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Static solutions of the combined Dirac-electromagnetic-gravitational field equations O'Hanlon, John David 1970

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STATIC SOLUTIONS OF THE COMBINED DIRAC-ELECTROMAGNETIC-GRAVITATIONAL FIELD EQUATIONS by JOHN DAVID 01HANLON B.Sc, National U n i v e r s i t y of Ireland, 1964 M.Sc, National U n i v e r s i t y of Ireland, 1965 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t hou t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT I t i s assumed that charged, spin-^, matter d i s t r i b u t i o n s can be described i n terms of a Dirac spinor f i e l d i n t e r a c t i n g with the ele c t r o -magnetic f i e l d and a scalar g r a v i t a t i o n a l f i e l d . The f i e l d equations and the energy-momentum tensor are found from an action p r i n c i p l e . The f i e l d s are not quantized. The f i e l d equations are examined and various l i m i t i n g forms discussed. This thesis deals p a r t i c u l a r l y with the time-independent spherically-symmetric case. Solutions are found for the exterior region of a charged g r a v i t a t i n g sphere. The behaviour of these solutions depend on the value of the charge-mass r a t i o . When this r a t i o has the value (4-TT Q ) z, where Q i s the g r a v i t a t i o n a l constant, the e n t i r e system can be solved a n a l y t i c a l l y . The ensuing solution, c a l l e d the Weyl-Majumdar solut i o n , i s obtained and discussed. When the charge-mass r a t i o i s smaller than (im Q ) 2 , normalised solutions are found which y i e l d e l e c t r o s t a t i c and g r a v i t a t i o n a l potentials singular at the o r i g i n . The matter density i s well-behaved everywhere. Normalised solutions were not found for l a r g e r charge-mass r a t i o s . The s i g n i f i -cance of the solutions, and the accuracy of the numerical technique are discussed. A l t e r n a t i v e Lagrangian densities are considered which may y i e l d non-singular solutions. i i ACKNOWLEDGEMENTS The author would l i k e to express hi s deep appreciation to his teacher and research supervisor, Dr. P. R a s t a l l , f o r h i s patience and many kindnesses i n seeing this work through to i t s conclusion. The author would also l i k e to thank Dr. H. Dempster of the Computer Centre for h i s h e l p f u l discussions regarding the numerical technique. Thanks are also due to the National Research Council of Canada for providing f i n a n c i a l assistance i n the form of N. R. C. Scholarships. i i i TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENTS i i i CHAPTER I INTRODUCTION 1 CHAPTER I I THE FIELD EQUATIONS 7 CHAPTER III TIME-INDEPENDENT SYSTEM 22 CHAPTER IV EXTERIOR FIELD OF A CHARGED SPHERE 31 CHAPTER V THE WEYL-MAJUMDAR METHOD FOR OBTAINING 38 STATIC SOLUTIONS OF THE FIELD EQUATIONS CHAPTER VI ASYMPTOTIC SOLUTION FOR THE DIRAC 51 WAVE-FUNCTION CHAPTER VII NUMERICAL RESULTS AND DISCUSSION 58 CHAPTER VIII ALTERNATIVE LAGRANGIAN DENSITIES 72 BIBLIOGRAPHY 8.1 i v LIST OF FIGURES Figure Page 1 Observed mass 61 2 Mass density 63 3A Metric function S = (- )^ 64-3B Metric function 65 4 Normalisation constant . . . . , 66 5 Radius of core 67 V CHAPTER I INTRODUCTION Our purpose i n this thesis i s to f i n d s t a t i c , s p h e r i c a l l y symmetric solutions of the combined g r a v i t a t i o n a l , electromagnetic and Dirac f i e l d equations. In other words we are trying to construct a c l a s s i c a l model of an elementary charged p a r t i c l e . We consider only unquantized f i e l d s , which means that the solutions cannot be expected to correspond to r e a l physical objects. However, i t seems sensible to investigate the c l a s s i c a l problem ( ' C l a s s i c a l 1 i n the sense that we do not consider pair creation) before attempting the much more d i f f i c u l t quantum one. If the c l a s s i c a l problem has solutions, they may help us i n the other case. There i s also the hope, as suggested by Dirac (1951), that the c l a s s i c a l solutions, i f they e x i s t , may give the correct value for e/m, where e i s the charge and m the mass of the electron. (Of course, we would not expect to obtain the values of e and m separately. Dirac believed the value of the e l e c t r o n i c charge e to be a purely quantum phenomenon and not derivable i n a c l a s s i c a l theory.) The concept of a body i n c l a s s i c a l f i e l d theory i s generally treated i n one of the following two ways. Either the body, or " p a r t i c l e " , i s considered as a s i n g u l a r i t y i n the otherwise s i n g u l a r i t y - f r e e f i e l d , or else i t i s assumed to be a mass of f l u i d obeying some more or less a r b i t r a r y equation of state. Both treatments are obviously unsatisfactory. In the f i r s t case we forego the p o s s i b i l i t y of saying much about the i n t e r n a l structure of the p a r t i c l e . _In the second case, we are permitted to have g r a v i t a t i o n a l and other f i e l d s which are regular everywhere, which i s a great advantage, but, unfortunately, other problems now a r i s e . The equation of state i s a r b i t r a r y , and the very concept of f l u i d , which i s -1-2. borrowed from macroscopic physics, probably has no place i n the micro-scopic domain. There e x i s t s , however, a t h i r d p o s s i b i l i t y . We can t r y to construct solutions of the f i e l d equations themselves which are l o c a l i z e d i n space, are regular everywhere, and which represent concentrations of mass, and possibly, charge. Such solutions were f i r s t studied i n d e t a i l by Wheeler and h i s co-workers (Wheeler 1955, 1962; Power and Wheeler 1957; B r i l l and Wheeler 1957). His i n t e n t i o n was to draw attention to, and to explore, the e x t r a o r d i n a r i l y r i c h physics of curved empty space. In this con-nection he used f i e l d s of zero r e s t mass, since only these had been geo-metrised. The solutions which he found, and to which he gave the name "geons", were smooth over the whole of 3-space, and represented objects which were extremely large. A c l a s s i c a l analysis was v a l i d only when the e l e c t r i c f i e l d strength £ was less than the c r i t i c a l f i e l d strength, £CI?IT = -m* / , of pair theory. This y i e l d s a mass of the order of C^/C Q ^2 £CRIT} ' V . ID ^ g and a radius ~ l o " cm. No such objects have yet been observed. The physics of smaller geons has not been investigated because quantum e f f e c t s would have to be considered, and as yet no s a t i s -factory quantum theory of g r a v i t a t i o n e x i s t s . In addition to being excessively large, geons are unstable, although their l i f e t i m e s can be very long. The geon, then, as envisaged by Wheeler, constitutes a geometrical model for mass, or, i n h i s own words " mass without mass". In order to create geons which are smaller i n both l i n e a r dimen-sions and mass than the above, i t i s necessary to re-introduce matter f i e l d s . This means that we now have to take into account a new parameter, m , the bare mass of the matter f i e l d . The analysis, however, remains B c l a s s i c a l i n the sense that the f i e l d s are not quantised (although Planck's 3. constant does appear i n the f i e l d equations). The neutral Klein-Gordon geon has been studied by Feinblum and McKinley (1968) and by Kaup (1968) . These authors examined the time-in v a r i a n t spherically-symmetric solutions of the coupled Klein-Gordon-E i n s t e i n equations. There are c e r t a i n differences i n th e i r approaches so we w i l l consider them separately. Feinblum and McKinley (1968) sought solutions that would correspond to a spectrum of bound states from a si n g l e unobservable "bare" mass, thus i n d i c a t i n g a set of observable " p h y s i c a l " masses. Since the problem i s too d i f f i c u l t to solve a n a l y t i c a l l y , they used a numerical technique. As boundary conditions they assumed that f or large values of the r a d i a l co-ordinate the metric should asymptotically approach the Schwarzschild metric, and the wave-function for the Klein-Gordon f i e l d should approach the one given by solving f or the zeroth-order approximation i n the g r a v i t a t i o n a l f i e l d . A large value f o r ^ , the r a d i a l co-ordinate, was chosen and a step-by-step i n t e g r a t i o n toward the o r i g i n performed. A -12 value of 1.28 x 10 g for the bare mass was taken. I t was found that for normalized functions the parameters involved i n the equations, v i z . the eigen-energy E and the normalization constant A for the wave-function, became so small that the equations became ill-behaved and solutions were not obtained. However, an unnormalized s o l u t i o n was found f o r the ground state. The metric proved to be well-behaved everywhere except at the o r i g i n . At this point the curvature tensor diverged. The authors a t t r i -bute this to the i m p o s s i b i l i t y of solving the eigen-value problem exactly by numerical methods. Kaup (1968), i n h i s discussion of t h i s paper, pointed out that the correct explanation for the occurrence of the diver-gence might be that they had used an i n c o r r e c t value for the bare mass. He based this a s s e rtion on a study of th e i r normalisation procedure. However, since this p a r t i c u l a r s o l u t i o n was unnormalised to begin with, the point seems academic. F i n a l l y , we note that the " p h y s i c a l " or observable mass of t h e i r s o l u t i o n was equal to 0.07 times the bare mass, or 0.9 x 10 ^ g , and i t s diameter was approximately 4 x 10 cm. In h i s own work Kaup (1968) obtained solutions which were better behaved. His normalisation was s l i g h t l y d i f f e r e n t from that used by the aforementioned authors. For boundary conditions he assumed that the metric approached that of Schwarzschild, and he obtained the asymptotic form of the wave-function by solving the Klein-Gordon equation i n the "Coulomb" p o t e n t i a l of the g r a v i t a t i o n a l f i e l d , i . e . the f i r s t - o r d e r approximation. Again only the ground state was considered. I t was found that there was an upper l i m i t on the value of the bare mass, which was m = 1.75 x 10 ~* g. This i s roughly of the order of ( -&C/Q-) D The solutions were then examined for s t a b i l i t y , and i t was shown that for Klein-Gordon geons adiabatic r a d i a l perturbations are forbidden. This means that they are therefore r e s i s t a n t to s p h e r i c a l l y symmetric g r a v i t a t i o n a l collapse. Although these structures are much smaller than the o r i g i n a l geons of wheeler, they are s t i l l too massive to be considered as models for any know p a r t i c l e s . Nevertheless they are of considerable i n t e r e s t . In this s p i r i t we w i l l study geons obtained by solving the Dirac-Maxwell-gravitational f i e l d equations. I t i s necessary to introduce the electromagnetic f i e l d because the Dirac equations describe a charged p a r t i c l e . This also means that we w i l l have to consider another para-5. meter, e, the electron,charge. I t w i l l be shown that we do not encounter the same d i f f i c u l t i e s with the normalisation as the aforementioned authors since, i n our case, the Maxwell equation ensures that a l l solutions w i l l be automatically normalised. Like Kaup ,(1968) we f i n d that there i s an upper l i m i t of 4-. 