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Astrophysics and Space Science 197: 179-212, 1992. © 1992 Kluwer Academic Publishers. Printed in Belgium.

THE PROPERTIES OF THE STRONG STATIC FIELD OF A COLLAPSAR IN GRAVIDYNAMICS
VLADIMIR V. SOKOLOV Department of Relativistic Astrophysics of the Special Astrophysical Observatory of the Academy of Sciences of Russia, Stravropol Territory, Russia

(Recei ved 2 December, 1991) Abstract. In a totally nonmetric model of the gravitational interaction theory (gravidynamics) a compact object strong field (a collapsar) - an analogue to the black hole in General Relativity (GR) - is investigated. In the case of extremely strong (for gravidynamics) collapsar field a region filled by matter (a bag) must have a radius equal to r
*

= GM/c2
mass is contained in the bag, the other one of its total energy (Mc2) is distributed in the form of a 'coat' in space around the bag, i.e., in the form of a continuous medium (as a relativistic 'gas') of virtual gravitons. The object must have a surface (the bag surface) with absolutel y definite physical properties. The potential of such a sur face is finite (* = -c2/2) and mass a particle in a bound state on the bag surface is t wo times less than mass of the same particle in a totally free state. The bag surface can undergo periodic oscillations (pulsations) with the period GM/c3 3 10-5 s. An energy density inside the bag with the extremely strong gravitational field or with an extremely dense 'coat' (shrouding the bag) is determined by gravitation theory constants only and depends on the distance to the bag center r in the following wa y: (r) = (c4/8G)r -2. In this case the bag matter is most probabl y in the state of quark-gluon plasma.

1. Introduction This paper continues the study of co llapsar physical properties in gravidynamics (GD), started in papers (Sokolov, 1992a, b; hereafter referred to as [P1] and [P2]). My purpose here is to elucidate such properties o f the co llapsar which must lead ult imately to abso lutely definite experimental tests allowing us dist inguishing the co llapsar in GD fro m the co llapsar ('black hole') in GR. Like previous papers, here I emphasize the connection o f such objects problem in GD with the problem o f localizat ion and the sign o f gravitat ional field energy. In [P2] the fo llowing was used as an init ial idea of the co llapsar fro m the po int of view of the dynamic field theories: The GD co llapsar is a bound, spherically-symmetric object, a 1


'bag' filled by particles and fields together with the gravitational field surrounding this bag. Particles, bound in such a bag, wit h nonzero (may be) rest masses, mo ve in such a way t hat both the bag itself and the field around it look like so me stationary state in which the continuous exchange between the bag and the surrounding fie ld occurs. The processes of the gravitat ional self-act ion pla y an essent ial and even determining ro le near the surface o f the bag wit h dimensio n GM/c2. Here the quantit y M determines first of all the total energ y (Mc2) of the whole configuration ('the field + the bag = the collapsar'), i.e., the co llapsar total mass. At a given and constant value Mc2 and wit h a constant energy in the bag, such an exchange must lead to settled/steady-state (definite and constant) values o f energymo mentum-tensio ns for the static field in vacuum around. For reasons ment ioned later over and over again, we shall deal only with macroscopic (non-quantum) objects of the stellar mass order or greater. For such bags the stationar y exchange, the bag the field, means the existence in vacuum of the static and sphericallysymmetric gravitat ional field (as for the field of a point particle with the mass M), corresponding to a static (more exact ly, stationary) and a spherically-symmetric matter distribut ion inside the bag. Unlike GR, in GD there exists a possibilit y (in principle) for transit ion into such a stationary state with the spherically-symmetric and compact bag with the dimensio n o f only several kilo meters for stellar mass collapsars. In GD such a transit ion - a sphericallysymmetric co llapse - is accompanied by an energy loss o f the co llapsing body in the form o f scalar radiat ion or scalar gravitons. Wit h all this, a changing o f the energy-mo mentum tensor trace of the who le co llapsing system (the 4-scalar of the who le configuration) is the source of scalar waves. In this connect ion, one can pick out and then sharply dist inguish between two problems concerning co llapse o f a body: The first one is the problem o f the co llapse process per se. I.e., this is the problem o f the descript ion of a non-stationary process (explosio n) acco mpanied by an essent ial loss o f init ia l total energy o f a co llapsing system in the form o f gravitat ional waves. In particular, in who le such a problem will require also the allowing for effects of the braking of falling matter by radiat ion arising at the collapse. The second problem may be reduced as a matter of fact to the just ification of the ver y possibilit y o f a steady (stationary) state existence, when one can neglect the radiat ion from the system as much as at least its influence on the value of total energy (Mc2) o f the co llapsar beco me insignificant. In this paper and in previous ones [P1, P2] the question is already basically on a result 2


