IC/80/15B

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

GEON-TYPE SOLUTIONS

OF THE NON-LINEAE HEISENBERG-KLEIN-GORDON EQUATION

Eckehard W. Mlelke

and

INTERNATIONAL Reinhard Scherzer

ATOMIC ENERGY

AGENCY

UNITED NATIONS

EDUCATIONAL,

SCIENTIFIC

AND CULTURAL

ORGANIZATION 1980MIRAMARE-TRIESTE

IC/60/158

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIOHAL CENTRE FOR THEORETICAL PHYSICS

GEOH-TYFE SOLUTIONS OF THE HOH-LIHEAR HEISEHBERG-KLEIH-GORDOH EQUATION •

Eckehard W. Mielke

International Centre for Theoretical Physics, Trieste, Italy,

and

Reinhard Scherzer

In3t i tu t fiir Beine und Angewandte Kernphysikder Chriatian-Albrechts Universitat Kiel, 2300 Kiel ,

Federal Republic of Germany,

MIRAHABE - TRIESTE

Octoter I960

• To be submitted for publication.

ABSTRACT

As a model for an "un i ta ry" f i e l d theory of extended p a r t i c l e s -Je

consider the non- l inear Klein-Gordon equation - associa ted with a "squared"

Heiseaberg-Pauli-Weyl non- l inea r spinor equat ion - coupled to s t r o n g g r a v i t y .

Using a s t a t i ona ry spher i ca l a n s a t i for the complex sca l a r f i e ld as well as

for the background metr ic generated v ia E i n s t e i n ' s f i e ld equat ion , we a re

able to 3tudy the effects of the scalar self-interaction as veil as of the

classical tensor forces. By numerical integration we obtain a continuous

spectrum of localized, gravitational solitons resembling the geons previously

constructed for the Einstein-Maxwell system by Wheeler . A self-generated

curvature potential originating froa the curved background partially confines

the Schrodlnger type wave functions within the "scalar aeon". For zero angular

momentum states and normalized scalar charge the spectrum for the total

gravitational energy of these solitons exhibits a branching vith respect to

the number of nodes appearing in the radial part of the scalar field.

Preliminary studies for higher values of the corresponding "principal quantum

nuaber" reveal that a kind of fine splitting of the energy levels occurs,k I,

which may indicate a rich, particle-like structure of these quantized geons.

A fundamental theory of matter based on the quark hypothesis has to

accommodate an in-built mechanism of (at least partial) confinement of the

constituent field in stable particles, otherwise they should be observable at

2)some detectable rate . In order to circumvent the Paul! exclusion principle,

these fundamental fermion fields are assumed to obey paraetatistics, or

equivalently, have to carry, besides, flavour, additional colour degree3 of

internal freedom. These colour models aro .! I :;t inguished by the binding

i.e. 10mechanism of quarks in hadrons.(Whether thi.; is mediated by scalar vector{see e.g. Ref.5) or tensor "gluons" 6''7'.

In "quantum chromodynamics" (QCD) , nowadays the moat prominent model

for strong interactions, the dynamics of the mediating vector gluons is

determined by an action modelled after Maxwell's theory of electromagnetism.

The resulting model is a gauge theory of the Tang-Mills type . However itQ)

is known that in such sourceless non-abelian gauge theories there are noclassical glueballs which otherwise would be an indication for the oceurr«nceof confinement in the quantized theory. (The phenomenological consequences ofthe possible existence of glueballs in QCD have been di3cused by Robson •)The reason simply being that nearfey small portions of the Yang-Hills fieldsalways point in the same direction^ internal 3 p a c e amj therefore must repeleach other as like charges. Nevertheless, vector gauge fields might be an

f-)important ingredient of any model in order to explain saturation

The confinement i tself , according to the proposals of an unconventional

scheme termed "colour feometrodynamics" {CGMD) * may be achieved by

strongly 1 ^ ' ' 1 5 ' >l(>> interacting massless tensor gluons, their dynamics

presumably being determined by Einstein-type field equations. CGKD is a

GL(2N,C) gauge model in curved space-time vhich nay be regarded as a generalization

of Einstein's gravity theory. The la t t e r corresponds to a gauging of the covering

group SL(2,C) of the Lorentz group. Since CGMD i s , in general, based on a

Riemann-Cartan space-time , Cartan's notion o f torsion is known to

induce non-linear spinor terms into the Dirac equation. This has a profound

effect on the "fundamental" spinor fields

(1.1)

-1-

-2-

distinguished by S colour (or flavour) internal degrees of freedoE. It car.12)

also be shown for this GL(2N,C) gauge invariant generalisation *" that these

quart type fields have to satisfy the Heisenbers-Pauli-Weyl

linear spinor equation

l I'

~0 (1.2)

generalized to a curved space-time (compare also with Ivanenko ). Here

L are space-time dependent generalizations of the Dirac matrices ya tensored vith

the U(n) vector operators X (generalized Gell-Mann matrices). Essentially

the modified PlancK length

jf = ti*k

of strong gravity lM,15).l

aelf-interaction in (1.2).

Old k/crf IO~'cm (1.3)

occurs also as the coupling constant of the

If we transfer the ideas of Mach and Einstein to the microcosmos, the

curving-up of the hadronic background metric should be self-consistently

produced by the stress energy eontent TU(I)J) of the spinor fields (1.1)

via the Einstein equations

J*. -v »

with "cosmological term" (we employ the sign conventions of Tolman 7- In

thiB new geometrodynamical model extended particles owninginternal symmetries

should be classically described by objects which closely resemble the geona and2k)

wormholea of Wheeler . In some sense this approach is also related to the25)

issue to vhich Einstein and Rosen addressed their 1935 paper .

"Is an utonistic theory of matter and electricity conceivable

which, while excluding singularities in the field, makes use of no

other fields than those of the gravitational field (g ) and those

of the electromagnetic field in the sense of Maxwell (vector

potentials tf )7"

-3 -

The geoit, L.t;. :i "gravitational electromagnetic entity" was originally

devised by Wheeler ' ' to be a self-cor.sjstent, non-singular solution of the

otherwise sourcefree Einstein-Maxwell equations having persistent large scale

features. Euch a geon provides a well-defined model for a classical "body"

in general relativity. If spherically symmetric geons would stay completely

stable objects they could acquire the possibility to derive their equations

of motions solely from Einstein's field equations without the need to

introduce field singularities. In a sense this approach also embodies the

goals of the so-called unitary field theory *

Geons, as we are using the term, are gravitational solutions, which

are held together by self-generated gravitational forces and are composed of

localized fundamental classical fields. The coupling of gravity to neutrino29)

fields has already been considered by Brill and Wheeler . The latter work

lays the appropriate groundwork for an extension of their analysis to non-

linear spinor geons satisfying the combined equations (1.2) and (I.U). In

this paper, however, we have avoided algebraic complications resulting from

the spinor structure as well as from the internal symmetry by considering

rather non-linear scalar fields coupled to gravity. In order to maintain a

similar dynamics we assume - as in a previous paper (Ref.30, hereafter referred

to as I) - a self-Interaction of these scalar fields which can "be formally

obtained Ijy "squaring" the fundamental spinor equation (1.2). "Linear" Klein-

Gordon geons have been previously constructed . However, we view the

additional coa-linearity of the scalar fields as an important new ingredient

for our model.

The precise set-up of this theory is given in Sec.II. For the intended

construction of localized geons, the stationary, spherical ansatze of I are

employed for the scalar fields, whereas the metric is taken in its general

spherically symmetric canonical form. As in the case of a prescribed

Schwarzschild background - analysed in I - the curved space-time effects the

resulting Schrodinger equation for the radial function essentially via an

external gravitational potential.

The stress energy content of these scalar solutions determines the

currature via Einstein's field equations. In Sec.II we review the spherical

symmetric case and include also a method which enables us to incorporate non-

zero angular momentum states into this framework hy averaging the stress energy

of these scalar fields over a spherical shell.