4 x 10 ^g on the value of the bare mass. One further difference between our approach and that of Kaup (1968) and Feinblum and McKinley (1968) i s that we do not use the E i n s t e i n theory to describe the scalar e f f e c t s of the g r a v i t a t i o n a l f i e l d . We use instead the^theory of gravita-tion ( R a s t a l l 1968a, b) which i s simpler and more tractable i n many respects. A l l of these points w i l l be discussed i n more d e t a i l i n the body of the text. One of the reasons why this problem i s p h y s i c a l l y i n t e r e s t i n g i s that there may be solutions for only c e r t a i n values of e/m. To get a d i s -crete set of values one imposes d i f f e r e n t i a b i l i t y conditions on the f i e l d s (consider the example of the hydrogen atom). For a p a r t i c u l a r s o l u t i o n to be p h y s i c a l l y meaningful we demand that i t be c o r r e c t l y normalised and that the energy density be everywhere f i n i t e . We also require that the e l e c t r o -s t a t i c and g r a v i t a t i o n a l f i e l d s be well-behaved i n the region outside the l o c a l i s e d matter d i s t r i b u t i o n ( p a r t i c l e ) . Since there i s no way to measure these f i e l d s inside the p a r t i c l e , there i s no physical reason for demanding r e g u l a r i t y i n this region. Imposing this extra condition may lead, as indicated above, to a set of dis crete values for the r a t i o e/m. There i s one case i n which the above equations can be solved analy-t i c a l l y . That i s when we assume that the component § o o of the metric "Censor i s a function only of the e l e c t r o s t a t i c p o t e n t i a l Ao. This method of solving the equations of e l e c t r o - g r a v i t a t i o n a l theory was introduced by Weyl (1917) and further investigated by Majumdar (1947). The method i s v a l i d only for the case of s t a t i c f i e l d s . Majumdar showed that, i n matter-free space, the f u n c t i o n a l r e l a t i o n s h i p must be of the form coo = A + BA 0 4 H T T Q C - ^ A " , (1.1) where A and B are constants, G i s the g r a v i t a t i o n a l constant, and c the speed of l i g h t i n vacuo. When the constant B i s so chosen that the r i g h t -hand side becomes a perfect square, then (1.1) i s ca l l e d the Weyl-Majumdar r e l a t i o n (WMR). The WMR can be used to sim p l i f y considerably systems of equations of the type described above (see, for example, Das 1962, 1963; De 1965, 1969; Mukherjee, 1963).- Solutions of the Klein-Gordon-Maxwell-Einstein field equations have been found; by Das and Coffman (1967) f o r the case when the WMR i s assumed. They showed that, s t a r t i n g from any given s t a t i c , purely g r a v i t a t i o n a l universe, one can construct universes corresponding to solu-tions of the above equations, provided only that a single d i f f e r e n t i a l equation i s s a t i s f i e d . The WMR was found to imply an equality between the charge and mass parameters of the theory. Starting from the well-known Schwarzschild universe, they obtained solutions corresponding to p a r t i c l e s -5 -33 of mass ™ 3 x 10 g and radius ~ 2 x 10 cm. The energy E of the matter 2 f i e l d was found to be equal to the bare mass m^ , or E = m^C , thus giving a binding energy zero to the Klein-Gordon p a r t i c l e . The metric obtained has a co ordinate singularity at s p a t i a l i n f i n i t y which made i t s physical i n t e r p r e t a t i o n d i f f i c u l t . Other solutions were found which also had s i n g u l a r i t i e s at f i n i t e values of the r a d i a l co-ordinate. In this thesis we w i l l also, f o r the sake of completion, consider the Weyl-Majumdar problem for the scalar g r a v i t a t i o n a l f i e l d . We w i l l f i n d r e s u l t s s i m i l a r , i n many respects, to those of Das and h i s co-workers. The method i t s e l f , i t s drawbacks and advantages, w i l l be discussed. To sum up: i n the following we examine the possible states of a Dirac p a r t i c l e at r e s t i n i t s own e l e c t r o s t a t i c and g r a v i t a t i o n a l f i e l d s . In the f i r s t part the f i e l d equations are derived and examined; i n the second part we investigate the solutions. CHAPTER I I THE FIELD EQUATIONS We assume that space-time i s a four-dimensional pseudo-Riemannian manifold of s i g n a t u r e + 2, which obeys the Lichnerowicz d i f f e r e n t i a b i l i t y c o n d i t i o n s , and that there e x i s t co-ordinate systems, c a l l e d Newtonian char t s , i n which the m e t r i c tensor has components of the form (2.1) where L a t i n i n d i c e s range from 1 to 3, Greek i n d i c e s from 0 to 3. The f u n c t i o n (J> (OC'.OL'X*2°) i s c a l l e d the g r a v i t a t i o n a l p o t e n t i a l , fyo i s a constant, and C E i s the n a t u r a l speed of l i g h t . I n the above, and i n what f o l l o w s , we use the procedure and n o t a t i o n of R a s t a l l (1968a, b) . The geometry of our space i s determined by the s i n g l e r e a l f u n c t i o n tj> , which i s a r b i t r a r y up to the a d d i t i o n of a constant. The meaning of (j>0 i s roughly the f o l l o w i n g : S p e c i a l Newtonian charts always e x i s t whose tangent v e c t o r s are ortho-normal w i t h respect to the m e t r i c c^ v at any p o i n t where the p o t e n t i a l has the value <j>0 . Charts of t h i s k i n d are c a l l e d (J>0 -charts and are i n general determined up to a s h i f t of o r i g i n and a constant orthogonal transforma-t i o n of the s p a t i a l co-ordinates ( R a s t a l l 1968a). I t i s c l e a r from (2.1) that i f a (j> 0-chart e x i s t s , then a (j)^-chart a l s o e x i s t s , f o r any constant (J)^  . The p h y s i c a l p r e d i c t i o n s of the theory, however, should depend n e i t h e r on the choice of the constant fyQ , nor on the p a r t i c u l a r -7-C^>0 -chart once t h i s constant i s chosen. cj>p - Quantities I t i s possible to define a new metric tensor f i e l d r; i n the following way. Let jo by any point i n space-time, then, by our f i r s t assumption, there e x i s t s a cj>p -chart on some neighbourhood of p . The metric tensor (^p) i s defined at p by r e q u i r i n g where *9eyi = - So/*, and are the tangent vectors of the (j>0 -chart at p . Since the (j>o -charts cover space-time, i t follows that /) i s defined g l o b a l l y . I t can also be shown that V) depends only on the choice of (j>0 and not on the choice of (j>o-chart ( R a s t a l l 1968a). The metric VJ can be used to define cJpo- lengths and times i n analogy with natural lengths and times i n s p e c i a l r e l a t i v i t y . Consider two neighbouring points, XL > a n d "X- + i n three-space, which have the same time co-ordinate t = X° /CE • The distance between them i s given by d £ E = \l( V dx" dx*) , **dp {-C e- 2(<j>-<Ul V d^dq : 1 ( 2.3) This i s the "natural" length. The meaning of the subscript E w i l l be explained shortly. The (|>o -length, on the other hand, i s defined as 9. d k = \l r)ji„ A x ^ J ^ ) \J dx^dx" (2.4) S i m i l a r l y f o r times. The i n t e r v a l between the two time s t s t + cit , at the one space-time point X1 i s given by dcE = Q-1 \IC- V d W x O t = sue)? { Cf2 C<|>-<)Ol . (2.5) This i s the "natural" time. The <JJo -time, however, i s defined by = cr1 \j (- dz* ) } dt. (2.6) A word i s now i n order concerning our notation. Wherever the subscript £ appears, i t means that the quantity subscripted i s measured i n natural or "experimental" u n i t s . I f i t does not appear, then the quantity i s measured i n cJ)Q -units (to be defined below) and i s , there-fore, a QSo - quantity. The one exception to this r u l e i s the g r a v i t a t i o n a l p o t e n t i a l (j) . (j) i s always measured i n natural u n i t s . The natural and (|> - units of length and time are rel a t e d accord-ing to the following expressions dJ2e = S~L dJe, (2.7) dte = S dt , (2.8) which are derived from (2.3) - (2.6), and where R a s t a l l (1968a) has shown that i f we add to the above a change also i n 10. the u n i t of mass of the form c 3 Tn£ = b m , ( 2 . 9 ) then the equation of motion for a p a r t i c l e , written i n <)p0 -units, becomes formally i d e n t i c a l to the corresponding equation i n s p e c i a l r e l a t i v i t y . We summarise as follows. Let Q.E be any quantity measured i n natural units and l e t i t s dimensions be TQel = [ LI'M^T^J; then i t s value i n dp -units i s Q , where Q . QtS1*-*-^ . ( 2 . 1 0 ) Quantities measured i n natural units i n general do not depend on the choice of dp0 . However, i f the quantity Q i s a tensor, then i t s components obviously w i l l depend on the p a r t i c u l a r co-ordinate system used, regard-less of the dimensions of Q. . In this work our convention w i l l be that (2.10) holds only for i n v a r i a n t quantities. Thus, i f Q. i s a tensor at the point p , and ( e„ , C, , P2 , ) i s a basis of the tangent space at p , then Q = e ^ ® e v ® ® e P ® e * : , (2.11) and (2;10) applies to the tensor i t s e l f , and not to i t s components i n d i v i d u a l l y . The meaning of <|>0 -quantities w i l l become evident i n the next section, where we discuss the action p r i n c i p l e . The Action The Dirac theory of the e l e c t r o n i n f l a t spacetime has been studied i n great d e t a i l by many authors (see, for example, Rose 1961; Corinaldesi and Strocchi 1963). The f i e l d equations may be deduced from a v a r i a t i o n a l p r i n c i p l e with an a c t i o n i n t e g r a l of the form A = I l F (Y>Ao) d * X , (2.12) where and Xp - %{nfos + rty-^^vM, (2.i3) i n = " 4 l>v> ^ (2.14) ilNT = C . (2.15) «£p ' Xn' a n (* ^*'NT a r e> respectively, the Lagrangian densities for the Dirac f i e l d , the Maxwell f i e l d , and the interaction. The notation i s the usual one: ^ i s the Dirac wave function, Ayu i s the electromagnetic tor p o t ential, the Dirac matrices are related to the Minkowski vec metric tensor V)'** by and ^ = i L|> V° 5 where Vp i s the spinor conjugate to ^ We can introduce the interaction with the gravitational f i e l d i n two ways: ( i ) We can write out the Lagrangian density i n a generally co-variant form, derive the f i e l d equations, and then take into account the parti c u l a r form of the metric (2.1). This method has the advantage of being unambiguous, and shows off the geometrical character of the gravi-t a t i o n a l interaction. ( i i ) We can use the procedure, or "prescription", sketched out by Rastall (1968b). This method i s applicable only to those f i e l d s whose Lagrangian densities are known i n the special r e l a t i v i s t i c l i m i t , which i s the case with (2.12). We take the flat-space Lagrangian density, (the meaning of the subscript f w i l l be explained l a t e r ) , which i s a function of the f i e l d components ^tn and th e i r derivatives 'Irnyu. > and we make the following r e - i n t e r p r e t a t i o n s . A l l quantities are now to be considered as <j>0 - q u a n t i t i e s . We must f i r s t however, replace ^mjO by ( C _ > ) ^ £ . The co-ordinates ( X 1 , t ) are re-interpreted as (po -co-ordinates, the components of the f i e l d s and the parameters of the theory (such as & , TO , C ,$L ) are assumed to be measured i n d> 0-units. Once this r e - i n t e r p r e t a t i o n has been made, we can derive the f i e l d equations i n the usual way. So f a r we have not mentioned the g r a v i t a t i o n a l f i e l d i t s e l f . This i s included i n our theory by the addition of a term C£Q to the Lagrangian density of the other f i e l d s «£f . We make the assumption that i t i s possible to write the t o t a l Lagrangian density for the coupled Dirac-electromagnetic-gravitational f i e l d s i n the form where <fp i s the Lagrangian density of a l l the f i e l d s other than the g r a v i t a t i o n a l f i e l d , and £.