r a t i o n , o r : dA * ± / ^ J = ±J*A%+**' , w h i c h g i v e s i n . 2 ) JAM ct,~s**~.+ . B e f o r e g i v i n g t h e d i f f e r e n t s o l u t i o n s t o t h i s i n t e g r a l we w i l l a t t e m p t t o f i n d a t l e a s t a p a r t i a l i n t e r p r e t a t i o n o f b and a. The c o n s t a n t b was d e f i n e d as where Y i s t h e c o n s t a n t o f g r a v i t y . F rom t h e p r e v i o u s chapterH = ^ f ( ^ ' J 3 where f(tc)yO i s t h e d e n s i t y a t t i m e to. The d i m e n s i o n o f b i s (time)"* as c a n e a s i l y be v e r i f i e d . The meaning o f a i s o b t a i n e d f r o m e q u a t i o n I I . 1 ) and b y a p p l y i n g t o i t t h e c o n d i t i o n fc(t*)=l . We had:/T 'fiB)<Yb , w h i c h g i v e s : (gf J - -^r. j where v i s the v e l o c i t y . Hence a t * i = o^ I I I . l ) becomes: AT T=./f0 (6f~*-J w h i c h c a n be i n t e r p r e t e d as t h e k i n e t i c e n e r g y p e r u n i t mass o f m a t t e r a t f s ^ . I f we c h o o s e "to=0, t h e n a g i v e s a measure o f t h e e n e r g y p e r u n i t mass o f m a t t e r a t " C r e a t i o n . " I t a l s o f o l l o w s t h a t a must h a v e d i m e n s i o n ( t i m e ) - 2 . The f u l l m e a n i n g o f a a n d b w i l l become a p p a r e n t i n C h a p t e r V. - 1 1 -We r e t u r n now t o t h e i n t e g r a l I I I . 2 ) and d i s c u s s i t s t h r e e d i f f e r e n t s o l u t i o n s a c c o r d i n g t o w h e t h e r I I I . 2 . H y p e r b o l i c u n i v e r s e . The i n t e g r a l h a s in.3) t = y*K„*--jhJz C j/z J F o r t h i s c u r v e t o s t a r t a t t h e o r i g i n o f a p l o t o f R ( t ) v e r s u s t we c h o o s e s u c h t h a t a t £ = 0 , t h a t i s : ^ To g e t a q u a l i t a t i v e p i c t u r e o f t h e c u r v e we n o t e t h a t f o r s m a l l R we have:Jg; ^ Jj[ 3 3 1 ^ t h e c u r v e b e h a v e s l i k e : ft.*- (^^~J3 • On t h e o t h e r h a n d f o r l a r g e R we h a v e d&-J£ . T h e r e f o r e : /?- JGLi~t O ( ' ) . A l s o , s ince**?©, and &>o , we havej3^;>o, a l w a y s . Hence R ( t ) i s a l w a y s c o n c a v e downwards w i t h no p o i n t s o f i n -f l e c t i o n s . At/?=C, oo . ( s e e g r a p h I I I ) I I I . 3 . E l l i p t i c u n i v e r s e . H e r e we h a v e , «.<o. The s o l u t i o n i s : I I I .S) t - ,4 . -^" 'MM/ + c* • A g a i n we c h o o s e C 2 s u c h t h a t at/*-o , /?~o : - 1 2 -i s a maximum a t t f = - - . Thus a t t= "t , ^  - o . a. max' Jrfr B e c a u s e o f t h e c o n d i t i o n o f r e a l i t y we. must t a k e ^ = -+J^+<1 b e t w e e n ~k~0 a n d i - " ^ ^ J arid b e t w e e n t m a x and th e ti m e a t w h i c h t h e s e c o n d z e r o o f R o c c u r s vie must t a k e |j^f - - A a . The two b r a n c h e s a r e t h u s s y m m e t r i c a l a b o u t i = ^ " m a x * S u b s t i t u t i n g /<? ~— ^ i n t o t h e e q u a t i o n x^ re o b t a i n : I I I . 7) ^ ^ a . ^ c 2.<xS-^ S i n c e we ha v e symmetry a b o u t t w o v , t h e s e c o n d z e r o o f iliCLA, R o c c u r s a t : ^ = ^ C * « = ~ "~7=~ • ( s e e g r a o h I V ) III.J+. P a r a b o l i c u n i v e r s e . T h i s i s t h e s i m p l e s t o f t h e t h r e e c a s e s , s i n c e a ^ o . We h a v e : ^ * + Cj T h e r e f o r e : i n . 8 , n - d ^ f , S i n c e o.tisD , / ? = c 5 , we h a v e : III.9) C3 - O , A t / ? * ^ , - ^ . A l s o ^ i s al w a y s c o n c a v e dovjnward and 0 a t # = e P . ( s e e g r a p h V) P o r the p u r p o s e o f c o m p a r i n g . a l l t h r e e c u r v e s , g r a p h V I h a s b e e n p l o t t e d w i t h a r b i t r a r y v a l u e s f o r a and b, t h a t i s , )a\ = l (,o.V0~"*, J r = 3 (io\rs)~l . -13-CHAFTER I V : ANTI-MATTER. IV. 1. A l t e r a t i o n o f t h e g r a v i t a t i o n a l f i e l d e q u a t i o n . We a r e now p r e p a r e d t o t a c k l e t h e m a i n p u r p o s e o f t h i s work, n a m e l y t o c o n s i d e r c o n s e q u e n c e s o f t h e p o s s i b l e p r e s e n c e o f l a r g e amounts o f a n t i - m a t t e r i n t h e u n i v e r s e . I n p a r t i c u l a r x^ e w i s h t o i n v e s t i g a t e a u n i -v e r s e c o n s i s t i n g o f e q u a l amounts o f m a t t e r and a n t i -m a t t e r as p r o p o s e d b y M o r r i s o n and G o l d i n 195>6^ -^ « A u n i v e r s e i n w h i c h a n t i - m a t t e r h a s p o s i t i v e g r a v i t a -t i o n a l mass c o u l d be d e s c r i b e d b y t h e m o d e l s t r e a t e d p r e v i o u s l y p r o v i d e d t h e s m a l l e s t u n i t s o f m a t t e r o r a n t i - m a t t e r a r e a t l e a s t o f t h e s i g e o f g a l a x i e s , so t h a t no a n n i h i l