IL NUOVO CIMENTO VOL. 51 A, N. 1 1 Maggio 1979

The Gravitational Charge �89 as a Unifying Principle in Physics.

L. MOTZ

Ruther]ord Observatory, Colu~bia U'nivcrsity - .New York, -IV. Y.

J . :El'sTEIN

Physics Department, Bloon~]ield College - Bloom]icld, iV. J.

(ricevuto 1'1 Febbraio 1979)

Summary . - - We show in this paper tha t the basic quantizat ion con- dit ion 4Gm2/c=nh, which was deduced previously from the Wcyl principle of gauge invariance, as well as from the general-relat ivist ic 2-body problem, leads to a par t ic le with gravi ta t ional charge �89162 I t has the following propert ies : 1) I t is a mini-black-hole; 2) i t gives a na tura l gravi ta t ional cut-0ff for otherwise divergent self-energy cal- culations; 3) i t represents a single quantunl of action h and hence is an enscmblc of fluctuating gravi ta t ional fields in i ts lowest quantum state ; 4) since its Bekenstein entropy is just k, to within a factor of the order of 1, i t gives a physical meaning to the otherwise mysterious Boltz- mann constant ; 5) it leads to a s ta t is t ical mechanics of the interior of a macro-black-hole of mass M in thc sense tha t i t yields the quant i ty ~V = M/m as the number of dist inct part icles within the black hole and the quant i ty N')-/2 as its number of excitable interior degrees of freedom; 6) i t yields a simple formula for the square root of the fine-structure constant (accurate to bet ter than 1 par t in 1000) as the rat io of the number of degrees of freedom of the electromagnetic field with source to the number of degrees of freedom of the gravi ta t ional field with source; 7) i t leads to a formula for the masses of leptons, mesons and baryons; 8) i t permits one to express both the weak- and strong-coupling constants as small powers of the fine-structure constant .

1 . - I n t r o d u c t i o n .

1"1. The W e y l gauge invarianee. - T h e g r e a t sc ient i f ic ques t , f r o m t h e a d v e n t

of E i n s t e i n ' s g e n e r a l t h e o r y of r e l a t i v i t y to t h e p r e s e n t , ha s b e e n for a s ingle

p r i n c i p l e t h a t can u n i f y t h e v a r i o u s forces t h a t g o v e r n t h e d y n a m i c s of o b j e c t s

88

TIIE GRAVITATIONAL CIIARGE � 8 9 AS A UNIFYING PRINCIPLE IN PIIYSICS 8 9

from the subnuclear to the cosmological level. Shortly after the publication of Einstein's early papers on general relativity various physicists attempted the unification of gravity and electromagnetism. But, except for the work of Weyl, who introduced the concept of gauge invarianee, the results were more or less of a formal nature with the gravitational and electromagnetic fields appearing as distinct, noninteracting components of the general field. One can indeed derive Maxwell's and Einstein's field equations from such a general field, but such derivations reveal nothing new about the nature of the grav- itational and electromagnetic fields, nor do they reveal any interaction be- tween these two-component fields.

Weyl's principle of gauge invariance differs substantively from the other unified field theories in that it is based on the physical principle of the non- integrability of length, which is directly related to a complex Abelian gauge field. The introduction of such a gauge field into the Riemannian affine ge- ometry of general relativity generates an additional vector field, which can be identified with the 4-vector potential, and, by imposing the condition of gauge invariance on the curvature tensor, introduces an additional second- order antisymmetric tensor which has the formal properties of the electro- magnetic field. Serious objections were raised to Weyl's theory when Einstein and others pointed out that the nonintegrability of length necessarily leads to many-valued structures for each particular kind of atom, so that the distinc- tive features of the spectra of all the atoms of a given kind would not exist. I t was pointed out by one of us (LM), as also by WEYL (1) himself some 10 years after his original gauge theory (*), that the introduction of an imaginary gauge gets rid of this nonuniqueness difficulty. Indeed as shown by LM (2) the intro- duction of a complex phase factor of unit absolute value in the transport of a length permits one to deduce the relativistic Hamilton-Yacobi equation (the gauge divided by i becomes the classical action), the Lorentz force, the Bohr- Sommerfeld quantum condition, or the Klein-Gordon relativistic wave equation for a charged particle moving in an electromagnetic field.

Which of these four equations we deduce depends on how rigidly we impose the condition that a length (the size of a particle) remain unchanged during parallel transport. If, following W]~YL, we define the change in 1 (4-length) by the equation

(1) d log 1 --~ u .dx . ,

where dx~ are the infinitesimal components of the displacement, then the za

(*) L.M. wishes to thank C. N. YANG for calling his attention to Weyl's 1929 paper. (') II. W~:YL: Zeits. Phys., 56, 330 (1929). (2) L. ~IOTZ: a) Phys. l~ev., 119, 1102 (1960); b) l~l~ys. Rev., 126, 378 (1962); c) Phys. Rev., 113, B 1622 (1964), with A. S~IZER; d) Nuovo Cimento, 26, 1 (1962); e) •uovo Cimento, 6 B, 95 (1970).

90 I,. MOTZ and a. EPST]~IN

are the components of the 4-vector potent ial of a field generated by the con- dition of gauge invariance. I f 1 is the complex quan t i ty A exp [iW], where A and W are real space-time functions of the co-ordinates x,, then 1 can change ei ther by a change in A or in the dimensionless funct ion W. Since only A is physically measurable, any coustraint on the change in A during a parallel t ranspor t must generate a subclass of functions W, as shown in ref. (~), t ha t are of great physical significance.

F r om a classical pre-Weyl point of view the only physically meaningful condition tha t we can impose on A is tha t dA ~ 0 at every point. This con- dition is implicit in :Newtonian mechanics which assumes tha t only those paths are realizable in na ture along which the dimensions of a part icle do not change. I t is easy to show (~) tha t this most rigid constraint leads immediate ly to the relativistic Hamfl ton-Jacobi equat ion for a charged part icle in an electro- magnetic field defined by the vector potent ia l

(2) Au i tic �9 6

where ti,, e and e have their usual meanings. The invar iant action S of this Hami l ton-Jacobi equat ion is just hW. I t is no tewor thy t h a t h drops out of the final cquat ion as it must, since the classical path, by definition, is one at each point of which all elements (position and momentum) of the part icle are pre-

cisely defined. I t is also no tewor thy (2,) tha t the relativistic Hamil ton-Jacobi equat ion can be cast algebraically (no physical assumptions involved) into an operator form tha t is exact ly equivalent to the Klein-Gordon relativistic equa- tion. This formal similarity between the classical and quantum-mechanical descriptions of a part icle in terms of identical differential operators shows us how the constraint on dA must be relaxed to pass f rom the t t ami l ton-Jacobi to the Klein-Gordon equation. We must replace dA ~ 0 by

where Q is the densi ty of classical paths per unit action aud the integral is

extended over all permissible classical paths tha t can eonnect the initial and

final positions of the particle. This is equivalent to Feynman ' s (~ sum over

paths ~ procedure and leads to the Klein-Gordon equation when in t roduced

into (1). Tha t the probabi l i ty ampli tude (wave funetion) of a particle should be

generated together with the eleetromagnetie field by a Weyl-gauge transfor- mat ion in this manner supports Weyl 's dictum tha t (~ ... this new principle of gauge invariance, which stems not f rom speculation bu t from experience, points forcefully to the conclusion tha t the electric field is a necessary concom- i tant , not of the gravitat ional , bu t of the ma t t e r wave field represented by ~ ~) (~).

T I t E G R A V I T A T I O N A L CIIAICGE �89 AS A U N I F Y I N G P I C I N C I P L E I N P I I Y S I C S 91

The two constraints on dA described above are bu t the two extremes of other possible constraints which lead to different properties of the orbit. Thus the

rf

constraint ~ d l o g A = 0, where the integral extends from au initial position r i

r~ to a final position r,~ leads to the Lorentz pondermot ive force, and the con- straint ~d log A - - 0 leads to the Bohr-Sommerfcld quan tum condition for a closed orbit.

