IL NUOVO CliViENTO

VoL. 8 A, N. 3

1 Aprile 1972

Elementary Particles and General Relativity (*). D. K. R o s s Physics Department, Iowa State University - Ames, Ia.

(ricevuto il 5 Gennaio 1971)

Summary. - - An elementary particle is found to adjust its radius out to the point where the gravitational potential is minimum or to where the effective gravitational force is zero. By means of this principle, the usual well-known relationship between the mass and radius of the electron is found through the electromagnetic interaction and the Reissner-NordstrSm metric. Much more interesting results for the proton are found in the case of the strong interactions. Predicted radii agree well with experinlcntal values. These relatively simple results suggest that general relativity may play a role in elementary-particle structure.

1.

-

Introduction.

Numerous a t t e m p t s have been m a d e to c o n s t r u c t distributions of charge a n d mass which resemble the e l e m e n t a r y particles observed in n a t u r e using general-relativity theory. EINSTEIN (1), for example, in 1919 considered static spherically s y m m e t r i c a l distributions of electricity held together b y g r a v i t a t i o n in a modified version of t h e field equations of general r e l a t i v i t y . H e f o u n d t h a t all these distributions were in e q u i l i b r i u m w i t h n o t h i n g to single o u t a particular distribution. I n more r e c e n t times, a t t e n t i o n has shifted to t h e geons first considered b y WI~EELER (8). Geons are ((elementary particles ~) in t h e sense t h a t t h e y are composed e n t i r e l y of e l e c t r o m a g n e t i c and g r a v i t a t i o n a l

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (1) A. EINSTEIN: Sitz. Ber. Preuss. A/cad. Wiss., 349 (1919). (2) J . A . WREELER: Phys. Rev., 97, 511 (1955); E. A. POWER and J. A. WHEELEI~: Rev. Mod. Phys., 29, 480 (1957). 603

604

D . K . ROSS

energy i n t e r a c t i n g t h r o u g h E i n s t e i n ' s field equations. W i t h a t y p i c a l size of 10 ~2 m a n d a m a s s of 10 ~ g, geons are quite unlike t h e e l e m e n t a r y p a r t i c l e s of physics. T h e p r e s e n t p a p e r does n o t c o n s t r u c t p a r t i c l e s in t h e a b o v e sense, b u t finds an i n t e r e s t i n g r e l a t i o n s h i p b e t w e e n t h e m e a s u r e d m a s s a n d radius of the electron a n d p r o t o n o b s e r v e d in n a t u r e . T h e s e r a t h e r simple considerations a n d t h e new principle we i n t r o d u c e suggest t h a t general r e l a t i v i t y m a y h a v e a larger role to p l a y in e l e m e n t a r y - p a r t i c l e t h e o r y t h a n h i t h e r t o was t h o u g h t . This p a p e r is divided i n t o two p a r t s . W e t r e a t e l e c t r o m a g n e t i c forces in general r e l a t i v i t y in t h e following Section a n d t h e m o r e i n t e r e s t i n g strongi n t e r a c t i o n forces in Sect. 3.

2. - Electromagnetic

forces.

L e t us consider t h e solution of E i n s t e i n ' s field e q u a t i o n s for a churged m a s s point. This line e l e m e n t was f o u n d b y I~EISSNEI~ (3) a n d b y N O ~ D S ~ 5 ~ (4) to be (1)

ds2=

(

1--2m°~ -e~dt ~ r r ~]

(

1--

r

-~-r ~]

d r ~ - - r 2 ( d O ~ - s i n ~ O d ~ 2)

'

where ml = ~ M / c ~ a n d l ~ n/4~eoC ~. T h e q u a n t i t y ~ is t h e g r a v i t a t i o n a l constant, M t h e m a s s of t h e particle, a n d e t h e charge in ~ K S units. This m e t r i c was derived b y s u b s t i t u t i n g the e l e c t r o m a g n e t i c e n e r g y - m o m e n t u m t e n s o r for a s t a t i o n a r y p o i n t charge into t h e E i n s t e i n field equations. W e can i d e n t i f y t h e g r a v i t a t i o n M p o t e n t i a l as ~2

(2)

(p~,

= ~ (g0o-1),

where

(3)

g0o= (1-2m° + e,2 T

T2 ] "

This definition of g r a v i t a t i o n a l p o t e n t i u l is m e a n i n g f u l only w h e n one is dealing w i t h an e q u i l i b r i u m configuration us in t h e p r e s e n t case (5). T h e g r a v i t a t i o n a l p o t e n t i a l is p l o t t e d as a f u n c t i o n of r in Fig. 1. W e notice t h a t t h e p o t e n t i a l

(s)

H. R]~ISSNER: Ann. der Phys., 50, 106 (1916).

(4) G. NORDSTR(J~I: Verhandl. Koninkl. Ned. Akad. Wetensehap., A[del. •atuurk., 20, 1238 (1918). (5) B. K. HARRISON, K. S. THORN]~, M. WAKANO and J. A. WI~]~L]~R: Gravitation Theory and Gravitational Collapse (Chicago, 1965), p. 21.