£ i s the "purely" g r a v i t a t i o n a l part, being a function only of tj> and i t s d e r i v a t i v e s . For the time being, we are mainly concerned with £>f . We have outlined above two methods of deriving i t . In our case, both methods give the same r e s u l t . The general r e l a t i v i s t i c formulation of the Dirac equation has been investigated by many authors (see, for example, the review a r t i c l e s 13. of B r i l l and Wheeler 1957, and of Bade and Jehle 1953). We w i l l not go into any d e t a i l s here but we w i l l simply state the r e s u l t s . The general r e l a t i v i s t i c a ction i s given by A F = J <£ F E f ? <PX , (2.19) where <£ F £ i s the Lagrangian density for the coupled Dirac-Maxwell f i e l d s i n t e r a c t i n g with the g r a v i t a t i o n a l f i e l d . In (2.19), and i n the next few expressions, we have added the subscript E for l a t e r convenience, can be s p l i t up into i t s component parts, and f - r-> A \ / t <**\nrt - E J £ (2.22) t>x>t i s the (generally covariant) Lagrangian density for the Dirac f i e l d . The matrices s a t i s f y V f + T V = 7%^ > (2.23) which i s the generally covariant g e n e r a l i s a t i o n of (2.16); the covariant derivatives of the wave function are given by VA - \ % - , (2-24) where the J^u are the Fock-Ivanenko c o e f f i c i e n t s ( B r i l l and wheeler 1957); = ( ^ E V - A E ^ U J V ) where the AE^ U are the electromagnetic potentials; and the current density J E ^ j i s given by V" - ie.C ' P.^^ • < 2- 2 5 ) 14. The equations of motion derivable from the action (2.19) are the following, rCV. - i | A R H + /e We = 0 > (2-26) (2.27) and i t can be shown from (2.26) and i t s adjoint that the current Jg obeys the equation of continuity, ~ b v (NF8^rHO = 0 . (2.28) At this stage the metric form (2.1) is substituted for the (as yet) unspecified . The Fock-Ivanenko co-efficients are calculated in the usual way (see the above references). However, we do not need individual expressions for the lyu , since they appear in the equations only in the form ^ i ^ u , and i t can easily be shown that rf> = ia^C^M ^ r ^ ) , (2.29) where § , ^ = c>c}> jlbX^ , and that 9 = d« t ( V ) = -S-* • (2-30) The matrices Yy* = ^ / U v^ V a r e related to the f l a t space Dirac matrices Yyu by Yii = S"1 %\ , Vo = S Yo (2.31) Using (2.29), (2.30), and (2.31) we can rewrite the f i e l d equa-tions (2.26) and (2.27) as (2.32) Vs-Vw)= cr's-'Jf . (2 .33) For the sake of l a t e r convenience we l i s t the connections between the quantities encountered above and the corresponding cj)^  - q u a n t i t i e s . These expressions are derived by using the formula (2.10). * -c z 1 • • hs~' > e - e E S , J = A . = A. , F = • t - (2.34) We can apply the " p r e s c r i p t i o n " outlined above to obtain the Lagrangian density from the s p e c i a l r e l a t i v i s t i c £ p . We- obtain = - II f > v P v = i (E 2- 8 2 ) , <2-36> i l H T = C-' J ^ A ^ , (2.37) where we have written = S " ' F ; k 0 . (2.38) In the a p p l i c a t i o n of the p r e s c r i p t i o n to ot^ we have considered the 16. e l e c t r i c and magnetic f i e l d components Efe and Bfc to be the basic f i e l d s , rather than the potentials . I t i s quite easy to see that the dp^  - £f written above i s the same as the generally covariant «Lpg (equations (2.20) through (2.22)) except f o r a fac t o r , i . e . o£p = £FE S"2 . (2.39) This means that the action i n t e g r a l A F = J if d^X (2.40) i s the same, and therefore we have the same f i e l d equations, (2.32) and (2.33) . The t o t a l Lagrangian density i s given by (2.18) and the action i n t e g r a l for the coupled Dirac-electromagnetic-gravitational system i s A = J C £Q+ tf) d * X • (2.41) I t i s now necessary f o r us to specify £<; , the purely gravita-t i o n a l part of the Lagrangian density. R a s t a l l (1968b) has shown that a p a r t i c u l a r l y simple form f o r o £ § i s given by 4 = K S-2 y V ^ M , (2-42) where K = — (8-irQf} , and Q£ i s the Newtonian g r a v i t a t i o n a l constant. This choice f o r <£Q ensures that the R a s t a l l theory w i l l give the same astronomical predictions as the E i n s t e i n theory. Other choices f o r «£Q are possible, of the form where (j>( i s a constant independent of the choice of Newtonian chart. I f oi. 0 , then, as R a s t a l l (1968a) has shown, the p e r i h e l i o n advance of test p a r t i c l e s w i l l not be the same as i n the E i n s t e i n theory. 17. The g r a v i t a t i o n a l equation i s derived from the action p r i n c i p l e i n the usual way. V a r i a t i o n of (J> y i e l d s the Euler equation $£ H _ 1 ( "1 - n = >^d? ^ *M>V " ; ( 2 - 4 3 ) where £ = t>p + «CQ > and, e x p l i c i t l y ||s = -IK { Wc)> - S-4 (*M> - atftotf) , ( 2 . 44) - - Or1 (S * F £ o l FE£>^  + 2 S 5F E ijF E tj^ = - C E ' 7 ( E % B 2 ) (2.45) I t i s easy to show, using (2.35) and i t s adjoint, that O£D + <£lNT i s equal to zero whenever the f i e l d equations are s a t i s f i e d . We make use of t h i s f a c t to obtain - 5> D & UJe + r IVB ] S~* (2.46) where = + i?e(&.Cf) ' A ^ . There i s no p a r t i c u l a r s i g n i f i c a n c e i n the f a c t that we have used the natural wave-function Lp^ . Exactly the same r e s u l t s are obtained using a Lagrangian density <£ D + jdiNT written e n t i r e l y i n terms of -quan t i t i e s . We can use the vanishing of j t D + £ I N T t o s i m p l i f y (2.46); + /'»P.Y, ] . ( 2- 4 7 ) The g r a v i t a t i o n a l f i e l d equation becomes 1 8 . + * o c E y s (q>nvp - D*H> riy +/« W) . <2-48> We s h a l l see l a t e r that there a simple connection between the right-hand side of this equation and the energy-momentum tensor of the coupled system, the f i e l d equations f or which are now given by ( 2 . 3 2 ) , ( 2 . 3 3 ) and ( 2 . 4 8 ) . Conserved Quantities and Normalisation The action i n t e g r a l ( 2 . 4 1 ) i s invar i a n t under gauge transformations of the f i r s t kind, H^= Y , + = , ( 2 . 4 9 ) where o( i s a r e a l function of the co-ordinates. I f we make the above transformation, i t i s easy to show that SA = i*CE i v ° ( ^ COU^S"*) = 0 , ( 2 . 5 0 ) where we have assumed that o( vanishes on the surface S enclosing the space-time volume V . Since <=(. i s otherwise an a r b i t r a r y function, ( 2 . 5 0 ) gives us the continuity equation ( 2 . 2 8 ) . This can be written i n the form where J<* - _ i e C tyV • The existence of the continuity equation ( 2 . 5 1 ) means that our theory can be given a p r o b a b i l i s t i c i n t e r p r e t a t i o n . I f we integrate ( 2 . 5 1 ) over the three-space volume V3 ( X° = constant), we obtain, 19. which y i e l d s , i f we assume that J /Q_ vanishes s u f f i c i e n t l y f a s t at s p a t i a l i n f i n i t y , The p r o b a b i l i t y density D i s defined as D = ip> S " 3 ^ , (2.53) and i s a p o s i t i v e d e f i n i t e quantity. Equation (2.52) can be written The expression i n s i d e the brackets represents the t o t a l natural charge of the system and i s independent of the choice of c})^ - chart. In keeping with our i n t e r p r e t a t i o n of D as a p r o b a b i l i t y density, we normalise the wave function (J> as follows, J D d ' * = 1 > ( 2 . 3 4 , the i n t e g r a t i o n being over a l l three-space. I t follows that the t o t a l natural charge of the system i s equal to 6g , and i s a constant of the motion. Energy-Momentum Tensor We define an energy-momentum tensor i n the following way. The action i n t e g r a l (2.41) i s in v a r i a n t under space-time translations of the kind Of/1 + , where the are constants. In the usual way, by performing an i n f i n i t e s s i m a l t r a n s l a t i o n and s e t t i n g SA =0 we can show that " ^ I n O 1 ^ " ^MS>i] = D (2.55) 20. (2.57) where £ = o£ ( ^ m > ^ f , m ) a n d t h e \ m are the f i e l d components. We therefore define which i s the (mixed) <j>o -energy-momentum tensor. By means of (2.35), (2.36), (2.37) and (2.42), we can e a s i l y derive the e x p l i c i t form for - i S - 2 (<rF^Ac ) V - V v F ' c F , c ) The electromagnetic part can be symmetrized i f we note that The second term on the left-hand side has the divergence = " A v . \ r , (2-58) by the Maxwell equation (2.33). We obtain our symmetrized energy-momentum tensor T by adding to (2.57) the divergence-free term given by (2.58), or - i*c E s ( ? - Vyipyy) (2.59) 4-° The component I 0 of (2.59) i s given by f ° 0 = -K{( M O * + + (2.60) 21. and i s a p o s i t i v e define quantity. Using (2.59) we can rewrite the g r a v i t a t i o n a l f i e l d equation (2.48) as follows, ^ i s the trace of the energy-momentum tensor. I t i s to be noticed that, unlike the case f o r the neutrino and the electromagnetic f i e l d s , the trace of that part of T^j, corresponding to the g r a v i t a t i o n a l f i e l d does not vanish. In f a c t and this vanishes only i f ^ - f y i s a n u l l vector. CHAPTER III TIME-INDEPENDENT SYSTEM We are attempting to construct a time-invariant model for the electron, and therefore we assume that a l l p h y s i c a l l y measurable quan-t i t i e s are time-independent. The s t a t i c system i s characterised by the following: A J = o Ifc = % *c|, OEt/O > (3.1) Ao > S > and 'Xg are functions only of the space co-ordinates 0C L , and E i s a r e a l constant which represents the energy of the matter f i e l d . Using (3.1), the f i e l d equations (2.32), (2.33), and (2.48) reduce to the following set l*ECeV '•-*•(£ * e E A O ; X E + *CE)fJ(*i- I C ^ ^ X E ^ , ( 3 2 ) ~>j (S" a dj Ao) = - ir eS- a^ EY 07C E , (3.3) + C x C e 7 ) - , S - 1 { i ( E - . e E A o ) X E Y ° X E + i r n E C £ 2 . (3.4) The remaining Maxwell equations merely state that J ' = 0 • (3-5) These equations can be s i m p l i f i e d s t i l l further. We choose the following representation for the Dirac matrices r . i P , -22-23. Wc 0 where p = / I 0 0 O \ 0 \ 0 0 o 0 -I 0 \ 0 o o -I j t (3.6) and the G- are the (two-by-two) P a u l i spin matrices. We also write VV7 (3.7) where li. and V are two-component spinors. Using (2.31), (3.6) and (3.7), the Dirac equation (3.2) can be written as a pair of two-component spinor equations, ( E 4 e E A 0 - m E c e 2 S ) U + i - f c c c S ' ^ j ^ j V = 0 , ( E 4 e E A 0 + m £ c e 2 S ) V + U c t $V; 2hU =0 (3.8) In terms of this new notation, the Maxwell and g r a v i t a t i o n a l equations (3.3) and (3.4) take the form ^fc^M = eeS-MlaP+IVl2) , ( 3.9) Equations (3.8), (3.9) and (3.10) represent the time-invariant system, where as yet we have made no assumptions about any p a r t i c u l a r s p a t i a l symmetry. We w i l l use these equations when we discuss the Weyl-Majumdar solutions. However, we are more concerned with the s p h e r i c a l l y symmetric 24. case. Spherical Symmetry In this case we assume that a l l p h y s i c a l l y measurable quantities are functions only of the r a d i a l co-ordinate r . We write everything i n terms of a new co-ordinate-system, the i s o t r o p i c r a d i a l co-ordinates, i n which the l i n e element takes the form ds2 = - S^Cdx.