1"2. Weyl principle and strong gravity. - Tha t there is no formal difference between the classical-relativistic Hamil ton-Jacobi equat ion for the action and the Klein-Gordon wave equat ion for the quantum-mechanical ampli tude of a part icle in an electromagnetic field, both of which can be deduced from Weyl 's principle in a )I inkowski space, suggested tha t the int roduct ion of the Weyl principle into a Riemannian space would generate a wave equat ion for a charged

part icle interact ing with bo th the gravi tat ional and electromagnetic fields. Such an equat ion would thus reveal the nature of the relationship between

gravi ty and electromagnetism and would be the first step in the construction of a unified field theory. This program was developed by LM (see paper d of ref. (~)), who showed t ha t the int roduct ion of the Weyl principle into the Einstein field equations leads to the second-order equat ion for a two-component spinor

where R ~lh ~ is the Gaussian curvature at the point occupied by the particle, Fu~ is an ant i symmetr ic field tensor equal to curl A~,, and a , ~ =- i (7,7 ~ - 7~7,). We obtain the Feynman-Gel l -Mann form of the Dirac equation (3) by placing R-2/6h 2 equal to the reciprocal of the square of the Compton wave-length. This then gives the basic equat ion

(5) Gm ~ .~ hc

for the square of the gravi tat ional charge.

The manner in which (5) was derived from Weyl 's principle led to an inter-

pre ta t ion of this condition which is close to Salam's subsequent theory of

strong gravi ty (*). The essence of this in terpre ta t ion is contained in the following paragraphs taken f rom LM~s 1962 paper.

(s) R. P. FEYNMAN and M. GELL-)fANN: Phys. Rev., 109, 193 (1958). (*) Privat(~ communication to L.M. front SALAM. T h e rcl;ttionship between Strot~ff Gravity and L.M.'s work has been explored in papers by K. P. SINnA and C. SIVAItA~ ~nd in Sivaram's unpublished thesis. Also private communication from C. SIVARAM.

92 L. ~OTZ and J. EPSTEIN

(( There are two ways to interpret this conclusion. Either we must assume the proper density within the particle to be extremely large (of the order of 1048g/cm3), or we must suppose that the •ewtonian gravitational constant changes abruptly as we pass into the interior of a particle. We shall adopt the latter hypothesis, for not to do so would lead to a model of a particle in which the mass is distributed over a sphere of radius 10 -:~ cm, which bears no re- lationship to any of the fundamental lengths.

If we assume that the proper density of a particle is that associated with a constant mass distributed over a sphere of radius ~/mc, then the gravitational constant inside the particle must have the value ~e/m ~ ~.

Thus strong gravity need not be introduced as a distinct theory, unrelated to previous theories, but can be deduced from the fundamental concept of gauge invariance.

This concept of strong gravity (or /-gravity) based on this idea of a dis- continuity in the ~ewtonian gravitational constant across the boundary of a subatomic particle was pursued subsequently by many authors, most notably by ABDVS S,~Lh~. Many striking results can be obtained in this way, as shown particularly by a recent work of Sivaram and Sinha (4). We believe that intro- ducing a discontinuity in (7 is just a formal way of describing a phenomenon of much deeper significance. This becomes clear if one adheres to the strictly physical (i.e. operational) viewpoint when considering gravitational interactions~

just as EI~STEI~ did when considering the noncovariance of Maxwcll's equa-

tions to Galilean transformations. The ad hoe hypothesis that saved the day then was the ]5orentz contraction. Einstein's analysis of the operational meaning of simultaneity led to the real solution and, of course, has had profound and revolutionary consequences.

We believe that introducing a discontinuity in G is such an ad hoe hypo- thesis and that full understanding can come only from an analysis of the physical- operational meaning of this constant. This too has profound consequences and is central to what follows.

How does one actually measure (7? One measures the gravitational force

(or potential energy) between two bodies of known (( mass ,) (that is inertia).

The strength of the interaction at a known distance gives the square of the

gravitational charge q~. This in turn is equated to (Tm ~ owing to the observed universality of acceleration (principle of equivalence). Thus the constant G

represents physically the ratio of gravitational charge squared (force or po-

tential energy) to inertia squared. G is no more a distinct physical enti ty than is the (arbitrary) constant in Coulomb's law. The physical entities are force (po- tential energy) and inertia. This point seems trivially obvious when thus de-

(4) C. SIv~A~ and K. P. SI~[A: -prog. Theor. _Phys. (Japan), 55, 1288 (1976); .Pro- ceedings o] the International Con]erence on Frontiers o] Theoretical .Physics (New Delhi, 1977); .Phys..t~ev. D, 16, 1975 (1977).

TIIE GRAVITATIONAL CHARGE �89 AS A UNIFYING PRINCIPL~ IN PHYSICS 93

lineated, bu t this slight change of viewpoint is very important . W h a t is the situation from the operational approach when we (~ observe ~ an apparent increase in the value of G? Clearly there is too much force or potential energy to be accounted for by the observed quant i ty of inertial mass. How does this occur?

If one accepts the general-relativistic picture of gravitat ional charge, a sharp discontinuity in the value of 6t seems untenable, for it means a sudden discontinuity (of magni tude ~ 10 4o) in the curvature of space-time around the particle. F rom the topological standpoint, this is quite strange, unless one takes the view tha t the interior of the particle is to be removed from consi- deration in the theory altogether, bu t then it is impossible to speak of the value of G in the interior at all.

The operational viewpoint, however, shows tha t an apparent increase of G must occur in any t ight ly bound gravitat ional system, whether it be two quarks or two galaxies. This follows from nothing more than the equivalence of inertia and energy (that is from special re la t iv i ty-- -~t imate ly from the constancy of the speed of light)! I f two massive objects bind at sufficiently close range, so tha t the binding energy is a measurable fraction of the rest energies, an ap- preciable decrease in the inertial masses is observed. The gravitat ional charges, however, have not decreased because they measure the potential energy of the bond (the energy radiated away), and this clearly must be put back in toto if one is to separate the objects. Thus, a t close range, the potential energy is larger than Gm21/r, where m I is the inertia.

The casual observer says t ha t a new force has come into play at short range; the thoughtful person, surmising tha t the force is wholly gravitational, says tha t G has increased, bu t the person who holds carefully to the physical-op- erational viewpoint knows tha t the observed m I must be (not <~ might be ~)) too small. The potential energy is, in fact, Gm~/r, where m o is the original mass when the objects were far apart , and which may be orders of magni tude larger than ml, if the binding is large enough. Thus the observed value of G (defined by PE equal to Gm~jr) must increase whenever the binding energy approaches the magni tude of the original inertial masses.

But , one objects, does this not affect the principle of equivalence on which general relat ivi ty itself is founded? Indeed it does. The principle of equivalence must now read as follows:

The gravitat ional mass (charge) and inertia of a particle are strictly pro- portional and this proport ionali ty is universal for all bodies in a region of space of uni]orm curvature. The ratio of gravitat ional to inertial mass ( ~ square root of the lqewtonian constant) of a particle varies directly with the local curv'~ture (external gravitat ional field).

I n an external field, the gravitat ional mass does not change, bu t the inertial mass decreases in direct proportion to the local curvature. In fact, the value

~ L. MOTZ and Z. EPSTEIN

of the propor t ional i ty of gravi tat ional mass to inertial mass (the value of G) is a direct measure of the external f i e ld - - tha t is of the local curvature.

This modification ((, extension ~ is perhaps bet ter) of the principle is forced upon us unambiguously by the equivalence of inertia and energy. Rigid adherence to an unmodified principle of equivalence contradicts the equivalence of inertia and energy. The only reason this is not well known is tha t it is only in the subnuclear domain tha t one finds binding energies which ~rc substantial compared with the rest energies. This si tuation will always produce an ap-

paren t ly new short-range force, which in the present case we believe to be the subnuclear (( strong ~> force.

The requirement tha t physical entities be given operational meanings (physical veritiability), so crucial in the origins of both the theory of re la t iv i ty and quan tum theory , once again shows its great construct ive power.

