Volume 33A, number 6 P H Y S I C S L E T T E R S 30 November 1970

T H E P R O T O N A N D N E U T R O N M A S S E S A N D A C O N J E C T U R E F O R T H E G R A V I T A T I O N A L C O N S T A N T

I . J . GOOD Virginia Polytechnic Inst i tute and State Universi ty, Blacksburg, Virginia, USA

Received 24 September 1970

Some formulae are suggested connecting the velocity of light, Planck's constant, the f ine-s t ructure constant, the gravitational constant, and the masses of the electron, proton and neutron. They are accurate enough to suggest that Eddington's fundamental theory is likely to be partly right.

If E d d i n g t o n ' s f u n d a m e n t a l t h e o r y [1] con ta ins any e l e m e n t of t r u th , the not ion of "phase s p a c e " in h i s s e n s e m u s t be i m p o r t a n t . In the s p i r i t of the g e o m e t r i z a t i o n of p h y s i c s it i s t h e r e f o r e n a t u r a l to c o n j e c t u r e that the v o l u m e of a s p h e r o i d in th i s p h a s e s p a c e should be r e l a t e d to the fundamen ta l p h y s i c a l cons t an t s . In th is no te s o m e f o r m u l a e a r e m e n t i o n e d that suppor t t h i s c o n j e c t u r e . A l though t h e s e f o r m u l a e a r e b a s i c a l l y " n u m e r o l o g i c a l " , wi th only the g l i m m e r - ings of a t h e o r y , i t shou ld hot be f o r g o t t e n that , f r o m the point of v i e w of t h e o r e t i c a l p h y s i c s , the whole point in the c o s t l y m e a s u r e m e n t of p h y s i - ca l c o n s t a n t s i s in the hope of u l t i m a t e l y f inding r e l a t i o n s h i p s b e t w e e n them. N u m e r o l o g y i s good when it i s s i m p l e and a c c u r a t e enough to c o m - p e n s a t e fo r the l ack of a suppo r t i ng f u l l - d r e s s t h e o r y , e s p e c i a l l y if i t s e e m s to point t o w a r d s s o m e of the f e a t u r e s tha t such a t h e o r y should have .

In E d d i n g t o n ' s t h e o r y the "phase s p a c e " has R 1 = 10 =22(22 + 1) /2 r e a l a n d I 1 =6 = = 22(22 - 1) /2 i m a g i n a r y d i m e n s i o n s ; the "double phase s p a c e " has R2 = 136 = 42(42 + 1)/2 r e a l and 12 = 120 = 42(42 - 1) /2 i m a g i n a r y d i m e n s i o n s ; and the "quad rup l e p h a s e s p a c e " has R3 = 32896 = = 162(162 + 1)/2 r e a l and 13 = 32640 = 162(162 - 1) /2 i m a g i n a r y d i m e n s i o n s . B a s e d on t h e s e phase s p a c e s , t o g e t h e r wi th s o m e o b s c u r e a r g u m e n t s , Eddington m a d e e s t i m a t e s of v a r i o u s d i m e n s i o n - l e s s p h y s i c a l c o n s t a n t s inc lud ing the m a s s r a t i o r n p / m ¢ of the p r o t o n to the e l e c t r o n , the g r a v i - t a t i ona l cons t an t G and the n u m b e r N of p ro tons and e l e c t r o n s in the u n i v e r s e . It i s p l a u s i b l e to a s s u m e that the n u m b e r N ' of p r o t o n s , e l e c t r o n s , a n t i p r o t o n s and p o s i t r o n s i s 2N, o r at l e a s t that N ' i s jus t a s f u n d a m e n t a l a cons t an t a s N. Eddington m a d e u s e of a c o r r e c t i o n f a c t o r / 3 =

= 137/136 (Bond ' s cons tant ) but I sha l l r e p l a c e th i s by fl' = 1/(136a) w h e r e a is the f i n e - s t r u c t u r e cons tan t fo r which the c u r r e n t b e s t o b s e r v a t i o n a l e s t i m a t e i s g iven [2] by a = 1/ (137.036 02 + + 0.00021).