*? + S 2 (dT^rMB^ + Y'sU'edip1) , (3.11) where the r e l a t i o n s h i p s between the spherical co-ordinates T , 9 , <p and the Cartesians OC , X , X are the same as i n f l a t space. We make use of the following i d e n t i t i e s = X ( l - V * ) - T * ( r * V ) , (3.12) where V i s the gradient vector, and te-r)(€-Jn= r i + i ? ( i x l ) , (3.i3) where £ = -1T X V i s the usual angular momentum operator. Using (3.12) and (3.13) we obtain $•2 = ir " T *r i • (3.14) The Dirac equations can be written (3.8) i n Hamiltonian form V V ) W ) » (3.15) where W , the Hamiltonian, i s given by H = - eE Ao + p mECe7<s -UctSa-(TC|r + 7 "T^W > (3.16) and < K i s the angular momentum operator f o r the Dirac f i e l d : 25. X = p * 0 > (3.17) and (3.18) In t h i s co-ordinate system i t can be seen that, except for the factors i n S, H has the same form as the s p e c i a l r e l a t i v i s t i c Hamiltonian. I t i s easy to show that the three operators J , J 3 , and <X commute with H and with each other and therefore determine three constants of the motion. We choose a representation i n which the operators H, (K , J 2 and 3j are diagonalised. I f we follow c l o s e l y the flat-space treatment of the same problem (the modifications are obvious), then i t can be shown (Corinaldesi and Strocehi 1963) that the eigen-functions of these operators are given by (3.19) The eigenvalues of the operators H , <J^  , T and J3 with respect to the above eigenf unctions are, respectively, E , | , j ( j + | ), and mj. In the f i r s t case we have i . e . (3.20) 26. \~U7r- ] 7e (3.21) We have put S. = J - J . In the second case k = -(j +5 ), i = j +5 , and the expressions f o r the Aj are the same except that we must replace I by (I - I ). To obtain the equations for the r a d i a l functions £(r) and g(r) we expand the Hamiltonian equation (3.15), f I 1 (3.22) where we have replaced cK by i t s eigenvalue k and we have used the r e s u l t ( C o r i n a l d e r i and Strocchi 1963) that A j (3.23) From (3.22) i t i s easy to see that the r a d i a l equations are as follows *ck(£«£)G - - S - * ( E + < e A.-«.c,»S) F , (3.24) (3.25) 2 7 . where we have written ^ = T"'F and g = -IT"' Q . I t follows from our assumption of s p h e r i c a l symmetry that the expressions s-'TCV T - ' C F ' I X / ^ l ' t Q ' I X i ' " ^ ) ' ) , (3.26) which appear i n the g r a v i t a t i o n a l and electromagnetic f i e l d equations, must be functions only of the r a d i a l co-ordinate. This means that 1 ^ j^) anc* I X/W^  I are constants, i . e . are independent of the angular co-ordinates 0 , ( p . We can use t h i s f a c t to determine the allowed values of the eigenvalue k . For the f i r s t group of solutions ( S"2 ^ e '^ ) , where V = " ( j + z ) = - Jl J i t c a n D e shown, by considering the properties of the s p h e r i c a l harmonics Y^ "^  , that only the f i r s t case j = i f u l f i l l s the above condition. We have S. = I , mj = * 2 and ( 3 . 2 7 ) For larger values of J these expressions w i l l i n general be functions of 9 and ( p . S i m i l a r l y , f or the solutions ( S~* Xg^* ), where k = j + 2 = JL + | , only the case J = i , Jl = 0 i s allowed. Equation ( 3 . 2 7 ) i s again v a l i d i n this case. To sum up, we have two solutions, of d i f f e r i n g p a r i t y , which obey the c r i t e r i a of time-invariance and s p h e r i c a l symmetry; I 9« ( 3 . 2 8 ) 2 8 . where k = - t = - I , and ( 3 . 2 9 ) where k = t + I = I The electromagnetic and g r a v i t a t i o n a l f i e l d equations ( 3 . 9 ) and ( 3 . 1 0 ) can be re-written i n terms of the r a d i a l co-ordinates. Using the above r e s u l t s , we have, 1 d (Q-? d A o A r1 dr V * dr J = |* $>'7y-2 ( F 2 + 0 , ( 3 - 3 0 ) - OnrKCB 1)-' S- 7T- 7 { (E+? eAo - i ^ c E ' S ) F ? + ( E H ? E AO + i m E C / S ) G 2 ] . ( 3 - 3 1 ) The system of time-invariant s p h e r i c a l l y symmetric equations i s now given by ( 3 . 2 4 ) , ( 3 . 2 5 ) , ( 3 . 3 0 ) , and ( 3 . 3 1 ) . We have four equations for four unknown functions of T . In the following chapters we w i l l attempt to solve these equations using a v a r i e t y of methods, including numerical i n t e g r a t i o n . For the sake of future convenience we make a t r a n s i t i o n to dimensionless notation. A l l lengths w i l l be written i n terms of the (bare) Compton wavelength of the Dirac p a r t i c l e , •£ (n\ eC E )-1 , and a l l energies i n terms of the (bare) r e s t energy ^^d^ • Furthermore, we define and w i l l use l a t e r the following dimensionless constants 29. ot = e6' /(mrtce') , £ = E / O e C e 2 ) . (3.32) t i s , e s s e n t i a l l y , the square of the bare mass written i n r e l a t i v i s t i c u n i t s , oL\ i s the fi n e - s t r u c t u r e constant, and £ i s the dimensionless energy eigenvalue. To achieve a dimensionless notation, the following substitutions are made, A 0 => Q 0 = e e A 0 / ( w e c e 2 ) , F = > F = [ e £ V C 4 1 r m E c ; ) ] + ^ F , Q = > G, = Y => f = (3.33) The f i e l d equations become UH)Q= -S- 2 ( ^ O . - S ) F • (3.34) [ f f l r - SJ(e+Q.*S)5 . (3.35) + H o i ' 1 S " 7 ( f + Qo + 2 S ) Q * . (3.37) The normalisation condition (2.54) can be expanded to y i e l d 30. which, i n the previous notation, becomes We can use the Maxwell equation (3.36) to integrate this expression and to obtain the normalisation condition i n the form of conditions on the boundary values of the g r a v i t a t i o n a l and electromagnetic p o t e n t i a l s : CHAPTER IV EXTERIOR FIELD OF A CHARGED SPHERE In this section we obtain solutions for the g r a v i t a t i o n a l and e l e c t r o s t a t i c f i e l d equations i n the region outside a s t a t i c s p h e r i c a l l y symmetric d i s t r i b u t i o n of charged matter. E s s e n t i a l l y , we are solving the problem of a charged point p a r t i c l e . The corresponding Riessner-Nordstrom s o l u t i o n i n general r e l a t i v i t y i s well known. I t w i l l be shown that for points very distant from the centre of the sphere, the two theories y i e l d metrics which agree up to f i r s t order i n r _ l , where X i s the r a d i a l co-ordinate. Only i n one case, when the g r a v i t a t i o n a l and e l e c t r o -magnetic f i e l d s are r e l a t e d i n a s p e c i f i c way, do the two theories pre-d i c t the same space-time structure. The s o l u t i o n obtained i s examined for various values of the r a t i o ^ ( i f T T GjE M E Z ) ' , where e e i s the charge and ME the g r a v i t a t i o n a l mass of the sphere as seen by a d i s t a n t observer. I t i s found that, when this r a t i o i s greater than one, the metric i s well-behaved only outside a c e r t a i n radius, the "Schwarzschild radius" of the e l e c t r i c charge. On the other hand, when the r a t i o i s less than one, this s i n g u l a r i t y does not occur and the metric i s well-behaved everywhere except at the o r i g i n . In this l a t t e r case the e l e c t r o s t a t i c p o t e n t i a l i s everywhere f i n i t e . The region of space i n which the asymptotic solutions are v a l i d i s c a l l e d the " e x t e r i o r region". The solutions of the Dirac equation i n which we are most interested are l o c a l i s e d i n the "strong" sense, i . e . the matter density contains a factor of the form e x p C - a ^ T ) , where a?' i s a constant which depends on the binding energy of the f i e l d . 32. 2 Therefore the exterior region i s that part of space f o r which 1 » Q. . In t h i s region the matter density i s so small that i t has a n e g l i g i b l e e f f e c t on the e l e c t r o s t a t i c and g r a v i t a t i o n a l f i e l d s . The s i t u a t i o n can be compared to that of a very t h i n atmosphere surrounding a very dense body. Because the g r a v i t a t i o n a l and e l e c t r o s t a t i c f i e l d s are long-range, the e f f e c t of the dense concentration of matter, which may be at a distance, completely overshadows that of the small amount of matter i n the neighbour-hood. However, the inverse problem i s quite d i f f e r e n t . The "athmosphere" i s very much influenced by the e l e c t r o - g r a v i t a t i o n a l f i e l d s . This problem w i l l be examined i n the next chapter. In the exterior region, the equations for the system are d r l 6r) = ( d r J , d r V b d F r / = 0 . (4.2) (4.2) i s integrated immediately to y i e l d where i s a constant of i n t e g r a t i o n . Inserting (4.3) into (4.1), we obtain d T V d r J = Cj1* (4.4) Writing Z = S - ' , U = ( 4tr Q £ /A* C E~ ) r - ' , (4.4) becomes " (&V +1=0 d 2 i dU.* ~ ^cUU + I = u • (4.5) The general s o l u t i o n of (4.5) has been given by Kamke (1943) and i s 1 = a " l S m ( a U + ^ ) , (4.6) where a, h- are constants to be determined from the bouadary conditions. In terms of UL , and using (4.6), (4.3) becomes 33. M - f - ^ ^ a* f4 7<> d a V H n / suvHau-ttr) > ^ ' } which i s integrated to y i e l d A 0 = c, + a ( ~ - J 2 cot Cau^ I ) , ( 4 < 8 ) where C, i s a constant also to be determined from our boundary conditions, which we obtain by assuming that f o r very large values of r , the gravi-t a t i o n a l and e l e c t r o s t a t i c potentials have the form 6 = - + o ( r - z ) . (4.9) T We r e c a l l that, f o r large Y (small <f) ), we have 1 = S" 1 = I - CE~2 $ . (4.10) Expanding (4.6), we obtain 2 = + — o>s (a/ a - a s u ^ C ^ * ) ^ + 0(T-»), ( 4 . 1 D where we have retained the X-2- dependence for future comparison with the Riessner-Nordstrom metric. In the same way we can show A0= c, + ac*U(^)* -§ +o(t») . (4.i2) I t i s clear from (4.9), (4.10), (4.11) and (4.12) that a = svcxb- , C, = - Cosh 34. -I ^ • l^T/ f ) • l - e ? " ) ( 4 - 1 3 ) where we must take, i n a l l cases, the p o s i t i v e square root. Inserting these values for the various constants into the s o l u t i o n (4.6) and (4.8) gives e , CASE ONE: 0 < CoS-t < I In this case a , I; are r e a l , and MTr£eM e 2 < C E Z . The metric i s well-behaved and regular outside a c e r t a i n radius X0 , given by x 0 = ( — j (4.i6) where we have assumed, without loss of generality, that 0 < Ir < "H/l. . j . The metric function "H. = (~^°° )* has zeros at T 0 and at a count-able i n f i n i t y of points i n X < T0 . For X » To , the expansions (4.11) and (4.12) are very accurate. Using (4.13), these become Z + 0 C T-») , (4.17) A o = - W r + O C r - ^ • (4.18) In general r e l a t i v i t y the corresponding Riessner-Nordstrom s o l u t i o n i s given by 35. A - - ?e -L For large r , we can see that the solutions are equivalent up to f i r s t order i n Jr - 1 . For smaller T , and for a nonzero cos b, the theories p r e d i c t d i f f e r e n t g r a v i t a t i o n a l f i e l d s . CASE TWO: ) < CoS lr In t h i s case the constants 3. , Jr axe pure imaginary, and f£*< <ni£jtM*. The solutions f o r "Z and r\0 are then ? = . a? C a ' i L 4 V ) , (4.20) + ^U^X* oAJk C a ' u . V ) , (4-21) where a ' = - i i = (coflt - I ) z , = , and Cosir i s given by (4.13). The metric function U has no s i n g u l a r i t i e s except at the o r i g i n , and i s regular everywhere. For large TC the expansions (4.17) and (4.18) again are applicable provided the factor ( I — CbS*>(r ) l i s replaced by An important s p e c i a l case of t h i s group of solutions i s when we put 6E = 0 . (4.20) reduced to ? = *%p ( Qt MeCe-7 I " 1 ) , (4.22) and i s the e x t e r i o r metric function f o r a point p a r t i c l e with no charge. (4.21) becomes simply A 0 = 0 . CASE THREE: COS& = I In t h i s case fg1 = itn QeME2 • This s i t u a t i o n occurs i n most of 36. the Weyl-Majumdar models. A simple c a l c u l a t i o n shows that (4.14), (4.15) reduce to * = I + <ll2le ^ (4.23) where QeV\e Q{X = ( ^ e C f 4 / 4TT ) * i s the Schwarzschild radius of the sphere. The l i n e element i s given by where d-fL7' = Sun29dcp* + d B 2 . I t i s i n t e r e s t i n g to note that general r e l a t i v i t y predicts the same r e s u l t s as the above. The s o l u t i o n (4.23) has been studied by Bonnor (1960, 1964), and by Papapetrou (1947). CASE FOUR: CoS \j = 0 In this case M e = 0 . Bonnor (1960) has pointed out a very peculiar feature of the Riessner-Nordstrom s o l u t i o n i n general r e l a t i v i t y . If we put Y\t = 0 and at the same time r e t a i n £e £ 0 , we obtain a s o l u t i o n for a charged point p a r t i c l e which has no g r a v i t a t i o n a l mass. The g r a v i t a t i o n a l p o t e n t i a l i s of order X - ' 2 , and i t appears as i f the e l e c t r o s t a t i c energy does not contribute to the g r a v i t a t i o n a l mass. A s i m i l a r s i t u a t i o n occurs i n the present theory, f o r , i f we take M e =0 , the solutions (4.14) and (4.15) become i = cos C r, /r ) , (4.26) A ° = ton Cr./r) , (4-27) where r, = ( QEee* A T T C V ) * . (4.28) 37. For X » X , , the expansion i n ~£ y i e l d s no dependence and there-fore the system exerts no long-range g r a v i t a t i o n a l a t t r a c t i o n . 