These points were discussed in depth by the present authors in the years following the ]962 paper. This led LM in ]972 (5) to re interpret the quanti- zation condition (5) as representing a fundamenta l particle (christened (, uni ton )~), of mass ~ V/~c/G ~ ] 0 5 g. The quantizat ion condition was re- derivcd there in a s traightforward manner, wi thout recourse to gauge in-

w~riancc, by analyzing the general-relativistic two-body problem and quan- tizing the component of angular momentum along the line joining the two particles. One finds, to the sixth order, tha t in such a system this component is exact ly 4Gm~/c, so tha t the quan tum condition becomes

(6) 4Gm2/c --:- nh .

For n----1 we obtain

(7) Gm ~ = ~ e ,

which is a more specific version of the quant izat ion condition derived in the ]962 paper. The factor �88 turns out to be crucial for tile accurate predict ion of numerical results. As shown in ref. (5), only one uni ton in about 1017--1019

protons is needed to solve several cosmological problems, including the ab-

sence of the predicted solar neutr ino flux, the missing mass needed to close the Universe and to stabilize galactic clusters and the quasar energy source (we are seeing these as they were at a t ime when there were m an y more free unitons than now which were rapidly condensing into protons, releasing 1019GeV

per pro ton !). Tha t some multiple of V / ~ plays a fundamenta l role in the s t ructure of

ma t t e r has been suggested by o thers - - in par t icular COSTA DE BEALqKEGARD and MARKOV (o). Thus COSTA DE BEAI;REGARD states tha t (, In terms of inter-

(5) L. ~OTZ: NUOVO Cimeuto, 12 B, 239 (1972). (s) O. COSTA DE BEAUREGA]~.D: Compt. Rend., 252, 849 (1961); M. A. ~AI~KOV: Sou. Phys. JETP, 24, 584 (1967).

THE GRAVITATI()~AL CIIAItG~ ~ ~ / ~ AS A UNIFYI.N'G PRI-N'CXPLE I~ PHYSICS ~

actions t/c plays the same role in the case of gravi ty as e 2 does in the case of electromagnetism )>. And fur ther on, (< we propose therefore to consider tic as the square of a universal gravitat ional charge r or, more exactly, to relate tha t gravi tat ional charge m (having the dimensions of mass) to ?/e via G m ~ - - t~c >>.

:Note tha t COSTA D~ BEAUREGARD equates the gravi tat ional charge to the inertial

mass m of the part icle ra ther than to V ~ m as we do. MA~Kov refers to such particles as (, Maximons >> and suggests tha t their masses << are the upper limits of possible masses of micro-particles >>. Considerations such as these led us to postulate tha t the const i tuent quarks in the interior of hadrons are in fact unitons. In a previous paper (~) we show tha t Lagrange 's analysis of the grav- i tational three-body problem, when combined with the Pauli principle~ requires the quarks to be in a linear ar rangement (gravitat ionally bound), ro ia t ing about the center of mass at the middle quark.

We show that , if there are three distinct types of quarks, having the

quan tum numbers of the fundamenta l representat ion of SU3, then there are 18 distinct states of the linear ro ta tor which have exact ly the quan tum num-

bers of the 8- and 10-dimensional representations. If, following the reasoning above, we demand tha t the gravi tat ional charges be large (----~ ~/ti-e) and

tha t the inertial masses be small when bound (--= �89 then the lowest Bohr orbit is the pro ton radius, and a simple classical calculation gives the magnetic moments of bo th the p ro ton and neut ron (not just the ratio), all to within a few percent. We show tha t the S U3-symmetry is based on the internal mo- tions of the const i tuent quarks; in part icular, the (< strangeness >> is identified with the general relativistic precessional motion of the bound quark.

Since the center quark in the linear ro ta tor is in an energy state different from tha t of the end quarks, the statistics difficulty, inherent in all models which pu t the quarks in the same ground-state energy, is avoided. Thus there is no need to introduce para-Fermi statistics (color) in this model. We believe tha t this is a considerable simplification, since the experimental evidence use(][ to support the colored-quark hypothesis is ambiguous at best. The evidence usually cited is based on the ratio of cross-sections for ba ryon product ion to tha t for muon product ion in e+e - scattering, and the ~o decay rate.

W ha t appears to be a strong argument for colored quarks is presented in the assertion tha t a colored-quark model predicts the value 2 (before the dis- covery of charm) for the rat io

a(e+-[-e - ~ hadrons)/a(e+-4-e - ~ ~t+-]- ~- ),

whereas the Fermi quark model gives the value ,~. The val idi ty of this ar-

gument must be questioned from a theoretical point of view, since it is based

(~) J. EPSTEIN aud L..'~[OTZ: NUOVO Cime~zto, 38A, 345 (1977).

96 T.. MOTZ and J. EPSTEIN

entirely on the application of QED to the process e + e - ~ ?--> q~l ~ hadrons. I t is argued that each possible pair q~l contributes an additive term e ~ (where

q

eq is the fractional electric charge on the quark) to the ratio, and the total contribution is the sum of these individual terms. Thus for ordinary Fermi statistics (before charm) one gets -32-, whereas for colored quarks the result is three times as large. If colored charmed quarks are also taken into account, this calculation gives 10/3. But all of this is based on the assumption that the process is entirely electromagnetic in nature which is certainly not valid if the dominant quark interactions are gravitational with each quark and anti- quark having one unit of gravitational charge. Using the same argument as above, we then find that the total contribution is 3 without charm and 4 with charm. These values are in much better agreement with the experimental dat~ as discussed by ELLIS (s) and RICHTE~ (~).

The same kind of argument, as we have just presented against using the a-ratio to support colored quarks, can be marshalled against the argument that the amplitude for the ,~0_> 2V decay requires colored quarks. The con- ventional quark analysis assumes that the decay amplitudes are to be deter- mined entirely by QED which for Fermi statistics necessarily leads to values that are too small by a factor of three. Again calculations based on gravitational interactions give the right order of magnitude without color statistics.

Since color is unnecessary in this model, there is no need to impose the requirement of color singlets and thus no need to dread the discovery of a free quark. Indeed we should look forward to such a momentous event. That this has not yet occurred is due not to a law of nature, but to the 1019 GeV we must use to create quarks in the laboratory.

2. - Gravitat ional , q u a n t u m and t h e r m o d y n a m i c properties o f un i ton-quarks .

We have seen that the fundamental equation Gm2.~ ?ic/4, as deduced from the general-relativistic equation of motion of the two-body problem, implies the existence of a basic unit of gravitational charge (?go)t/2, which, owing to its magnitude, must play a crucial role in the structure of matter. We see from this that gravity, far from being negligible in particle physics, is the dominant force there. That gravity must be taken into account in the dynamics of elementary particles has been viewed with less than enthusiasm by most physicists and with considerable hostility by some, but recent develop- ments in black-hole physics and in supergravity gauge theories have evoked renewed interest in the gravitational field as a suppressor of divergences in

(s) J. ELLIS: Proceedings o/the X VII International Con]erence on High.Energy Physics (London, 1974). (9) B. RICIITER: Xobel Address, reprinted in Science, 196, 1286 (1977).

' r i l e GRAVITATIONAL CIIAIr 1o ~r AS A UNIFYING PRINCIPLE IN I)IIYSICS ~7

self-energy calculations. Thus ISmXM, SAL~X3r and STIr ( lo) have showli

tha t , in a gravi ty-modif ied q u a n t u m elect rodynamics with specially chos(qt

te rms t h a t b reak the e lect romagnet ic and gravi ta t ionM symmet r i e s in ils

Lagrangian , cut-off te rms appear whietl can make the self-energies of eleetr(ms and I)hotons finite. MOI.'F~XT (11) and S[v;tRA.~r (~2) have also proposed gravi ta- tiona.1 models of part icles which are free of self-energy infinities. Wi thou t going

in{o tile kind of detai led calculations found in the above, papers , we shall show

by a very general a rgumen t tha t a grav i ta t iona l cut-off t ha t suppresses self- energy infinities mus t lead to our fundament:f l equat ion (6) or (7).