a t i o n p r o c e s s e s w i l l h a v e t o be t a k e n i n t o a c c o u n t i n -the d y n a m i c a l e q u a t i o n s . However, M o r r i s o n and G o l d , a b a n d o n i n g th e p r i n c i p l e o f e q u i v a l e n c e , p r o p o s e d t h a t a n t i - n u c l e o n s h a v e n e g a t i v e g r a v i t a t i o n a l r e s t mass. I n a d d i t i o n , t o p r e s e r v e c h a r g e symmetry, t h e y a r g u e d t h a t a l l o t h e r f o r m s o f e n e r g y ( t h a t i s , e l e c t r o n s , p o s i t r o n s b i n d i n g e n e r g y ) o u g h t t o be a t t r a c t e d g r a v i t a t i o n a l l y b y b o t h n u c l e o n s and a n t i - n u c l e o n s . A t p r e s e n t i t i s I m p o s s i b l e t o t e s t t h i s h y p o t h e s i s i n the l a b o r a t o r y b e c a u s e o f t h e s h o r t l i f e o f a n t i - m a t t e r i n t h e p r e s e n c e o f m a t t e r ; one c a n n o t make an a n t i - p r o t o n l i v e l o n g enough t o see i t r i s e o r f a l l u n d e r g r a v i t y . • Q u e s t i o n i n g t h e P r i n c i p l e o f E q u i v a l e n c e r e q u i r e s -Ik-e x a m i n i n g t h e e x t e n t t o w h i c h t h i s p r i n c i p l e i s e s t a b -l i s h e d t o d a t e . The t e r m " P r i n c i p l e o f E q u i v a l e n c e " i s , u n f o r t u n a t e l y , u s e d n o t a l w a y s w i t h t h e same m e a n i n g a t t a c h e d t o i t . The G e n e r a l T h e o r y o f R e l a t i v i t y r e s t s v e r y h e a v i l y on two v e r s i o n s o f t h e E q u i v a l e n c e P r i n c i p l e , t h e s t r o n g and t h e weak one, as p o i n t e d o u t b y D i c k e 9 I n t h e s t r o n g v e r s i o n i t i s assumed t h a t t h e l a w s o f p h y s i c s a r e i n d e p e n d e n t o f time and p o s i t i o n i n a f r e e l y f a l l i n g l a b o r a t o r y . I n t h e weak f o r m on t h e o t h e r h a n d i t i s assumed t h a t i n a u n i f o r m g r a v i t a t i o n a l f i e l d , a l l b o d i e s f a l l w i t h e q u a l a c c e l e r a t i o n i n d e p e n d e n t l y o f t h e i r s t r u c t u r e . T h i s was t h e s t a t e m e n t t h a t E o t v o s t r i e d t o j u s t i f y I n h i s famous e x p e r i m e n t . The s t r o n g and t h e weak f o r m o f t h e E q u i v a l e n c e P r i n c i p l e a p p e a r q u i t e d i f f e r e n t . A c t u a l l y t h e y a r e r e l a t e d as c a n be s e e n f r o m t h e f o l l o w i n g c o n s i d e r a t i o n s . The d e f i n i t i o n o f t h e s t r o n g v e r s i o n o f t h e E q u i v a l e n c e P r i n c i p l e i m p l i e s t h e p o s i t i o n and time c o n s t a n c y o f b o t h t h e s t r o n g and t h e weak i n t e r a c t i o n s , i n c l u d i n g g r a v i t y . T h i s s t a t e m e n t i s p a r t i a l l y s u b s t a n t i a t e d b y e x p e r i m e n t . Prom t h e a c c u r a t e E o t v o s e x p e r i m e n t s i t c a n be d e d u c e d t h a t n u c l e a r b i n d i n g e n e r g y i s p o s i t i o n i n d e p e n d e n t t o w i t h i n one p a r t i n one h u n d r e d t h o u s a n d . F u r t h e r t h e c o u p l i n g c o n s t a n t o f e l e c t r o m a g n e t i c i n t e r -a c t i o n , A , i s known t o be c o n s t a n t w i t h g r e a t a c c u r a c y , by observation of l i g h t from distant galaxies,' Thus, at l e a s t f o r the strong Interactions, we expect p o s i t i o n independency. I f t h i s were not. so, the following s i t u a -t i o n would a r i s e . F i r s t one must keep i n mind that a l l matter consists of atoms.composed of protons, neutrons, and electrons, held together almost exclusively by the strong forces. Now i f an atom i s displaced, a change i n i t s binding energy and hence i n i t s t o t a l r e s t energy, ( that:is, binding energy plus -energy due to mass }' should r e s u l t . As a consequence the I n e r t i a l mass p f the atom w i l l a l t e r . Then, since binding energy and t o t a l r e s t energy are equivalent to i n e r t i a l mass, one should expect a. v a r i a t i o n i n the r a t i o of binding energy to t o t a l r e s t energy of the atom; But t h i s i s contrary to the experimental data given above. On the other hand when we consider the weak interactions and gravity, the s i t u a t i o n i s quite d i f f e r e n t . Their contribution to the binding energy of atoms i s so n e g l i g i b l e that i t cannot be measured. AT so one Unless one wishes to assume the main part of the t o t a l r e s t energy also to be caused by coupling v i a s e l f -energy e f f e c t s . The mass difference between iso t o p i c spin multiplets at l e a s t can