Our geons contain a fixed (quantized) scalar charge. By imposing this

restriction in Sec.V we not only fix an otherwise undetermined scale of our

geons but may also increase their stability. The main concern of Sec.V i s ,

- I t -

however, to contrast two notions of energy for our gravitational solitons,l) tb-e field ener^ of the general relativistie scalar waves and 2) the total

32)gravitational energy of such an isolated system . In order to probe ourconcepts ve construct in Sec.VI & simplified geon by considering radially

constant scalar solutions owning the particular constants admitted by the non-

linear self-interaction. Outside a ball of radius C_ the scalar fields are••/ is

diseontinuously set to zero. Although this procedure is rather artificial,

we thereby obtain a ""bag"-like object having inside a portion of an

Einstein microcosmos and outside a Schwarzschild manifold as background

ap&ce-t ine.

In general, the resulting system of three coupled non-linear equations

for the radial parts of the scalar and the (strong gravitational) tensor

fielda has to be solved numerically. In order to specify the starting values

for the ensuing numerical analysis we derive in Sec.VII asymptotic solutions

at the origin and at spatial infinity. Sec.VII is then devoted to a discussion

of the numerical results. Preliminary speculations are offered with the aim

to interpret particles as "quantum geons".

Sec.VIII concludes the paper with a prospective overview of other

developments concerning "gravitational solitons".

(8.3)

ia chosen to be similar to that used in I. Such a model has recently been

treated in 1+1 dimensions according to quantum field theoretical methods

Variation for £JCK,,/jf(» yields the non-linear KLein-Gordon equation

[a u /Jew*)] <r =where

(2.U.)

(2-5)

denotes the generally covariant Laplace-Beltrami operator. When (2.1*) laout

explicitly written/for thechoiee (2.3) of the self-coupling i t will be referredto as the non-linear Helgenberg-Klein-Gordon equation

I I . THE MODEL

Following the outline given in the Introduction we may consider as a

simplified model a theory consisting of H complex scalar fields

(2.1)

Their dynamics i9 governed by the Lagrangian density

(2.2)

defined on curved pseudo-Riemannian spaee-tiae with metric tensor f . In

to obtain a similar dynamical probl™ 13 in the non-linear spinor theory

given by (1.2) the self-interaction potential

IS 6O.6)

In I i t has been shown that (2,6) is formally similar to that obtained by

"squaring" the fundamental spinor equation (1.2). This is part of the

motivation for considering a If I -term in the corresponding Lagrangian

density (2.2).

Although the resulting quantum field theory, contrary to the \<f\

model, would not be renormallzable according to standard criteria of perturbation

theory we include in this paper the additional |*pj self-interaction

term. For a semiclaaaical approach it may be instrumental for the construction

of quasiatable, spherically symmetric and localized solutions. At least In

flat apace-time,Anderson has shown by means of a phase-space analysis thatfor

8 s -f (2 .T )

-6-

particle-like (stable) solutions can exist. CO is the ratio "between dynamical

mass and the "bare" mass **. of this model (see Sec.III). Intuitively, ve

auspect that the stability of these solutions (and possibly also their degree

of "confinement") is enhanced by the attractive forces exerted on them via the

coupling to strong tensor gluona ' . Geometrically speaking, this would

correspond to a curved background manifold.

the badcgroun

(HW, p.5OU)

In our model this curving up of

the background Is self-consistently produced from the stress energy tensor

Cf) * - £JLHHS

* a., . - • ' « , (2.8)

of the scalar fields tf via Einstein's field equations ( l . i t ) . In effect,our geometrodynamical model is then completely determined by the Lagrangiandensity

UK 6

since (2.6) and (2.9) can be derived from it by a variation for

and for

III.

(2.9)

SPHERICAL SCAIAK WAVES IH A CURVED BACKGROUND

As a semiclassical model for a particle we are considering sptericalgeon type ' solutions which minimize (2.9). More precisely,we are looking forspherical wave configurations which solve tbe HKfi equation (2.6) in a s t a t i c ,spherically symmetric background space-tiise which in turn is determined by ( l . b ) .

vei l 3 'As is veil known 3 ' ( see , e.g. MTW, p.59k and box 23.3) a canonical

form of the general (flimensionless) line element for this background reads:

2trIF* it*

- e(3.1)

if the sign conventions of Tolman 3' are adopted. Here V = V (̂ )

X S /l(?) are functions which, depend solely on the dimensionless

"Schwarzschiia type" (MTW, p.T2l) radial co-ordinate

and

M*cr * Ul (3.2)

The determinant of this metric Is given by

5lh<? . (3.3)

For the construction of spherical scalar waves, we take up the veil-knownfact that solutions of the free, linear^ Klein-Gordon equation can be expandedIn terms of spherical harmonics Y^(^.f) which are elgenfunctions

t Ax * LCC + OT, Y"(e,0

* fife % Vof the Laplace operator 4 g on 2-sphere S2. Although a non-linear theory

in general does not respect such an expansion, the non-linear terms of the

field equation (2.6) admit the two dist inct separation anaStss of I ;

X - l - 1ce>

elsewhere

(3.5)

(3.6)

Due to the familiar addition theorem (Landau and Lifschitz,

-8-

L

I

a 3elf-i.nteracti.0n given by a polynomial in |c#| =q = 1remains spherically symmetric as required by separability.

(3.7)

P 1 flp^

In order to see how space-time curvature affects the vave equation it

is instructive to define

and #.respect to the new co-ordinate $« / Mie Schrodinger type vfave function

th.e structure of the Laplace-Beltrami operator is similar to that of a

coaformally flat Klein-Gordon operator with an external potential .

Consequently, the stationary ansatia (3.5) and (3.7) together with

the property (3.11) cast the HKG equation (2.6) into the Schrodinser type

equation

(3.8)

(in Borne equations belov abbreviated by H or F, respectively) and tointroduce Wheeler'a tortoise co-ordinate" a* (MTW, p,663) via thedifferential form 3 "

* CJL-v)/SL ,

5 s e 6

Then the line element (3.1) takes the form

(3.9)

(3.10)

It resembles the metrical ground form of a space-time vith two conformally39)flat portions. Then a kind of "conformal change" of the

Laplace-Bertrami operator (2.!*) aay be calculated with the formal result

CY-X)/Z\e ) (3.11)

-9-

;F - -=-|- F + 1 - e co

with an effective curvature potential (compare with MTW , p.868)

implicitly given by

(3.12)

(3.13)

Here and in the following the factor

denotes the dimensionless ratio between the Planck mass M» and the "bare

mass" it of the a "constituent" fields. So far for the formal

aspects of the theory. For the ensuing numerical calculations, however, it

is more convenient to use the equivalent radial equation

e\z) R(3-15)

written explicitly in terms of 6 . (The dash denotes differentiation vith

respect to ^ .) It may he obtained from (3.12) and (3.13) by resubstitutiona,

or more directly from (2.6) and the original ansatze (3.5) and (3.6) founded

on the background (3.1). It generalizes Eq..(3.9) of Kaup;

linear Klein-Gordon equation. -10-

31) derived there from a

IV. THE EINSTEIN FIELD EQUATIONS

By applying Hachian ideas to the microcosmos, the strong gravitational

background vill be determined from the stress energy content of the scalar

waves via the Einstein equations (1.1*).

With respect to the diagonal metric of (3.1) these equations reduce to

(see Tolman 2 3 , p.

e V- 1 U. it.v'-x'

and

Here

gravitationa.1 fields from the field equation (2-9)- Cue to the contracted

Bianchl identity V G = 0 there exists the same relation between the

diagonal elements of the Einstein tensor G . Therefore it is enough to

consider the two remaining equations (It.l) and (H.3) only. (Bef.^l, p.1*88).

For the supposed spherically symaetric background (3.1) we notice that

the non-trivial ansatz (3-5) would lead to an inconsistency in the

gravitational field equations. The reason being that the corresponding

stress energy tensor (2.8)#in the scalar case given by

would also depend on the angular distribution of the solutions, contrary to

the Einstein tensor.

