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Astrophysics and Space Science 201: 303-319, 1993. © 1993 Kluwer Academic Publishers. Printed in Belgium.

MASSES OF MACROSCOPIC QUARK CONFIGURATIONS IN METRIC AND DYNAMIC THEORIES OF GRAVITATION

V. V. SOKOLOV and S. V. ZHARYKOV Special Astrophysical Observatory of Russian Academy of Sciences, Nizhnii Arkhyz, Stavropol Territory, Russia

(Recei ved 26 August, 1992) Abstract. Within the bounds of the general relativity and in gravidynamics, spherically-symmetric configurations are considered with the limit equation of state (P = ( - 4B)/3) and with the density increasing to the center. It is shown that unlike GR, where the existence of strange stars only is permissible (u-, d-, -quarks), in the consistent dynamic theory of gravitation the existence of stable configuration with ~ r
-2

(quark star) is

possibl e with a 'bag' out of quark-gluon plasma which includes all possible quark flavors (u, d, s, c, b, t, .. .). The total mass of such a compact object with the bag of the radius of 10 km (whose surface consists of the strange self-bound matter) must be 6-7 M.

1. Introduction By the metric theory we mean here, first of all, general relat ivit y (GR) and all versio ns o f gravitat ional theories which proceed from Einstein's principle of equivalence. A project of theoretical model o f gravitat ional int eraction based on the consistent applicat ion of dynamic principles (gravidynamics) is presented in previo us papers (Sokolov, 1990, 1991, 1992a, b, see also the references therein). In gravidynamics (GD) the law of equivalence o f inertial and gravitat ional masses is certainly true, but here 'the principle of equivalence' is not used in any way which we consider, fo llowing Fock (1961), only a kinemat ical consequence of the fundamental law: mi = mj. Accordingly, in the consistent dynamic theory o f gravitation the field is not reduced to the space-t ime metrics which, as in Maxwell's electrodynamics, can always be described by Minkovsky's metric tensor. In the suggested report we try to answer the question why in GR only quark configurations, consist ing o f strange matter (u-, d-, s-quarks) - 'strange stars' - are permitted, while in GD quark-gluon plasma of analogous objects may consist of quarks of all possible types - 'quark stars' proper. As will be seen fro m the following, the peculiarit ies o f GR and GD beco me most essent ial when we consider the utmost in-ho mogeneous quark 1


configurations in both theories. There are already many calculations o f quark configurations (strange stars) within the bounds o f GR (i.e., the calculat ions on the basis of Oppenheimer-Vo lkoff's (OV) hydrostatics equations; cf. Haensel et al., 1986; Alcock et al., 1986; Benvenuto and Horvath, 1989; Krivoruchenko, 1987; üvergÅrd and üstgaard, 1991). The same calculat ions were carried out recent ly in the SAO o f Russian AS. But unlike other groups, we were interested first of all in the way the modern (dynamic) theory of strong interactions - quantum chromodynamics (QCD) - 'works' in the condit ions of the strongest gravitat ional field of a co mpact object with a mass of M. Up to now the corresponding observat ional information still does not exclude alternat ives to GR. Ult imately, our basic purpose is a test o f the gravitat ional interact ion theory, elucidation of observat ional consequences and obtaining est imates allowing to compare GR and GD in a descript ion of the same object - compact quark configuration.

2. Utmost Inhomogeneous Quark Configurations in GR This section addresses the calculat ions o f purely quark configurat ions described by the limiting equation of state PQ = 1/3 ( - AB), where is the total energy densit y ins ide a huge quark bag, 4B/c2 =
QG P

(1) is a macroscopic

densit y on the surface (PQ = 0) o f the bag consist ing of the quark-gluon plasma (QGP). Equation (1) is the limit to which tends the corresponding total equat ion of state (Alcock et al., 1986; Haensel et al, 1986),describing the media consist ing o f quarks with masses tending to zero. Quark-gluon interactions stay in the lowest order in ac (i.e., 2c/ should be sufficient ly small so that it remains the first term o f expansio n in t he expressio n for thermodynamic potential). Below, we speak about sufficient ly co ld (catalyzed) quark matter at temperatures not more than (for example) 1010 K when this matter is already a degenerate Fermi fluid. Thus in this paper the quest ion is on a totally cooling down quark star whe n electrons are absent in the picture, and wit h no electrons, there is no corresponding neutrino flux. As a matter of fact, subject to the above remarks, we shall use here an asymptotic MIT bag model (üvergÅrd and üstgaard, 1991), where the bag constant B is a measure o f confinement strength. Integration of equat ions of hydrostatic equilibrium (OV equat ions) gives, in particular, the relat ion between the mass and the radius of the compact object (Figure 1). Here the value B is chosen as B = 67 MeV fm-3. This corresponds to the macroscopic densit y on the surface (PQ = 0) of QGP-bag equal to
QGP

1.7 n

ucl (nucl = 2 . 8 x 1 0

14

g cm-3).