31 CHAPTER V THE WE YL-MAJUMDAR METHOD FOR OBTAINING STATIC SOLUTIONS OF THE FIELD EQUATIONS Before proceeding with our i n v e s t i g a t i o n of the system of equa-tions (3.34)-(3.37) i n the general case, i t i s i n s t r u c t i v e to examine the one case i n which the system can be solved a n a l y t i c a l l y . Consider a d i s t r i b u t i o n of charged matter i n equilibrium. Pro-vided no other forces are present we can say that the equilibrium i s maintained by a balance of e l e c t r o s t a t i c and g r a v i t a t i o n a l forces. Such a balance implies, f o r time-invariant systems, a r e l a t i o n between the e l e c t r o s t a t i c and g r a v i t a t i o n a l p o t e n t i a l s . Weyl (1917) postulated that this r e l a t i o n could be i n the form of a functional r e l a t i o n s h i p between the component c j e o of the metric tensor and the e l e c t r o s t a t i c p o t e n t i a l Ao• Assuming t h i s , he showed that 9oo = A + 8 A 0 + A: , (5 . D where A and B are constants. He obtained this r e s u l t by studying a x i a l l y symmetric electrovac universes i n general r e l a t i v i t y . Majumdar (1947) extended this work to the case where there i s no s p a t i a l symmetry, and showed that (5.1) remains v a l i d . I f the constant B i s chosen so that (5.1) reduces to a perfect square, then the whole system of the com-bined Einstein-Maxwell f i e l d equations reduces to a si n g l e Laplace equation. These ideas were ca r r i e d over to the case of non-empty space by Das (1962) who showed that, i n c e r t a i n cases, the imposition of the Weyl-Majumdar r e l a t i o n , -38-39. 8-- { A* ±(^)*A.]* , (5-2) implies, and i s implied by, the equality of the dimensionless charge and mass parameters of the system i n question. In this chapter the Weyl r e l a t i o n (5.1) i s examined i n the frame-work of the scalar theory of g r a v i t a t i o n . A l l of the r e s u l t s and a l l the examples shown here have counterparts i n general r e l a t i v i t y . F i r s t of a l l , the r e l a t i o n analogous to (5.1) v a l i d f o r the scalar theory i s obtained. An attempt i s then made to understand the physical meaning of the constant B . This i s f i r s t done for the s p h e r i c a l l y symmetric case, for which the s o l u t i o n i n the e x t e r i o r region i s given i n Chapter IV. I t i s shown that, i r r e s p e c t i v e of the charge-mass r a t i o , every (exterior) s o l u t i o n obeys the r e l a t i o n (5.1), and that 8 = ffTT QE ME Cf1 ee~' , where ME i s the t o t a l g r a v i t a t i o n a l mass, and ?e the t o t a l charge of the sphere. This r e s u l t i s then extended to the general case where i t i s shown that, even without s p h e r i c a l / symmetry, B represents the mass-charge r a t i o of the system. I t i s i n t e r e s t i n g to note that, with this value for B , (5.1) reduces to (5.2) only i f eE = ± ( 4 i r q E y M E . (5.3) This i s probably why, i n previous work on this problem, the assumption (5.2) always led to the r e s u l t (5.3). — .. D i f f i c u l t i e s a r i s e when we attempt to extrapolate (5.1) into the i n t e r i o r region. I t turns out that, both for the Klein-Gordon and Dirac f i e l d s , imposing (5.1) i n this region leads to an over-determined system of equations, unless (5.3) i s v a l i d . 40. The Weyl-Majumdar r e l a t i o n for our theory i s obtained by consider-ing the free-space e l e c t r o s t a t i c - g r a v i t a t i o n a l equations. These are, from (3.9), (3.10), ^ m ( S - 2 ^ » A 0 ) = 0 > (5.4) ^ 4 = ^ $~2 C^f\oT (5.5) We assume that (J) i s a function only of A 0 , which means that = ()>" (B» Ac)* + fc)wAo j (5.6) where the prime denotes d i f f e r e n t i a t i o n with respect to A 0 • Inserting (5.6) into (5.4), (5.5) y i e l d s ~^m™A 0= l ^ ' c r 2 C^ml\^ f (5.7) and hence f + 2 c - ( 4 ' ) a = , 2 i s s - » The s o l u t i o n of (5.9) i s given by 9* = A 4 BAo + ^ A* > (5.9) (5.10) -e which corresponds to the r e l a t i o n (5.1) i n general r e l a t i v i t y . The constant A has no physical s i g n i f i c a n c e , and, without loss of generality, may be taken to be equal to 1. The constant B i s at present undetermined. I f , however, i t i s so chosen that (5.10) becomes a perfect square, then the f i e l d equations (5.7), (5.8) reduce to a single Laplace equation. In Chapter IV we obtained solutions for A0 and S i n the exterior region of a s p h e r i c a l l y symmetric charged matter d i s t r i b u t i o n . I f <?E i s the charge, and Me the g r a v i t a t i o n a l mass of the sphere, then these 41. solutions are given by (4.14), (4.15) i n the case where 3- = , by (4.23), (4.24) i n the case where a. = 0 , and by (4.26), (4.27) i n the case where Cos I t i s easy to show that a l l these sets of solutions obey the r e l a t i o n (5.10). In the f i r s t case S* - I • A . * ttS A- , ( 5 . n , where ^ if IT CE ME4 > i n the second case S 2 = ( I ± Ao) 1 , (5.12) where =• itxv Qe rig* , and i n the l a s t case S 2 = I + A (5.13) where ME = 0 . We see that, i n a l l cases B = c (5.14) If we assume that (5.11) i s v a l i d even i n the presence of charged matter, then this implies a simple r e l a t i o n s h i p between the charge and mass d e n s i t i e s . In such a case the f i e l d equations can be written ^ ( S _ 2 b m A 0 ) = c r , (5.15) ^ " " " ^ ~ ^ ^ ( ^ O 1 ^ p , (5.16) where 6~, f> are r e s p e c t i v e l y the charge and g r a v i t a t i o n a l mass d e n s i t i e s . For the moment we have l e f t these quite general. Using (5.10), we e a s i l y obtain <T . s - f - ' ( ^ f ) ' < 5 - 1 7 ) which y i e l d s , (5.18) 42. Previously we have obtained the expression (5.14) for B i n the case of spher i c a l symmetry. Using (5.18) we can extend this r e s u l t to the more general case. From (5.15), (5.16), by considering the f i e l d s f a r from the source, i t can be shown that the charge and g r a v i t a t i o n a l mass para-meters are given by e£ = JVd3x , MeC£* = J { S - M ^ A f t y + f } d SX . (5.19) Using (5.15), (5.18), we obtain Provided that Ao^mA& f a l l s o f f s u f f i c i e n t l y r a p i d l y at s p a t i a l i n f i n i t y (and this i s always true for a l o c a l i s e d d i s t r i b u t i o n ) , i t follows from (5.19) that (5.20) y i e l d s B = 2V (JE ME Ce 'ef ' which i s again (5.14). As ,a f i r s t example we consider a scalar (Klein-Gordon) matter f i e l d . For such a system the time-independent f i e l d equations are The Klein-Gordon wave-function i s given by where 06 i s r e a l and E represents the energy of the matter f i e l d . For such a system 43. <5" " ^7T, C E » < * O X ! . (5-22) Inserting (5.22) into (5.18), and can c e l l i n g out the common factor 9( 2, we obtain - ^ CM,C« ,4e.Ao)CE«*eA.) (5-23) Comparing (5.23) with (5.11), and equating the c o e f f i c i e n t s of the various powers of A 0 , we f i n d that the only Weyl-type s o l u t i o n f o r the K l e i n -Gordon f i e l d i s the Weyl-Majumdar s o l u t i o n with E - M«c,* -«.c.« - (5.24) The binding energy, defined by E g = ( -TOgC/ - E ) i s equal to zero, and the system of equations (5.21) can be reduced to two equations, " c W X = 0 , ^mrnCS-') = - e E S " ^ V (5.25) Similar r e s u l t s hold f o r the Dirac f i e l d , but the d e r i v a t i o n i s more laborious. The time-independent f i e l d equations (for the Dirac-e l e c t r o s t a t i c - g r a v i t a t i o n a l f i e l d s ) are given by (3.8), (3.9) and (3.10). In our present notation " ^ e t f S-' ( l U | * - I V l 7 ) . (5-26) The r e l a t i o n (5.18) becomes 44. S ' ^ E - I V e 4f e A e XlUiS. WI 2 ) = ^ e C E 2 S - ' C l U - l ' - l V l 2 ) , (5.27) and t h i s expression holds f o r a l l X . For points very d i s t a n t from the centre of the matter d i s t r i b u t i o n AQ ~ 0 , S ^ 1 , and the Dirac f i e l d i s e s s e n t i a l l y free. I t can be shown that, as we approach s p a t i a l i n f i n i t y 1 U l 2 - l V l 2 = ^ O W ' + I V I O , (5.28) and hence, i n this region (5.27) y i e l d s IE -M e C e * = E , (5.29) which means that E = MeCe2 . (For a demonstration of (5.28) see, for example, C o r i n a l d e r i and Strocchi 1963 p. 156, and our own re s u l t s i n Chapter 6). Inserting (5.29) i n (5.27) we obtain = TYieCe* S-'( ml'-IVl*) • (5.30) Comparing (5.30) with the corresponding expression f o r the Klein-Gordon f i e l d , we f i n d that the wave-function does not cancel out. This i s due to the i n t e r a c t i o n between the g r a v i t a t i o n a l f i e l d and the spin of the Dirac p a r t i c l e . Since S i s a function of A e only, (5.30) gives us an expression r e l a t i n g Ao , \ul^ , and IVl* . There i s no reason why the Dirac equations (3.8) should be consistent with t h i s r e l a t i o n and there-fore, i n general, we have an over-determined system. To show this we w i l l examine i n d e t a i l the above equations f o r the case of spher i c a l symmetry. I f we write 45. tS± = C F ^ Q 2 ) ; (5.31) the f i e l d equations (3.24), (3.25), (3.30) and (3.31) become « - f O F - - ( i ( (5.32) C-UfOG, = ( f r - i ) F , (5.33) - HS-V, , (5.34) + * £ & ( n < r--pO (5.35) The Weyl condition (5.30) can be written <*6~+ = p<5~- (5.36) M u l t i p l y i n g (5.32) by («C+p)Q, (5.33) by (o( - J3 ) F , and subtracting we get - I af ' -r f < • « • - - - o • . < 5"> If instead we add, we get = 2. C e l l - p * ) FQ } (5.38) where M F*Q2 = C5+2- 6^ . A combination of (5.36), (5.37) y i e l d s The solutions 8 =0 , <5V = 0 are t r i v i a l . We r e c a l l that 6>S ^  <5"_* and therefore p ^  < . Assuming p £ 0 > (T+^ 0 , from (5.39) we obtain P ~ M y y " + I / ; (5.40) where JUL i s a constant of int e g r a t i o n . Using (5.11), (5.31), (5.40), and 46. r e c a l l i n g that k. = | for s p h e r i c a l l y symmetric solutions, we obtain the following expression which i s v a l i d f o r a l l T , (5.41) There are two ways of looking at this equation. I f yW ^ 0 then we must assume i t i s an ordinary equation and solve accordingly f o r Ao • (Recall that S i s a function only of Ao .) On the other hand, i f ft = 0 , then the expression on the right-hand-side i s a constant, which means that i f we are to obtain a n o n - t r i v i a l s o l u t i o n for Ao we must assume that (5.41) i s an i d e n t i t y i n A 0 and equate the c o e f f i c i e n t s of the various powers of Ap . In the f i r s t case,y«-£0 , we f i n d = " eE U + v r* 0^17; ) > ( 5 - 4 2 ) where f> = * , £ = E CmtCe")"' and S = bit Qefi-tee~'Z . S must always be p o s i t i v e and therefore the solutions f o r S , A 0 are as follows * l i - i f ' J > A. . - ^ ( | - ^S) (5.43) These expressions are regular f o r a l l T . However, when we use (5.43) to solve f o r 6 + , 6 * _ i n d i v i d u a l l y , we obtain expressions which are not compatible with the second of the Dirac equations (5.38). The proof of this i s long and tedious so we w i l l not reproduce i t here. I t i s enough to state that ju. ^ O does not lead to a consistent set of solutions. However, i f />• = 0 , then we can look upon (5.37) as an i d e n t i t y i n Ao • Expanding S by means of (5.11) and equating c o e f f i c i e n t s of the various powers of Ao , we obtain 47. e £ * = QtE7Ce~Ht (5.44) Ao i t s e l f remains undetermined and we now have the correct number of equations and unknowns. The r e s u l t s (5.44) are the same as were obtained for the Klein-Gordon f i e l d , and correspond to a s o l u t i o n of Weyl-Majumdar type. Returning b r i e f l y to the case of no s p a t i a l symmetry we f i n d that the conditions (5.44) imply, from (5.11), (5.26) s* - ( i ^ A.y, E(IUIMVI') = EClTil -IVl3) • ( 5 . 