of the collapse - on the collapsar and its properties. In particular, we want to prove here that the collapsar, and, more correctly, the bag, may have also a so lid surface. More exact ly, it means that the outer border of the bag may, for some reason or other, be in the stable equilibrium state; i.e., it (the border) is static, for example, for the bag radius R = r GM/c2. (That is impossible, in principle, in GR at r = 2r * already.) Suppose that the co llapsar wit h such a bag radius R = r * really exists, i.e., it is stable. Then this object in GD is defined first of all by its constant 4-scalar - the trace of energymo mentum tensor (EMT) of the whole configuration (because everything had co llapsed and stabilized already). Ult imately, it means (as it was mentioned above) that the collapsar has the mass M. I am going to explain what I mean more exact ly below. The point is that (firstly) the field of a 'po int' with the mass M at r >> r * is spherically symmetric and by force o f correspondence principle this field is almost Newtonian one, i.e., it is determined by the same M. As a matter of fact (as [ P 1 ] has shown), Newtonian field o f a massive 'point' consists by half of the field energy densit y
00
*



in vacuum and it invo lves the

repulsio n scalar field wit h the spherically-symmetric potential

(r )

fM , 8 a r

determined by the quantit y M also. (The notation is anywhere the same as in [PI, P2].) The scalar (r) is a so lut ion o f Poisson's equat ion for the massive 'po int' at the center (at the origin of coordinate):
f T, 2ac 2

(1)

where one must take the quant ity T = Mc2(r) (the trace of EMT of the who le configuration) as a source. The correspondence principle and the spherical symmetry o f the field po int 'acted' here. But on the other hand (and secondly) the spherically-symmetrical repulsion potential (r) defined by the quantit y M, will determine as before the scalar component of the collapsar field up to the bag surface o f radius R = r * . The field o f itself does not contribute in the source ­ in the right-hand side o f spherically-symmetric equatio n (1) ­ by force of the condit io n ikik = 0 for the field EMT ik, i.e., of the massless gravitat ional field condit io n. That is why at least out of the bag, in vacuum, the trace of the collapsar EMT will be zero as before. And inside the bag, by force of Po isson's equat ion feature, T may be a funct ion only of r. Thus, for the 3


funct ion T(r) we have T(r) = 0 at r > R = r * and T(r) 0 at r R = r * (so, -funct ion became 'more definite'). Here the spherical symmetry and lack of the field mass (the zero-mass field condit ion) acted. All in all, one can say that correspondence principle (the spherical symmetry of massive 'point' field and the massless gravitational field condit ion) allow us present ing a 'half' (at least) of static vacuum fie ld of the collapsar as the scalar source T of two forms T = Mc2 (r) and T = T(r), (2b) (2a)

where in the first case (2a) one can take all r greater than r * , and in case (2b) T(r) 0 at r r * , i.e., inside the bag it is already a nonzero (posit ively defined) spherically-symmetric funct ion up to r = r * . Accordingly, (2a) and (2b) mean that the integratio n alo ng vo lume for T gives always the same result equal to Mc2 for any surfaces in vacuum surrounding the bag co mpletely. Or, in other words, one can say t hat if an object with the bag of radius o f R = r
*

is

stable, then at any distance fro m the bag in vacuum we deal wit h the object (collapsar) having a definite rest mass M (and total energy Mc2) at the usual Newtonian field of the po int with mass M at infinit y, i.e., at r >> GM/c2. I emphasize once more that here the quest ion is basically on co llapsar properties in vacuum, more exact ly out of the bag - the sphere filled by matter. All the time I say about the vacuum potential, vacuum gravitat ional field, excluding the bag itself. Certainly, it simplifies considerably (and makes deficient in so me degree) the collapsar problem, reducing it to the revealing of only gravitational properties of these objects. However, allowing for inevitabilit y of so me simplifications, one must say that the study of gravitat ional properties is first of all the study o f a matter behavior in the collapsar field. In particular, one can speak about the matter behavior on the bag surface that is determined by gravitat ional field of the collapsar near this surface. Though the most reliable conclusio ns of the paper concern the collapsar properties in vacuum, including the bag surface, one can (and must) go on, trying to understand how the bag itself is constructed from the po int of view of GD. (I repeat again: we exclude here the quantum size o f the bag.) Out of the sphere of radius R = r * the gravitational field remains the same, passing, in particular, to the spherical Newtonian field of po int mass M at r >> GM/c2. The matter (substance) distribut ion inside the bag must have spherical symmetry. 4