2


Usually all calculat ions o f this kind (see, for example, Haensel et al, 1986; or Alcock et al, 1986) are interrupted near point C in Figure 1. This corresponds to the OV limit for a compact object with the equation o f state (1). All configurations lying to the left of po int C turn out to be unstable wit h respect to small radial perturbations. Branch C-D in Figure 1 is closest to the black ho les - 'permanent ly' unstable objects. Such an instabilit y is described in many manuals on GR (Zel'dovich and Novikov, 1971; Shapiro and Teukolsky, 1983). In Figure 1 the dotted line shows the M/R connection for neutron stars. This connect ion is close to that given by Bethe-Johnson's equation of state. The same connect ion M/R at B = 67 MeVfm-3 corresponds at first (i.e., for small masses 0.5 M) to neutron stars, and then to neutron stars with a growing (with further mass increase) quark nucleus, arising ins ide the compact object when its central densit y exceeds the value 4B/c2 (see, in detail, Haensel et al, 1986).

Fig. 1. The mass-radius relation for purely quark configurations with equation of state (1). The dotted line shows the M/R relation for neutron stars. Point C correspods to the OV-limit, point D indicates the positon of the utmost inhomogeneous quark configuration in GR. The Black Hole region is shaded (see the text), B = 67MeVfm-3.

From Figure 1, it is seen that at the same mass, neutron stars are more extended, or less compact objects, than purely quark configurations, are calculated for limit equation o f state (1). In GR only black ho les wit h infinite gravit ational red shift z can be more compact objects. Thus, if Equat ion (1) is really the limit of the equation of quark matter state at superhigh densit ies (>
nucl)

, then it means that, in the bounds o f GR, simply, there are no

more co mpact hydrostatic equilibrium configurat ions than those shown in Figure 1. In this 3


sense, purely quark configurations corresponding to the curve AC in Figure 1 are the limit objects for GR. Here we can speak of hydrostatic equilibrium stable compact objects with the surface (z is a finite value) instead of the event horizon. But the basic difference between purely quark configurations and neutron stars, which is important for us to underline here, is the fact (and it is seen in Figure 1) that the quark matter or strange matter in GR is self-connected. Such an object is really a huge quark bag whose hydrostatics at small masses (A branch in Figure 1) is guaranteed only by strong (color) interact ion. At M M and at masses close to the OV limit the curve in Figure 1 'turns' to black ho les due to GR effects. Hydrostatic calculat ions wit h the use o f OV equatio ns give the fo llowing dependence on the value of B for the maximum mass of purely quark configurat ions

M

max

67MeVfm 3 (C ) 1.85 B

1/ 2

M

(2)

The analogous formula for the OV limit was obtained by Haensel et al. (1986) as a result of numerical calculat ions including the equat ion of state (1). From this paper and also fro m the calculat ion by Alcock et al. (1986) it fo llows that the basic parameter which determine s the value of masses and radii o f such dense (>
nucl)

and co mpact objects - quark

configurations - is the value B. Our calculat ions have confirmed the conclusion that the crucial factor is the cho ice o f the macroscopic densit y value QGP-bag. In other words, equation of state (1), in which quarks are considered in the limit as almo st free noninteract ing massless part icles at the calculat ion o f the hydrostatics o f objects wit h such high densit ies, gives approximately the same values o f the 'observed' parameters as the equation o f state does, allowing for a finite value of the constant of co lour interaction and nonzero mass o f s-quark. As it will be seen from the fo llowing, this circumstance can be direct ly interpreted in the bounds o f consequences of QCD and macroscopic properties o f QGP. The cho ice o f 4B/c2 densit y on the surface (PQ = 0) of the macroscopic quark bag can be determined fro m the fo llowing reasons. The upper value of the constant B is determined by the condit ion of the self-connection of quark (strange) matter at zero pressure P, formulated by Witten (1984). This condit ion demands that the corresponding energy per baryo n at P = 0 should be less than energy per baryo n for the most stable non-strange matter - crystal iron: i.e.,
QG P

>

nu cl

at the surface of the

4


4B B EQ ( P 0) 860.6MeV 3 ns 67MeVfm

1/ 4

930.4MeV (56 Fe)

(3)

Hence, for the B constant, we obtain that B 91.5 MeV fm-3, or for macroscopic densit y of plasma on the surface of the QGP-bag, we obtain
QGP

2.3

nucl.

It means that at a big leap

of densit y on the bag surface so me hadron configuration would be energet ically more preferable than QGP. The lower value of B is determined by the fact that with decreasing dimensio ns and mass of the bag (A branch in Figure 1) we shall co me ult imately to a model o f bags (MIT) or to a 'macroscopic' configuration with baryon number A 100 (consist ing most ly o f u- and dquarks?). Thus, the masses of macroscopic quark configurat ions must be connected wit h hadrons mass spectra by the (semi-empiric) relat ion (Chodos et al., 1974) B B
MIT

= 0.13 | V | 67MeVfm-3,

(4)

where | V | is the energy densit y o f QCD-vacuum equal to 0.5 GeV fm-3 (Novikov et al., 1981). Thus the densit y at the surface of the macroscopic quark bag is so mewhere within t he limits 2.3
nucl



QGP(PQ

= 0) 1.7

nuol.