To demons t ra te this we consider a 1)~rtiele of mass mo emi t t ing and reab- sorbing a vi r tual t)hoton of f requency r. Dur ing the, process of v i r tua l emission

the part icle acquires a recoil veloci ty v, su(.h t ha t

h v 7~ o V (8) - my

c V l ~,@ (eonserva | ion of n l o m e n t u m ) ,

where m is the mass of the p~rticle during the recoil.

Solving for (v/c) 2, w e obtain

(9) (v /c) ' - \ h2v', -~- 1) - l .

Thus during the vir tual emission the mass of the recoil pa r t ich~ becomes

t 2 o i ( 1 0 ) m = c---72 1 ~ - I F ' v ~ .

ISIIAM, SAI~A.M and STICATIIDE~ (lO) ]lg~ve shown t ha t for a part icle of m~ss m

the gravi ty-modif ied quan t um eh,ctrodylmmics gives a cut-off a t the frc- quency ~, such t ha t

e a]i, (11 ) hv ~ : t G n i "

Subst i tu t ing this in (10) and dropping (mo c:/hv) '~ in the l)arenthesis on the right- hand side, we o b t a m

(7) G m ~ ~ hc, '4

for the cut-off mass of the recoil pa r tMe .

(10) C. ISIIAM, A. ~ALAM and J. S'rRATIIDEF: Phys. Rer. D, 3, 1805 (1971). (l~) j . MOFFAT: Phy,~. Re,'. D, 17, 1965 (1978) . (12) C. SIVAI~AM: unpul)lished Thesis, Indian lnstit.utc of Scien(~c (1977) and references therein.

7 - 1l N u o r o C i m e n t o &.

98 L. MOTZ and J. EPSTEIN

We can see physically why this condition should apply if we note that this is essentially the requirement that the recoil particle be a black hole. I t is obvious that, if, owing to its recoil velocity, the mass of the particle becomes so large that the particle becomes a black hole, it can no longer emit virtual photons so that a gravitational cut-off is attained. Again we see the critical role played by gravitation in subatomic phenomena because of the equivalence of inertia and energy.

An important question which arises concerning the uniton is whether such a particle can be stable against radiative decay via the Hawking-radiation process, which goes extremely rapidly for black holes of small mass. We shall show, using very general arguments, that the Hawking process, by its very nature, cannot occur for unitons. The quantum radiation from a black hole, as deduced by H A W ~ G , is the sum of many discrete quantum processes, each one of which represents a single quantum transition from a higher to a lower quantum state. I t is obvious that such radiation nmst cease when every pos- sible radiator in the black hole is in its lowest quantum state, just as the ra- diation from an atom stops when every electron in the atom is in its lowest quantum state. We shall now show that a uniton is the lowest quantum state that can occur in a single gravitationally bound structure. We may express this differently by saying that the uniton cannot decay radiativcly by a quantum transition because it is itself a single quantum of action. The only way it could radiate would be by spontaneously disappearing from one point of space-time and reappearing just as spontaneously at some other point, which, of course, would violate all conservation laws. In this respect it is like the hydrogen atom in its ground state which has an infinite lifetime precisely because it represents a single quantum of action, and the quantum theory requires each radiative process to consist of a single quantum of action.

To prove our assertion we start from Weisskopf's (~s) expression for the self-energy of a charged particle arising from the random fluctuations of the field whose source is the charge itself. If e is the electric charge on the particle and a is its radius, Weisskopf's exact expression for this energy is

(12) Wf~uo = l i m 2 e 2 h / m c a ~ . a--)'0

To apply this expression to the uniton in which the dominant fluctuating fields are gravitational rather than electromagnetic, we replace the square of the electric charge by the square of the gravitational charge G m ~. We thus obtain

(13) Wtz~o ~ 2 G m 2 ~ / m c a 2 �9 g r a y

(la) V. WEISSKOI'F: .Phys. Rev., 56, 72 (1939).

TIlE GRAVITATIONAL CIIARGE ~ ~v/~-~ AS A UNIFYING PRINCIPLE IN PIIYSICS 99

] f this is to represent a single quan tum of action and hence the quantum ground s ta te of the particle, we must have

(14) 2Gm2h/mca ~ = 2uhv,

where v is the f requency of tho field fluctuations. I f we follow Weisskopf and set v----c/27ea and then place a = h/me, we obtain

(/m'- = ~ c / 2 ,

which is our basic quan tum equat ion to within a factor of �89 We consider now the thermodynamic properties of uuitons and show,

start ing from our fundamenta l relationship (7), tha t the cn t ropy of a single uni ton is of the order of k, the Bol tzmann constant . We believe this to be o[

extreme importance, since it relates this somewhat mysterious constant , which plays such an impor tan t role in statistical mechanics and in physics in general, to the other basic constants of nature. We shall demonstra te this by applying the Bekenstcin (~4) expression for the en t ropy of a black hole, which can be

wri t ten as

(15) Sbh----- ( ln2/Src)kc3h-~G-~A ,

where A is the area of the black hole. I f we place A = 4~(2Gm/c2) ~- (the con- dition for a black hole), we obtain

(16) Sbh ---- (In 2)(k/h)(4Gm~/c)/2.

Using eq. (7) tha t defines the uniton, wc obtain

(17) S.,,,to , = ( l a 2 ) k / 2 ,

which is a most remarkable relationship, indeed, even with the factor (ln 2)/2.

Since B E K ~ S T m ~ himself is not quite sure of this numerical factor and states

tha t it ma y be off by a factor of 2, it is not unreasonable to say tha t the Bol tzman~ constant is just the en t ropy of the basic gravi ta t ional charge.

We obtain an equation quite similar to (17) if we s tar t out f rom the Sackur-

Tetrode formula, which we write for a single particle in thc form

(18) S = k In [(kT) t V(2x~m)lge~/(S~'~h3)],

(,4) j . D. BEKENSTEIN: ]~hys. Rev. D, 7, 2333 (1973).

10[} L. MOTZ and J. EPSTEIN

where V is the volume occupied b y the particle, g is its stat ist ical weight and e is the base of the na tura l logari thms. I f we assume the uni ton to be a Fe rmi part icle of spin ,~ and if there are n dist inct types of unitons, we m a y place

g = 2n. Following BEKENSTEI.N again we place kT.-= zh./2uc, where z is the surface g rav i ty of the uni ton and obtain

( 9 ) S..iton --- k ln [{ h~t'~ ~ -~ra~ ] �9 [\2nc] 87~1i ~ (2n) (2z~m)~ el ,

where a is the radius of the uniton. Since ~ := Gm/a ~, we obtain

I ( G m 2 ~ ! ) .~ na 3 ] (20) S.~to~ : k In [~27t.m a -~ 3 ~ (2~m)~ ~ ~- k In nf.,

3~"

or, since e 5_~ ]48.4,

] ( ' . ) S~,~o~ ~ , In L 9z2 k In (n/6�89

where we have used eq. (7) to replace Gm2/c by ]//4. Since we do not know

how m a n y independent quark states there are, we do not know what value of

n to use in this formula, bu t for n- - - -4 we obtain ~ value _~ [,~ ln(2 .66)Jk , which is quite close to the value obtained f rom the Bekenstcin formula. Fo r

n ---- 5 we obtain % v.flue _~ k In 2. T h a t the Bekenstcin and Sackur-Tet rodc f~rmulae should give similar values for S is quite dramat ic .

The appl icat ion of the laws of the rmodynamics to individual quarks leads

to a stat ist ical mechanics of black holes tha t , to some extent , implies the ex- istence of discrete, gravi ta t ional ly in teract ing entities in the interiors of black holes. To see this we introduce the to ta l ~,q'avitation~l binding energy E of the black hole, which we write in the fo rm

(22) E = ~(GM/I"~2)MR,

where is tile mass and R is the radius of the black hole. ]f the black hole

c. nsist cf N dist inct particles, each of mass m, we m a y write (22) in the fo rm

(using the Schwarzschild value for 1~)

(23) E--: �89 2 - -- (GM/R2)GN2m2/c2.