For localized solutions the spherical, asymmetry of the scalar waves

(3.5) ia expected to be negligible sufficiently far away from the centre

of the geon. Therefore it ia physically Justifiable not to discard ansatz

(3.5) but rather consider the Einstein equations (lt.l)-(lt.3).with respect to

an averaged stress energy tensor

p.1*88). Suitable is an average

property

a s proposed by Pover and Wheeler (see

over a spherical 3hell defined by the

o e

(U.6)

denotes a dimensionless "cosmological" constant and the dash again means

differentiation with respect to £ .

Although this set of equations may look like an overdetermined

system, a simple argument sbovs that this is not the case.

According to second-order variations! principles which can be

itO) >generalized to a curved space-time there exist the conservation lawvt VYv T ^ = 0 for the stress energy tensor (2.B) provided that the matter fieldequatlolia(2.i*) hold.. In our case, this law relates the two tangential

0tensionsc

knovledge of

= T.*to the radial tensions 1 T , T n and d T /dr. Thisf , * r 0 r= T," Is not instrumental for the determination of the

The evaluation of ^ T u . V ^ w i l 1 'be facilitated by employing the identity

=* 0-11-

which results from the application of Stoke'5 theorem.

With respect to the ansEtze (3.5) and (3.6) the averaged radial tensions

come out as

(it.8)

and

vhere the spherical average of the Lagrange function is explicitly given by

x ~

("4.10)

In order to bring the radial Einstein equations in close analogy withthose known for a perfect fluid (Ref.23, p.2lfM we may formally introduce In(U.8) the dimensiooless proper density ? 0 0 of the scalar field. It turnsout to be explicitly given by

Furthermore, the dimensionlesa proper "nydrostatic pressure" p- of a

scalar field can (implicitly! be defined by (U.9).

By subtracting (l(.l) from (1».3) and evaluating the combination

f- =

ve obtain

(fc.ia)

Ct-13)

whereas in this notation (U.3) is equivalent to

-fU.Ik)

These equations are generalizations of those considered by Kaup

Ua)and e.g. by Kodama et aX.

31) for a l inear

for a non-l inear Klein-Gordon f i e l d .—4.

It should be noted that f t . l i t ) Is a l inear d i f f e r e n t i a l equation In • .

As la w e l l known the general formal so lut ion can be written as

& — I ~ * ~ 3 ? C * i i 5 )

where

(it.

denotes a mafla function. The meaning of the latter terminology will lieilluminated If ve consider the Einstein equations outide the region where"natter" fields vanish, i . e . R = 0 in our case. Then tbe non-linearequations (U.13) become linearized and the vacuum Einstein equations admit the.(with respect to the canonical metric (3.1)) unique set of exact solutions

- 1 3 - -Ht-

- A.- e. U.IT)

exteriorThey descritie the/Schvarzschild-de Sit ter geometry for a. mass distributionlocated at the Origin. Here CiS 0L {to) ia the parameter measuring thegravitational mass ot M* at spatial infinity.

V. GRAVITATIOHAL ENEBCT OP GEOMS WITH QUANTIZED CHAHGE

In order to associate some quantum meaning to the time-dependent

localized solutions of the HKG equation the Bohr-Sommerfeld quantization rules

may be imposed. For a field theory with infinitely many degrees of freedom

this semicl&seical quantization condition reada

j1s1

<d0 a,(5.1)

the time integration being performed over the 3emlperiod ffji of the solution.

In • curved space-time the canonical conjugate field momenta are defined {see,

e.g. Fulling UU)) by

•jr „

In a. s ta t ic background and for the stationary anBatze {3.2) or (3.3) ovnlng

the semlperiocl If A • - eo u 0 /ft i t is not difficult to see that the Bohr-

Sommerfeld oondition (5.1) is equivalent (see also Ref.l45, Sec.3.6) to tbe

charge quantization

( 5 .3 )

vhere

-15-

in a curved space-time ia- the conserved tota l charge of the complex scalarfields. The condition (5-3) may also increaae the stabi l i ty of theae

26)"quantum geons" provided that this stabilizing device for non-linear

*̂5)seiolclassical field theories applies also in curved space-time.

By insertion of the ansatae (3.2) or (3.3) we obtain the expression

11

(5 .5)

For fixed A and pre-aasigned «u this condition normaliiea the a prioriarbitrary Planck length £* with regard to the coupling constant Jt of ournon-linear model. On the other hand, i f we fix this ra t io to be e.g. Jlmljtm 1as we will assume in our numerical calculations, the condition (5.6) determinesthe physically immaterial i n i t i a l constant C^ appearing in the asymptoticsolutions (7.1) discussed in Sec.VII.

Our normalization ) i s the same as that used by Kaupdeviates from the condition suggested by Feinblum and HcKinley

3D but

In a curved space-time the energy concept is known to be rather subtle^

Let us recall that for matter fields coupled to gravitation the locally

conserved it-momentum is given by

the integration being performed over a space-like hyperaurface. Differently as

in the case of flat Minkovski space, in (5.6) tht stress-energy pBeudotensor

t » y of Landau and Lifshltz (WW, p.!*66) must be included in order to

account for the contribution from the gravitational f ield.

-16-

For a quasistatic isolated systen and .A. = ° Tolrr.an "' Uee alsoRef.23, p.235) has derived the following equivalent expression for its totalenergy

(5 .7)

is operationally more useful

This result which is exact in the static casef, in particular in numerical

calculations, since the volume integral has to be extended only over the region

actually occupied by the "matter" fields.

In our construction of localized aeons we can satisfy the criteria for

the applicability of (5,7) if we require the radial part of the scalar field

to be exponentially decreasing in the asymptotic region & -¥ °* (see Sec.VI).

Thereby the gravitational background field (3.1) tends sufficiently

fa»t to that of a Schwarischild geometry given by (U.1T) with .A • 0 . By

construction the total mass of our geon is then known to be ot H* (compare

with Eq.(6.2) of Ref.Ul) and the Einstein relation

M'c1 (5.8)

holds in a rest frame.

Using the static background (3-1) and the formal relations (It.8) andthe Tolman energy (5-7) can be cast into the form

(5.9)

23)(Tolman , p.248). With respect to a non-linear scalar field theory defined

by (2 .2) , the to ta l energy is equivalent to

the integration over angular variables formally being absorbed in the averages

defined earl ier by (h.6). After inserting the ansatse (3-5) or (3.6) we obtain

tfec mam «a»iici t result

(5.U)

The model dependent rat io J[*IJL may be eliminated by the previously derived

normalization condition (5-5)- Then in view of (3.1U) the formula

JL 1fin '

finally determines the total gravitational mass ctM* of a "scalar neon withquantized charge". It is interesting to note that gravity alone modifiesthe "bare" mass p. by 2l(» , vhereaa a "mass renormalUatic-n" on this aemi-classlcal level is due to the (non-linear) self-interaction. The relation(5-12) may be contrasted with the curved space-time definition (see, e.g.BefAU)

£1 e (5.13)

of the field energy of the N scalar "constituent" fields <P' of the geonwithout the contributions from the self-consistently generated gravitationalfield.

Inserting (U.9) together with (4.10) and then substituting thenormalisation condition (5.4) yields the "mass formula"

-aft-

J£ (JV,««•)" *

0. As a precautionary measure for the case that this

(5-lU)

expression diverges at the origin •, we have introduced a "cut-off" length p > 0itt (5.1U), enabling us to study the "regularized"instead. After subtracting the boundary term

field energy E,

VI. GEON WITH A CONSTANT "BAC-LIKE" CORE

Before we turn to a numerical analysis of the spherical Einstein-Klein-Gordon system it is instructive to study an exact solvable geon containing aconstant scalar "core" of radius ( . . In order to obtain this highly idealizedconfiguration one has to note the fact that the non-linear equation (3.1ci) forthe radial distribution of the scalar field <q admits for L = 0, besides zero, theconstant solution

(6.1)

5.15)

fron (5-13) and then using the normalization condition (5.5) we may alternatively

consider vhst we call the normalized energy

£ - F, =• k { co

+ ST T**^ S 2 ^ (5.16)

which should be compared vith (U.9) of I. Furthermore, it could be phyBiaally

interesting to study the binding energy

e*- 4 /I/f «

(5.17)

also in a curved space-time (3.1), provided that the f.- component of themetric tensor is also a constant, i .e.

e - (6.2)

35)In the case that this condition holds Anderson's analysis of the classicalphase apace (R.E1) of the HKG equation can be applied to a certain extent.The main result is that stable (particle-like) solutions can exist ifthe parameter B^ . employed already in (2.7) satisfies the condition

n -«" (6.3)

Then the stable equilibrium points in the phase space of the asymptotic

version of (3-15) correspond to the constant solution (6.1) with the negative

sign of the square root. This solution which the geon may adopt in its

interior gives rise to a constant scalar density

of a scalar particle within a geon as a function of CO and A . FollowingRef.31 we «ay define it as the energy of a "free" scalar particle from whichits energy contribution to the total geon mass at rest is subtracted. A"free" teat particle means free of gravitational and self-interaction. Inview of (5.16) the corresponding normalized energy is 6) Me which is thefactor appearing in the phase of the ansatze (3.5) or (3.6).