be explained by the (Q)>(9) difference i n coupling to the electromagnetic f i e l d . - 1 6 -c a n n o t make any d e d u c t i o n s f r o m t h e E o t v o s e x p e r i m e n t as f a r as t h e c o n s t a n c y o f t h e s e i n t e r a c t i o n s i s c o n -c e r n e d . To t e s t w h e t h e r the g r a v i t a t i o n a l c o n s t a n t and the weak c o u p l i n g c o n s t a n t s v a r y w i t h t i m e , D i c k e i n v e s t i g a t e d t h e c o n s e q u e n c e s o f s u c h a t i m e d e p e n d e n c y b y e x a m i n i n g g e o l o g i c a l , a s t r o n o m i c a l , and b i o l o g i c a l d a t a . He c o u l d f i n d no e v i d e n c e i n f a v o u r o f t h e i r c o n s t a n c y . a c t i o n s , t h e r e i s no d e f i n i t e e x p e r i m e n t a l s u p p o r t f o r t h e s t r o n g v e r s i o n o f t h e E q u i v a l e n c e P r i n c i p l e . However, i f we n e g l e c t the c o n t r i b u t i o n s o f t h e weak i n t e r a c t i o n s and g r a v i t a t i o n t o the b i n d i n g e n e r g y o f atoms, we a r r i v e a t a weakened f o r m o f t h e s t r o n g v e r s i o n o f t h e E q u i v -a l e n c e P r i n c i p l e t h a t i s b a c k e d b y the a c c u r a t e E o t v o s e x p e r i m e n t , and o t h e r s . I t t h e n a p p e a r s t h a t t h e weak f o r m o f the E q u i v a l e n c e P r i n c i p l e i s a n a p p r o x i m a t i o n o f the s t r o n g f o r m o f t h e E q u i v a l e n c e P r i n c i p l e . i s f o r t h e weak f o r m o f t h e E q u i v a l e n c e P r i n c i p l e , l e t us d e s c r i b e t h i s e v i d e n c e . T h u s , e x c e p t i n t h e c a s e of t h e s t r o n g i n t e r S i n c e . the o n l y s u b s t a n t i a l e v i d e n c e a v a i l a b l e The d i a g r a m r e p r e s e n t s a c r o s s -s e c t i o n o f a s i m p l e E o t v o s v a r i o u s s u b s t a n c e s a r e s u s -p e n d e d f r o m e n d s a and b . -17-The d i f f e r e n c e i n h e i g h t o f w e i g h t s a t a and b a l l o w s f o r t h e s p a t i a l v a r i a t i o n o f t h e e a r t h ' s g r a v i t a t i o n a l f i e l d . I f t h e r e i s a s t r u c t u r e d e p e n d e n c y f o r d i f f e r e n t b o d i e s i n the e a r t h ' s g r a v i t a t i o n a l f i e l d , t h e n t h i s w o u l d show I n a d e f l e c t i o n o f t h e beam, s i n c e t h e g r a v -i t a t i o n a l c o n s t a n t c a n t h e n be w r i t t e n ft' - ( I + ^) > X b e i n g a c o n s t a n t , and Yo t h e g r a v i t a t i o n a l c o n s t a n t f o r a s t a n d a r d s u b s t a n c e s u c h as w a t e r . F u r t h e r , a d e f l e c t i o n i n t h e h o r i z o n t a l p l a n e s h o u l d a l s o be n o -t i c e d due t o t h e s u n ' s o r moon's f i e l d when the beam i s p l a c e d p a r a l l e l t o a m e r i d i a n . I f f o r two s u b s t a n c e s s u c h as p l a t i n u m and m a g n a l i u m ( % */t /f £ fo 6 ^7$ J a t ends a and b t h e g r a v i t a t i o n a l c o n s t a n t s a r e YA J Yg. , then^Xfl+Xjexia = ^0(1^X4) ' T h e n o b v i o u s l y - = ~X*. - y-g, . E o t v o s f o u n d t h a t w i t h i n e x p e r i m e n t s . ! ^* ~% l i m i t s X<i-Tg. <• fULio" • Hence i t f o l l o w s that; the vreak f o r m o f t h e E q u i v a l e n c e P r i n c i p l e i s s a t i s f i e d t o w i t h -i n one p a r t i n t w e n t y m i l l i o n . -* I t s h o u l d be m e n t i o n e d t h a t E o t v o s p e r f o r m s h i s e x p e r i m e n t s on what B o n d i ^ 0 ) c a l l s t h e p a s s i v e g r a v i t a t i o n a l mass o f a body. A c c o r d i n g t o B o n d i a bod y ' s p a s s i v e g r a v i t a t i o n a l mass i s t h e mass a c t e d upon b y t h e g r a v i t a t i o n a l f i e l d s . On t h e o t h e r h a n d , h e c a l l s the mass f r o m w h i c h g r a v i t a t i o n a l f i e l d s o r i g -i n a t e t h e a c t i v e g r a v i t a t i o n a l mass. The p o i n t t o n o t e i s t h a t E o t v o s s e t s up the a p p a r a t u s i n s u c h a way as t o d e t e c t a r o t a t i o n o f t h e t o r s i o n b a l a n c e beam c a u s e d b y t h e a c t i o n o f t h e g r a v i t a t i o n a l f i e l d o f t h e e a r t h , t h e s u n a n d / o r t h e moon. - 1 8 -I n c o n n e c t i o n w i t h t h e above e x p e r i m e n t s D i c k e r e p o r t s t h a t he i s p r e s e n t l y w o r k i n g on two E o t v o s t y p e e x p e r i m e n t s t h a t r e p r e s e n t an o r d e r o f m a g n i t u d e improvement o v e r E o t v o s f i n d i n g s d - O . I t must be p o i n t e d o u t , however, t h a t t h e above m e n t i o n e d t e s t s f o r