Elsewhere further (in the GD) I shall endeavour, in the st yle o f all modern theories o f dynamic field, to adhere to the idea that an 'elementary' po int GD object (at r >> GM/c2) is the collapsar indeed (at r >/~ GM/c2) and that it consists itself o f interacting (another 'fundamental') po int objects. I.e., we shall always consider, at least for macroscopic collapsars, that inside the sphere of radius R = r * there are bound particles of rest masses m*
a

0 interacting by means of so me massless fields. In accordance with that one can say that inside the bag there is matter with the equat ion of state not harder than p = /3 (Landau and Lifshitz, 1973). In other words (prior to possible vio lat ion of this requirement by quantum corrections generally speaking) the EMT trace of such an interacting particles system must correspond to the trace of the bound particles only. I.e., the spherically-symmetrical function T(r) in (2b) inside the bag (at r R = r * ) can always be presented as

T ( r ) * c

2

1 v2 / c

2

,

(3)

where * = m*a (r ­ ra) and v2 are nonzero functio ns (depending on r ) inside the bag only. Certainly, in principle, at sufficiently small and even quantum dimensions of the bags the EMT trace of massless fields ins ide the bag becomes nonzero for sure because of the allowance for quantum corrections. But we emphasize everywhere that here the question is on the objects wit h sufficient ly large (not less than several kilo meters and more) dimension o f the region filled by matter, where the corrections can still be terrifically small. Anyway, we shall endeavour to use condit ion (3) for the EMT trace of the collapsar unt il obvious contradict ions arise in physical properties o f the bag and the field around. (In such a situat io n the allowing for quantum effects inside the bag becomes inevitable.) Eventually, in the following sections we shall answer to questions arise already in [P1]: What is the radius o f the bag (in kilo meters) for the last stable state with (still) static field outside the bag? How much energy is contained in the bag and how much in surrounding field? What is the potential on the surface o f such a bag equal to? Do the collapsar properties depend on the value o f M in fact or not (like properties o f black ho les in GR). As we shall see later, all these questions can be reduced to one: has the GD collapsar a surface and what are its properties? In our previous paper [P2], supposing that the energy-mo mentum-tensio ns of both components of gravitat ion - tensor and scalar ones - are equal to each other in every point out of the bag, we obtained for the co llapsar field in vacuum t he 4-potential ik, whose nonzero components of the form

5




00



GM / f (1 1 r* ) , r 2r

(4)

11
where as before r * GM/c2.

22

33

GM / f (1 1 r* ) , r 6r

In this paper I shall endeavour to answer the question about a spherically-symmetrica l equilibrium configuration - the collapsar; i.e., to understand the static field properties in vacuum up to the sphere r = r * , proceeding from the fact that it is this potential whic h determines these properties. For that I will have to invest igate the motion o f test particles in potential (4) at least out of the sphere r = r * . In other words, here we approach the situation near the gravitat ional radius r * from the point of view of equations of motion of particles in a given field o f the form (4). Eventually we must have a self-consistent and complete descript ion of gravitat ional properties o f the collapsar fo llowing, on one hand, equat ions o f motio n and, on the other hand, field equat ions, constituting the basic system of equat ions o f the GD [P1]. 2. Test Particle in a Given Field of Form (4), Particle Mass in GD In this sect ion, with a view to main quest ion about the possibilit y in principle of stable equilibrium on the sphere r = r * , we shall look how the particle mass changes when 'immersed' in field (4), stipulat ing the possible role of gravitat ional radiation. For the total energy o f part icle in the given field (4), using the general correlat ion obtained (Baryshev and Sokolov, 1983) we have mc
2 2

E=

1 v / c mc
2

2

(1

f 0k u c2

k

1 v2 / c2 )



1 v2 / c

2

[1

r* r (1 * )] r 2r

(5)