(5)

Accordingly, maximum mass o f purely quark configuration must be in the limits 1.58 M Mmax(C) 1.85 M. This is another consequence of the calculat ions o f the kind (Haensel etal, 1986; üvergÅrd and üstgaard, 1991): namely, the applicat ion of the modern pheno meno logy of strong interactions reduces considerably t he value o f the OV-limit. In the old pheno meno logy with Yukawa's potential and the exchange o f vector mesons, where the equation of the t ype of P = is used as a limit equation o f state at /c2
nucl,

the

corresponding value of the OV-limit is more than 3 M (Rhoad-es and Rumni, 1974; Shapiro and Teukolsky, 1983). To understand why, in GR, only strange stars are possible it is necessary to apply to the profiles o f (energy) densit y the corresponding hydrostatically balanced quark configurations. In Figure 2 the behaviour of (energy) densit y is shown inside the quark bag as dependent on the distance from the center of a spherically-symmetric configuration. This densit y profile corresponds on the OV-limit - i.e., this is (r)/c2 for the last stable hydrostatically balanced configuration which can exist in nature, if the equation of state (1) for QGP is true and, of course, if GR is true. Thus, GR, together with the limit equat ion o f state (1), imposes the limitation on the maximum achievable densit y QGP (densit y in the center): 5




GR

10

nucl

28 x 1014 g cm-3

(6)

and, consequent ly, on numerical value of the OV-limit.

Fig. 2. (Energy) density profile for the last stable (OV-limit) spherically-symmetric configuration corresponding to point C in Figure 1. The distance r from the center of QGP-bag is measured in km, total density in g cm-3, B = 67 MeV fm -3.

All other stable configurat ions on the basis o f Equation (1), lying to the right of point C, are even more ho mogeneous. In the limit, as was ment ioned above, at r 0 (branch A in Figure 1) we deal ult imately wit h the model o f MIT-bag with an abso lutely homogeneous profile (r) at baryo n number A 100. We emphasize here once more that purely quark configurations, corresponding to points on the curve fro m A to C in Figure 1, are the most ho mogeneous ones fro m possible stable compact configurat ions corresponding to OVequations (Alcock et al., 1986). After all, we can calculate the profile (r)/c2 o f the utmost inhomogeneous hydrostatically balanced configuration corresponding to point D in Figure 1. This profile is shown in Figure 3. The densit y in the center of such an object tends to infinit y and falls, with r increase, very clo se to a law (r) ~ r-2 . But such configurat ions, according to GR, are never realized in nature since they are most ly unstable wit h respect to small radial perturbat ions (Zel'dovich and Novikov, 1971) and 6


during the t ime o f the order of R/c they must collapse into black ho les. But nevertheless we consider the situat ion in more detail, since it is such a configurat ion (forbidden in GR with a big 'margin') which can be realized as a stable stationary state in the bounds o f the dyna mic alternat ive to GR - in gravidynamics.

Fig. 3. (Energy) density profile for the utmost inhomogeneous hydrostatically-balanced configuration with equation of QGP state (1) calculated in the bounds of GR. The density on the QGP-bag surface is taken to be equal to
QGP

= 1.7 nucl (B = 67 MeV fm -3)·

For macroscopic densit y at the bag surface, in accordance with condit ions (5), we choose for definiteness some average value
QGP(PQ

= 0) = 2

nucl

5.6 x 1014 g cm-3.

(7)

In accordance wit h QCD (the theory o f co lour interactions) and also in accordance wit h what is known about expected properties o f QGP (Emel'yanov et al., 1990; Co llins and Perry, 1975), it can be considered that the bag surface consists mainly o f the lightest u- and d-quarks which co me first in the state of deconfinement at such a gigant ic macroscopic densit y. It can be interpreted (analogously to usual plasma) that the free path length l (relat ive to color interact ions) o f u- and d-quarks in such a plasma with
Q GP

beco mes either equal or even

much greater than lc 1 fm; lc being the characterist ic radius of strong interaction or the radius o f confinement. In the end, l can simply become a macroscopic value co mparable wit h the bag dimensio n. These are just the quarks for which at
QGP

the equation of gas state of

asymptotically free quarks (1) turns out to be true, since for them we can assign mu md 0 7


(at such characterist ic transmissio ns o f mo mentum in QGP which correspond to particle interact ions in such dense matter). A heavier s-quark (ms 200 MeV) at as a result of equilibrium reactions of the type u+du+s (see the review by Haensel, 1987). Color interaction of this ('more non-relat ivist ic', than u and d) quark is st ill rather strong ( > 0.45) at
QGP QGP

exists in plasma as heavy 'admixture'

and that is why the s-quarks must have

smaller mean-free-path in QGP than 'massless' u- and d-quarks. So one can say that a heavier s-quark arising at weak interactions at mainly in the vo lume o f the confinement l
3 c QGP

is st ill

1 fm3. Densit y boundary over which the
QGP.

deconfinement or almost total 'defreezing' of s-quark occurs is rather close to 'defreezing' of j-quark occurs when the macroscopic densit y beco mes greater than
QGP