t h e E q u i v a l e n c e P r i n c i p l e a p p l y t o one k i n d o f m a t t e r o n l y , n a m e l y , m a t t e r p r e v a i l i n g i n o u r p a r t o f t h e u n i v e r s e . I t w o u l d be d i f f i c u l t t o t e s t the E q u i v a l e n c e P r i n c i p l e f o r a n t i - m a t t e r b y u s i n g p r e s e n t day e x p e r i m e n t a l t e c h n i q u e s . T h e r e i s n e v e r t h e l e s s one p o s s i b l e a p p r o a c h t o t e s t - t h e p o s t u l a t e d a n t i - g r a v i t y o f a n t i - m a t t e r b a s e d on f i e l d t h e o r e t i c a l a r g u m e n t s . . The g e n e r a l b e l i e f i s t h a t t h r o u g h i n t e r a c t i o n w i t h t h e meson f i e l d t h e n u c l e o n i s d i s s o c i a t e d p a r t o f t h e t i m e i n t o n u c l e o n -a n t i - n u c l e o n p a i r s . L e t u s d e n o t e t h e d i s s o c i a t i o n c o n s t a n t b y 9, T h e n t a k i n g t h e h y p o t h e s i s o f M o r r i s o n and G o l d i n t o a c c o u n t , t h e i n e r t i a l mass I and t h e w e i g h t ¥ o f a c o m p l e t e atom a r e : 1 = A f\ + ZTI , and where A i s t h e mass number, M t h e n u c l e o n mass, Z t h e a t o m i c number, and m t h e e l e c t r o n mass. T a k i n g t h e r a t i o E o f t h e i n e r t i a l t o g r a v i t a -The much weaker e f f e c t o f e l e c t r o n - p o s i t r o n vacuum p o l a r i z a t i o n h a s b e e n n e g l e c t e d s i n c e i t w i l l n o t a f f e c t the argument. -19-t i o n a l mass and c o n s i d e r i n g o n l y f i r s t terras i n & we o b t a i n : E = I + &{ I - Zfcj . I f dP were known a c c u r a t e l y we c o u l d p e r f o r m an E o t v o s t y p e e x p e r i m e n t on two s u b s t a n c e s d i f f e r i n g i n 2f v a l u e s and t h u s o b t a i n a d i f f e r e n c e i n t h e i r E v a l u e s . On the o t h e r h a n d s i n c e f o r n u c l e o n s & i s u n c e r t a i n , t h e E o t v o s e x p e r i m e n t s w o u l d s e t an u p p e r l i m i t t o O • As an example c o n s i d e r C a r b o n and U r a n i u m 11, /\ = 3.3lJ. T h i s ' g i v e s : e < k f ^ k i - °-  0 0 0 7 7 • Thus t h e v i r t u a l n u c l e o n - a r t t i - n u c l e o n f o r m a t i o n w o u l d be no g r e a t e r t h a n ,077% i f a n t i - n u c l e o n s h a d n e g a t i v e g r a v i t a t i o n a l mass. S c h i f f ( 1 2 ) a r g U e s t h a t on g r o u n d s o f t h e vacuum-p o l a r i z a t i o 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 and t h e E o t v o s e x p e r i m e n t s , p o s i t r o n s c a n n o t h a v e n e g a t i v e g r a v i t a t i o n a l mass. T h i s , however, does n o t i n v a l i d a t e M o r r i s o n and G o l d ' s h y p o t h e s i s , f o r i t p o s t u l a t e d t h a t p o s i t r o n s a r e a t t r a c t e d b o t h t o n u c l e o n s a n d a n t i - n u c l e o n s f o r r e a s o n s o f c h a r g e symmetry. On t h e o t h e r h a n d , as S c h i f f a l s o p o i n t s o u t t h e vacuum p o l a r i z a t i o n b y t h e meson f i e l d i s l i t t l e u n d e r s t o o d and c a l c u l a t i o n s b a s e d on c o n v e n t i o n a l meson f i e l d - t h e o r y c a n n o t be c o n s i d e r e d as r e l i a b l e . T h u s , on g r o t u i d s of t h e p r e c e d i n g g e n e r a l a r g u -ments one c a n n o t e x c l u d e the h y p o t h e s i s o f M o r r i s o n -20-and G o l d . I n the p r e s e n t work an a t t e m p t i s made t h e r e -f o r e t o i n v e s t i g a t e c o s m i c c o n s e q u e n c e s o f a p a r t i a l v i o l a t i o n o f t h e E q u i v a l e n c e P r i n c i p l e , w h i c h was f i r s t c o n s i d e r e d b y M o r r i s o n and G o l d . T h i s r e q u i r e s a d r a s t i c change i n t h e f i e l d e q u a t i o n II.3). C o n s i d e r -i n g o n l y volume e l e m e n t s c o n t a i n i n g l a r g e numbers o f g a l a x i e s and s u p p o s i n g t h a t t h e s m a l l e s t u n i t s o f m a t t e r o r a n t i - m a t t e r a r e o f ' t h e s i z e o f g a l a x i e s , t h e a v e r a g e g r a v i t a t i o n a l r e s t mass d e n s i t y , a p p e a r i n g as f i n I I . 3 ) , s h o u l d be z e r o . B e c a u s e o f t h e i r m o t i o n , however, b o t h g a l a x i e s and a n t i - g a l a x i e s s h o u l d c o n t r i b -u t e a t e r m t o t h e r i g h t h a n d s i d e o f I I . 