We can now show tha t this necessarily leads to the Bekenstein t empe ra tu r e

�9 f the black hole, provided Gm2= ~c/.l, which means t ha t 2~" is just the number

( f uni ton quarks in the black hole. To do this we app ly the equipar t i t ioa

theorem to the black hole and place E - nkT/2, where n is the to ta l numbe r

THE GRAVITATIONAL CHARGE ~ AS A UNIFYING PRINCIPLE IN PHYSICS 101

r ~S of dcgrec~ of freedom tha t contr ibute to the energy E of the black hole. thus have

(24) n k l /.. == (GM/R2)(N'~/c)(Gm2/c).

We

If now we place G m2/c = hi4, we obtain

(25) n k T -- c 2

( ~ . - the surface g rav i ty ) ,

which (except for the factor 1/2 ) is the Bekenstein temperature , provided

(26) n -= N~'i2.

The appearauce of N~/2 instead of N may be somewhat surprising at first sight, bu t a li t t le thought ~il l show this must be so, since the unitons in the black hole in teract gravi tat ional ly in pairs and there are just N ( N - - 1 ) / 2

interact ing pairs (N is negligible with respect to N ~) (*). Tha t N 2 appears in this natural way, entirely as a consequence of the equipart i t ion theorem, the value of the basic gravi tat ional charge, and the laws of thermodynamics is very gratifying.

One could, of course, introduce a par t i t ion funct ion of the form

~ e x p [-- e,/g] for a black hole, where g is the mean energy per degree of freedom i = 1

and, hence, of the order of BM2/2N2R. The en t ropy would then be

<27) , ~ = k[,~ lo~ ~ e~pE-- ~,/1 + ~/k~'],

where E, the total energy of the black hole, equals g(~/~'g)(h)g ~ exp[- -e , /g ] ) .

l,f we place n =- (M/m) 2, where m is the mass of the uni ton and place E / k T = n,

(27) becomes

s~ ~_ k(~ M,/a.+)(, -!. log ~ e~p r- ~,/~l) �9 i = I

Replacing Gm ~ b y tic/4 and making the reasonable assumption tha t inside a

black hole e~ >~g for i > 0 and eo = 0, we obtain

(28) Sbh = kGMz/hc .

(*) We wish to thank Dr. t)~1~cus for a discussion of this question. For a full "~nalysis of this point see The Ma~y Body l>roblem, edited by J. Pb:RCUS (New York, N. Y., 1963), p. 163.

102 L. MOTZ and J. EPSTEIN

For M ~ c2R/2G (a black hole) we then obtain the Bekenstein value of the en t ropy

(29) Sbb ~_ ke3A/(47t~G).

I t thus appears tha t the interior of a black hole behaves statistically as though i t consisted of M / m particles, each of mass (~e/~G)i, and each inter- acting gravi ta t ional ly with all the others to give ( M / m ) 2 internal degrees of freedom.

3. - Calculat ion o f the f ine-structure constant and the mass formula .

COSTA DE BEAUltEGARD (6) has suggested tha t (( the three quanta of mass,

mo (electron mass), m~ (pion mass) and M (nucleon mass) are created b y a resonance in the electromagnetic, the gravi tat ional and the mesonic fields respectively ~), and has emphasized the striking fact t ha t these masses are related

to each other by the simple equations

(30) e2/me -~ 2hc/m~: -~ g2/M ,

where g~ ~ 13A ~v is the square of the strong-coupling constant . We believe tha t eqs. (17) are not mere numerical coincidences, bu t stem from a profound relationship between the way the uni t gravi tat ional charge and the way the uni t electric charge came into existence via the fields to which they are coupled. Since the uni t electric charge is e and the uni t gravi tat ional charge is 1 (/ge)~ and their rat io is �89 it is clear tha t the solution to the puzzle of the nu- merical value of the fine-structure constant will be found if we unders tand how these two basic charges arise f rom and are coupled to their respective fields. Though we are unable to answer this question completely at this t ime, we

believe tha t we can show, by taking into account the degrees of f reedom of the gravi tat ional and electromagnetic fields in the presence of a particle, why

the fine-structure constant has the value 137 -1 . I f we accept the modern point of view tha t gravi tat ional charge and electric

charge appear as the consequences of the spontaneous breakdown of the sym-

metries of sourceless fields, we may reasonably assume tha t the rat io of the two charges must equal the ratio of the excitable degrees of f reedom of the fields to which they are coupled. This in a sense is equivalent to Costa de Beau- regard 's assertion tha t a quan tum of mass (in our case a quan tum of gravi ta t ional charge) mus t be considered as a (( resonance ,) of a field. However , our concept

differs f rom his in tha t we introduce only one gravi tat ional cha rge - - the charge V ~ / 2 - - a s the (( resonance ~) of the gravi tat ional field, and t rea t the charge e as the (( resonance ~ of the electromagnetic field.

THE GRAVITATIONAL CHARGE ~ % / ~ AS A UNIFYING PRINCIPLE IN PHYSICS 103

Costa de Beauregard's choice of the word (~ resonance ~ is a happy one for it carries with it the concept of equipartition of energy among the resonating degrees of freedom of the fields. This prompts us to relate the ratio unit gravitational charge: unit electric charge to the ratio number of degrees of freedom excited in gravitational field of a particle: number of degrees of freedom excited in the electromagnetic field of a particle.

The electromagnetic fields are the independent components of the anti- symmetric tensor F~, and these are the first derivatives (curl) of the potentials A~. The divergence of the field, which vanishes identically in empty space, is proportional to the current density 4-vector in the presence of sources, The gravitational case is more complex, since the potentials constitute a sym- metric tensor and not a vector. The field quantities (first-order derivatives of the potentials) are the Christoffel symbols

(31) / • / t v ~-- 1 (gQ~,~ + go~,~-- g~,~),

where the comma denotes ordinary differentiation. Since these symbols are symmetric in #, v, there are 40 independent fields. The derivatives of these field quantities (divergence) are the l~icci tensor which vanishes in empty space and is proportional to the matter tensor in the presence of sources.

Thus, at first sight, one might expect the ratio of charges V~/2e to equal 40]6. But ~0 and 6 are the degrees of freedom of sourceless fields and we are concerned here with a source. How to count the additional degrees of freedom introduced by the source particle is, therefore, a critical question.

In the electromagnetic case it might seem that the particle introduces the components of the current ~-vector, raising the number of degrees of freedom

to 10. Careful analysis, however, shows that the degrees of freedom stemming from the motions of the source have been counted twice. All Lorentz frames are already included in the 6 degrees of freedom of the F ~ . In the rest frame of the particle there are only 3 independent F~v, since the magnetic compo- nents vanish. Lorentz transformations produce the other three. In the rest frame of the particle there is only 1 component of the current jo -~ ~; a Lorentz transformation gives the other three. Assigning a count of 6 to the fields and

to the current counts the velocity degrees of freedom twice. If one assigns 6 to the field components, the only new degree of freedom introduced by the particle is the charge density Q, giving a total of 7. Alternately, one may con- sider the fields in the rest frame (3 components) and include the velocity degrees of freedom in the current vector, giving again a total of 7 components.

This procedure gives us insight into the gravitational case as well. The matter tensor has in general 10 components, but again taking the ~0 field components of the F ~ and the full 10 of the matter tensor, we count the degrees of freedom associated with the motion of the source twice. We must consider

104, L. I~IOTZ and J. EPSTEIN

either the Christoffel symbols or the mut ter tensor in the rest frame, not both.

The si tuat ion in the gravi tat ional case is fur ther complicated by the fact th~tt the mot ional degrees of freedom of the gravi tat ional fields include more than the Lorentz transformations, since they are derived from potentials which arc covar iant with respect to general co-ordinate transformations.