(the notation is anywhere the same as in [ P1]). It follows from (5) that at a given particle velocit y v its total energy E is minimum at the sphere of radius of r = r * . As an example one can consider the (parabolic) case o f the part icle falling wit h the zero initial velocity v0 at r and conserved total energy. In other words, we neglect effects o f 6


gravitat ional radiat ion. Fro m (5) we have for the particle energy E |

r

= mc2 = const. Then

for maximum velocit y which the particle can achieve at such a falling to the centre of sphere r = r * , we have 1 mc 2 , 2 22 1 v / c at r = r * and at v0 = 0 for r = . Of course, everywhere here the fulfillment of the condit ion m << M is meant, for field (4) can be assumed given. We can note obvious difference o f result (6) fro m what has been well known for a lo ng time in the analogous case in GR. In the same situation wit h a part icle falling to the centre, in GR (in the absence of radiation) the particle velocity beco mes equal to the velocit y o f light at 2r * (where M is the mass of black ho le in GR). The velocity v
ma x

mc

2

v

ma x

=

3 c/2 0.86 c .

(6)

= 0.86c (at v 0 |

r

= 0) must be understood as some limit but never

achieved velocit y o f a particle falling (parabo lically) to the centre. Generally speaking, at mot ion wit h a large velocit y in the given field (4) with a sufficiently large gradient o f potential nearby the sphere r = r * , in principle it must arise gravitat ional radiat ion, taking away a part of energy of the falling particle. The velocit y, really attainable at r = r * , must be less than (6). Thus, let us remark here that the total energy of a part icle, generally speaking, can decrease at its 'immersio n' to field (4) down to the depth r r
*

i.e.,

E|

r

= mc2 > E |

r r*

In part icular, the energies o f the same (?) part icle, rest ing at first at infinit y and having co me to rest for some reasons (including the loss for gravitat ional radiation), 'stuck' to the sphere R = r * , differ two times, as it fo llows fro m (5) as

E|

r

= mc2 ,

E|

r = r*

= Ѕ mc2 .

(7)

Thus, before we reach bottom of the potential well in field (4) to the full stop, to the 'merging' with the bag at the sphere r = R = r * , the particle must lose a half of its init ial rest mass for gravitational radiation (and, possibly, fo r other forms of radiation). I.e., in GD the mot ion of test particles in field (4) occurs with changing rest mass of these particles, as was noticed already in [ P 1 ] . 7


In the fo llowing it will be convenient to use an analogue of Newtonian potential in GD, corresponding to tensor potential (4):

GD f 00 (r )

GM 1r (1 * ) . r 2r

(8)

The behavior of this funct ion is drawn in Figure 1. At r >> r = GM/c2 equat ion (8) passes to Newtonian formula, and on the sphere r = r GD|
r = r*
*

we have

= - c2/2 ;

and one can say that there is no 'Newtonian' potential deeper than - c2/2. Total energy (5) of a particle in the given field (4) can now be rewritten by 'Newtonian' potential

E=
where it is seen that the quantit y m
GD

mc

2 2

1 v / c

2

(1

GD ), c2

(5')

is the 'potent ial' energy (the total energy minus mc2) o f

a test particle of mass m, resting (v = 0) at a given distance r from the centre. So far as the main purpose here is the possibilit y o f a static (v = 0) in field (4), this more accurate definit ion of the meaning of the quantities m
GD

and GD is essential for the following.

8


Fig. 1. The gravitational field potential () in three theories (in units of c2). The solid thin line is the collapsar potential in gravidynamics GD, solid thick line is the 'potential' of GR, dotted line is the Newtonian potential.

The same energy can be written by the 0-component of the 4-vector Ak nkun A0, A = A) introduced in [ P 1 ] as A0 = =

( =



00 2 2

, A = A =
2 2

1 v / c

1 11 v c 1 v2 / c

2

, (9)

E=
where e fm. Now the difference between the
GD

mc

1 v2 / c

+ e

and -component of the 4-vector Ak is well seen (0-

component of the Ak is an analogue o f -component of the field in electrodynamics). The value o f at r = r * depends, in particular, on the particle velocit y, that is not very convenient. For example, if we assume that the limit value of velocity, which a part icle can achieve falling fro m infinit y to the centre with the zero init ial velocit y, does not exceed the value (6) v 3 c/2 then the maximum 'depth' for A0 = -component would be equal to f |
r=r* max

=

= - c2 .