3 ) , r e p r e s e n t i n g t h e g r a v i t a t i o n a l e f f e c t o f t h e k i n e t i c e n e r g y d e n s i t y o f t h e u n i v e r s e . S i n c e i t i s known f r o m e x p e r i m e n t t h a t a n t i - m a t t e r h a s p o s i t i v e i n e r t i a l mass, we c a n r e p r e s e n t t h e k i n e t i c e n e r g y d e n s i t y b y 7~~g (we c o n s i d e r o n l y n o n - r e l a t i v i s t i c a p p r o x i m a t i o n s ) , where f i s now t h e i n e r t i a l mass d e n s i t y o f b o t h m a t t e r and a n t i - m a t t e r . A c c o r d i n g t o t h e h y p o t h e s i s o f M o r r i s o n and G o l d t h i s k i n e t i c e n e r g y d e n s i t y c o n s t i t u t e s p o s i t i v e g r a v i t a t i o n a l mass d e n s i t y / ^ ^ f• Hence t h e f i e l d e q u a t i o n II.3) c h a n g e s i n t o : i v . D r y = ««*irfl. e q u a t i o n s II.1) and I I . k ) r e m a i n i n g u n c h a n g e d . I V . 2 . G e n e r a l s o l u t i o n o f the c o s m o l o g i c a l  E q u a t i o n . 21-The e q u a t i o n s f o r an i s o t r o p i c u n i v e r s e a r e now: I I . 3 ) j IV.3) becomes u p o n s i m p l i f i c a t i o n : i v . k ) 1 = - t«lA f P u r t h e r , a s i m p l e i n t e g r a t i o n o f iV.2) g i v e s ' J > ^ ) = j ^ ^ . i } where the c o n s t a n t M h a s t h e same meaning as i n p a r a -g r a p h I I . 3 . S u b s t i t u t i n g t h i s v a l u e o l f f t ) I n t o I V . l j J , one o b t a i n s : tt = -XHA.(& o r s i m n l y : (-fa) where O b v i o u s l y , p i s a p o s i t i v e d i m e n s i o n l e s s c o n s t a n t . E q u a t i o n IV.5) c a n now be e a s i l y i n t e g r a t e d i n t h e f o r m % ~ ~~P ^ / w h i c h i s e q u i v a l e n t t o : l v » 6 ) MT ~ °^ * ^ > where << i s a c o n s t a n t o f i n -t e g r a t i o n w i t h d i m e n s i o n o f (time)""*", t h e meaning o f w h i c h w i l l become c l e a r l a t e r ( see p a r a g r a p h V . 3 ) . The s i m p l e s t way t o s o l v e I V . 6 ) i s t o i n t e g r a t e t e r m b y t erm: 1 ~ J € ^ «K » where C.'.is a c o n s t a n t o f i n -t e g r a t i o n , j u t J ( , - * + "Jit*- %f«3*j'* ' i> i v.7) U i i K - f & R - J t l - J — - r + C_ . G T C L j t h . U : Case. 3, cL=*o . - 2 2 -The c o n s t a n t C i s s o c h o s e n t h a t a t / = o , lZ=°. However i t c a n n o t be e v a l u a t e d d i r e c t l y . The method f o r i t s c a l c u l a t i o n w i l l be shovm i n t h e n e x t c h a p t e r , appen-d i x A. L e t i t s u f f i c e h e r e to g i v e i t s v a l u e a s r IV.8) C = ft (s&p-.WtfJ , w h i c h i s d i m e n s i o n l e s s . The c o m p l e t e s o l u t i o n o f e q u a t i o n I V . £ ) c a n now be w r i t t e n : IV. The shape o f t h e c u r v e o f e q u a t i o n IV.9) c a n be d e t e r m i n e d i f one n o t e s t h a t for/?=°°, /?= °< . T h e r e -f o r e f o r l a r g e R, R h a s t h e f o r m R-o(t -f c , c b e i n g a c o n s t a n t . F u r t h e r f o r , H . G r a p h V I I , V I I I , a n d IX i l l u s t r a t e t h e s e q u a l i t a t i v e f e a t u r e s . As i n c h a p t e r I I I , h e r e one a l s o h a s t h r e e p o s -s i b l e m o d e l s , a c c o r d i n g t o w h e t h e r o O o ,<* = o , o<<o. I V . 3 . T y p e s o f u n i v e r s e s . C a s e l:o(>^. T h i s c a s e r e p r e s e n t s e x p a n s i o n and i s shown on g r a p h V I I . The c u r v e a p p r o a c h e s t h e s t r a i g h t l i n e R=*i+c a s y m p t o t i c a l l y . C a s e 2 : < ^ ° . T h i s c a s e i s i l l u s t r a t e d b y g r a p h V I I I w h i c h shows c o n t r a c t i o n , o f t h e u n i v e r s e . I f t h i s c a s e w o u l d c o r r e s p o n d t o o b s e r v a t i o n , we c o u l d f i t i t w i t h t h e o b s e r v e d ' r a t e o f c o n t r a c t i o n o f t h e u n i v e r s e , and t h e n c h o c s e some z e r o o f t i m e , w h i c h i s t a k e n a r b i t r a r i l y h e r e . -23-C a s e 3"d.=s0. If<<=-o , we. h a v e : ^ j = o , t h e r e f o r e R:= c o n s t a n t = k. T h i s o b v i o u s l y i s a s t a t i c u n i v e r s e a s s e e n f r o m g r a p h I X . I t i s h o w e v e r i n u n s t a b l e e q u i -l i b r i u m . . To show t h i s - , we e x p a n d R i n t e r m s o f °C a b o u t <K= Oln. a T a y l o r s e r i e s : %(«.)= &(o) + ^jf* + ^J^***' ' " F r o m above we have foro(.