4 5 ) ,2 Obviously IVl = 0 , and the equations for the system become I <5j~dj U = 0 , (5.46) ~^m(S-')= - Mil Q t E Cr" S ^ I U ) ' . (5.47) There i s a close r e l a t i o n s h i p between (5.46), (5.47) and the correspond-ing equations f o r the Klein-Gordon f i e l d (5.25). We have a d i f f e r e n t power of S on the r i g h t hand side of (5.47) but this i s due to the inter-action of the spin with the g r a v i t a t i o n a l f i e l d . The equation (5.46) can be solved. We can write 6Ai = <5Y + r ~ r ) (5-48) where K = 6"- + I i s the angular momentum operator. We have only to consider the following equation ( + Y " r ) ^ = 0 > (5-49) 48. for the two-spinor U. . The solutions can be expressed i n terms of the eigenfunctions of K ( C o r i n a l d e r i and Strocchi 1963) r«j)+ t » f c,+ y J ( 5 . 5 0 ) where the C~ are normalisation constants. The eigenvalues for M are k + = j + 7 = 4 + I , and the functions ^ obey the r a d i a l equations (5.51) ( r ) r = o Y ; - U . (5.52) Inserting the value for l l l l "* into the second equation of (5.46), we obtain a countable i n f i n i t y of possible equations for S~' , each one corresponding to a d i f f e r e n t angular momentum state. This i s i n contrast to the si n g l e equation described by Das (1962), who assumed that XL was constant. For the case of sph e r i c a l symmetry we must have k = — \ , and the only s o l u t i o n of (5.51) which tends to zero for large T i s given by r ( 5 . 5 3 ) 49. where | q i s a constant and -k = - i = - ) . From (5.53) we have |U| = (nn^-1-J--H , and the equation for S _ l becomes 1 d r / = - ^ T b > (5.54) where A = Qe E Ce"11 . Writing 2. = S~' , = T _ l , (5.54) can be s i m p l i f i e d A p a r t i c u l a r s o l u t i o n of (5.55) i s given by This s o l u t i o n diverges for large T and must be discarded. The general s o l u t i o n (of (5.55)) can be written down i n terms of e l l i p t i c i n t e g r a l s . Transforming (5.55) yet again, we f i n d B i s a constant of i n t e g r a t i o n and i s determined by the normalisation. We require T dhr T - - D = 0 > < 5 ' 5 8 > and hence, i f we write 1a = ~i(y-b) , B = | A ^ , 3 . (5.59) (5.57) can be transformed to i n t e g r a l form. I t becomes •3 dt J. (5.60) where t = 2"Z0-' and t 0 = T.0~ . Since 2. i s always p o s i t i v e , so i s t , and therefore 2. ^  for a l l ^  . We use the general formula 50. dt ^3 orr a + i - 1 k' (5.61) where k = Son. 7 5 ° , to obtain a formal s o l u t i o n for our problem. We have, then , (5.62) i where <f3 6 ? H = cn' (5.63) The s o l u t i o n (5.62) has the correct asymptotic behaviour f o r large T : 1 tends to I f o r Y•=? oo . However, for small T , the function becomes p e r i o d i c and there are s i n g u l a r i t i e s i n the metric. The system lias j from ( 5 5 « , (5 .59) , 0 0 , ( 5 . 6 4 ) To sum up: we have shown that solutions of the Dirac-Maxwell-gravitational f i e l d equations can be obtained by using the Weyl-Majumdar method and that these solutions imply that the metric tensor i s not regular over the whole range. •CHAPTER VI .ASYMPTOTIC SOLUTION FOR THE DIRAC WAVE-FUNCTION In deriving the asymptotic forms f or A 0 and S (Chapter IV) we assumed that i n the exterior region the matter density i s n e g l i g i b l e . In the present chapter we w i l l demonstrate the v a l i d i t y of that assumption by f i n d i n g the asymptotic form of the Dirac wave-functions. These solutions w i l l be used i n the numerical i n t e g r a t i o n i n Chapter VII to determine the boundary conditions on the wave-function. The Dirac equations i n dimensionless notation are given by (3.34) and (3.35). We consider terms i n the i n t e r a c t i o n only up to f i r s t order i n T-' . Using (3.32), (3.33) and (4.9) we obtain a „ - - i + 0 ( f - * ) , i , s-1- i + £ + o ( r* ) • (6.D Inserting these expressions f o r S and Q,0 into (3.34), (3.35), we f i n d where £c e v. ' ' I t i s i n t e r e s t i n g to note that, even when we consider only the T - ' dependence of the g r a v i t a t i o n a l f i e l d , we cannot simply replace the f l a t --51-(6.2) (6.3) (6.4) (6.5) 52. space expression ( F_ + 6E A 0 ) by ( E + CE A 0 £ CgT2 )> where Ece"2<|> i s the g r a v i t a t i o n a l p o t e n t i a l energy of the p a r t i c l e i n the f i e l d of the source: the i n t e r a c t i o n i s more complicated than a simple Coulomb p o t e n t i a l . In order to i l l u s t r a t e this point, to which we w i l l return l a t e r , we have l e f t the mass of the source i n i t s o r i g i n a l form and only at the end of our calc u l a t i o n s w i l l we make the s p e c i f i c a t i o n that corresponds to the s e l f - f i e l d of the Dirac p a r t i c l e . If we write F = F' ea+C-O » Q = Q ' ^ ( - c r ) , (6-6) where C = C\~i. ")* p , the equations (6.2) and (6.3) become, Our next step i s to construct power series solutions f o r F and Q' . We require a l o c a l i s e d s o l u t i o n (bound state) which means that F , G, must approach zero f o r large T . We write F* = < r » ? a; <y" , C = crpr 4n crn (6.9) r n.-6 where a j ^ 0 , C £ 0 . Substituting (6.9) into (6.7) and (6.8) we obtain, by equating c o e f f i c i e n t s of <y v - +P~' (6.10) 53. For V = 0 we have CP + k ) l j - < a; = o , (P -Oa*' + «2&0' = o , <6-n) which implies p = + ( k z - (6.i2) The equations (6.10) lead J. to the following recurrence r e l a t i o n f o r the &'v t j v + I* - k + « 3 * O - O C v + p - k ) j q . = \ ) + 1 + £ ot.\TPiT + C i - O C v - t - g - t e - i ) 7 » (6.13) A s i m i l a r recurrence r e l a t i o n f o r the &w i s obtained by replacing ^i/T^p" + C l - O C v + p - k ) For very large V , we have from (6.13) and t h i s would imply that the functions F' , Q' increase l i k e exp ( 16" ) for large o~ . This contradicts our hypothesis of a l o c a l i s e d matter d i s t r i b u t i o n . Therefore the series must terminate at some f i n i t e value of T I , say r»' , and this means that <V+| = = 0 . (6.16) Putting V = TI' + | , n' i n (6.10) and (6.14) re s p e c t i v e l y , and solving for p we obtain (6". 17) 54. On the other hand, we have, from (6.12) (6.18) From (6.17) and (6.18) we can obtain an expression f or £ . However, before continuing with the present problem we s h a l l consider, as a c o r o l l a r y and an example of the above work, the "hydrogen-atom" problem i n g r a v i t a t i o n a l theory. We tre a t the cen t r a l body as a f i x e d point-p a r t i c l e of mass and we consider only terms up to order T - 1 i n the i n t e r a c t i o n . (6.17) and (6.18) apply with c< = 0 . We must take the p o s i t i v e sign f o r p i n (6.18). Since £ i s contained i n the expression f o r p , we have a much more complicated system than the one which occurs i n the e l e c t r o s t a t i c case. The eigenvalues are given by the solutions of the following equation, which i s t h i r d order i n Z i 4 v>v - { * 2 r , y + (fe» -if*)' + ytf-rfO + / k + {aovy - xlf- tixy - aft'- n * ) / I e* + { cV-rt*)2 - 4h*y ] = 0 , (6.19) where ytx = i\e ME n\e -fc Ce~] . The eigenvalues are l a b e l l e d by the values of the angular quantum number -k and the r a d i a l quantum number 711 , which refer s to the number of nodes of the wave-function. In contrast with the usual ( e l e c t r o s t a t i c ) hydrogen atom problem, there i s no natural way to define a " p r i n c i p a l quantum number". For Tl' = D , we have E*>° = ivryf . C6-20) B r i l l and Wheeler (1957) have also considered this problem. They examined the behaviour of a Dirac electron i n a Schwarzschild gravita-t i o n a l f i e l d . Up to order T _ l , i t can be shown that t h e i r r a d i a l 55. equations are equivalent to ours. However, i n solving t h e i r equations, they used a d i f f e r e n t approximation, and neglected one term of order T~' , which we have included, and therefore t h e i r r e s u l t s d i f f e r from ours. E x p l i c i t y l y , t h e i r r a d i a l equations were of the form (Equation (39) of th e i r paper) -e ~kx - * c = o , (6.2D d r ' where £ V = € * = ) — 2Qe MBCe~2T~' • Expanding, and keeping a l l terms i n t"' , We obtain j f c ( E + . . A . -<(> + «.c,') F - £ .±Q . <K'£--.- - 0 , (6.23) where <j> = - Qe M EEc e- 7r-' i s the g r a v i t a t i o n a l p o t e n t i a l energy of a p a r t i c l e of energy E • From (6.6) we have <J>E"'~ '= |r e £ ^-t1",^ C , (6.24) again to f i r s t order i n f 1 . B r i l l and wheeler dropped this term since, i n t h e i r case, the c o e f f i c i e n t of T"1 Q was small compared to k. In our case, on the other hand, we have included a l l terms of order T - > , i n order to obtain a more general r e s u l t . For the case of a Dirac electron i n i t s own g r a v i t a t i o n a l f i e l d , we make the assumption that Me = E C ~* , or, i n other words, that E i s the t o t a l energy of the Dirac f i e l d . On t h i s point Kaup (1968) and Feinblum and McKinley (1968) d i f f e r e d . Kaup defined N e as follows 56. where N = J f i J V x . <6-26> TYl i s the i n t e g r a l of the zero-zero component of the energy-momentum tensor of the Klein-Gordon f i e l d , and N i s a conserved quantity. (For the case of a charged f i e l d J ° i s the charge density.) He found that, i n general, with the d e f i n i t i o n (6.25), MfC^^E . Feinblum and McKinley, however, used a d i f f e r e n t normalisation, which generated a d i f f e r e n t Nl , and therefore Kaup's r e s u l t s do not hold i n t h e i r case. They simply assumed, but did not prove, that E = MEC e 2 . In the present theory, i f we use (6.25), then i t i s easy to show, from (2.54), (2.57) that and therefore MgCg2 = E • Unfortunately, due to the complications of the f i e l d equations, there i s no simple way to prove that the Mf as defined by (6.25) i s i n f a c t the t o t a l g r a v i t a t i o n a l mass of the system. We simply assume that i t i s . For a s p h e r i c a l l y symmetric s o l u t i o n k = - | and hence 7\' = 0 . From (6.17) and (6.18) we obtain I - of - £ 7 + l c ( t £ J - = 0 , (6.28) where t i s given by (3.32). The s o l u t i o n of (6.28) i s 57. £ a = 0 0 - ' { ( I - + 4 c* ) * • - I + 24t]r (6.29) Hence, i f we know t , which i s e s s e n t i a l l y the square of the bare mass of the Dirac p a r t i c l e , we can calculate £ . I t i s remarkable that only for one value of Tne , given by l e t t i n g o(. = t , do we obtain <£ = I , and therefore only i n that case does the binding energy of the f i e l d vanish. (This i s i n f a c t the Weyl-Majumdar s o l u t i o n which has been examined i n a previous section, and i n which case i t i s possible to obtain an a n a l y t i c s o l u t i o n and to prove that E = \^EC^ • ) From (6.9), (6.10), the asymptotic solutions for P , Q. are F = Q„' 0-£*) ? / 2 f>* e a ^ ( - NTT^ p ) , (6.30) where £ and p are given by (6.29) and (6.17). In the following chapter we w i l l use these expressions as boundary conditions for the numerical i n t e g r a t i o n . CHAPTER VII NUMERICAL RESULTS AND DISCUSSION The non-linear system of equations ( 3 . 3 4 ) - ( 3 . 3 7 ) has been solved by numerical methods. B r i e f l y , our procedure was as follows. We f i r s t chose a value ^ of the r a d i a l co-ordinate so large that the asymptotic solutions obtained previously were v a l i d . A step-by-step numerical i n t e g r a t i o n toward the o r i g i n was then begun. In the i n t e g r a t i o n process, the four equations ( 3 . 3 4 ) - ( 3 . 3 7 ) were replaced by the s i x f i r s t - o r d e r equations, ff • -JQ . S - 5 ( f - - S ) F , (7.!) 4f " f F + S-2(£+ a 0 + sK , ( 7 . 2 ) c\a0 = Q a p - a M ( 7 . 3 ) 6} I 3' > 5 ? = r1** , (7-4) 4* = S - * ( F % Q * ) , ( 7 . 