Le t the v o l u m e of unit s p h e r o i d s in phase s p a c e , double phase s p a c e and q u a d r u p l e phase s p a c e be deno ted by V 1, V 2 and V3. F o r e x a m p l e

V 1 = ~5 /120 , V3 = ~68 /68!

The fo l lowing n u m e r o l o g i c a l r e s u l t s a r e e x - t r a o r d i n a r i l y a c c u r a t e . F i r s t

m p / r n e ~ I I I 2 V 1 = 1836.118 , (1)

the o b s e r v e d va lue be ing 1836.109 =~ 0.011. F o r m u l a (1) i s e q u i v a l e n t to the r e m a r k by Lenz [3] that m p / m e ~ 67r5, and b e g i n s to g ive i t a g e o m e t r i c a l mean ing . Next (2)

(mn - r a p ) l i n e ~ V 1 / ~ ' = 2 .530884 + 0 .000004

the o b s e r v e d va lue be ing 2 .53090 + 0 .00025 so eq. (2) i s aga in wi th in e x p e r i m e n t a l e r r o r . E q u i - v a l e n t l y

m p / ( m n - rnp) ~ I l l 2 f l ' = 725.4848 + 0,0011 (3)

the o b s e r v e d va lue be ing 725.48 :~ 0.06. By ana logy with f o r m u l a (1) t h e r e migh t be a

p a r t i c l e whose m a s s i s e i t h e r m l = 1112 I3 V 3 m e o r m 2 = I213V3.me o r m 3 = I 113V3me . Although m3 s e e m s a p r i o r i a l i t t l e l e s s l ike ly than m l o r m 2 to c o r r e s p o n d to r e a l i t y , i t wi l l be u s e d h e r e b e c a u s e it g i v e s m o r e i n t e r e s t i n g r e s u l t s . S ince m 3 / m e ~ 10 -57 the p a r t i c l e migh t have a d i a m e t e r about 10 -19 t i m e s that of the e l e c t r o n , that i s , about 10 -32 cm. It migh t t h e r e f o r e be r e l a t e d to the " w o r m - h o l e s " ' o f W h e e l e r [e.g.4] which have

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d i a m e t e r s of o r d e r ,/G t[/c 3 ~ 10 .33 cm. Be tha t a s i t m a y , it i s r e m a r k a b l e tha t the d i m e n s i o n - l e s s c o n s t a n t

~ - ~ c ~ / r n 3 ~ N ' = 3 × 136 × 2256 (4)

t he le f t s i d e b e i n g (4.727 76 ± 0.001 1) x 1079 and the r i gh t s i d e 4.724 32 × 1079. The d i s c r e p a n c y i s 3.1 s t a n d a r d d e v i a t i o n s but the r a t i o of the l o g a r i t h m s of the two s i d e s of eq. (4) is about 1 . 0 0 0 0 0 2 7 so the d e g r e e of a g r e e m e n t s e e m s b e t t e r than the d e g r e e of d i s a g r e e m e n t , e s p e c i a l - ly a s p u b l i s h e d va lue s of t he s t a n d a r d d e v i a t i o n s of p h y s i c a l c o n s t a n t s a r e o f t en too low. M o r e o v e r , s i n c e , by eq. (3), m n = m p (1 + 1 / I l I 2 f l ' ) , wi th in e x p e r i m e n t a l e r r o r , it m igh t not be too ad hoc to w r i t e m~ = m 3(1 + 1 / R 1R 213 ' ) . Then

c / G / ' m ' 3 ~' 3 x 136 × 2256 , (5)

the e r r o r be ing only one p a r t in a m i l l i o n if the b e s t c u r r e n t e x p e r i m e n t a l v a l u e s f o r / ~ , c and G a r e a s s u m e d . The r a t i o of the l o g a r i t h m s of the

L E T T E R S 30 November t970

two s i d e s of eq. (5) d i f f e r s f r o m uni ty by l e s s lha~t one p a r t in a h u n d r e d mi l l i on . If eq. (5) i s e x a c t - ly t r u e then the p u b l i s h e d s t a n d a r d dev ia t ion f o r t he e x p e r i m e n t a l va lue of G = (6.673 2 ± 0.0031) x 10 -8 c m 3 / g s e c 2 can be r e d u c e d by a f a c t o r of 50. On the o t h e r hand , if eq. (4) is exac t , we would have

2 G : (6 .68293 ± 0.00006) x 10 -8 c m 3 / g s e c (6)

R e . f e r e , t c e s

[1] A. S. EddingIon, Fundamental theory. (University P r e s s , Cambridge, 1946).