=o , fl = -h . T h e r e f o r e : R(o)= -k A l s o ££J? = f \£ • U s i n g t h e p r e v i o u s r e s u l t o f H(o)~ A : F o r t h e h i g h e r d e r i v a t i v e s : Hence: IV.10): RfaJ {U**)* • E q u a t i o n IV.10) shows t h a t " f or^i=0 we g e t t h e s t a t i c -case., B u t f o r a n o n - v a n i s h i n g o< , R goes t o i n f i n i t y . T h e r e i s t h e r e f o r e u n s t a b l e e q u i l i b r i u m . P r e s e n t day e x p e r i m e n t a l o b s e r v a t i o n i n d i c a t e s t h a t t h e u n i v e r s e can o n l y be d e s c r i b e d b y an e x p a n -d i n g m o d e l . H e n c e , o f t h e three- m o d e l s d i s c u s s e d , h e r e , t h e o n l y one t o be c o n s i d e r e d i s t h e one w i t h p o s i t i v e * * . -2k-CHAPTER V: COMPARISON WITH EXPERIMENTAL EVIDENCE. V . l . I n t r o d u c t i o n . I n t h i s c o n c l u d i n g c h a p t e r i t w i l l be shown t h a t o u t o f t h e t h r e e c o s m o l o g i c a l m o d e l s d i s c u s s e d i n c h a p t e r t h r e e , t h e e l l i p t i c m o del i s t h e o n l y one c o m p a t i b l e w i t h p r e s e n t day a s t r o n o m i c a l d a t a . T h i s m o d e l w i l l t h e n be f i t t e d t o t h e a c t u a l u n i v e r s e a n d compared w i t h t h e m o d e l c o n t a i n i n g a n t i -m a t t e r . I n o r d e r t o c a r r y out t h e above p l a n i t i s n e c e s -s a r y t o d e t e r m i n e the two p a r a m e t e r s o c c u r r i n g i n t h e e q u a t i o n f o r R ( t ) . T h i s i s r e a d i l y done b y e m p l o y i n g the e x p e r i m e n t a l l y known p a r a m e t e r s V . l ) m j and ^ X = 8JJ±1 , 2^ b e i n g t h e p r e s e n t t i m e . I f one a d o p t s M c V i t t i e ' s n o t a t i o n , ( 1 3 ) ^ i s the H u b b l e c o n s t a n t , t h a t i s , t h e c o n s t a n t o f p r o p o r t i o n a l i t y i n the v e l o c i t y - d i s t a n c e r e l a t i o n f o r n e b u l a e . I t s r e c i p r o c a l h a s t h e d i m e n s i o n of t i m e , and i t I s i n c a s e o f n e g a t i v e h2 an u p p e r l i m i t t o the age o f the u n i v e r s e . h i h a s t h e r a n g e o f v a l u e s : " ^ T h i s a p p e a r s i m m e d i a t e l y i f one p e r f o r m s t h e f o l l o v i n g i n t e g r a t i o n : ^olA * - A , , w h i c h l e a d s t o tfA- U,)~', where t h e d e f i n i t i o n  0 Ufa) - I a t the p r e s e n t t i m e "*»«. i s a d o p t e d . F u r t h e r , i f the same r a t e o f e x p a n s i o n s i n c e " C r e a t i o n " i s ; assumed, and si nee h £ i s known t o be n e g a t i v e , t^ = (f/Jsets an u p p e r l i m i t t o the age o f t h e u n i v e r s e . -25 O. ffC (/o'y-sj'' £ J L o . l i l C i o ' y i J ' . h 2 , on the o t h e r hand, i s t h e a c c e l e r a t i o n p a r a m e t e r , t h a t i s , i t m e a s u r e s t h e a c c e l e r a t i o n o f t h e e l e m e n t s i n t h e u n i v e r s e , and h e n c e h a s d i m e n s i o n s o f ( t i m e ) " 2 . I n t h e f o l l o w i n g d i s c u s s i o n h 2 w i l l a p p e a r o n l y i n t h e d i m e n s i o n l e s s number , t h e v a l u e o f w h i c h i s -(3±.V) o r -(t-Ct.S), as c a l c u l a t e d b y M c V i t t i e ( 13) and Sandage (^-k) r e s p e c t i v e l y . ~J* w i l l t h e r e f o r e be c o n s i d e r e d t o have t h e r a n g e : — 3-8 -jr*. - I-? t I t I s now p r o p o s e d t o o b t a i n a minimum and maximum c u r v e f o r e a c h m o d e l , c o r r e s p o n d i n g t o t h e e x p e r i m e n t a l l y known u p p e r and l o w e r l i m i t s o f hj_ and h 2 . V . 2 . E x c l u s i o n o f t h e h y p e r b o l i c and p a r a b o l i c  m o d e l s . I n c h a p t e r t h r e e t h e s o l u t i o n t o e q u a t i o n 11.13) gave t h r e e c o s m o l o g i c a l m o d e l s , f o r a = o,<a->° , a-<.o9 I t i s now s i m p l e t o show t h a t t h e models w i t h ayo , OL = O , a r e i n c o m p a t i b l e w i t h a s t r o n o m -i c a l o b s e r v a t i o n . P o r t h i s p u r p o s e , e q u a t i o n s 11.13) and I I I . l ) w i l l be s u b s t i t u t e d i n t o t h e e q u a t i o n s V . l ) w h i c h d e f i n e hj_ and h 2 . Two e q u a t i o n s r e s u l t : - = ^ L ^ , Rp d e n o t i n g R(iAJ, t h a t i s , t h e v a l u e o f R a t t h e p r e s e n t t i m e ^ , The s e c o n d e q u a t i o n i m m e d i a t e l y g i v e s : V.2) 4 — M fir , - 2 6 -w h i c h u p o n s u b s t i t u t i o n i n t o t h e f i r s t e q u a t i o n g i v e s : «. - Hh f^i +• x ^) o r a l n n l y : E q u a t i o n V . 3 ) i n e v i t a b l y l e a d s t o t h e c o n c l u s i o n t h a t a ^ o , since-/,**© , and X ^jjTi.  3 " *  < 0 r e p r e s e n t s o f c o u r s e t h e e l l i p t i c m o d e l , e q u a t i o n I I I . 5 ) The u n i v e r s e c o n t a i n i n g a n t i - m a t t e r c a n t h e r e f o r e be compared o n l y w i t h t h e e l l i p t i c u n i v e r s e . V . 3 . C o m p a r i s o n o f the e l l i p t i c w i t h t h e a n t i - m a t t e r m o d e l . C a l c u l a t i o n s o f and /S o f e o u a t i o n IV.9) l e a d t o : V.Ij.) <*- « < * ' fi - -where t h e d e f i n i t i o n fi.