]~LEMI~IqTARYPARTICLESAND GENERALI~]~LATIVITY

605

or

L

T

P

r

Fig. 1. - The gravitational potential as a function of the radius for a charged mass point with the mass and charge of the electron.

goes t h r o u g h a m i n i m u m a t a r a d i u s r~. A t t h i s p a r t i c u l a r r a d i u s t h e effective g r a v i t a t i o n a l force g i v e n b y (4)

Fo~, = -- V~,,~

is zero. The r a d i u s r~ f o u n d b y t a k i n g t h e d e r i v a t i v e of (2) w i t h r e s p e c t to r a n d s e t t i n g t h e r e s u l t e q u a l to zero is

e2 (5)

r~ = 4UeodgC~.

I f we n o w l e t M a n d e be t h e m a s s a n d t h e c h a r g e on a n e l e c t r o n r e s p e c t i v e l y , we see t h a t r~ is j u s t t h e c l a s s i c a l r a d i u s of t h e e l e c t r o n ro. T h i s h a s t h e n u m e r i c a l v a l u e 2 . 8 . 1 0 - 1 ~ m . The f a c t t h a t t h e R e i s s n e r - N o r d s t r 6 m m e t r i c e x h i b i t s such a n e q u i l i b r i u m p o i n t is w e l l k n o w n . This result suggests, however, the

general principle that a particle assumes a radius such that the effective gravitational force is zero at that radius. U s i n g t h i s p r i n c i p l e we c a n c a l c u l a t e t h e r a d i u s of an e l e c t r o n g i v e n i t s m a s s a n d charge as a b o v e . This p r i n c i p l e m a k e s v e r y g o o d sense h ~ t u i t i v e l y since i t s a y s t h a t a particle samples the geometry of this space in order to know hov large to be. Since no o t h e r l e n g t h s t a n d a r d s e x i s t for a n e l e c t r o n b u t t h e m i n i m u m in t h e c u r v a t u r e of space a r o u n d i t , i t is n o t s u r p r i s i n g t h a t we get t h e c o r r e c t a n s w e r for t h e r a d i u s . T h e a c t u a l n u m e r i c a l r e s u l t is n e i t h e r n e w n o r v e r y i n t e r e s t i n g in t h e e l e c t r o m a g n e t i c case since we c a n w r i t e d o w n t h i s r a d i u s for an e l e c t r o n f r o m d i m e n s i o n a l a r g u m e n t s alone t o w i t h i n a n u m e r i c a l c o n s t a n t . L e t us n o w a p p l y t h e p r i n c i p l e s u g g e s t e d b y t h e a b o v e to t h e s t r o n g i n t e r a c t i o n s . I n t h i s case t h e r e s u l t s a r e n e w a n d m u c h m o r e interesting.

606

D.rK. ROSS

3. - S t r o n g - i n t e r a c t i o n f o r c e s .

Consider n o w a p o i n t m a s s w i t h a ¥ u k a w a p o t e n t i a l field. :For definiteness we shall t a k e t h e p o t e n t i a l to be (6)

~b(r) -- g exp

[---rm],

where we t a k e t h e s t r o n g - i n t e r a c t i o n coupling c o n s t a n t g°-/lgc~14.5 a n d m -~ m=c/h. The q u a n t i t y m= is t h e m a s s of t h e pion. This expression is directly analogous to --G

(7)

U(r) -- 4xeor

in the e l e c t r o m a g n e t i c case. W e would n o w like to calculate t h e m e t r i c for this Y u k a w a particle. L e t us first calculate t h e e n e r g y d e n s i t y in t h e Y u k a w a field since we will n e e d this to s u b s t i t u t e i n t o E i n s t e i n ' s field equations. T h e t o t a l p o t e n t i a l e n e r g y of a g r o u p of objects i n t e r a c t i n g t h r o u g h t h e Y u k a w a p o t e n t i a l is

f f ~x~x'l exp [--mlx--x'lJd~xd~x ', W=~1 J3

(8) where

O~(x) is

t h e (~Y u k a w a charge d e n s i t y ,. W e can r e w r i t e (8) as

if

W=-~

(9)

q~(x) q~(x)d3x

where (10)

¢(x) =fqo(x')

is the ¥ u k a w a p o t e n t i a l . (11)

exp [-- m Ix - - x' [] d3x,

~ow

V'( exp[-mr] r )

-- V2(exp r[-- mr])

which is just exp[--mr][m2/r--4~(5(r)]. and using (11) gives (12)

Ix -x'l

+ exp [--mr]V~(1) ,

A p p l y i n g V ~ to b o t h sides of (10)

V2 ~b(x) ----m2q~(x) -- 4 ~ ( x ) .