[2] N. Baraseh-Schmidt , A. Barbaro-Gal t ier i , C. Brieman, S .E .Derenzo , L.R. Pr iee , A. Rittenberg, M. Roos, A.t t . Rosenfeld, P.Soding, C.G. Wohl, Par t ic le proper t ies (Janu'~ry 1970) Lawrence l{adia- tion Laborator3, Berkeley, California.

[3] F. Lenz, Phys. Rev. 82 (1951) 554. [4] J. A. Wheeler, "Curved empty space- t ime as the

building material of the physical world: an a s s e s s - ment", in Logie methodolog3' and philosophy of Scienee (Stanford University P r e s s , 1962) 361-374.

C O N T R A C T I O N D E S P H A S E S /3 E T 5 D U P L U T O N I U M S O U S L ' E F F E T D E L ' A U T O I R R A D I A T I O N A 4 . 2 ° g

J. J A C Q U E M I N * et R. L A L L E M E N T ** Sert,ice du Plutonium, Centre d 'Etudes Nu('16aires de Fonlenay aux Roses , France

Regu le 27 August 1970

Les correlations entre |es mesures de variations de longueur et de r6sistance en fonction du temps d'autoirradiat ion fi 4.2 OK nous ont amen6s "t penser q u c [e s phases ~ et 5 du plutonium devaient se eont rae ter sous l 'effet des dommages caus6s par l ' i r radiat ion. Nous awms mis en 5videnee exp6ri- mentatement cette contraction, par t ieul iSrement importante pour la phase 5.

Sous l ' e f f e t de la d 6 s i n t d g r a t i o n c~ s p o n t a n ~ e , le p l u t o n i u m 239 et s e s c o m p o s 6 s s u b i s s e n t une i r r a d i a t i o n n a t u r e l l e dont l e s c o n s 6 q u e n c e s s u r un g r a n d n o m b r e de p r o p r i ~ t ~ s son t m a i n t e n a n t b i e n connues .

L e s v a r i a t i o n s de r d s i s t a n c e 61ec t r ique in - d u i t e s p a r ce p h 6 n o m ~ n e p o u r l e s p h a s e s a , fi et 6 du p l u t o n i u m ont 6t~ p a r t i c u l i S r e m e n t b ien 6 tu- d i ~ e s [1]. On sa i t que l ' a l l u r e d e s v a r i a t i o n s de r ~ s i s t a n c e n ' e s t a b s o l u m e n t p a s la m e m e pour la p h a s e a e t p o u r l e s p h a s e s / 3 et 5. Dans le

* Boursier de these 3~me cycle - SPu-SECBPu. ** Agent CEA - SPu-SPuL

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p r e m i e r c a s la r 6 s i s t a n c e d ' un fi l de p lu ton ium c~ a u g m e n t e en fo n c t i o n du t e m p s , pu i s p a s s e p a r un m a x i m u m . Dans le d e u x i ~ m e c a s , e l l e aug- m e n t e c o n t i n u e l l e m e n t [2].

Nous a v o n s p r o p o s 6 une i n t e r p r e t a t i o n de ce p h d n o m 6 n e d a n s un a r t i c l e d6jfi publ i6 [3], dont l e s a r g u m e n t s son t l e s s u i v a n t s :

On sa l t que le v o l u me du p lu ton ium a u g m e n t e s o u s l ' e f f e t de l ' a u t o i r r a d i a t i o n b i en p lus l e n t e - meri t que la r 6 s i s t i v i t ~ . La r ~ s i s t i v i t 6 p f in i t p a r d e v e n i r c o n s t a n t e a l o r s que le vo lume V a u g m e n t e e n c o r e . Dans c e s c o n d i t i o n s l e s v a r i a t i o n s de la r d s i s t a n c e R d 'un f i l de l ongueur L e t de s e c t i o n S s ' 6 c r i v e n t :