8ISO

-20-

aa well as to a. constant "pressure" term

If we abaort (6.U) into an effective "cosmologicftl"

(6.5)

(6.6)

solutions of {It.lb) which are regular at the origin read

si * (6.7)

From e » const, I.e. Eq.(6.2), we can infer that the constant radial

solutions (6.1) exist only In (a portion of) an Einstein "microcosmos" (compare

with Tolaaa 2 , $135 and fl39). (Note that (a.6) admits also non-trivial1«7)

radial solutions in an Einstein Universe. These exact solutions however

are notgeon-type solutions, i.e. they do not satisfy the Einstein equations

(1.1*) at the same tine.)

The remaining radial Einstein equation (U.13) yields the "equation of

state"

h * (6.8)

for the "density" in terms of the "hydrostatic pressure" pQ . After

insertion of (6.U) and (6.5) into (6.8) we obtain

{ 6. 9 )

which, in view of (6.1), determines • V (W) as a function of W and TV. .

Our geon construction may now follow closely those by which

Schvargschild (see Hef.23, $96) obtained the exterior and interior solutions

for & spherical star consiting of an incompressible perfect fluid of constant

property density £ Q ( r To this end we may use (6.l) together with the condition

(6.2) and the resulting metric function (6-7) as solutions for the Interior

OS <} * gg of aballJTaen the curvature potential (3.13) associated Vith this

metrical background is also a constant, i .e . acre precisely

In order to interpret the interior solution (6.1) as a kind of "bound state"

within a negative potential produced hy a gluonlc "bag" of tensor forces we

have to require -A.eff > 0.

In view of (6.6) and (6.8) this condition can be satisfied for

ba&^ 2 ̂ 0 * ° o n l T - S u c h a non-sero "bag constant"-A^ in necessary for

the interior of the geon in order to compensate for the "vacuum" pressure (6.5).

of the "<viark-type" scalar fields. A similar mechanism haa been proposed in

the phenomenological MIT "bag" model of hadronB ' (If -A. would be

zero, the condition (6.8) is the same as that for a random distribution of23)electromagnetic radiation (Tolman , p.217) except for the Bign.)

Outside the constant bag-like core of radius y_ we may simply continuewith ft » 0 ( i f In this Idealized construction we are contented vith solutionsvhich are only piece-wise dlfferentiable and continuous) and obtain for

/V.,^. =• 0 a Schwarzschild solution (lt.lt) for

(6.U)

ThiB crude geon construction allows us to evaluate the total charge (5. K) in

closed fora:

Insertion of (6.1), (6.2) sad (6.7) into (5.5) yields

(6.12)

The remaining integration can be performed vith the aid of an integral

representation (see Eq,. (30) of Bef.ItT) of the hypergeometric function. The

charge quantization (5-3)then leads to the normalization condition

-21- -22-

If •= l/3V X z

for simplicityFor-A = 0 and assuming herej/also at = 3 ^

'8

(6.13)

the above result can be used' to

determine the scalar density (6.U) respecting the condition (6.8) as

VII. LOCALIZED GEON SOLUTIOHS FROM NUMERICAL INTEGRATION

According to our . introductory remarks we will reserve the term "geon-

type solution" (compare Sec.VIII for other notions) for configurations resulting

from a self-consistent coupling of fields to gravity in which both the "matter"

waves andJmetric tensor are (non-singular) and sufficiently localized. A

precise criterion for localization depends crucially on the circumstance of

whether or not a cosmologlcal constant -A. is included. For the present we

put .A, = 0 and may then require for localized, spherically symmetric scalar

geons the following set of

eo 3

In a similar way we may calculate the already normalized expression (5.12)

for the gravitational mass otM*.

Let us consider the case 6 = 1 . The insertion of (6.1) and further-

Sore {6.9) into (5.12) yields

(6.15)

a result which could also be inferred from the comparison of condition (6.8)

with the equivalent expression (5-9) of the Tolmao energy.

Since for y > J f i the space-time geometry Is determined by the

Sehvarzsehild solution (U.17) with_A. _,. " 0 and ct given by (6.15), an

"observer" placed outside the core wi l l regard this gravitational hound state

of scalar fields as an object having the mass ot. M*. For_/t, = 0 thebag

hadronic environment, i.e. the strong curvature generated by the "tensor gluons"

f-.j inside the "constant core" gives rise to such a strong "Archimedes' »T)effect on the scalar constituent fields, that their binding energy (5.lit)

2freeones equal to the rest mass UUc of a self-interacting"quark".

This can be summarized in a Wheeler-type phrase: A constant'*bag-like"

geon may have "mass without mass" (Ref.3T, P-25).

a) Asymptotic solutions at spatial infinity

We proceed similarly as in Sec.III.c of paper I

radial solutions which behave asymptotically as

30)and consider

(T.I)

If \u\<-i , such boundary conditions for the numerical integration would

necessary lead to exponentially decreasing Yukava-type solutions (Bee also

Bef.US) irrespective of the parameter 6" • Since the>sealar waves would

already be sufficiently localized, the equations (U.13) and Ct.lU) pass into

the Einstein vacuum equations. Given the canonical form (3.1) these yield

uniquely the Schvarzschild solutions (U.9) with A • 0.

Therefore

y ^ (7.2)

will hold. The asymptotic forms (7.1) and (7.2) are then inserted Into- (3.15)

in order to determine tf

coefficients of the 1/9 - expansion we obtain

onVgrounds of self—consistency.

6- - - 1 ffi

By equating the

(7.2)

This result being independent of the quantum number L of angular momentum

agrees for the plus sign with that obtained by Kaup

- 2 3 -

Let us turn to the

b) Asymptotic solutions at the origin

Guided by the "constant core" case analysed in the preceding

section ve found it reasonable to assume

y 2s. 0 '** C ^ C «=• con if.

in the vicinity of the origin.

0 ) Suppose ve find

- c0 (7.5)

Then from (it.lU) and (U.13) i t follows that the radial metric function behaves

(T.6)

at the origin. This corresponds to the exact result (5-7). Then the radial

equation (3.15) takes the asymptotic form

c V*(I R+ - Xcf?z + 1 - e - V c ^ ^ (7.7)

(compare with tq,.(3.11) of t). A familiar argument expanded in I yields

~ C' 5 (7.8)

as asymptotic solutions regular at the origin. The system of approximate

solutions (7.M, (7.6) and (7.8) turns out to be consistent.

OJJ ) Another set of asymptotic solutions can be obtained by proposing

instead of (7.5) the trial function

(T.9)

for i » 0.