(tn)=- I i s a d o p t e d . (see a p p e n d i x B f o r d e t a i l e d c o m p u t a t i o n s . ) S i m i l a r l y one f i n d s : The a d o p t i o n o f t h i s d e f i n i t i o n h a s t h e f o l l o w i n g a d v a n t a g e . The s c a l e f a c t o r R ( t ) i s r e l a t e d t o t h e r a d i a l d i s t a n c e r o f a g i v e n e l e m e n t o f t h e u n i v e r s e b y t h e e q u a t i o n ( s e e p a r a g r a p h I I . 3 ) s "2"* = :2? ( M t ) r i s t h e p o s i t i o n v e c t o r o f t h e e l e m e n t a t t = t„ , and t h u s i f R(if)= I , ( h e r e t„ = t A ) I t g i v e s t h e r a -d i a l d i s t a n c e o f any p a r t i c l e , a t t h e p r e s e n t t i m e . I t i s t h e n c l e a r t h a t t h e above d e f i n i t i o n e n a b l e s one t o d e s c r i b e t h e p a t h o f any p a r t i c l e t h r o u g h o u t I t s e n t i r e h i s t o r y , i f o n l y i t s p r e s e n t d i s t a n c e = i s known. ~ ^ r " v ' . a » ( r f a p k X : E l f t j t t i c a n d a . n f l - m a . t - t e r m o d . e £ s -- 2 7 -E q u a t i o n s V.ij.) and V . 5 ) i m m e d i a t e l y s u g g e s t a mea n i n g f o r t h e c o n s t a n t s , a, b . Thus «* i s d i r e c t l y r e l a t e d t o t h e H u b b l e c o n s t a n t h i , and w o u l d be e q u a l t o i t i f t h e d e c e l e r a t i o n h 2 were z e r o ; f m e a s u r e s the d e c e l e r a t i o n ; a w o u l d e q u a l h i f o r v a n i s h i n g d e c e l e r a t i o n ; b i s t w i c e t h e d e c e l e r a t i o n . The minimum and maximum v a l u e s o f , f$ , a,b, c o r r e s p o n d i n g t o minimum ai d maximum h i and h 2 a r e now computed. T h e y a r e : 1, M „ j3 a. j ^ minimum .1)4.6 - 3 . 8 3 . 2 5 x 1 0 - 3 3 . 8 - . l b , 3 .l6JEj> v a l l i e s maximum v a l u e s . 2 3 2 - 1 . 8 3 . 8 2 x l 0 - 2 1 . 8 - . l i j . 0 . 1 9 6 T h e s e v a l u e s a r e t h e n s u b s t i t u t e d i n t o t h e i r c o r r e s p o n d i n g e q u a t i o n s r e s u l t i n g i n a minimum and maximum c u r v e f o r e a c h m o d e l . The c u r v e s a r e p l o t t e d on g r a p h X. The ag e s o f t h e u n i v e r s e f o r t h e m o d e l s a r e g i v e n b y th e a b c i s s a ' s f o r t h e o r d i n a t e / ^ / . Thus f o r t h e e l l i p t i c model t h e l o r e s e n t age o f t h e u n i v e r s e l i e s b e t w e e n 2 . 1 and 2 . 7 b i l l i o n y e a r s , a n d f o r t h e mo d e l c o n t a i n i n g a n t i - m a t t e r , b e t w e e n 1 . 3 and 1 . 9 5 b i l l i o n - 2 8 -years. Hence, for a given and the anti-matter model leads to a much younger universe than the cor-responding matter universe. This result considerably weakens the hypothesis of Morrison and Gold who considered a universe which (a) consists of equal amounts of matter and anti-matter, and in which (b) the anti-matter has negative gravitational rest mass. Por, there i s independent astronomical, and even geological evidence that the age of the universe must be in excess of at least 9 2x10 years. Thus, for instance, the age of the earth i s estimated to be between 3.k. and 5x10^ years, the ages of meteorites between k.£ and 5>xl0^ years, and the ages o f bhe globular clusters NGC £272 9 (17) 9 NGC 6205 are given as 5>xl0 years and 2x10 years respectively. Analytically the present time t for the e l l i p t i c model is given by: t Ji I Then for maximum hj_ and h g , ^ *'v«° years, and for minimum h^ and h 2, ir-t.7fio^ years. Similarly for the anti-matter model: •> which for corresponding minimum and maximum hi and h 2 fixes the age of the universe between 1.95>xlO*9 year's and 1 .3x10° years. -29-O b v i o u s l y , t h e r e s u l t o f t h e p r e s e n t w o r k d o e s n o t e x c l u d e t h e p o s s i b i l i t y t h a t o n l y one o f t h e h y p o t h e s i s i s t r u e , n a m e l y ( a ) o r ( b ) . B u t a u n i v e r s e i n w h i c h b o t h ( a ) a n d ( b ) a r e t r u e , seems t o b e i n c o m -p a t i b l e w i t h o b s e r v a t i o n a l d a t a a v a i l a b l e a t p r e s e n t . -30-APPENDIX. This appendix is added with the intention of presenting some of the details of calculations of chapters four and five, which otherwise would have distracted too much from the main line of the argu-ment. A. Determination of C. C is the constant of integration occurring i n equation IV .7). Under the transformation A 1) ' * - / • > equation IV.7) becomes: _ <— I ^  /I A.2) x = A where x and y are now dimensionless variables, and the constant A i s : A.3) /j ~ £ - -J*/* The transformation A.l) has the advantage of permitting quick calculations of -ooints of equation IV.7) for any ^  andfi , once equation A.2) is fixed. It.also allows the evaluation of C, which by a direct substitution of /{=><?, fi***? Is rather d i f f i c u l t to de-termine . Suppose now that equation A.2) is subjected to the condi tion: y - ( at * = Then A assumes the value -.5713. Further, from a graph of A.2) with this value -31-o f A,- i t i s f o u n d t h a t ^ o a t = ~ • W # I f new .lk'8'8 i s added t o a l l v a l u e s o f A . 