The

+ msc2/l3c 9.1 x 1014 g cm-3,

(8)

in accordance wit h the interpretation of chemical potential, as the change o f energy densit y at the unit change o f concentration of particles o f a given kind. In other words, at > s, in every 'cell' o f 1 fm3 vo lume, there is already more than one s-quark and it beco mes just as 't ight' for them as it was at
QGP

for u- and d-quarks. Yhen the co lour interaction of s-

quarks must beco me weaker ( <0.45). Accordingly, at QGP densit ies greater than s, squarks also can be considered relat ivist ic (ms 0). As a matter of fact, the confirmat ion o f such a logic o f'defreezing' o f s- (and heavier) quarks is the fact noted in the quoted paper by Haensel et al. (1986) and mentioned before: namely, that the determining parameter of the equation o f state for QGP is the value B or the densit y value at which QGP is formed. The direct calculat ion o f the hydrostatically balanced configuration wit h the limit equat ion of state (1), not only reproduces the results of that paper, where the authors, besides the B, allow also for c = 0.45 and msc2 = 200 MeV, but varying 4B/c2 value in (1), one can reproduce, with an acceptable precisio n, almost all the results o f calculat ions by Benvenuto and Horvath (1989) wit h other parameters c and ms. Hence, we conclude that indeed the densit ies massless part icle) must be close
QGP

and s (at which -quark can also be considered a free

QG P

~ s. I.e., Equation (1) can actually be applied wit hout

paying attention to the essent ial difference in masses between u-, d-, and s-quarks. Or, in other words, the fo llowing interpretation of the results of all the ment ioned calculat ions is possible. At
QGP

s , the heavier quark is present in QGP as a 'heavy

admixture' and does not distort strongly the limit equation (1). At > s heavy quarks, as well 8


as lighter ones, are also in the state of deconfinement and so here Equat ion (1) beco mes applicable again. From what is said above, it beco mes clear why in GR only strange stars are possible. As has been noted, the last stable hydrostatically-balanced configuration (see Figure 2) imposes limitation on densit y (6). Only then can one speak of strange stars, following the same logic, the deconfinement ('defreezing') of even heavier c-quark (mcc2 =1.4 GeV) must occur at densit ies greater than the limit densit y GR. In every 'cell' o f l3c volume there can be at least one c-quark if the macroscopic densit y turns greater than c s + msc2/l3c 33.9 x 1014g cm-3. (9)

In that case the color interaction beco mes so weak that the corresponding c beco mes even less. Therefore, one can consider for c-quark, that mc 0. Of course, for the configuration in Figure 2, which is st ill attainable in GR, so me admixture o f c-quarks can appear in plasma near the center as a result of some (d + u d + c) weak processes analogously wit h the admixture of s-quarks at < s. Returning to the utmost inho mogeneous configuration wit h the densit y profile in Figure 3, at > s in plasma there must be a lot of relativist ic charmed quarks (mc 0) and like the case o f s-quark the properties o f QGP are described ult imately by limit equat ion (1). But all the configurat ions wit h the densit y in the center greater than pc, as well as the ut most inho mogeneous configuration, are to the left of the OV limit in M/R curve (point C in Figure 1). Hence, if GR remains true even in such a strong gravitat ional fie ld then in nature there exist only strange stars as maximum co mpact statio nary objects with the surface (i.e., with a finit e z) but not with the event horizon. ('Charmed stars' do not exist in GR.) The utmost inho mogeneous configurat ion (point D in Figure 1) in which all quark generat ions would be 'defrozen' is not realized either, according to GR ­ it is 'eaten up' by black ho les. By reasoning of this sect ion, we tried to make more concrete analogous speculations expressed by Alcock et al. (1986), in connection with their use o f the same equation o f state, independent of the number o f particle flavors. Since if instead of the full expressio ns we use their limit (1), then, strict ly speaking, everything that is said about quark masses, quark flavors, chemical potentials, number densit y o f different flavors of quarks, densit y () at which 'the appearance' of the next flavor occurs and even the use of so me finite value c , is now only interpretation in the bounds o f perturbat ion QCD. The results o f calculat ion o f hydrostatically equilibrium configurat ions are determined in the end only by the value 4B/c2 or the energy densit y at which the limit (1) can be used. Certainly, one should try to understand why the results o f such calculat ions differ in less 9


than 4% fro m the result s of calculat ions wit h the help o f full expressio ns. That is why where we use the notion of 'defreezing' of the next flavor, meaning first of all the fact that for this 'new' flavor, at the given densit y, the condit ion l > lc begins to be fulfilled. Besides, in so-called 'full expressio ns' the used values of quark masses and the value of c are fixed. At the same time it is known that quark masses are measured at definite effect ive transferred mo mentum Q. Numerical values which are usually used concern the distances between the quarks of the order of 10
-14