Assuming that e ft 0 for £ -» 0 such that

(7.10)

ve obtain from (U.lU) the result

e — (7.11)

This together with (7.M satifies also (U.13). Furthernore, the insertion of

(7.11) into the radial equation (3-15) yields

R'±5

(T.ia)

Its integration results in (7.9), the integration constant already being

determined by (7.10). The sets (7.<0, (7.9) and (7.10) of asymptotic solutionsU6)

have earlier been discussed by Feinblum and HcKinley

In. spite of the fact that in the latter case the radial part of thescalar waves i s logarithmically divergent at origin, ve should not disregardthese solutions. For a more precise reasoning ve have to also take thestrongly deformed Bpace-tine manifold at the origin into account. Thiseffect l>econies more transparent i f we consider the function Fo^$#) definedvia (3.8) in terms of the "tortoise"co-ordinate (3.9). In view of (T.M and(7.12) the latter acquires the asymptotic behaviour

near the origin. By applying 1'Hospital' a rule ve find that

(7.13)

IT.HO

-26-

tends to zero for^-*>0. Therefore the charge integral (5.5) should be

bounded even at the origin. Since the subsidiary conditions of bounded square-

integrability and fast decrease at spatial infinity turn out to be fulfilled t

solutions with the asymptotics 0 in curved space-time can also be regarded

a3 "eigensolutions", according to the criteria of quantum "echanics (see § 16

of Ref.38). With respect to the formal SchrSdinger-type equation (3-12) the

dynamics corresponds to the motion of a "particle" in a centrally symmetric

field characterized by a centripetal potential

(7.15)

near the origin (compare with Ref.38, p.109).

With th is information at hand we have performed the numerical cal-culations on a DEC-PDP-10-computer using single precision BAG and IMSL Librarysubroutines. The evaluation of the functions R^($), e * and e*1 '^ hasbeen done in double precision mode. For a l l calculations the free parametersof the model have been fixed according to :

Then for each given at and n.^ the system of ordinary differential equations(3.15), (U.13) and (U.II4) has been numerically solved by Hunge-Kutta formulasof order 5 and 6 as developed by 3.H. Verner and coded by Hull and co-workersin the IMSL siibroutine DVERK. The global error of the solutions has beenestimated to be less than 1.10 . Using the asymptotic conditions (7.1) and(7.2) as start ing values the integration has been performed going from f^a= 30backwards to zero. Self-consistent solutions are constrained by two additionalconditions. F i r s t , they have to ful f i l l the charge normalization (5,5) andsecond, they have to reproduce the parameters OL chosen for the i n i t i a lconditions consistent with the Tolaan integral (5.12) for « . This has beenachieved by an i terat ive least squares f i t using the HAG subroutine E0UFAFwhich is based on a method due to Peckham . Thus the parameters at and C^have been determined by minimizing the sura of squares

V(7.16)

where J denotes the j t h iteration step, ct,_1 is the result of the (j-l) t h

atep which fixed the initial conditions (7.1) and (7.2) and Q., * are cal-

culated from (5.5 ) and (5.12), respectively.

Using appropriate starting values for C ^ and at , a convergence of

(7-16) better than l.io" has been obtained by the method resulting in a

relative error in Ot and Q of less than 1.10 . The numerical integration

of Q, 0C and E has been performed by using the HAG Library procedure

D01GAF which estimates the value of a definite integral(when the functional

is specified numerically) using the method described by Gill and Miller

The maximal relative error should in no case be larger than 1.10

As can be deduced from the asymptotic solutions (7.10) and (7-12) of

the second set 0 , the energy expression (5-ll») contains a term I n £ Q

which diverges for - Therefore in Fig.I* we have computed only the

finite part of E, i.e. E , corresponding to the "cut-off" parameter f_ = 0.001.

So far our method of integrating backward has produced solutions

belonging solely to the set O-j. of asymptotic solutions at the origin. In

this case we found solutions with and without nodes. The number of node*

of the radial part R*($) of the wave function for finite values of £

(excluding the point ^ = 0) may be used to define aa a radial quantum number

n as in the non-relativistic Schro'dinger theory (Ref.38, p.109). Prom the

theory of the hydrogen atom we suspect the relation

*- L- A (7.17)

to hold (Kef.38,p.123) where n would denote the "principal quantum number of

the geon"

Figs.l and 2 reveal that our numerical results interpolate rather well

between the asymptotic solutions at infinity and at the origin (set 0._). In

the case without nodes (Fig.la) the radial scalar function B^tt) Joins smoothly

the asympotic solution (T-9 ) with the localized solution (7.1). Both metric

functions show a Schwarzachild-tyjie behaviour for large e . For small 5 ,

e becomes constant (Figs.lb, 2b) similar as in the constant core case,

whereas e tends to zero in accordance with (7-11). The latter function

develops inbetween a noticeable peak (Figs.lc, 2c) which corresponds to the

confining barrier in the effective curvature potential (3.13) Figs.le, 2e).

An interesting phenomenon can be observed on the level of the

Schrodinger-type vave function *'0{^)t) being defined vith respect to the

"pseudo-flat" space-time (3.10). F*(j*) is concentrated (see Figs.Id, 2d)

-28-

-27-

within the negative veil of V ff(**) with its maximum close to the zero of

the potential. For Bmaller values of o» this maiimum is shifted by the

barrier of the curvature potential closer to the origin. This seems to

indicate a self-generating effect of the geoaetro-dynamical confinement

mechanism (being here only partial)! This confinement scheme and its

13) extension including colour may be compared with, e.g. the MIT "hag"proposed•odel (see also Hasenfratz and Kuti for a reviev). There an ad-hocIntroduction of a "vacuum" pressure term is needed to compensate forthe outside directed pressure of the "quark gas". In contrast to thisphenomenological device our approach ratlwr resembles Creutz's 5 3 ' re-construction of a hag model from local non-linear field theory. Similar to hia,the "core" of our "bag" i s produced by employing the stable(quantun-mechanicallyIieastable solution)of the KKG equation for an extended part of the space.Surrounding this "core" i s a transition region,the "skin" of the "bag" consistingof an exponentially decreasing lukawa-type radial solution for the scalar

field and a Coulomb type potential for the "tensor gluona".

The total gravitational mass (5.12) as measured at infinity exhibitsa branching for the zero and one node solutions with respect to i ts (^-dependence(Fig.3). For n • 0 and low A) we may understand the qualitative behaviourof U(U) by comparing i t with (see Ref.51")

(T.18)

hut for higher values of to Eq.(5.13) tends to

* ~ /J* (7.19)

The resulting predictions for the zero and the maximum of the Tolman energy,

i.e.

W. f ' 0. 7\and

(T.20)

(7.21}

(7.22)

respectively, agree quite well with our results (Fig,3).

Already for n • 1 and higher node solutions we found parts of several

sub-branches in tf(ft>), i.e. the Tolman energy of these aeons Satiates in a noticeable

way (not shown in Fig.3). In a preliminary study we could distinguish the

corresponding solutions among others by the number of nodes in H".

Further analysis is in progress in order to understand these highly

interesting instances of a possible "fine structure" in the energy levels

of the aeons. In view of these rich and prospective structures does there261

exist the speculative alternative to interpret quantum geons as extended

particles capable of internal excitations?

VIII. OTHER GRAVITATIOHAL SOLITOHS

To some extent Wheeler's ' geon concept has anticipated the, (non-1*5)

integrable )Boliton solutions of classical non-linear field theories.

As mentioned in the Introduction a geon or gravitational soliton originally

was meant to consist of a spherical shell of electromagnetic radiation held

together by its own gravitational attraction. In the idealised case of a thin

spherical geon the corresponding metric functions have the values e c • —

well inside ande*e = 1 - jr well outside the active region. The trapping

area for the electromagnetic wave trains haa a radius of

This result has been confirmed by applying Ritz variations! principles

Constructions with toroidal or linear electromagnetic wares have been

given by Ernst

Vheeler .

55)vhereas neutrino geona have been analysed by Brill and

56)Brill and Kartle > u ' could even demonstrate the existence of puregravitational solitonS. By expanding the occurring gravitational waves in terasof tensor spherical harmonics i t can be shown that the radial functionexperiences the same effective potential (3.13) except that an additionalfactor =• appears in front of the contribution from the background metric.