First , leave all the motional degrees of freedom in the fields (40 compo-

nents) and look at the ma t t e r tensor in the rcst frame. Since the components of Tt,~ are proport ional to (Ju# u P, where u P is the 4-velocity, the only surviving

component , ve ry much as in the electromagnetic case, is Too - : ~ (1 component) , giving a to ta l of 4]. We shall recalculate this number, as in the electromagnetic ease, also by allowing the full freedom in the ma t t e r tensor (:10 components) and calculating the number of independent field components when the motional degrees of freedom which are counted in the ma t t e r tensor are removed.

We iirst look ,~t the metr ic tensor in thc part icle rest fr~me in a completely stat ic situation. In these co-ordinates the metr ic has the form

(32) g#~ =

fgo0 0 0 0 \

0 gll g~ g 3 1

0 g21 g2~ g2a /

0 g31 g3~ g33/

with all components independent of Xo. The definition (31) of FQuP then gives 24 indepcndel~t, nonvanishing components. This form of g~,, is, however, clearly to() restrictive. The original degrees of f reedom counted in the full 40 compo- nents of the fields include rotat ions as well as linear motions and only ttle la t ter have been pu t back in the ma t t e r tensor. Including the angular motions (centrifugal and Coriolis terms) in the metric results in (15) g0o and the off-

(liagonal terms gt2, g~3, g.-3 having ~t dependence on Xo. I t is then straight- forward to show from the definition tha t there are 31 independent, nonvanishing

F o~,,. These added to the 10 components of the ma t t e r tensor give again a

to ta l of ~:l. Thus we again arrive at 4] degrees of freedom for a gravi tat ional field

with source. ] t is interesting tha t the motion of the source (the space com- ponents of the velocity) involves three degrees of freedom for the electro-

magnetic field and 9 for the gravitat ional. Clearly this is in t imately related

(15) C. W. MrSNER, K. S. TI~OR~']~ and J. A. ~VtlEEI.EI~.: Gravitation (San Francisco, Cal., 1971), p. 450.

THE GRAVITATIONAL CHARGE �89 ~r AS A UNIFYING PRINCIPLE IN PIIYSICS 105

to the in teract ion of the field with the veloci ty components , t ha t is to the

4-force. WEYL has explained this us follews (~6) :

(~ The expression for the 4-force which the gravi ta t ional field exerts on a

moving m,

- - m F ~ u#u" ,

(u ~ is the 4-velocity) is complete ly analogous to the expression for the force

which the e lect romagnet ic field exerts on a charge e:

. _ e ~ " ~ 1 ~ U ~" "

lh)wcver , whcreas this expression depends on the 4-vclocfly linearly, the former

depends on it quadr~tical ly. I u Iqewtonian ~ a v i t a t i o n a l theory, the force is

independent of the veloci ty of the mass on which the force acts. B u t it was probable f rom the very s ta r t t ha t in a field theory of g rav i ta t ion a velocity- dependent correction would appear jus t us in e lcct rodynamics the velocity-

dependent Lorentz force e(v • of the magnet ic field H on ~ moving charge

appears. The Einste in theory leads in par t icular to a correction which is qua-

drat ic in the velocity. This requires tha t the gravi ta t ional field have not jus t

three or, with the in t roduct ion of t ime as a four th world co-ordinate, 4 com-

ponents bu t 4(4.5/2) : 40. The na ture of these dependences on veloci ty is

in accord with the fact t ha t the e lect romagnet ic field is derived f rom ~ po-

tent ial whose components ~ are the coefficients of an invar ian t linear dif- ferential form ~v~(tx ~, whereas the components of the poten t ia l of the grav-

i ta t ional field are the coefficients of a quadra t ic differential form g~kdx~dx k,). Thus, if lhc rat io of free grav i ta t iona l cha.rge to free electric charge equals

the r~t.io of degrees of f reedom of the fields, one would cxpcct

41 (3',~) :j -~- .

This relat ion is accurate to be t te r than 1 pa r t in a thousand.

Since the rat io of charges is one-half the square root of a 1, we obtaill

(33a) a-1 ==_ 4 .

I t is wor th noting t h a t .1, V/iw is the gravi ta t ional charge of a fundamen ta l

part icle with no electric charge (pure grav i ta t iona l charge). This value is

(16) I[. W~YL- Raum, Zeit ~en(1 Materie (Berlin, 1923), p. 226.

106 L. ~orz and J. EPST~rS

modffied slightly by the presence of electric charge~the amount is roughly calculated in our previous paper (7). The charge e represents the case of pure electric charge with no mass present ( i .e . the electron). The small mass of the electron apparently arises only from the energy of its electromagnetic field. The value e of the electric charge is modified considerably by the presence of a unit gravitational charge, resulting in charges of ~e and 2e/3 on the quarks. The mechanism through which this occurs is again the Weyl gauge invariance which predicts the existence of particles of charges e/3 and 2e/3 (with Land6 g factors of 2 and 1, respectively) as shown in ref. (7). I t is clear from the count of fields that the charge e is fundamental; the fractional charges arise from the interaction of the electromagnetic field with the very large gravitational charge V~-o/2.

4. - Mass f o r m u l a for leptons , m e s o n s and baryons .

If our assumption that the uniton-quark is the basic ingredient of all mas- sive particles (except the electron) is correct, then the ratio of masses of such particles to the mass of the electron should be expressible as small integral powers of �89 V'~-~/e. We shall see that this is indeed so for the rouen, for the lowest-lying mesons, and for the lowest-lying members of the octet and decimet. The nature of this power law is already indicated in eqs. (30). If we place g2 = 13.4]iv and note that 13.4 = (9/8)(]gc)�89 we have from (30)

m:lmo = m ./mo = 9(1v / )3

Proceeding in the same way with the muon, we have

m /m. = 6(l V c1 ) 2.

(m~v = nucleon mass).

The mass of A ++, the lowest member of the decimet, is given by

(36) = 12 (�89

The striking features of these expressions are the small integer exponents and the integer coefficients, which are integral multiples of the exponents. Thus all of these cases can be combined into a single general formula

(37) m , / m . = nk[ ( l i e )~ /2e]" , k > n .

In the accompanying table the values of the masses of various particles as predicted by (37) are compared with the observed values.

TIIE GRAVITATIONAL CHARGE � 8 9 AS A UNIFYING PRINCIPLE IN PHYSICS 107

m = nk [�89 ~/Vc/e]~m,

(n, k) Predicted mass Particle Discrepancy (hleV) from observed value (%)

n : 2 (2, 2) 70.02 net observed

(2, 3) 105.04 ~• 0.58 light

(2, 4) 140.05 r:• 0.34 heavy

n = 3 (3, 3) 922.24 ~N ~ 1.71 light

(3, 4) 1229.65 A 0.19 (<<width)

(3, 6) 1844.48 D 1.71 light

(3, 10) 3074.14 ~(3095) 0.6 light (<width)

(3, 11) 3381.54 ~(3400) 0.55 light (<width)

(3, 12) 3688.96 ~(3684) 0.13 heavy {<width)

(3, 13) 3996.38 ~(3950) or ~(4030) ~ 1

n : 4 (4, 4) 9596.56 Y(9.46) 1.4 heavy

I n the case of known un i ta ry multiplets, such as the meson octet and baryon octet and decimet, our formula gives the mass of the lowest-lying member of each multiplet . The other states lie higher in energy because of the internal angular momenta , I , Y, via the Gell-Mann-Okubo relation. One does not

know whether the various J /~ states are angular -momentum levels above the ground s ta te ~(3095) or deducible f rom our formula as they appear to be. I t

is interesting t ha t the spacings are obtained b y varying k b y one unit . I t ap- pears f rom the table t ha t there may be a restr ict ion on the range of k, which can be wri t ten as

(38) n ~ k ~ n a / 2 .

This would be strikingly confirmed if there are no J /~ states above (3, 13). At higher energies one must go to n ~ 4, whose lowest member is (4, 4) = Y. This is quite analogous to the restrict ion on the quan tum number 1 in a tomic theory.

We shall reserve the full discussion of the significance of the integers n

and k for a fu ture paper, bu t one can already see f rom the formula for the

pion and the nucleon tha t n is probably related to the number of quarks and ant iquarks in the particle.