cm, that in our case means a definite macroscopic

densit y . The less the distance between interacting quarks, i.e., the greater macroscopic densit y, the less quark masses can be. But this simply means that there must exist a dependence of c on (more details will be in the next section). In other words, mq(c) = mq[c()] = mq(); and then at calculations of hydrostatically equilibrium configurat ions only macroscopic densit y 4B/c2 becomes really the basic parameter. Then even in the case of the limit (1) it is not absolutely necessary to require that, for all 's, te fulfillment of the equalit y c = 0. For u- and d-quarks defreezed at >4B/c2 at a given p, we can put mu,d()0, then c()0. To have a possibilit y to use the perturbation theory the value c(2c/<1) must be sufficient ly small. At densit ies < s (8), the heavier quark s is present in plasma, but for it l is st ill rather small (llc) for its contribution into pressure could change the equation of state essent ially. At densit ies > s in the full equat ions besides decrease of c(), the value ms may also tend to zero, that leads in turn to the disappearance o f corresponding ('massive') terms in the full expressio ns. In the end we shall be even closer to the limit (1). All this reasoning could be illustrated by corresponding calculat ions, but the matter is that nobody know today the precise form o f the dependence o f c() and mq() and we ca n only guess (see the next sect ion) how c will behave at high and superhigh ( >>
nucl)

macroscopic densit ies. However, it may be the essence o f differences between quark configuration calculat ion results in the bounds o f asymptotic MIT bag model, i.e., with the help o f Equation (1), and ones in the bounds o f the Perturbative QCD model (üvergÅrd and üstgaard, 1991). In conclusio n o f this sect ion which has been dedicated to the pure quark configurations in GR. We emphasize once more that the co mpact object consist ing of strange matter with the equation o f state very close to (1) is the last opportunit y o f stable state after which only black ho les with z fo llow.

10


3. Quark-Gluon Plasma in Gravidynamics 'Quark stars', considered here as objects consist ing of QGP including all possible flavors (u, d, s, c, b, t,...) of quarks, turn out to be unstable in GR. One can say that such objects must not exist in nature according to all versio ns of GR as well in which there are 'frozen stars'. A 'quark star' as a limiting (in several senses) stable object with the total mass MQ with the QGP-bag surface (z ) of the radius o f R
QO P

GMQ/c2 (inside the Schwartzschild sphere

according to GR) can exist if we adhere to the dynamic, totally non-metric description o f gravitat ional interact ion. In gravidynamics - GD (unlike geo metrodynamics: GR) - the profile o f the total (energy) densit y (r)/c2 o f analogous quark configuration does not terminate in a vacuum (see Figure 4). Around the macroscopic quark bag (QGP-bag) a fur-coat exists - the 'gas' o f virtua l gravitons whose energy densit y has to be allowed for in the equation of state. Here the total mass MQ of the object entering the determinat ion of R words, the mass is determined by Newtonian rules. If
QG P

GMQ/c2 should
QG P

be found in 'lo ng-wave limit ' (like classic charge of electron); i.e., at r >> R

or, in other

Fig. 4. Density profile for a quark star in GD (solid line). Fat rectangle shows 'the background' created by gluons (see the text) distributed homogeneousl y (?) in the bag with RQ
GP

10 km. 'Vacuum' around the bag is

filled by a 'gas' of virtual gravitons (the fur-coat) with energy density (r). The densities are indicates at which 'the defreezing' of s, c,b,t, ... quarks occurs. The arrow indicates the density at which the perturbative QCD vacuum must be t otally restored (cm 0.7). The dotted line represents the density profile of analogous quark configuration in GR for 4B: c2 = 2
nucl

00

(B = 78.5 MeV fm -3).

11


the object is described by 4-potential (4) fro m the paper by Sokolov (1992b) wit h the radius of the bag R = R
2 QGP,

then MQ consists by half on the 'coat' mass. Thus for the total energ y

MQc of the quark star in GD one can write ½ MQc2 (the bag with R = R
QGP)

+ ½ MQc2 (the 'coat' in vacuum) = MQc2 .

Thus, the mass o f the who le configurat ion is determined by integration fro m its center (r = 0) and up to r = o. (This is an essent ial diffe rence from the definit io n o f mass in GR.) Strict ly speaking, this is the determinat ion of mass of any objects in GD. By force of this fact in GD there is no event horizon that is ult imately the result of total refusal o f geo metric pheno meno logy constituting the base of GR. The bag surface can undergo some unselected sphere r = 2GMQ/c2 as a result of the relativistic collapse, but the whole gravitat ing object, including its gravitat ional 'at mosphere' (or coat), can never be found under this 'horizon'. For the object, with limit achievable parameters, which a co ld quark star if R
2 QG P

=

2GMQ/c 10 km seems to be in GD, the equat ion o f state inside and outside the bag is the same: i.e., P= 1/3 . (10)

Inside the bag the total pressure is a sum o f two ('part ial') pressures. First of all it is the pressure PQ = 1/3 ( ­ 4B). Strict ly speaking, this formula gives the pressure of degenerate Fermi-gas of free and massless quarks. Then the contribut ion o f gluons (with so me admixture of gravitons?) in the limit under invest igat ion is determined by the equation PG = 1/3 (4B). (11)