ELEMENTARY

PARTICLES

AND

GENERAL

607

RELATIVITY

We can now solve (12) for @~(x) and substitute this ~g(x) into (9) to get (13)

W----3

[--V2~b(x) + m2~b(x)]

d3x "

Integrating the first term b y parts gives

if and our energy density for the ¥ u k a w a potential field is

(15)

u -- s ~ [Iv~l~ + m".l¢l~] "

For a point Yukawa charge ¢(r) is given by (6). Substituting this into (15) gives 1 g2exp[--2mr](1 2m ~_ 2m".) u(r) ---- ~ r~ ~ -k -7"

(16)

This is our result for the energy density in the Yukawa field. Now let us find the metric. If we assume a time-independent, spherically symmetric solution, the metric can be written as

exp iv(r)]

(17)

g~

=

I

t -- exp [~(r)] --

--r". sin" 0/

where v(r) and i(r) are yet to be determined. in the present case can be written as

(is)

r 2

The energy-momentum tensor

I u(r)e~ t 0

0

o~ where u(r) is the energy density given in (16) and we have assumed that the particle is stationary. Substituting (17) and (18) into the field equations of general

608

D.K.

~oss

r e l a t i v i t y gives

u(r) = exp [ - - 2] \ r 2

(19)

- - e38~

(20)

r~- ~ exp [ - - 2 ]

(21)

~'~.'-- (v')2--2~ " - ~

1

r]

1 T2

' 2

(v'--).') = 0,

r

where t h e p r i m e s denote differentiation w i t h r e s p e c t to r. W e can let - - e x p [--~(r)] a n d w r i t e (19) in t h e f o r m

ry'@ y =

(22)

1 --~

exp [ - - 2 m r ] (1~

y(r)

~- ~2 m - +2m~) ,

where ~ ~ -~ 7¢g~/0. W e can n o w easily solve (22) since it is linear in y. T h e e x a c t solution is

(23)

m~

y ~ exp [-- ;t(r)] = 1 2 m l

6~

+ - r - exp [ - - 2 m r ] -~ ~ exp [-- 2 m r ] , T

where m~ is a n i n t e g r a t i o n c o n s t a n t r e l a t e d to t h e m a s s of t h e p a r t i c l e . E a r i n g f o u n d ~(r), we can n o w solve (20) for ~(r). W e can w r i t e (24)

~'(r) = ~

,

where y is g i v e n in (23). D u e to t h e c o m p l e x i t y of (23)~ we c a n n o t solve (24) exactly. W e shall a s s u m e t h a t (25)

-ml - (<1 , r

-m~ - ~<1

and

r

<<1 ~

"

These ratios are all of order l 0 - ~ for v a l u e s of r of i n t e r e s t to us. I f we assume (25)~ we can i n v e r t y to first order in these q u a n t i t i e s a n d t h e n easily solve (24). T h e r e s u l t is (26)

exp [~(r)] ----1 2 m l r

_~ ~ exp [ - - 2mr] 2r 2

to first order.

Combining (17), (23) a n d (26) t h e n gives us t h e line e l e m e n t for a mass p o i n t w i t h ¥ u k a w a (~charge ~) d s 2 = ( 1 - - 2 r m l + ct exp [ ~ 2mr]~ 2r 2 ]

(27) _

1 _2

e~dt ~_

!_{_mgexp[--2mr]~__~exp[_2mr] T

dr~--r~(dO2-~sin~Odq)2).

E L E M E N T A R Y PARTICLES AND GENERAL RELATIVITY

60~

NOW let us i d e n t i f y ~gr~ with goo as in (2) a n d write the g r a v i t a t i o n a l potential as (2s)

w'"

=

-~L7

2r.

j "

A t this point, let us specialize to the proton. T h e n in correspondence with l~lewtonian g r a v i t y , we m u s t h a v e m, ~ z M , / c 2 for our integration c o n s t a n t where M , is the mass of the p r o t o n . We notice t h a t ~%,~ goes t h r o u g h a mini m u m exactly as in the electromagnetic case. W e shall now a p p l y our principle a n d say t h a t the p r o t o n adjusts its radius R out to the p o i n t where the gravitational p o t e n t i a l is a m i n i m u m or to where t h e effective g r a v i t a t i o n a l force is zero. This gives 2 nM. ~ exp [-- 2mR] mc~ R-~ e--i- + /~ + ~ exp [ - - 2 m R ] =

-

(29)

-

0,

where we divided t h r o u g h b y c2/2 in t a k i n g the d e r i v a t i v e of (28). Since a = = ~g2/ca a n d m = m,~c/~, (29) gives _~ i m p l i c i t l y as a f u n c t i o n of the mass of the proton, the mass of the pion a n d the coupling c o n s t a n t of the strong interactions. We can rewrite (29) as (30)

2Mpc2 = g2 exp [ _ 2 m R ] [ m + 1 ] .