Formulae for E, obtained above, differ only in notation. It is more essent ial to select a suitable and convenient formalis m, allowing us comparing results of GD wit h conclusio ns o f GR and Newtonian theory. In that case the -potential turns out to be inconvenient because o f its dependence on the particle velocit y, as was said above. In the following I shall endeavor to present the particle motion, analogously to classical mechanics, as a mot ion in potential (8) drawn in Figure 1. In Figure 1 the classical Newtonian hyperbo la (- GM/r) is drawn by the dotted line as a 'gauge' curve. It is well seen that sooner or later at the approach to the centre the potential
GD

goes to the left fro m the classical hyperbola because of the influence o f gravitat ional fie ld energy-tensio ns cont inuously distributed in space around the bag (for more details see [P2]). The corresponding analogue o f Newtonian potential in GR must go to the right from the 'gauge' hyperbo la ­ GM/r, for its derivat ive would inevitably beco me the infinit y in the po int r = 2GM/c2. Generally, very essential is the fact (I shall imply it elsewhere below) that the quantity M at r >> GM/c2 means the same in all three theories (GD, Newton, GR) that, of course, is a

9


consequence of the correspondence principle. It is always the mass measured by usua l astrophysical methods, or the mass in the usual dynamic sense, i.e., the mass not only as a gravitat ional 'charge'. The details, defining more exact ly this notion of the co llapsar mass, arise at the approach to the sphere r = r * . Ultimately, papers [P1, P2] and this one are dedicated in any way to the clarification of the meaning of the quant it y M. In part icular, when the static field o f an object with R = r
*

is spoken here about, I

always mean the existence of the classic limit for this field, i.e., the mass can always be understood in the usual classic sense as the mass of a matter point (see Introduction). If such a limit does not exist for some reason or other (for example, at r 0 and T 0, see [ P 1 ] ) , then, accordingly, one may not already speak about a static, or stationary at r = r
*

,

gravitat ional field. But eventually, then one may not already speak also about a point with the mass M. We shall return to this fact later. As was marked before (see (7)), the motion of a test particle in field (4) occurs in such a way that, generally speaking, the particle mass in the equations o f motion cannot be already assumed constant. And for the fo llowing it is very important to realise exact ly the two mo ments. On the one hand, at the 'merging' (or at full stop on the sphere r = r * ) of a test particle with a gravitat ing centre the energy lost for radiat ion must be large enough. It cannot be neglected if we consider, for example, the fall to the centre (and accelerat ion in field (4) near the sphere r = r * ) of the particle with an arbitrary velocit y. On the other hand, we can be interested in the possibilit y o f a steady state, in part icular for r = r * , when obviously there is no radiation. Is such a state of rest (v = 0) of the test particle on the sphere r = r * possible? We can also consider cases o f stationary states of particles near r = r
*

with velocit ies and accelerations st ill small enough for it would be

necessary to take into account the radiation. So far as gravitational radiation is still negligible at velocities v2/c2 << 1 and even at (v2/c2)2 << 1, then in the overwhelming majorit y o f practically important cases one may use the equatio ns o f mot ion in the field (4) with a given mass o f test particle, i.e., with a definite rest mass (m*) which does not change during the mot ion. It is also considered below in detail. This last statement agrees wit h what was said in Introduction about the collapse and collapsar. Ultimately, the very process of the pass of a particle in a stationary (bound) state at r = r * (the fall to the bottom of the potential well in Figure 1) can be still considered not in the who le vo lume. Below we shall basically speak about possible steady stationary states both 10


of a co llapsar and test particle, when there is no radiat ion already, or when it is sufficient ly small - analogously to quantum mechanics of atom when one studies the stationary states of an electron at first. One must keep in mind that any object before reaching ('merging' with) the sphere r = r * changes so mehow its structure so that its total rest mass would beco me two times less than the rest mass of the same object at infinit y. Thus, the mass in GD depends on distance at which it is measured. Such a result was met already in theory wit h the self-act ing field o f gluons in QCD for quarks strongly bound inside hadrons. Hereafter I shall endeavour to emphasize analogous properties of these two theories wit h massless and self-act ing gauge quanta: GD with two types o f real gravitons and QCD with 8 types o f gluons. 3. The Force Acting on a Test Particle near the Sphere r = GM/c2 and the Possibility of Equilibrium on this Sphere Let us begin wit h the simplest case - statics: i.e., let us assume that a test particle rests already on the sphere surface o f the radius R = r * . What forces act on it in that case? It is to the point to remember here an analogous problem for the black ho le in GR about the particle, 'lowered by the rope': Is the rest possible at r 2GM/c2? Using the vector A = -A and = A0 from (9), one can write down for the force acting on test particle in a given field, in the form known fro m electrodynamics:
dp v e(E [ H ]) , dt c