Hereafter, we shall proceed from the assumption which is apparent ly true in the case of huge ( >>
nu cl)

macroscopic densit ies in question. We consider that gluons which are

almost free and almost non-interact ing with each other (at the total densit y increasing to the center) can be distributed inside the bag wit h constant and posit ive densit y 4B. The interact ion o f gluons beco mes essent ial far from the center, maybe even near the very 'wall' of the macroscopic bag, where (as a result of that) the energy-mo mentum tensor trace o f massless gluon field beco mes non-zero... We do not know so far at what distance fro m the bag wall it will occur, that is why we choose here 'the simplest' limit case (11). As a result of it, in the sphere r = R energy densit y in the bag wit h R
QG P QG P

the sum of pressures is equal to (10). The total

10 km decreases from the center according to the 12


equation

(r )

4B 2 RQGP r c2

2

(12)

(Sokolov, 1991, 1992b), if the equation o f state inside the bag is taken in form (1). The QGPbag in GD turns out to be connected only by colour forces ('wall' o f bag), the gravitat ion inside the bag is 'switched off for such a limiting object which the quark star is. It can be assumed that in GD, in that case, QGP is in a totally (at R = R
QGP)

stress-free, self-bound

state when there are no forces binding the bag besides co lour ones. Then the distribution e(r) is here maximum inho mogeneous. It differs radically fro m an analogous case in GR at totally ho mogeneous distribut ion of densit y (d/dr0) when M0 also. Outside the bag (in 'vacuum') 'a gas' st ill remains fro m virtual gravitons with the same equation of state (10). Positive energy density 00() of the gravitational field falls here fro m the value 45 (at the boundary o f the two 'mediums': QGP - 'vacuum') according to the law

00 ( r )

4B 4 RQGP r c2

4

,

(13)

which is connected with the fact that at distances r GMQ/c2 the self-act ion of gravitons arises (scalar and tensor gravitons, tensor and tensor ones; see Sokolov, 1990, 1991, 1992b). Of course, it is not excluded that in QCD-theory which would be more correct than the bags model, the contribut ion of quarks and gluons in the total energy densit y (12) inside the QGP-bag could be distributed in an abso lutely different manner than we assume here. But (most probable) these are such contributions of fermio n and boson co mponents at the huge (/c2 >> nucl) densit y which increases towards the center ( ~ r -2), that as a result only suc h limit equation of state (10) conforming to the rest of physics turn out to be true. Strict ly speaking, fro m the very beginning, the question in (1) was only on quarks since here there is no explicit contribut ion into pressure which corresponds to gluons. To make sure of that it is sufficient to look at the full equat ion of state (Alcock et al., 1986) before the pass to the limit (1). Thus, in the case o f quark configurations in GR one cannot say about QGP: in such a plasma there are simply no gluons, if using the limit equat ion (1) as the equation o f state. In GD we do deal wit h QGP since in GD we try to account explicitly for the contribution of bosons, at least in the case of asymptotically free gluons wit h the equat ion o f state (11). The last can be justified apparent ly only in the case o f macroscopic densit y of QGP (12) increasing to the center, when the 'constant' of color interaction decreases sufficient ly quickly with the increase o f fro m the wall of the bag towards its center. Below, in this sect ion, it will be said that it is possible, in principle, at such a profile of (r) as (12). 13


To obtain the total (observable) mass of such a quark configurat ion (unlike what was in GR) at the integration of /c2 it is necessary to allow for the energy
00

of the gravitat ional

field itself. As a result, the mass of a quark star in GD can be expressed in terms o f the energy densit y value at the boundary between QGP and 'vacuum' as
2 nucl M Q 6.64M ; 2 4B / c
1/ 2

(14)

and the same restrict ions (5), which were ment ioned above, fix the mass and the radius o f the cold quark star in GD in the limit s 6.21 M MQ 7.25 M , 9.16 km R
QGP

10.69 km .

(15)

The lowest value of the total mass of the configuration and, accordingly, the lowest value of the QGP bag radius (as was said above) fo llow fro m the condit io n that at the densit y (12) on the bag surface equals /c2 2.3
nucl

(3,5). This surface consists from strange self-

connected (i.e., stable at PQ = 0) matter. Since increases towards the center of the bag and, consequent ly, all other kinds o f quarks beco me defrozen, then in GD (unlike what was in GR) it is necessary to speak not about a strange star, but about a cold quark star with the strange surface, if the densit y on this surface does not exceed 2.3pnucl. If we assume that so me other condit ions o f the type o f (3) are possible but at PQ 0, when at bigger densit ies on the QGP surface, already more massive quarks than s-quark beco me 'defrozen', then according to (14) the masses o f corresponding meta-stable quark configurations will be less than 6M down to the values M 1.4 M. But in any case, Witten's condit ion (3) at PQ = 0 fixes so me maximum mass ( > 6 M) of the most stable limit quark configurat ion in GD. Whether such a limit object can exist allowing for astrophysica l reasons is another quest ion. But such a limit can be a consequence in principle of GD and QCD, if GD gives a more or less correct descript ion of the strong gravitation. Thus, as fo llows fro m a brief review o f properties of co mpact objects, co llapsars (Sokolov, 1991) - from masses and radii o f such objects in binary systems such as Cyg X-l, A0620-00, LMC X-l, LMC X-3 - can readily be considered as 'candidates' into quark stars of GD with the strange surface. Some properties of the quark star which could lead to corresponding observat ional manifestations are dis cussed in more detail in t he quoted review. But apparently, the basic observational consequence confirming the versio n of GD u QCD suggested here could be indeed the existence of a selected mass value of co llapsars (or candidate into black ho les) 6 - 7 M (15). Since in GR there is not preferable mass values o f black ho les for all masses of 'candidates' greater than OV-limit, then one will have to invent some astrophysical (e.g., evo lutionary) arguments explaining the mass o f a 't ypical' co llapsar 14