In a relevant paper the coupling of linear Klein-Gordon fields togravity has been numerically studied by Kaup . Moreover, the problea ofthe stability of the resulting scalar geons with respect to radial perturbation*

-29- - 3 0 -

is treated. I t i s shown that such objects are resistant to gravitationalcollapse (related works include Kefs.58, k6 and U6). These considerationsare on a semielassieal level. However, using a Hartree-Foek approximationfor the second quantized two-body problem i t can be shown that the same

coupled Einstein-Klein-Gordon equations apply.

As an important new ingredient, Kodano. et a l . U2)have considered a

real scalar field with a. Cf self-interaction as a source for the gravitationalfield. In this preliminary study the Klein-Gordon operator corresponding toflat Space-time is assumed.

A general re la t iv i s t ie Klein-Gordon field with an effective <g self-interaction for an interior ba l l has also been analysed by this group .In order to avoid a singular conitguration at origin a repulsive (or "ghost-like")scalar field has been chosen as a source e'f Einstein's equations.In a further step Kodana constructed a spherically symmetric kink-type

solution for a repulsive scalar field with Ct self-coupling (compare alsowith Ellis ' ) - As is common for kinks, the radial function at spatialInfinity i s chosen to be a constant characterizing th i s non-linear model.If we want to transfer the method to our case, we may use instead of (2.3)the symmetric self-interaction

The additional constant in (8.1) necessarily eliminates the gravitational

source which otherwise would occur for the constant aolution

;f(8.2)

characterizing the kink solution asymptotically. In flat Minkowski space

It is a conjecture that (8.2) constitutes a first approximation to the vacuum

expectation value ^o} j | 0 ^ of the corresponding quantun field.

For the general re la t iv is t ic kink of Kodama the radial solution

becomes ierc at a certain radius rB at which the background geometry develops

a Schwarzachild-type horizon. (Geon-type solutions exhibiting an event

horizon may be termed "black solitons" .) The boundary condition at r ,

however, allows an extension of these solutions into a 3-manifold consistingof two- asymptotically Euclidean spaces connected by an Einstein-Rosen bridgeArguments are given that this extended, non-singular configuration isstable with respect to radial osci l lat ions.

I t should be noted that such solutions cannot be constructed for the1 2"worMhole" topology R x S x S which would be obtainable by identifying the

asymptotically flat regions. The reason simply being that the radial functions

of the kink has an opposite sign in the other sheet of the Universe. Since thep

quantum-mechanical probability density l^fWj i s single valued and completelyregular also for the wonnhole topology, such scalar kinks provide interestingobjects for further studies.

Although we have no intention to give a complete review, we would liketo mention that other studies on geons involve massleas scalar fieldscoupled Einstein-Maxwell-Klein-Gordon systems ' or even combinedDirac-Einstein-Maxtfell field equations T 1 ' .

As a final observation we remark tha t , according to a result of Bri l l

25)

72)

a massless scalar field can be geometrized in the sense of the "already unified

field theory" or "geometrodynamies" of Rainieh, Misaer and Wheeler . Loosely,

speaking, this means that the scalar field can be completely read off from

the "footprints" it leaves on the geometry.

ACKNOWLEDGMEMTS

We would like to thank Professor H. Rufflni and Professor Abdus Salsafor useful h ints . One of us (E.W.M.) would like to express his sinceregratitude to Professors F.W. Hehl, Abdus Salam and J.A. Wheeler forencouragement and support, and would also like to'thank Professor Abdus Salam,the International Atomic Energy Agency and UNESCO for hospitality at theInternational Centre for Theoretical Physios, Trieste. TTCiiB work wassupported by the Deutsche Forschungsgemelnschaft. The computations wereperformed at the Reehenzentrum of the Christian-Albrechts-Universltat, Kiel.

-31- - 3 2 -

REFERENCES

1) J.J.J. Kokkedee, The quark Model (Benjamin, New York 1969).

2) R.N. Boyd,D. Elmore, A.C. Melissimoa and E. Sugarbaker, Phys. Rev.

Letters {*£, 3l6 (1978),

3) O.W. Greenberg and C.A. Nelson, Phys. Rev. 32£, 69 (1977).

h) D. Friedberg, T.D. Lee and A. Eirlin, Phys. Rev. D13., 2739 (1976);

Hucl. Phys. B115, 1; 32 (1976).

5) W. Marciano and H. Pagels , Phys. Rep. 36£, 137 (1978).

6) A. Salam and J.Strathdee, Phys. Letters §TB, 1*29 (1977).

7) A. Salam and J. Etrathdee, Phys. Rev. Dl8_, 1)596 (1978).

8) C.H. Yang and R.L. Mills, Phys. Bey. £6,, 191 (195M.

9) S. Coleman and L. Snarr, Commun. Math. Phys. £6_, 1 (1977).

10) D. Robson, Hucl. Phys. B130, 328 (1977).

11) E.W. Mielke, Phys. Rev. Letters 32., 530 (1977); 32., 851 (E) (1977).

12) E.W. Kielke, "Gauge-theoretical foundation of colour geometrodynamies"

(to be published in the Proceedings of the 1978 Clausthal meeting,

Ed. H.D. Doetiner).

13) E.tf. Mielke, 1CTP, Trieste, preprint IC/8O/5U (to appear in Int. J.

Sbeoret. Phys.);

1*0 C.J". Ishoa,' A. SalBurand J. Strathdee, Fhys. Rev. D3_, 867 (1971).

15) C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. D§_, 2600 (1973).

16) C.J. Isham, A. Salam and J. Strathdee, Phys. Rev. D9_, 1702 (1971*)-

17) F.E. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod.

Phya. ]t8_, 393 (1976).

18) H. Weyl, Proc. Hat. Acad. Sci. (USA) 15_, 323 (1929).

19) H. Weyl, Phys. Rev. JJ_, 699 (1950).

20) W. Heisenherg, Introduction to the Unified Field Theory of

Elementary Particles (Wiley, London 1966).

21) E. Schrodinger, Sitzungsber.Preuss.Ak.ad. ¥iss. Phys.-Math. Kl. XI_,

105 (1932).

22) D. Ivanenko, Sov. Phys. 13̂ , lUl (1938); Suppl. Nuovo Cimento 6_, 3̂ 9 )1957).

23) E.C. Tolman, Relativity, Thermodynamics and Cosmology (clarendon Press,

Oxford 193!t).

-33-

2\) J.A. Wheeler, Phys. Rev. 27_, 511 (1955).

25) A. Einstein and N. Rosen, Phya. Rev. J*e_» 73 (1935).

26) J.A. Wheeler, Rev. Hod. Phys. 33., 63 (1961).

21) R.J. Finkelstein, Phys. Rev. 72., 1079 (I9U9).

28) R. Finfcelstein, R. Lelevier and «. Ruderman, Phys. Rev. 83_, 326 (1951).

29) D.R. Brill and J.A. Wheeler, Rev. Mod. Phys. 2£., 1(65 (1957), (E) 33_,

623 (1961).

30) W. Deppert and E.W. Mielfce, Phys. Rev. D20_, 1303 (1979).

31) D.J. Kaup, Phys. Hev. 1J2., 1331 (1968).

32) K.C. Ttolman, Phys. Rev. 3^., 875 (1930).

33) A. Chodoa, R.L. J a f f e , K. Johnson, C.B. Thorne and V.P. Weisskopf,

Phys. Rev. DO_, 3^71 (197**).

31*) P. Boyanovsky and L. Masperi, Phys. Rev. D21., 1550 (1980).

35) D.L.T. Anderson, J . Math. Phya. 12_, 9U5 (1971).

36) C.V. Misner, K.S. Thorne and J.A. Wheeler, Gravi tat ion (W.H. Freeman

and Co. , San Francisco 1973). (quoted as(*TW).

37) J.W. Wheeler, Geometrodynamies (Academic Press , Mew York 1962)

(contains r ep r in t s of Refs.2l* and I t l ) .

38) L.D. Landau and E.M. L i f s ch i t z , Quantum Mechanics - t ion-Relat ivls t ic

Theory (Pergaiaon, London 1967).

39) E.W. Mielke, Gen. Rel. Grav. 3_, 321 (1977).