F r o m the expression for the nucleon mass we deduce another result which leads to a new dimensionless universal constant , containing the five basic constants of nature , m~, e, G, h and c. We assume tha t each uni ton-quark contr ibutes the same fract ion of its free mass to a nucleon when it binds w i t h

108 L. ]MOTZ and J. "EPSTIdIN

two other uni ton-quarks. I f we call this fract ion /, then f rom (34)

(39) 3 I ra = 9[(hc)i/2e] a m ,

where m is the free mass of the uniton. In t roduc ing the value of m explicitly,

we then obtain

I f this cons tant is meaningful a t all, its impor tance lies in the manner in which

it unites the grav i ta t iona l cons tant with the f ine-structure cons tant and the

mass and charge of the electron. To complete our formal ism we note t ha t we can obta in the mass of the elec-

t ron f rom our formula b y placing n - - k = I and (~c)~/2 = e, which seems to indicate t h a t the entire mass of the electron is electromagnetic, s t emming en-

t i rely f rom its electric charge. This differentiates it qual i ta t ively f rom the m u o n whose mass is a lmost ent irely gravi ta t ional . I f we place n = k----1 in our formula , bu t replace e by the gravi ta t ional charge G~m, we obta in the

mass of the uniton. We have here an interest ing complemen ta r i ty be tween

electric and gravi ta t ional charge which m a y have a deeper significance than

is obvious f rom the formalism. We can even t ake the neutr ino into account b y placing n = 0. One possible in te rpre ta t ion of this is t ha t the neutr ino is

a spin-�89 uni ton-quark of zero mass and charge. This is not in conflict with our basic quant iza t ion condition (6) f rom which we obtained the uni t of grav-

i ta t ional charge, for we could have chosen the s ta te of zero angular m o m e n t u m instead of the s ta te of uni t angular m o m e n t u m f rom which (7) was deduced.

5. - Uniton-quarks and the intcraction constants in weak and strong interactions.

The concept of two fo rces - -weak and s t r o n g - - i n physics evolved f rom tile

Fe rmi theory of be ta -decay and Yukawa ' s theory of nuclear forces and the

success of these two theories in account ing for numerous phenomena in their

respect ive domains has grea t ly s t rengthened the belief in the dist inct charac ter

of these two forces. This has been fortified b y the discovery of leptons and hadrons as two dist inct groups of particles tha t are governed b y the weak

and s t rong force, respectively. I n spite of this s trong evidence, to the con t ra ry

we believe t h a t the weak and the strong interactions are two facets of a single

f o r c e - - t h e s t rong g rav i ty s temming f rom bound q u a r k s - - a n d we shall give

our reasons for this short ly when we deduce the magni tudes of the weak and

s t rong coupling or in teract ion constants f rom our gravi ta t ion.d theory. Be-

fore doing this, we note t ha t a difference in the s t rength of two forces is, in

itself, an insufficient reason for assuming tha t these two forces are different.

TIIE GI'~AVITATIONAL CltARG:E �89 V/he AS h UNIFYING PlCINCIPLE IN PIIYSICS 10~

A good counter-example is gravi ty itself, which is ex t remely weak under or- dinary conditions (the gravi tat ional interactions of ordinary objects) and yet

is stronger than any other force near or on the surface of a black hole of suf- ticiently high density. Indeed the very character of the gravi tat ional force changes so drastically as we get close enough to such a black hole that , wi thout the knowledge of general relat ivi ty, we would n~tural ly conclude tha t the gravi tat ional force in the neighborhood of a black hole is some new kind of force and not a gravitat ionM force at all. This example shows how dangerous it is to conclude tha t the weak and the strong interactions are different forces simply because pa r tMes respond differently to them.

Since we are interested here only in order-of-magnitude calculations of the

coupling constants over the entire range of part icle interactions, we shall not

consider such refinements as CVC theory, the universal V -- A Fermi interaction, or the Cabibbo hypothesis. Fur thermore , we do not assume tha t nucleons are surrounded by clouds of pions, but are gravi tat ional ly bound-quark triplets which interact with each other and with pions via the gravi tat ional charges of the quarks. To keep things as simple as possible and show clearly the distinction between the magnitudes of the various coupling constants, we go back to

(xml s cxcellent analysis of these constants in his 1950 Silliman Lectures (17). Much has happened in particle physics since then, but , except for the above- ment ioned refinemeuts and the fall of pari ty, Fcrmi 's overall discussion is still Yalid and well suited to our type of calculation.

FF~I~H divided the six impor tan t e lementary-part ic le interactions tha t were known at the t ime into two groups of three, placing in the first group

those interactions whose interact ion constants have the dimensions of a charge and in the sccond group those interactions whose interactio~ constants have the dimensiol~s of energy times volume (the Fermi coupling constants). The constants of the first group are (in units cm ~ g~ s --~)

e.., (the Yukawa interact ion const 'mt) "~ 10 -8,

e (the e lementary eleetrie charge) ----4.8-L0 .lO,

e3 (the interact ion constant for the interactions between pions, muons "rod neutrinos) ~ 1 . 1 . 1 0 ~

The const,~nts of the second group are (in units erg cm 3)

gL (the interact ion constant for the be ta-decay process .W ~ I ) ~ -

+ e - - t - ~ ) = 2 . 1 0 - " ,

g.2 (the interact ion constant for the spontaneous decay of the muon ~- ~ e - + O o + v ~ ) = 3 .3 . t0 -49,

g3 (the interact ion constant for rouen captnre P - ] - i z - ~ . J V ~ - v ~ ) = 1.3.10 -49.

('v) E. FEI{I~II: Elementary Particles (London, 1951), p. 36.

llO L. MOTZ and J. v psT~i~

The values of these constants were not deduced f rom the basic theory (no one has ye t shown how this can be done), bu t were obta ined b y compar ing the observed rates of the various processes wi th the rates calculated b y s tandard

pe r tu rba t ion theory. The near equal i ty of the three Fermi constants and the

in t roduct ion of the two-component neutr ino theory (nonconservat ion of par i ty)

led F]~u and G E L L - M ~ (3) to the CVC hypothesis and the in t roduct ion

of the universal Fe rmi in teract ion ~ ( 1 -t- 75) (the V - A interact ion), f rom which

one obtains the values 1.415 .]0 -49 for gl and 1.435.10 -49 for g2 (again by com-

par ing observed and calculated rates). We shall now show tha t the five non-

e lect romagnet ic coupling constants can be expressed as the p roduc t of the

basic g rav i ta t iona l charge (?/e)tl2 and small powers of the f ine-structure con-

s tant . F r o m this one m a y infer t ha t the strong and the weak in teract ion are

in t ima te ly re la ted to each other and to the gravi ta t ional force. Our theory

thus leads to a unification of ~ a v i t y and the weak and strong interact ions in

a simple and direct way. S ta r t ing with e.~, we have

(41)

e~ = ~ a - t ( ? g c l 4 ) t = 1 . 7 . 1 0 - s ,

e = 2at(t/cI4) j = 4.8.10 -~o,

e3 = o~3(?gc/4) ~ = 1.1.10 - 1 5 .

Since the dimensions of the Fe rmi g-constants are energy • volume, we m u s t introduce into our fo rmula an addi t ional fac tor with the dimensions of the square of a length to express these constants . To do this we write the g-factors in the form [ G t m R ] . ~, where m is the mass of the uni ton and R is some character is t ic radius. Placing G m 2 ~ ?go/4 and R z ?g/Me, where M is

the nucleon mass wc obta in

(42) g, = ~2(~c/4)(?g/Me) 2 = 1.5.10 .49 .

I f instead of working with the g's directly, we work with the deve lopment

p a r a m e t e r fl which Fe rmi defines as

fl~ ~_ g~(m4c2/?g6) ,

where m is to be replaced b y m s for the two react ions governed b y gl and g2 and b y m~ for the react ion governed b y g3. In t roduc ing the value of g t h a t led

to (42), we obtain for fl the simple expression

(43) fl -~ a * ( m / M ) ~ ,

THE GRAVITATIONAL CHARGE ~ ~ AS A UNIFYING PRINCIPL~ IN PHYSICS l l l

which has the numerical value 0.8..10 - ~ for m---- m, and the value 1.7.10 '

for m = m~. We note a ra ther simple relationship among the three constants, e 2, e3 and g,

(44) 9 / ( h / M c ) ~ --: 2a - t e~ e3.