in these close binary systems. In what fo llows, we shall try to imagine how the 'running' constant c o f the strong interact ion could depend on the parameters of the QGP bag under considerat ion. And, most importantly, what could be the dependence of this value on macroscopic densit y so that we could make at least the rough agreement between the asymptotic MIT bag model and the interpretation o f results with the help o f the perturbat ive QCD which we ment ion here rather often? In Figure 4 we marked macroscopic densit ies above which the corresponding quarks are already in the state of deconfinement. The total densit y (r) increases deep into the bag and, consequent ly at the approach to its center the quarks beco me co mpressed st ill t ighter. In other words, when the macroscopic densit y exceeds a certain level, the corresponding quarks are situated relat ive to each other and interact with each other at distances l less than l 10
-13

cm. The same can be said about any particle of QGP. In particular, the known formula for the 'running' constant c of color interaction can be written in the form

c ( l 2 )

12 , (33 2n f ) ln(lc2 / l 2 )

(16)

where nf is the number o f 'defrozen' quark flavors; lc, the radius o f confinement; and l, the distance between two neighbouring strongly-interacting co lour particles (l l; l being the free-path length of quarks in QGP which can be much greater than lc). For example, if one demands that u- or d-quarks (nf = 2) could be considered almost free ( 0.45, see Haensel et al., 1986), it is necessary that the mean distances l, to which the quarks must be 'co mpressed' in such QGP would be about 0.25lc ~ 10-14 cm. One may think (see the previous sect ion) that these are just the condit io ns (l < lc , c < 1) that are realized in plasma first for u- and d-quarks at the macroscopic densit y /c2, greater than the densit y at the boundary o f the QGP-bag (i.e., greater than the densit y o f phase transit io n in the QGP state). If we apply the idea of coupling constants depending on densit y (which is widely used in 'standard' cosmo logy o f Big Bang for our case of macroscopic bag) one can try to parameterize the macroscopic constant of strong interaction by the equat ion

cm ( )

12 . (33 2n f ) ln( / 4 B / c 2 )

(17)

We emphasize here that p is macroscopic and, therefore, a certain mean densit y whic h does not exclude that microscopic fluctuation o f densit y in vo lumes o f ~ (l)3 can, generally speaking, exceed p dozens o f times. In part icular, if we remember here the attempts of getting 15


'hot' QGP on colliders in vo lumes o f the order of several fm3, then the corresponding macroscopic densit y (here the macroscopic vo lumes of averaging ~ 1 cm3 are also meant) will be simply zero. In that case there is no quest ion about any gravitational effects which beco me essent ial only at big macroscopic masses. Here we mean co ld catalyzed self-connected matter which must be the source of the gravitat ional field. On the other hand, the strong interaction which is realized here in big (macroscopic) vo lumes has evident ly the character of macroscopic co lor interaction, something like 'co lor gravitat ion' inside a huge self-connected QPG-bag. Formula (17) could describe just such a macroscopic interact ion, when the exchange by gluons between the elements of vo lume inside such a bag, situated at macroscopic distances relat ive to each other, beco mes essent ial. Thus here, in our case, it is necessary to speak already about QGP in astrophysica l condit ions which differ considerably fro m corresponding condit io ns available in experiments on the Earth. Of course, formula (17) can be considered still only as an attempt of so me rude extrapolat ion in the regio n o f superhigh ( >>
nu cl)

densit ies. In particular, the calculat ion o f

l - mean distance between quarks at a given >4B/c2, the mean free-path 1 and also other parameters of QGP will demand further study of microscopic properties of such plasma as it is made for ordinary plasma. Here we are to meet the same problems that exist in QCD and in quark bag models (in particular). Especially since the notion of the macroscopic QGP-bag ca n be used directly at ~ r -2. Really, by use o f formula (12), the macroscopic constant cm of colour interaction (of macroscopic vo lume elements situated at macroscopic distances fro m each other) inside the QGP-bag can be expressed in terms o f r - the distance from the bag center ­ as

cm ( r 2 )

12 . 2 (33 2n f ) ln( RQGP / r 2 )
QG P

(18) radius is provided here in the
c

Then, the color confinement in the gigant ic bag of R

same way as it was in the model o f the homogeneous (in densit y) MIT-bag wit h the 'radius' l

in formula (16). In particular, the values c and cm must be in approximately the same relat ion as the value | v | - the energy densit y of QCD-vacuum (obtained fro m the analysis o f sum laws) and the value o f the constant


(which is connected with hadron mass spectrum). It

naturally fo llows fro m the fact that in (17) we actually use directly the model of quark bags. If, finally, so me definite value o f densit y at the boundary of the QGP-bag in accordance with the restrict ions (5) is chosen, then in evaluat ion of value a
cm

one can use the equation

16


cm ( )

12 (33 2n f ) ln( / 2

nucl

)

.