Uo) H. Rund and D. Lovelock, J.TDer. Deutsch. Math.-Yerein Jjj., 1 (1972).

kl) E.A. Power and J.A. Wheeler, Rev. Mod. Phys. 29., U80 (1957).

U2) T. Kodamo, K.C. Chung and A.F. da F. Teixeira, Nuovo Cimento U6_, £06

(1978).

1*3) R. Jackiw, Rev. Mod. Fhys. )Q, 68l (1977).

UU) S.A. Fulling, Fhys. Rev. DJ_, 2850 (1973).

1*5) V.G. Makhantov, Phys. Rep. 35C, 1 (1978).

\6) D.A. Feinblum and tf.A. McKinley, Phys. Rev. 16B., lkk5 (1968).

1*7) E.W. Mielke, J. Math. Phya. 21., 5>*3 (1980).

1*8) A.F, da F. Teixeira, I. Wolk and M.M. Som, Phys. Rev. D12., 319 (1975).

•*9} T.E. Hull, H.H. Enright and K.R. Jackson, TR No.100, Dept. of Computer

Science, University of Toronto (19T6).

50) G. Peckham, Computer J. 13., MB (1970).

51) P.E. Gill and G.F. Miller, Computer J . 15_, 30 (1972).

52) P. Hasenfratz and J . Kuti, Phys. Rep. liOC, 75 (1978).

53) M. Creutz, Fhys. Rev. D10., 17^9 (197I4).

5M H. Ruffini and S. Bonazzola, Phys. Rev. l3j_, 1767 (1969).

55) F.J . Ernst, J r . , Phys. Rev. 105., 1662; 1665 (1957).

56) D.B. Br i l l and J .B. Hartle, Phys. Rev. 135_, B271 (196U).

57) T. Regge and J.A. Wheeler, Phys. Rev. 106., 1063 (1957).

58) A. Das, J . Math. Phys. ^ 1*5 (I963).

59) K.C. Chung, T. Kodama, and A.P. da F. Teixeira, Phys. Rev. Dl6, 21*12 (1977).

60) T. Kodama, Phys. Rev. Dl8, 3529 (1978).

61) H.G. Ellis, GRG 10, 105 (1979).

62) A. Salam and J. Strathdee, Phys. Letters 6lB, 375 (1976); 6 6 B , Ik3

(19T7).

63) T. Kodama, L.C.S. de Oliveira and F.C. Santos, Fhys. Rev. D19, 3576

(1979).

6U) S. Deser and J. Higbie, Ann. Phys. (N.Y.) 58_, 56 (1970).

65) J-D. Bekenstein, Ann. Phys. (N.Y.) 62_, 535 (I97U).

66) M. Curses, Phys. Rev. D15_, 2731 (1977).

67) M. Lunetta, I. Wolk and A.F. da F. Teixeira, Phys. Rev. D2^. 328l (i960).

68) K.A. Bronnikov, V.V. Heloikov, G.N. Shikil* and K.P. Staniukovich,

Ann. Phys. (H.Y.) Il8_, 6k (1979).

69) A.F. da F. Teixeiro, I. Wolk and H.M. Som, J. Phys. A9_, 53 (1976).

70) A. Banerjee and S.B. Dutta. Cboudhury, Phys. Rev. D15_, 3062 (19T7).

71) M.J. Hamilton and A. Das, J. Math. Fhys. l8_, 20S6 (1977).

72) D. Brill, [Tuovo Cimento Suppl. 2_, 1 (196!*).

Fig. 2

FIGURE CAPriONS

Scalar geon without node. The results of the numerical integration

are shown for the case t*IJL ' *. fi s 0-2,-A-= 0, £= £ = 1, n = 3,k * 1, L = 0. The solutions which depend on to and J or **,

respectively, are presented as relief.

a) Radial solutions

b) Time-like metric function e *' .

c) Space-like metric function e • .

d) Schrodinger-type wave function FQ(<") given in terms of the

"tortoise" co-ordinate • • .

e) Effective curvature potential V f f(^*) exhibiting a deep well

together with a confining barr ier .

(In Figs.la and Id only a few solutions are shown in order to

avoid a too strong screening in the 3D plot which would otherwise

occur. In Figs.Id and le the curve3 are plotted up to $ • .values

corresponding to ^ = 5-)

Scalar geon with one node- Same case and presentation as in F ig . l ,except for a braoder CO range.

Radial solutions exhibiting one node.b) Time-like metric function ec) Space-like metric function e * .d) Schrodinger-type wave function ^ ( ^ * ) having also one node

outside the origin (see the magnification (2x in F and 50x in

\*) of part of the r e l i e f ) .

e) Effective curvature potential V ,,_(ft*) with a deep well in a

confining barr ier .

(Again in Figs.2d and 2e the curves are plotted up to $* -values

correspondong to ^= 5.)

Fig.3 Tolman energy or Schwarzschild mass atM* as a function of c*J for

"quantized" scalar geons with "principal auantum number" n = 1,2.

Flg.U "Regularized" field energyof the scalar waves within scalar geons

as a function of CO , corresponding to a "cut-off" length f- - 0.001.

- 3 5 --36-

- 3 7 - - 3 8 -

- 3 9 -- U O -

" • • • ! ; :

-1*1--1(2-

Fia.2c

-1.3-

fip

-1+5-.1.6-

WL'Q BL'0 t*O O'O

-1*7- UJ

- u a -

IC/80/2

IC/80/3

IC/80/U

IC/eo/15

IC/80/21

IC/80/22

IC/80/29

ic/eo/32

ic/eo/33IBT.REP.1

IC/80/38

IC/80/39

IC/80/li3

IC/8OA7

IC/8O/U8

IC/80A9

IC/80/50

IC/80/52

IC/8o/5!«

CURRENT ICTP FREPRINTS AND INTERNAL REPORTS

A.P. BUKHVOSTOV and L.U. LIPATOV: Ins tanton-ant i - tns tanton in te rac t ion inthe 0(3) non-l inew a-model and an exactly soluble fermlon theory.

RIAZUDDIN; K •* 2i decays In non-relat iv ist ic quark-gluon model.

G. SENATORE and M.P. TOSI: Theory of the surface dlpole layer and ofsurface tension in l iquids of charged part ic les .

S. RAI CHOUDHURY: QCD effects in a model of non-leptonic hyperon decays.

SAKIX K. PAK: Electric charge as the source of CP violat ion.

T. KATTERMAHB and J. PRZYSTAWA: Locking-in and incommensurability ofthe structural transition in BaMnFi,.

E.W. MIELKE: On pseudoparticle solutions in Yang's theory of gravity.

ABDUS SALAM and VICTOR ELIAS: Induced Higga couplings and spontaneoussymmetry breaking.

A. AMUSA: Point-triton analysis of exchange free direct two-nucleon transferreaction cross-sections.

WITHDRAWN

ABDUS SALAM: The nature o f t h e " u l t i m a t e " e x p l a n a t i o n i n p h y s i c s(Herbert Spencer Lecture - 12 November 1 9 7 9 ) .

V. de ALFARO, S . FUBIHI and Q. FUELAN: C l a s s i c a l s o l u t i o n s and ex tended•upergraTity.

J.C. PATI: Unity behind diversity in Nature.

J . LUKIERSKI: Supersymmetric <J models and composite Yang-Mills theory.

W. KROLIKOWSKI: Are lepton and quark families quantized dynamical systems?

V. DEPPERT: The role of Interaction in neutrino s t a t i s t i c s .

D. KUSNO and M.J. MORAVCSIK: On the problem of the deuteron shearingcorrections - I: The conventional approach.

E.W. WELKE: The eight-fold vsy to colour geometrodynaffllcs.

ic/80/55 A.R. HASSAH: PSvonon- as slated transitions in crossed e lectr ic and magnetic

fields.

IC/8O/56 A.B. HASSAS: Tiro-photon indirect transitions in crossed e lec tr ic andmagnetic f i e lds .

IC/Bo/57 S. HAFISOS: Techniques of dimensional renormalization and application tothe two-point functions of QCD and QED.

IC/80/62 H.S. C1WIGIE: Qua.ntum chromodynamics - A theory of the nuclear force.