2 ) , one o b t a i n s t h e i n i t i a l c o n d i t i o n ^ - ^ a t * - ^ , w h i c h t r a n s f o r m s i n t o the d e s i r e d i n i t i a l c o n d i t i o n o f e q u a t i o n IV.7) , n a m e l y : a t t-=0. The c o n s t a n t A h e n c e changes t o : A -y-. Mrt - -• * 7 1 3 -f- -wS6 — — • <f**s . W i t h the h e l p o f e q u a t i o n A%'2)» C c a n t h e n be d e t e r -m i n e d : IV.8) c ~ ( ^ f i . B. C a l c u l a t i o n o f ^ and 0 . P o r t h e p u r p o s e o f c o m p u t i n g ^ and/? e q u a t i o n s V . l ) , . w h i c h d e f i n e hj_ and h^ , a r e employ e d . F u r t h e r , as s t a t e d i n p a r a g r a p h V . 3 t h e d e f i n i t i o n R(-bJ=/ h o l d s . One t h e n a r r i v e s a t t h e e q u a t i o n s : al 4 ^= -A and fa*-* =-^, s i n c e and lUt/iJ = -cLT—— . Upon t a k i n g l o g s , t h e s e two e q u a t i o n s t a k e on t h e f o r m : w h i c h a r e e a s i l y s o l v e d f o r < * a n d / ? . T h u s : B . l ) ^ -*i / D i r e c t e v a l u a t i o n o f x f o r J=- / i n A.2) gave jr- — MM. A t - ? - - ' , however, t h e s l o p e i s * * '%:2lf<>°. Thus an u p p e r l i m i t t o the i n t e r s e c t i o n o f t h e c u r v e w i t h the x - a x i s i s g i v e n b y «'--.w«-/liJ where 4 * - ^ ^ . B u t = and t h e r e f o r e &x=.ooooof. Hence a t ^ = o, x = - - m « , s i n c e t h e e f f e c t o f 4* i s l e s s t h a n 5> i n t h e s i x t h f i g u r e . C. P o i n t s f o r t h e maximum and minimum c u r v e s  o f b o t h m o d e l s . I n t h e c a s e o f t h e a n t i -m a t t e r u n i v e r s e , b e c a u s e o f t h e g r e a t r e d u c t i o n i n c o m p u t a t i o n , t h e p o i n t s o f t h e minimum a n d maximum c u r v e were c a l c u l a t e d w i t h t h e h e l p o f e q u a t i o n A.2) and t h e t r a n s f o r m a . t i o n , e q u a t i o n A . l ) . T h e s e c a l c u l a -t i o n s a r e summarized i n t a b l e I . G r a p h X I r e p r e s e n t s e q u a t i o n A . 2 ) . I n t h e c a s e o f t h e e l l i p t i c m o del t h e computa-t i o n s were s h o r t enough n o t t o w a r r a n t t h e u s e o f a t r a n s f o r m a t i o n i n t o d i m e n s i o n l e s s v a r i a b l e s . D i r e c t s u b s t i t u t i o n o f the v a l u e s o f a and b i n t o e q u a t i o n I I I . 5 ) l e a d t o t h e c o o r d i n a t e s t a b u l a t e d i n t a b l e l l . -33-TABLE I COORDINATE POR ANTI-MATTER MODEL GENERAL CURVE : MINIMUM CURVE MAXIMUM CURVE I-i7jiio y s iO»h) 0 0 0 0 0 0 0 0.1 0 0.38 0 0.18 0.0004 0.2 0.50 0.76 0.02 0.36 0.0025 0.3 2.93 1.14 0.11 0.54 0.0082 0.4 9.56 • 1.52 0.39 0.72 0.0190- 0.5 22.00 1.90 0.89 0.90 0.0353 0.6 1.66 1.08 0.0831 0.8 3.90 1.44 0.1488 1.0 7.00 1.80 0.2280 .1.2 0.3220 1.4 -34-TABLE I I COORDINATES POR E L L I P T I C MODEL MINIMUM CURVE A 0 0 0.19 . 0.2 o.54 0.4 1.01 0.6 1.76 0.8 2.75 1.0 4.91 1.15 MAXIMUM CURVE a = -. Wo (to''*y*>J 0 0 0.13 0.2 0.37 0.4 0.82 0.6 1.35 0.8 2.09 1.0 3.27 1.2 5.88 1.4 Maximum R, act time Maximum R^. • • -35-BIBLIOGRAPRY PAPERS (1) Milne, E.A. and McCrea, W.H. Quat. J. Math. £, 73, 193k. (2) Bondi, H. and Gold, T. Mon. Not. R. Astr. Soc. 108, 252, 19k8. (3) Morrison, P. and Gold, T. Essays on Gravity. New Boston, New Hampshire, 1956. Dicke, R.H. Revs. Mod. Phys. 2£, 355, 1957. (5) Eotvos, R.V. Ann. Physik. 68, 11, 1922. (6) Wapstra, A.H. and Nijgh, G.J. Physica, 21, 796, 1955. (7) Minkowsky, R. and Wilson, O.C. Astronora. J. 122, 373, 1956. (8) Peynman, R.P. and Speisman, G. Phys. Rev. .9k., 500, 195k. (9) Marshak, Okubo, and Sudarshan. Phys. Rev. 106, 599, 1957-(10) Bondi, H. Revs..Mod. Phys. 29, k23, 1957. (11) Dicke, R.H. Science 129, No. 33k9, 621, 1959. (12) Schiff, L.I. Proc. Nat. Acad. Sciences !£, No. 1, 69, Jan.,1959. see also: Schiff, L.I. Phys. Rev. Letters 1, No. 7, 25k, 1958. - 3 b -(13) M c V i t t i e , G.C. Handb. P h y s . L I I I , kk5, 1959. ( I k ) Humason, M.L. M a y a l l , N.U. and Sandage, A.R. t A s t r o n o m . J . 61, 97> 195". (15) W i l s o n , J . T . R u s s e l l , R.D. and F a r q u h a r , R.M. Handb. Physik.. X L V I I , 335, 1956. (16) U r e y , H.C. M a t u r e , L o n d . 17J?, 321, 1955. (17) Sandage, A.R. Mem. S o c , Roy. S c i . L i e g e l i t , 25k, 1953. (18) Baum, W.A. A s t r o n o m . J . £9, k22, 195k. 

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