5 ) dp + a S ' H ^ Q . ^ S ) ^ ^ ( 7 . 6 ) where , ^ are defined by ( 7 . 3 ) and ( 7 . 4 ) . These were integrated by a numerical method, the d e t a i l s of which we w i l l c o n s i d e r l a t e r i n the chapter. The integrations were repeated for various values of the para-meters &o and TT)E . In a l l cases we assumed that 6 E was the e l e c t r o n i c charge, or, i n other words, that o(. had the numerical value that i s usually accepted for the f i n e - s t r u c t u r e constant. - 5 8 -59. As boundary data we use the following 3> -df " p* a d_Q_o ^ (7.7) P/2 where A > define d by Al = aifi-E1) , can now be used instead of GU as a parameter of the theory. Previously we have shown that for s p h e r i c a l l y symmetric solutions h. = - 1 . Following a procedure s i m i l a r to that used by Bethe and Salpeter (1957, p. 153), we now determine which i s the correct sign to choose for various values of the i n t e r a c t i o n parameters. For the case of an e l e c t r o s t a t i c p o t e n t i a l alone the rule i s quite simple. If the Coulomb po t e n t i a l i s a t t r a c t i v e ( r e p u l s i v e ) , we must take k = + I (- | ). When we include the g r a v i t a t i o n a l i n t e r a c t i o n , c e r t a i n modifications are necessary. From (6.9), (6.10), we can write down three expressions for' the r a t i o , aL ? + k o ( -C £ C 2£-0 h: P - * = _ \ f T r 7 r (7.8) (7.9) (7.10) (7.10) i s obtained from (6.9) by s e t t i n g \) = I +V , where V = 0 . For bound states this expression i s always negative. Imposing this con-60. d i t i o n on (7.8), (7.9) leads to a contradiction i f we take the wrong sign for 4< . For t<o< there are two cases; ( i ) t < cC(2-5hY , where S = I - £ * 10~5 . In this case -I £ £ < 0 t£C2£+i")< <^  ' a n <^ s o w e m u s t take k = -I . _ i ( i i ) o( < C < ° ( • Here the denominator of (7.8) and the numerator of (7.9) are p o s i t i v e , while j8< - I . This means that we can take either sign for k , or, k = - ) . For Z>°( we have three cases; ( i i i ) o( <Z< 1-44. In this region o< p <) , the denominator of (7.8) i s negative, the numberator of (7.9) p o s i t i v e , and therefore ^  = +) . (iv) 1.44 ^. t K 4".00. Here -|< |3 <0 , the other quantities have the same sign as i n ( i i i ) , and we must take k = + 1 . (v) 4.00 < £ < £*o . In this l a s t case, where p< - I , i t i s easy to show there there i s no choice of sign for k which makes (7.8) and (7.9) compatible. Hence, there exists an upper l i m i t of 4.4 x 10 ^ grams f o r the bare mass. In F i g . 1 we have plotted M e against TA e , where the bar indicates we have used dimensionless notation, or -friE = \TT = [ -me ( 7 . i i ) The graph i s also divided into four regions which exhibit the various required values of & as outlined above. In region I, k = - I , i n II k = — I , i n III k = + I and i n IV there i s no allowed value for k . We see that no matter how large the bare mass may be, the observed mass i s always less than 2.2 x 10 ^ grams. Of course, as we have seen above, there are other reasons why we may not take an a r b i t r a r i l y large value for the bare mass. In his study of the Klein-Gordon geon, Kaup (1968) found that no solutions existed which had a bare mass larger than 1.76 x 10 ^ grams. (This corresponds to a maximum value of 1.70 x 10 grams for the observed mas s.) 62. In the present work, solutions were found f o r various values of 1AE and A l . As an example F i g . 2 shows the shape of the mass density (see expression (2.53)) for the case t = ir\* = 0.05 ( -fnE = 0.2236) and = 0.002. The normalisation i s defined by where we have included the factor Jfir i n our dimensionless, D . For a c o r r e c t l y normalised sol u t i o n , from (3.38), we need 71 = o< . — 1 In order to obtain normalised solutions we f i x e d the value of X. - T1\B and integrated the system of equations f o r the whole range of possible values of A l • We repeated t h i s process f o r several values of Z . The r e s u l t s of this study are summarised i n F i g s . 2 - 5 . We found that f o r a l l -fnE , A 7 > the mass density D has the general shape exhibited by F i g . 2. Normalised solutions were found i n the range o( < t < 4.00. For A1 small we found Tl><< , and for A l large T i <o( , For the case t = 0.005 the best solutions (from the point of view of the normalisation) were given by A l = i . 750461 x 10 , i n which case nr\ -o( - o.io4 x \ D - L } ( 7.13) and A l = 1.750465 x I O - 3 , when (7 . 1 4 ) or ==• 10 c^ , which i s as good an agreement as can be expected i n numberical work. For other values of "C comparable accuracy was obtained. Contrary to what we had hoped, however, we found that i n every case the e l e c t r o s t a t i c and g r a v i t a t i o n a l potentials were singular near the s i 1.0 0.8 0.6 0.4 0.2 0 F I G . 4 NORMALISATION CONSTANT 0 .02 .04 .06 .08 .1 0 .12 .14 .16 A1 o r i g i n . Considered as c l a s s i c a l models of charged p a r t i c l e s , these solu-tions corresponded to objects comprising a t h i n s h e l l of matter surround-ing a highly singular region of space. In the following d e t a i l e d d e s c r i p t i o n of our solutions, we s h a l l make use of the terminology of such a " s h e l l " model. Discussion of the Solutions The system of equations (7.1)-(7.6) with the i n i t i a l conditions (7.7) was integrated numerically using, as our basic method, a Runge-Kutta subroutine of order 4 (Fowler 1964). A second method, based on an extra-polation procuedure using r a t i o n a l functions due to B u l i r s c h and Stoer (1966), was used on c e r t a i n selected examples to check our r e s u l t s and to evaluate the errors. The Runge-Kutta method used i s that due to G i l l (1951). As step-siz e we used "R. = ^ /50 where j0 i s the r a d i a l co-ordinate. Carr (1958) has shown that for Tv small enough (but not too small, to avoid excessive round-off error) this program i s very accurate. The second method, trans-lated into Fortran IV by M. L e s l i e (1966) from the A l g o l procedure of B u l i r s c h and Stoer (1966); involved the use of an automatic step-size correction procedure. This meant that a f t e r each i n t e g r a t i o n step 4i was changed to the optimal step s i z e for the next i n t e g r a t i o n step. The pro-gram also contained a subroutine for c o n t r o l l i n g the accuracy of the com-puted values of the functions being integrated. If was one such function then the computation of ^ at each i n t e g r a t i o n step was repeated u n t i l two successive computed values of d i f f e r e d at most by an amount IS where £ = — 6 10 and 5 was of the order ^ . B u r l i r s c h and Stoer (1966) examined i n d e t a i l the errors involved using this method and showed that i t gave r e s u l t s superior to most other commonly used methods, including the Runge-Kiitta. We did not use it all 69. the time because of the high cost i n computer time and storage. Assuming that the d i f f e r e n c e i n the r e s u l t s obtained using the two methods was of the order of magnitude of the errors involved, we com-pared the integrated solutions for several cases. As expected we found that i n the e a r l y stages of the i n t e g r a t i o n the d i f f e r e n c e was n e g l i g i b l e . As p-=7 0 ) however, the error increases somewhat u n t i l f o r the l a s t few i n t e g r a t i o n steps we had l o s t three or four d i g i t s . This means that, estimated i n t h i s rough fashion, our solutions were accurate to at l e a s t four places of decimals. Hence no q u a l i t a t i v e errors i n the shapes of the solution-curves was indicated. For example, for the case ft\t = 0.6325, AT = 0.001, both methods gave the same value for (j> at j° = 86.938180, whereas at f> = 0.11258820 the Runge-Kutta subroutine gave <§ = 3.3022970 while the Bulirsch-Stoer procedure gave (j> = 3.3025360. Normalised solutions were found f o r values of the bare mass range from 0.1 to 1.0 ( i n units of 2.2 x 10 ^ grams). In a l l cases these solutions had the same basic properties. c ^ Figure 3A shows the form of the metric function b = (- 2ao ) 2 for several values of Al (with TtlE = 0.2236), while Figure 3B shows the behaviour of three representative cases for small . We have discussed elsewhere the form of S f o r l a r g e . I t i s seen that a l l the solutions have the same basic shape, and d i f f e r only i n the p o s i t i o n p = /°M1|Nj where S has a minimum and i n the radius >^ of the inner core. (The inner core i s defined as that region of space close to the o r i g i n where the matter density i s n e g l i g i b l y small.) Of course only the normalised solu-tions are of i n t e r e s t . We include these graphs merely to i l l u s t r a t e the r e s u l t obtained which was that, for constant /frlE , the solutions behaved i n a smooth way with v a r i a t i o n s i n A1 and exhibited no q u a l i t a t i v e differences. 70. F i g . 4 graphs ffi £ against the value of A1 which y i e l d s a c o r r e c t l y normalised s o l u t i o n . I t i s seen that, i n the region > 0.23 ^ A] increases f o r increasing T?jE . I t i s to be r e c a l l e d that elsewhere we have shown the existence of a maximum value for the bare mass. For 0.23 >tf)E>\£(" J AT increases with decreasing *ii\B and approaches i n f i n i t y asymptotically as —3» NTO? • For example when 7flE = .0946 we found that A]= - 1.2 x 10^ gave a c o r r e c t l y normalised s o l u t i o n . At the point 1D£ = \Jo( (which corresponds to the Weyl-Majumdar case) our asymptotic s o l u t i o n i s no longer v a l i d . We have discussed this case i n Chapter 5. F i g . 5 depicts the radius of the inner core i n each case i n which a normalised s o l u t i o n was found. For Tf\E > $o7/ we found that the radius of the core increased with increasing bare mass. For Tfl f < ^ we were unable to f i n d any normalised s o l u t i o n s . I t may be s i g n i f i c a n t that, i n this region, the e l e c t r o s t a t i c s e l f - r e p u l s i v e force, as estimated from the asymptotic forms, i s greater than the g r a v i t a t i o n a l s e l f - a t t r a c t i o n . The case /7f\e = 0 gave solutions which could not be normalised, and which had an o s c i l l a t o r y behaviour at s p a t i a l i n f i n i t y . This i s caused by the 2. r f a c t that the binding energy Eg = m e C E — E = - E i s i n this case nega-t i v e . The only exception i s the t r i v i a l case T^ECE 2 = E = 0. In our search f o r solutions with S regular at the o r i g i n , we constructed a power series s o l u t i o n for the whole system f o r small p and attempted, by a l e a s t square method, to f i t t h i s s o l u t i o n to our numer-i c a l l y integrated one. Of course, since our equations are non-linear, we could only derive the c o e f f i c i e n t s for the f i r s t few powers of f> , and therefore we had no proof that the series were convergent. As a working hypothesis, we assumed that they were. The r e s u l t s of our investigations were that we could not make such a f i t and that i t was hig h l y u n l i k e l y that a s o l u t i o n for S regular at the o r i g i n existed. This conclusion i s r e i n -forced by the smoothness of the curve i n F i g . £ which suggests that the radius of the singular core never shrinks to zero. . •': 71. Of course i t i s impossible, i n p r i n c i p l e , to prove a negative statement l i k e the above by using a numerical technique. The best we can do i s to show that, within the capacities of our method, no s o l u t i o n regular at the o r i g i n can be obtained, and no s o l u t i o n i s indicated. There i s always the p o s s i b i l i t y that the problem i s an eigenvalue one, giving a regular s o l u t i o n only for very precise values of the parameters. We investigated this possibility very thoroughly, going to seven places of decimals, u n t i l we reached the l i m i t of accuracy of the-computer. That i s , u n t i l the computer output became i n s e n s i t i v e to changes i n the input. We found no evidence of a regular s o l u t i o n . An attempt was also made to integrate outwards from y° = 0 assum-ing a regular s o l u t i o n . Unfortunately, due to the lack of any d e f i n i t e i n t i a l conditions, there were too many unknown parameters and the attempt was abandoned. In the figures we have concentrated mainly on showing the behaviour of the metric function S . The e l e c t r o s t a t i c p o t e n t i a l a 0 was found to have properties very s i m i l a r to 5 . The matter density D always has the form indicated i n F i g . 