(19)

Equations (18) and (19) should be considered here only as an attempt to make the interpretation by means o f perturbative QCD, to which we resorted in this and previous sect ions, agree with the asymptotic MIT bag mo del. Of course, such equations can be an approximat ion as the 'init ial' approximated equat ion (16) itself. The most probable, the more precise dependence acm(), will lead to an even more quick decrease of co lor forces in the direct ion from the bag wall towards its center. In the end we shall have to use only limit (1) for the co ld, catalyzed QGP. It may be, at this limit, the difference between MIT-bag and QCD approaches will decrease or disappear alt ogether. We can, therefore, speak alread y about a classical limit of QCD inside the macroscopic bag. But one way or another, at considerat ion o f macroscopic QGP (i.e., QGP in astrophysical condit io ns) and the use o f QCD, an abso lutely definite dependence for ac() or acm() will be needed for sure. Below we show by corresponding est imat ions the fact that the approximate formulae (19) suggested above does not contradict considerably to 'the standards' of QCD. If we consider that the perturbative vacuum is restored at such macroscopic densities
QCD

that the relat ion o f
MIT,

Q CD

to the densit y on the bag border is the same as the relat ion (4)

between | v | and B



QCD

2

nucl

V 2 nucl 43.07 1014 g cm BMIT 0.13

3

(20)

(when four kinds o f quarks are defrozen, see Figure 4), then Equation (19) yields cm 0.74 at =
QCD

. It is close indeed to the QCD-value Q

CD

c

(1 GeV) = 0.7. At the same time, for
QCD

strange matter (nf= 3) at macroscopic densit ies, when 1515 g cm-3 <

, the macroscopic
MIT c

constant of colour interaction turns out to be equal to cm 2.4. It is close to the value o f 1975).

= 2.2 determining the mass splitt ing of hadron multiplets (Bogolyubov, 1968; De Grand et al.,

Thus both formulae (19), and from estimat ion (20), it fo llows that the QCD-vacuum (perturbative vacuum) in strange matter is not restored yet (cm 0.87), which agrees wit h calculat ions in the bounds of GR carried out by Kondrat yuk et al. (1990). As follows fro m Figure 4 and formulae (19), (20) the perturbative vacuum is totally restored in the interior of a QPG-bag o f a quark star in GD at the depth RQ
GP

­ r 7 km, i.e., in the case of the most

stable (limit) quark configurat ion with the bag whose surface consists of strange matter.

17


4. Conclusions From formula (19) it fo llows, in particular, that in the very center of the QGP-bag wit h RQ
GP

10 km (i.e., for r = 10

-13

cm), the macroscopic constant of color forces is only about 3

constants of electromagnetic interact ion. The densit y p here must be about 5.4 x 1052 g cm-3, and the mass (in r = 1 fm sphere) must equal 7 x 1014 g. Of course, at mutual distances between QGP particles much less than lc (10
-17

cm) and, correspondingly, at densit ies >> 101

6

g cm-3 the constants of all the three fundamental interactions (strong, weak, and electromagnet ic) must in the end beco me indist inguishable from each other. Thus, in the interiors o f a GD quark star - a stationary stable object wit h densit y increasing to the center according to the law (r)~r
-2

- just the phys ical condit io ns can be

realized under which all the interactions unite in one fundamental interact ion. For sure, the constants of weak and electromagnetic interact ions inside the QGP-bag of the quark star can be also expressed in terms of macroscopic densit ies. And it means that the ideas of Grand Unification of all interact ions could be tested without resorting to cosmo logy of Big Bang, but studying the same physics of superhigh densities (or 'cosmo microphysics'), observing compact objects (bright X-sources, -ray bursts, remnants of supernova explosio ns, etc.) of stellar masses. Of course, it should be admitted here that GR describes erroneously the strong gravitat ional field of such objects. But ult imately, if we abandon the convict ion {a priori) of absolute correctness of GR and do not forget that only a sufficiently complete experimenta l (observat ional) study o f strong gravitational fie lds can affirm or refute this conviction, then all the preceding discussion can be considered as a possible alternative to black ho les o f GR. We recognize that so met imes statements of certain things look schematic here. The matter is that many QGP properties and QGP itself is only a hypothesis, although a hypothesis which fo llows naturally fro m the experiments on colliders and from the theory o f quarks and leptons. But even now from all what has been said, it is clear that a consistent direct allowance for localizable posit ive (like in the case of all other gauge fields) energy o f gravitat ional field changes completely the co llapsar physics. In particular, one of the observat ional (experimental) arguments in favour of such or similar physics would be the existence o f so me selected value for the co llapsar mass (or for 'the candidates in black ho les' of GR). Proceeding fro m the theoretical scheme developed here, we consider that the collapsar - a compact object wit h its mass exceeding, certainly, the pulsar mass (or OV-limit in GR) - can be ident ified wit h the (limit) cold quark configuration in GD whose mass is 6.27.2M. 18


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