I f we let M . ca = 938 MeV, g~-/hc -----14.5 a n d m = 0.688.10 ~5 m -~, w e f i n d t h a t R --~ 0.796.10 -~5 m for t h e radius of the p r o t o n . This is v e r y close to the value /~ = 0.78.10 -.5 m for the r.m.s, radius of the charge d i s t r i b u t i o n of the p r o t o n measured b y I~cALLISTEI¢ and ttOFSTADTER (6) in electron-scattering experiments. Since this result for R is not s i m p l y r e l a t e d to the pion C o m p t o n wavel e n g t h and since (30) is far f r o m obvious f r o m dimensional considerations alone, these results for the p r o t o n are m u c h more i n t e r e s t i n g t h a n the corresponding results for the electron. I t is w o r t h m e n t i o n i n g at this p o i n t t h a t the m e t r i c t h a t we h a v e used for b o t h the electron a n d p r o t o n a b o v e has been in t h e Schwarzsehild form. Since physical results can never d e p e n d on t h e p a r t i c u l a r f o r m of t h e meSric used, we should m a k e sure t h a t we get the above results using the isotropie form of the respective metrics. Since we f o u n d a n d used goo to first order in the small quantities in (25) only, and since the Sehwarzschild and isotropic metrics are identical for g0o to this order, we see t h a t our results are indeed i n d e p e n d e n t of the metric. Also we note t h a t since we w o r k only to first order in the small parameters in (25), the above results can be o b t a i n e d f r o m the Poisson e q u a t i o n with the energy d e n s i t y in (16) s u b s t i t u t e d in for t h e usual mass density. E v e n t h o u g h the Poisson equation gives t h e same result as our general-relativity calculation a b o v e (as we would i m m e d i a t e l y guess since the p a r a m e t e r s in (25) (6)

]:~- W. MCALLISTERand R. HOFSTADTE~: .Phys. Rev., 102, 851 (1956).

610

D.K.

~OSS

a r e so s m a l l ) , i t is c o n c e p t u a l l y b a r r e n a n d d o e s n o t s u g g e s t w h e r e f u t u r e t h e o r y m i g h t go. T h e g e n e r a l - r e l a t i v i t y f r a m e w o r k r e l a t e s a p a r t i c u l a r i n t e r a c t i o n t o t h e m e t r i c of t h e s u r r o u n d i n g s p a c e a n d s u g g e s t s t h a t a p a r t i c l e s a m p l e s t h e g e o m e t r y of t h i s s p a c e i n o r d e r t o k n o w h o w l a r g e t o b e .

4. -

Conclusion.

These results for the eleotron and proton suggest that general relativity may play an important role in elementary-particle structure. That elementary p a r t i c l e s a r e c o n s t r u c t e d u l t i m a t e l y o u t of %he g e o m e t r y of s p a c e - t i m e is a v e r y attractive idea. The electron and proton and their antiparticles are the only completely stable particles with mass, and the above simple considerations give u s t h e r a d i i of j u s t t h e s e . P e r h a p s t h e e l e c t r o n a n d p r o t o n a r e s p e c i a l b e c a u s e t h e y d e p e n d o n l y u p o n t h e f u n d a m e n t a l e l e c t r o m a g n e t i c or s t r o n g i n t e r a c t i o n , respectively, giving rise to them. Other particles are then excited states. We have used unquantized general relativity in the above. Quantized general r e l a t i v i t y w i l l u n d o u b t e d l y p l a y a r o l e i a d e e p e r c o n s i d e r a t i o n s of t h e p r o b l e m . ***

I w o u l d l i k e t o t h a n k D r . 1~. IZEA(30~/~ f o r s e v e r a l s t i m u l a t i n g

conversations

in regard to this work.

•

RIASSUNTO

(*)

Si t r o v a che una particell~ elementare aceomoda il suo raggio sino al punto in cui il 9otenziale gravitazionale ~ minimo od a quello in c u i l a forza gravitazionale effettiva nulla. I n base a quest0 prineipio si t r o v a la solita ben nora relazione fra la massa ed il raggio dell'elettrone per mezzo dell'interazione elettromagnetica e la metrica di ReissnerNordstrSm. Nel caso delle interazioni forti si t r o v a n o risultati molto pifl interessanti per il protone. I raggi p r e d e t t i eoneordano bene con i valori speriment~li. Questi risult~ti relativamente sempliei suggeriscono ehe la relativit~ generale pub avere un ruolo nella s t r u t t u r a delle partieelle elementari. (*)

Traduzione a cura della .Eedazione.

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