One might have expected this f rom the observat ion tha t the reaction P + ~ - ~ _ ~-2V-~%, governed by the interact ion constant g, can be considered as the result of the two successive reactions ~(" ~ I ) + ~ - and r : - ~ _ ~ - ~ - ~ governed by the constants e.. and e3, respectively.

I f we introduce our mass formula (37) into our expressions for the various coupling constants, these become simple powers of the fine-structure constant. :Expressing e~ and e3 as multiples of e, we obtain

(45) ez/e ---- (�89 e3/e ---- ~n/2.

The g-constants defined by (42) become

(46)

For the development parameters/3~ and /3 , which we obtain from (43) by placing m equal to m~ and m~ respectively, we obtain

(47) ~ = ( t ) " ~ ' , /3. = ~ ( ~ ) ' ~ ~

We consider these simple formulae, expressing the various coupling con- stunts as small powers of a, as strong evidence for the existence of unitons and for our model of hadrons. They also support our content ion tha t the strong and the weak interactions are different manifestat ions of the gravi tat ional force.

6. - S u m m a r y and conc lus ion .

In this paper, which is a cont inuat ion of three previous papers (.~,sa), we

have shown tha t the uni ton-quark (m = �89 V / ~ / G ) unifies m an y different aspects of physics ranging f rom black holes to e lementary particles. We have shown tha t gravi ty supplies the cut-off t ha t changes divergent field theories to con- vergent ones, and the correct cut-off f requency is obtained if the cut-off mass

is just t ha t of the uni ton-quark; the un i tomquark is stable against decay via the Hawking process because it is a gravi tat ional s t ructure in its lowest quan tum state with a single quan tum of action. Hence it is pro tec ted against a quan tum

l l 2 L. 1ROTZ ~I ld J , EPSTEIN

decay by the quantum theory itself. Using the Bekenstein expression for the en t ropy and tempera ture of a black hole, we show tha t the en t ropy of a free uni ton-quark is just the Bol tzmann constant k, giving a deeper significance to this otherwise somewhat mysterious constant. Our analysis indicates tha t there is more information in a macroscopic black hole than just the total mass, charge and angular momentum. The black hole behaves like a statistical gas of N---- M/m uniton-quarks with N~/2 interactions. We deduce the numerical value of the fine-structure constant within 7 parts in 10000 from the ratio of the number of degrees of freedom of a free gravi tat ional field (with source) to the number of degrees of freedom in a free electromagnetic field (with source). F rom this same ratio of fields we deduce a muss formula which gives the masses of leptons, mesons and baryons in terms of the mass of the electron and two integers. The formula is correct to within un error averaging ,~ few parts in a thousand. We deduce tha t the fraction of its free mass tha t ~ uniton- quark contr ibutes to a nucleon is ] = ~(a--'G~mJe); which is remarkable in tha t it is the first physically significant formula involving all the known fun-

damenta l constants. We conclude tha t the strong and weak forces are bu t diverse manifestat ions of the gravi tat ional force, since we can derive the strong and weak coupling constants f rom the fundamental gravi ta t ional charge.

If the model proposed in this and our previous paper turns out to be correct (an experimental search for such extremely massive quarks is current ly in progress by ULLMA.~ of Lehman College, N.Y.), we believe t h a t a large step in the direction of a purely geometric unified iield t h eo ry - - a s envisioned by E ~ N S T ~ - - w i l l have been accomplished. In a very concrete sense it seems tha t fundamenta l physics can, as a whole, be derived from a symmetr ic tensor in a 4-dimensional Riemannian continuum, if the requirement of gauge invariance ~ i n the Weyl sense~is imposed. The only parameters left free are the muss of the electron, which, it is hoped, will be calculated from the energy of the field, and the charge on the electron (or equivalent ly the value of Planck 's constant). All else can be calculated. The logic proceeds as follows.

The gauge invariance introduces a 4-vector, whose curl is the electromag-

netic-field tensor. The Einstein-]~icci tensor is enlarged to a gauge-invariant structure, from which an invariant Lagrangian is constructed. The variat ion

of this Lagrangian leads to the Dirac equat ion of a uni ton-quark of mass V/iZc/G and charge ~e or ~e (with Land6 g factor of 2 and 1, respectively).

The ratio of fundamenta l gravi tat ional charge to fundamenta l electric charge

(the fine-structure constant) is calculated from the ratio of the degrees of f reedom of a gravi tat ional field with source to the degrees of freedom of an electric field with source. I f the mass of the electron is assumed, vir tual ly all the other masses of particles states can be calculated from this same ratio and the Gell-Mann-Okubo formula for the internal angular momenta . The grav- i tat ional 3-body problem analysed by LAGRA.NGE and the Pauli principle lead to the linear model for the ba ryon states. This, in turn, gives the 18 dist inct

TIIE GRAVITATIONAL CItA:RGE ,~ ~r AS A UN'[FYING PRINCIPLE IN PIIYSICS ]1~

b a r y o n s ta tes a n d the correct m a g n e t i c m o m e n t s . One ca lcula tes i m m e d i a t e l y

the B e k e n s t e i n t e m p e r a t u r e for u macroscopic b l ack hole from the proper t ies

of the u n i t o n - q u a r k a n d expla ins the B o l t z m a n u c o n s t a n t k as the e n t r o p y

of a free u n i t o n - q u a r k . The coupl ing cons t a n t s are found also f rom the funda-

m e n t a l g r a v i t a t i o n a l charge.

T h u s o r d i n a r y g r a v i t y a n d the s t rong a nd weak forces are shown to s t em

f rom the genera l - r e l a t iv i s t i c t heo ry of g r~v i ty ( tha t is f rom geome t ry ) ; the

e l ec t romagne t i c force and the ])article wave field follow f rom the gauge in-

var iance .

�9 R I A S S U N T O (*)

Si nlostra in questo lavoro ehe lt~ condizione di quantizzazione di base 4Gm2/c = n h,

ehe 5 stata dedottn, precedentemente dal prineipio di Weyl d'i l tvarianza di gauge e dal problema generale a due corpi, porta, a una partieella con carica gra.vii.aziona]e 1o V~-c- I[~L le segaenti propriet'~: 1) ~ un mini buco nero; 2) d~ uu t~glio gravita.ziona.le n~tu- r;de degli al tr imenti divergenti c~dcoli di autoenergia; 3) rapprcscnta lln singolo quanto di ~rzioni h c quindi ~ uu insieme di campi gravitazionali f luttuanti nel sue state quartiico pifi basso ; 4) poi('h6 la sun entropia di Bckenstein ~ proprio k, fine ad un fattore d'ordine di 1 d;t un sigtlificato fisico alla al tr imcnti misteriosa costante di Boltzmann ; 5) porta ad una meccanica statistica dell ' interne di un macro buco nero di massif M, nel sense che dh hr quantit'5 _h; ~ Mira come mmmro di par~ieelle distinte nel buco nero e la quantit;'l. Nz/2 COlne sue numero di gradi di libert;r interni eeeitabili; 6) dh una fornmla semplice per la radice quadrata della costante di s t rut tura fine (con accur~tczz~r migliore di 1 partc su mille) come rapporto di ml nulnero di gradi di liberth del campo elcttromagne- rico con sorgente con il numero di gradi di libcrt;~ de1 campo gravita,zion,%le con sorgentc; 7) porta ad una formula per le masse dei leptoni, dei mesoni e deibar ioni ; 8) permet, to di csprimere le costanti di accol)piamento debole c forte colne piceole potcnze della costante di s t rut tura fine.

(*) Trad~tzione a eura della l:edazione.

Pe3roMe lle rlo.rly~lelto.

8 - II A'ltovo Cimento A.