IC/80/67 WITHDRAWN

IC/80/68 A. AYEKSU: pynamic dislocations in high quarts.

IC/6O/69 T. PERSI: Time dispersion relations and small-time behaviour In the decay1ST.HEP.• of an unstable system.

IC/80/70 M.F. KOTKATA, A.A. EL-ELA, E.A. MAHMOUD and M.K. El-MOUSLY: Electrical"transport and structural properties of Se-Te semiconductors.

IC/8O/7I A.Yu. OROSEERO and A.R. HiOKHLOV: Some problems of the statistical theory ofINT.REP.* polymeric lyotropic liquid crystal3-

IC/80/72 J.C. PATI and ABDUS SALAM: Quark-lepton unification and proton decay.

IC/80/73 ASHOK DAS: Are massless aupersyimiietric gauge theories really snassless?

IC/8O/7U G. HAIELLA: Chiral rotations and the fermion-boson equivalence In theIBT.REP.* Schwinger model.

IC/BO/75 J. SPA1EK and K.A. CHAO: Kinetic exchange interaction in a, doubly degenerate

narrow band and its applications to Fe^Cx^Sj and ^i-^x3?*

IC/80/76 L.K. SHAYO: The generalized pressures on oscillating cantilever pipesIHT.REP.* conveying inviscid fluid.

IC/80/77 E. GAVA, R. JENGO and C. QMERO: Finite temperature approach to confinement.

IC/80/78 A. AYENSU: The dependence of mechanical, electrical, thermal and acousticIHT.REP.* properties of tropical hardwood on moisture content.

IC/80/79 J.S. NKOMA: Effect of impurities on the two-dinensioneO. electron gaspolarliability.

IC/8O/8O A.M. JAYAHNAWAR and N. KUMAR: Orbital diamagnetlsm of a charged Brownianparticle undergoing birth-death process.

IC/80/81 L.K, SHAYO; On the solution of the Laplace equation In the presence of aIHT.REP.* semi-infinite boundary - The case of an oscillating cantilever plate in

uniform incompressible flow.

IC/BO/82 E.R. HISSIMOV: Higher quantum conserved current in a nev completelyXNT.REP.* integrable model.

• Internal Reports: Limited distribution.

THESE PREPRISTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586,

I-3I1IOO TRIESTE, ITALY.

IC/8O/85 Tfco RUIBAO and Pu FUCHO: A proof of the absence of modulated phase transitionand' spin density wave phase transition in one- and two-dimensional systems.

M/80/66 PuFOCHO: Density of s t a t e s , Polason's formula of summation and Walfisz'sIBT.HEP.* ronnula.

IC/80/87 A.K. BASDYOPADHYAY, S.K. CHATTERJEE, S.V., SUBRAMANYAM and B.R. BULKA:IBT.REP.* Electrical r e s i s t i v i t y atudy of some organic charge transfer complexes under

pressure.

IC/8O/88 In-Gyu KOH and Tongdui. KIM: Global quantity for dyons with various chargeHT.HEP.* distr ibutions.

IC/8O/B9 K.G. AKDEHIZ, M. GOODMAH and R. PERCACCI: Honopoles and twisted sigma madela.

IC/80/90 A.R. HASSAN and F. Abu-ALALLA: Exciton-polarlzations by two-photon absorptionHT.REP.* in eemiconductors.

IC/80/91 A.R. HASSAfi: Phonon-polaritons by two-photon absorption in so l ids .IKT.REP,*

IC/80/92 L. LOHCA and J. KOHIOR: Exact results for some 2D-Ialng models v l t hIIT.REP,* periodically distributed impurities.

IC/80/109IBT.EEP."

IC/80 / l l lIMT.REP.*

IC/80/U2

1C/8O/X13IHT.REP.*

P. FURLAH and R. RACZKA: On higher dynamical symmetries in models ofrelatlvistie field theories.

A. BREZIHI and G. OLIVIER: Localisation on veakly disordered Cayley tree.

S.K. OH: The BJorken-Paschos relation In the unified gauge theory.

A.A. FADLALLA: On a boundary value problem In a strongly pseudo-convexdomain.

IC/8O/9A

ic/80/95

ic/80/96

IC/60/97

IC/80/98HT.REP.*

IC/80/99HT.HEP.»

IC/80/100

IC.80/101HT.REP.*

IC/80/102

ic/80/103DTT-HEP.*

IC/30/105

IC/So/106HT.RKP.*

ABDUS SALAH and J. STRATKDEE: On Witten's charge formula.

ABDUS SALAM and J. STHATHDEE: Dynamical mass generation in (UT(l) x U (l))2.

I* n

I . S . CRAIGIE, f. BALDRACCHIHI, V. ROBERTO and H. SOCOLOVSKY: Study offactorisation in QCD vi th polarized beams snd A production at large p_.

K.K. SINGH: Renormallzation group and the ideal Bose gas.

HABESH DADH1CH; The Penrose process of energy extraction in electrodynamics.

0 . HUKHOFADHYAT and S. LUHBQV1ST: Dynamical polar l iabi l l ty of atoms- '

A. QADIR and A.A. MUPTI: Do neutron stars disprove multiplicative creationIn Dlrac's large ruutber hypothesis 1

V.V. MOLOTKOV and I.T. TODOROV: Gauge dependence of vorld l ines and invarianceof the S-matrix in r e l a t i v i s t i c c lass ica l mechanics.

Kb.I. PUSHKAHOV: Solitary excitations In one-dimensional ferromagnets atT * 0 .

Kb.I. PUSKHAROV and D.I. PUSHKAROV: Solitary clusters in one-ctimenstonalferromagnet.

T.V. HOLOTCOY: Equivalence between representations of conformal superalgebrafrom different subgroups. Invariant subspaces.

MEHBEL SACHS: On proton mass doublet from general r e la t iv i ty .

L.C. PAFALOUCAS: Uncertainty relations and semi-groups In B-algebras.

IC/BO/117 P, KAZEKAS: Laser Induced switching phenomena in amorphous GeSejt A

-phase transit ion model.

IC/80/118 P. BUDIHIs On splnor geometry: A genesis of extended supersymaetry.

IC/&0/119 G. CAMPAGHOLI and E. TOSATTT: ABF -intercalated graphite: Self-consistentband structure, optical properties and structural energy.

IC/80/120 Soe Iln and E. TOSATTI:. Core level shi f ts in group IV semiconductors and•emimetal*.

IC/80/121 D.K. CBATORVECI, 0. SEBATORS and M.P. TOSIi Structure of l iquid alkal i .1ST.REP.* metals aa electron-ion plasma.

IC/80/126 T*e Seventh Trieste Conferenea on Particle Ptaysles, 30 June - •» July 1980IHT.RKP.* (Contributiona) - Part I .

IC/80/13T The Seventh Trieste Conference on Particle Physics, 30 June - *• July 1980IHT.REP.* (Contributions)- Part I I .

IC/60/128 ?ABID A. KHVMJA: Temperature dependence of the short-range order parameterand the concentration dependence of the order-disorder temperature for I l -Ptand Bi-Fe systems In the Improved s t a t i s t i c a l pseudopotential approximation.

IC/SO/130 A. «IADIR: Dlrac's large number hypothesis and the red shif ts of distantIJIT.REF.* galaxies.

IC/80/131 S. ROBASZKIEWICZ, R. MICHAS and K.A. CHAO: Ground-state phase diagrain of

IBT.REP.* extended attract ive Hubbard model.

IC/80/133 K.K. 8IDGH: Landau theory and Giniburg criterion for Interacting bosons.

IC/8O/13U R.P. HAZOUME: Reconstruction of the molecular distribution function* from theIHT.REP.* s i t e - s i t e distribution functions In c lass ica l molecular fluids at equilibrium.IC/80/135 R.P. HAZOUME: A theory of the nematic liquid crysta ls .IRT.REP.*

IC/80/136 K.G. AKDEHIZ and H. HORTACSU: Functional determinant for the Thlrrlng modelIHT.REP.* with instanton _ —, _