2. CHAPTER VIII ALTERNATIVE LAGRANGIAN DENSITIES In Chapter I I we discussed how to introduce the gravitational interaction by a simple generalisation of the flat-space theory. We assumed that the t o t a l Lagrangian density for the system could be written i n the form t. = « £ F + o f q . , where depended only on the gravi-tational potential <j> and i t s f i r s t p a r t i a l derivatives. The " f i e l d " part < £ p was known i n the special r e l a t i v i s t i c l i m i t and i t s generali-sation was unambiguous. However, the one uncertainty i n our theory lay i n the choice of £>q . In the choice we made, we were guided by a desire for s i m p l i c i t y and by a desire to have a theory whose predictions for the perihelion advance of test p a r t i c l e s were the same as those of the Einstein theory. A p o s s i b i l i t y e x i s t s , however, that the study of the oblateness of the Sun may indicate an error i n the Einsteinian prediction for the perihelion advance of Mercury, and i n that case the choice of an alternative Lagrangian density for our theory would be j u s t i f i e d . In the context of the present work, moreover, the modified equations may lead to solutions which are regular everywhere. Consider, then, instead of (2.42) the gravitational Lagrangian density £Q - K s-* r v <iv 4>„ (><m-40), where \\ i s a constant. I f A = 0 then we are back to the case which we have studied already, and which predicts the same perihelion advance for t e s t - p a r t i c l e s as the Einstein theory. I f A = - ) , then, as Rastall (1968b) has shown, the energy densities of the gravitational and -72-matter f i e l d s behave i n the same way as sources of the g r a v i t a t i o n a l f i e l d . The p e r i h e l i o n advance of t e s t - p a r t i c l e s i s 87» less than that predicted by general r e l a t i v i t y and i s compatible with Dicke's measurements of the solar oblateness (Dicke 1967). F i n a l l y , with A = - 2 , the energy densities of the g r a v i t a t i o n a l and electromagnetic f i e l d s behave i n the same way as sources of the g r a v i t a t i o n a l f i e l d and the p r e d i c t i o n for the p e r i h e l i o n advance of t e s t p a r t i c l e s i s 167« less than occurs i n general r e l a t i v i t y . The modified g r a v i t a t i o n a l equation i s obtained i n the usual way by varying £ with respect to <^> . The v a r i a t i o n of £>p i s the same as before, but (2.44) becomes, from (8.1) $-4((j> - ^ U M l (8.2) (•6 ' » From (8.2) the g r a v i t a t i o n a l f i e l d equation (2.48) becomes' + ~ t 9 ( ^ r D 0 i p _D>r^p +,M")0. ( 8 - 3 ) The Maxwell and Dirac equations of course remain unchanged. The energy-momentum tensor i s altered i n a s i m i l a r way. Using (8.1), we f i n d f o r the symmetrized energy-momentum tensor - S-'(F' lF v t - l S"v FVF,f) - 5*cs(tprvvic-vrfry) , <8-4) 7%. which takes the place of (2.59), and - V . - i t ; ) . (8.5) and (8.6) replace equations (2.60) and (2.61). The g r a v i t a t i o n a l equation for the time-independent system i s now given by instead of (3.4), while (3.10) i s replaced by - ? CE-ie rA.) C W + N I ' ) - i m,c,"(lui '- lvl ')l . (8-8) For the case of sphe r i c a l symmetry i t can e a s i l y be shown that (3.31) becomes A ° T d y ^ d r / = - aire/ b tdy J + ( E + ee A 0 + i s ) Q 2 ^ , (8.9) or, i n dimensionless notation, i n place of (3.37) + « $ J ( f - f Q> + is) V . (8.10) 7g. Asymptotic Solutions In the region of n e g l i g i b l e matter density the equations for the system (4.1) and (4.2) become From (8.12) we get • = y " b T > ( 8 . 1 3 ) where i s a constant of i n t e g r a t i o n . Inserting (8.13) into (8.11) we obtain d r )" 2 b • <8'14> w r i t i n g y = S *^ , U. = ( XT QeyO3) M Ce"*» ") * T~* , (8.14) becomes a" - si sc*"° • 0 • where the primes denote d i f f e r e n t i a t i o n by IL . This equation i s v a l i d f or a l l A except A = 0 , which case has been studied previously. M u l t i p l y i n g across by ij' and integrating we f i n d where C| i s a constant. Integrating once again, we obtain ( c 1 < - ' | * y y * ) v * = c * + ( 8 - 1 7 ) where c z i s another constant, jm. , C, and Ca. are determined by consider-ing the forms of the e l e c t r o s t a t i c and g r a v i t a t i o n a l potentials very far from the centre. We require, for large T $ = - CeMa 1 + 0 ( T - * ) f (8.18) 76. A 0 = - Jj ^ + 0 <^ r" 2) > (8-19) where, as before, eE i s the charge and M£ the mass of the central body. As an example we consider the case A = - 2. From (8.17) we obtain (8.20) i n the' c ase where C, £ 0 , and = au. + C 3 (8.21) where C3 i s a constant, i n the s p e c i a l case C, = 0 . To determine C, > C 2 , C 3 we r e c a l l that = S~* and use (8.18). We f i n d that V KTUl* , C 3 I . (8.22) For the s o l u t i o n (8.21) we require further more , " = I ) (8.23) t T T / t and, as we w i l l see shortly, this means that the s p e c i a l case (8.21) corresponds to the Weyl-Majumdar solution. From (8.13) we have where (8.25) i n the f i r s t case, and S 2 = f l + IV.}' (8.26) 77. i n the second case. Solving (8.24) we obtain (8.27) for C| :£ 0 , and i n the Weyl-Majumdar case. For large T (small IL ) we expand these series and use (8.19) to derive the constants. We f i n d r _ \—Si f 0 C , C 2 2 " > ^ = ^ (8.29) We can substitute the value of yU thus obtained into (8.22) and (8.23) to obtain the f i n a l values f or the constants c, and C^ . . To sum up, the solutions obtained above are v a l i d i n the region where the mass density i s n e g l i g i b l e . We can use them i n the same way as the solutions of Chapter IV to determine the boundary conditions f or the e l e c t r o s t a t i c and g r a v i t a t i o n a l f i e l d s . Weyl-Ma jumdar Relation The Weyl-Majumdar r e l a t i o n v a l i d f o r the modified Lagrangian density i s obtained, as i n Chapter V, by considering the free-space e l e c t r o s t a t i c - g r a v i t a t i o n a l equations. These are, from (3.9), (8.8) (8.30) <P + — O ~TT S L o m A 6 j (8.31) 78. Using (5.6) we e a s i l y obtain (8.32) A . - f, { s-*-» - f - A f t / } («-3»-and hence . 2 •e which y i e l d s , f i n a l l y Z S ) = ' ~ / " Q ^ " • < 8- 3 4> We can solve t h i s equation i n the following way. L e t t i n g ^ = S^ 2 *^ (8.34) becomes Mul t i p l y i n g across by , and integrating, we obtain (8.36) where A i s a constant. The s o l u t i o n can be written as ^ — = 8 + Ao } (8.37) where 8 i s a constant also, and For example, i f A = 0 , (8.37) can be immediately integrated to y i e l d (5.10). On the other hand, i f A = - 2 , then (8.37), (8.38) become (8.38) 4 = ^ * = B +' Ao . (8.39) (8.40) Integrating (8.39), and solving for y we obtain y = -£ (BvX + A.tft ) , FCOSIGHTAO 4 CJ S^rvil 6ft A 0 (8.41) where F = A A - 1 Su4 (jl* B ) > G = A A"' Cosfl. 6ft B ) -Without loss of generality we can set F = I and obtain, f i n a l l y y = Cos^(\TtA 6) + G SA*JI (\aA0) (8.42) By methods s i m i l a r to those used i n Chapter V we can show that Q - ( 8 . 4 3 ) When Q = I , the Weyl-Majumdar s o l u t i o n i s involved and the system of equations (8.30), (8.31) reduces to the single Laplace equation T U ( S - * ) - 0 . (8.44) Conclusion In this work we have investigated and found s t a t i c , s p h e r i c a l l y symmetric solutions of the combined Dirac-electromagnetic-gravitational f i e l d equations.. We have shown the existence of normalised solutions which describe s p h e r i c a l , s h e l l - l i k e models for p a r t i c l e s . Unfortunately, these solutions involve e l e c t r o s t a t i c and g r a v i t a t i o n a l f i e l d s which are not regular at the o r i g i n . Our investigations have led us to the conclu-sion that such solutions do not e x i s t , at l e a s t within the framework of the present theory. I t i s possible that the choice of a d i f f e r e n t Lagrangian density for the g r a v i t a t i o n a l f i e l d might lead to equations which do pro-duce regular solutions. Another p o s s i b i l i t y i s that we might abandon the 2 assumption, E = M C , of Chapter VI. (In Chapter VIII, some a l t e r n a t i v e E E Lagrangian densities are investigated.) The 'solutions' found correspond to objects with mass of the 2 80. — 6 — 36 order of 4.8 x 10 grams, inner radius 7.5 x 10 cms, and outer radius -34 3.7 x 10 cms. They are thus much heavier and more compact than any of the known elementary p a r t i c l e s . Of course, since we are using an un-quantized theory, i t would be unreasonable to expect models which corres-pond to actual physical objects. Another possible explanation for the properties of our solutions i s that our ( i m p l i c i t ) assumptions regarding the topology of space i n the inner region are i n c o r r e c t . In this context i t i s i n s t r u c t i v e to consider the ideas of Wheeler (1968) on the geometrodynamical d e s c r i p t i o n of e l e c t r i c charge. In his view e l e c t r i c charges are nothing but sets of l i n e s of force trapped i n "wormholes". Our inner singular region should perhaps be replaced by the mouth of such a wormhole. Wheeler (1968) has also shown that, i f one deals i n distances of -33 the order of the Planck length, 1.6 x 10 cms, then, s t r i c t l y speaking, one should use quantum geometrodynamics, i f one knew how. In the present state of our knowledge, however, we can only explore, as deeply as possible, the c l a s s i c a l theory i n the hope that some day the r e s u l t s may be of use i n the study of the more complete theory. BIBLIOGRAPHY Bade, W. L. and Jehle, H. 1953. Rev. Mod. Phys. 25, 714. Bethe, H. A. and Salpeter, E. E. 1957. Encyclopaedia of Physics, 35, Part 1 (Springer, B e r l i n ) . Bonnor, W. B. 1960. Zeits fur Physik 160, 59. 1965. Mon. Not. R. Astr. Soc. 129, 443. B r i l l , D. R. and wheeler, J. A. 1957. Rev. Mod: Phys. 29, 465. C o r i n a l d e r i , E. and Strocchi, F. 1963. R e l a t i v i s t i c Wave Mechanics (North-Holland, Amsterdam). Das, A. 1962. Proc. Roy. Soc. A267, 1. 1963. J. Math. Phys. 4, 45. Das, A. and Coffman, C. V. 1967. J. Math. Physc. 8, 1720. Dicke, R. H. and Goldenberg, H. M. 1967. Phys. Rev. Letters 18, 313. Dirac. P. A. M. 1951. Proc. Roy. Soc. 209A, 291. Feinblum, D. A. and McKinley, W. A. 1968. Phys. Rev. 168, 1445. Kaup, D. J. 1968. Phys. Rev. _172, 1331. Kamke, E. 1959. Diffe r e n t i a l g l e i c h u n g a n Losungsmethoden und Losungen (Chelsea Publishing House, New York). Majumdar, S. D. 1947. Phys. Rev. 72, 390. Mukherjee, M. N. 1963. Nuovo Cimento _27, 1347. Papapetrou, A. 1947. Proc. Roy. I r i s h Acad. A51, 191. Power, E. A. and Wheeler, J. A. 1957. Rev. Mod. Phys. 29, 480. R a s t a l l , P. 1968a. Can. J. Phys. 46, 2155. 1968b. J. Phys. A. 1, 501. Rose, M. E. 1961. R e l a t i v i s t i c E lectron Theory (John Wiley and Sons, New York). Weyl, H. 1917. Ann. Phys. 54, 117. Wheeler, J. A. 1955. Phys. Rev..97, 511. 1968. Topics i n Nonlinear Physics (edited by N.J. Zabruski, Springer-Verlag, New York). 

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