(10)

where as usual E = -A/t - , H = rot A, and e fm. The difference fro m electrodynamics is (and it is essent ial) that there is a dependence on the particle velocit y in vectors E and H already. But as far as we assumed that a particle rests ('lowered by the rope' in field (4) - the case of statics), then at v = 0 the 'inconvenient' vectors A and H disappear fro m (10). (The vector A does not depend explicit ly on time by force o f the static character of potential (4) and, hence, A/t = 0, i.e., the field is given). As a result we obtain at v = 0: = - 00(r), E = + 00(r), A = 0, H = 0. (11)

I dist inguish here especially this case in connection wit h the quest ion posed at the beginning on the very possibilit y of statics at r ~/= r * . But, strictly speaking, case (11) corresponds to the fact that for the gravitat ional force in Equat ion (10) we neglected all the 11


ratio v/c power expansio n terms, beginning with v2/c2. As far as in GD (unlike electrodynamics - ED) there are no terms proportional to the first power of the ratio v/c (A/t = 0), then the case in question of 'statics' (11) allows, generally speaking, motions wit h low velocit y v/c 0, if only the second power of this ratio would be st ill sufficiently small: v2/c2 << 1. For example, 'statics' (11) can describe cases of motion with sufficient ly large velocit ies v -1

, but it depends on a concrete/specific proble m

Fe

d 00 (r ) d GmM GM / c 2 r0 m GD r0 2 (1 )r0 . dr dr r r

(12)

As usual, r0 is directed fro m the centre. In this formula there is no dependence on the particle velocit y but it is necessary to keep in mind that it is fulfilled till one can assume that v2/c2 << 1. Only in that case the influence of addit ions to the force (and in particular of 'magnet ic' addit ion) is st ill sufficient ly small. For example, if small additions to the force of the order of v2/c2 = 0.07 can be st ill assumed negligible in the condit io ns of a given problem, then it means that formula (12) can be used at the study of mot ions in field (4) with velocit ies vx 0.26c = 80000 km s-1. Thus, with corresponding restrict ions, formula (12) for the force can be applicable in many pract ical important cases of the particle motio n in a given spherically-symmetrical field of form (4). It is particularly essent ial if one means the cases of motion of macroscopic objects in gravitational field with velocit ies though big, but still sufficient ly far fro m ultrarelativist ic limit, when v c. Subsequent sections will show that in (12) the restriction must exist of r on the side o f small r < r * , connected with the impossibilit y o f applicat ion of the very 4-potential (4) at too 12


small r. But from (12) it is seen that at least at r > 0 the force act ing on the test particle is always finite and r = r * = GM/c2 it simply becomes equal to zero. Thus, the field described by vacuum potential (4) allows, in principle, the possibilit y o f stable equilibrium o f the particle on the sphere R = r * as it is seen fro m formula (12) and Figure 1. At a further decrease o f the distance to the centre (r < r * ), for example, by the compressio n of the central object (the bag) the repulsion must arise and, consequent ly, the point of minimum energy of particles (5) on the sphere r = r
*

is simultaneously the posit ion

of stable equilibrium for such particles. In other words, the vacuum potential (4) does allow the existence of the surface of the collapsar even at the bag radius R = r * , i.e., the bag boundary can be at r = r * in the state of stable equilibrium. It differs radically from what GR gives in the case. In GR (see Figure 1) the forces at distances r 2GM/c2 (i.e., even earlier and at the same M) will tend to infinit y in the frame of reference in question - no equilibrium is possible in principle. It is necessary to say that GD, consistent ly allowing for the energy (or the self-action) o f gravitational field, leads to rather unusual situation at r = GM/c2. The feature of GD is the fact that though we have an object with a strong gravitational field over its surface (00 Mc2/r3 and with a strong attraction over the sphere with R = r * ), but the surface it self o f such an object (the bag) at r = r * is in the region of total equilibrium o f gravitat ional forces, i.e., at r = r * upper layers of such an object do not press at all on lower layers. It is abso lutely different from what we have got used to in Newtonian gravitation, and fro m what GR gives, assert ing that the weight of upper layers o f contracting object is always only increasing at the decrease o f the dimensio n o f the object with a given M. Ult imately, all investigated modifications o f the equation of state of co mpact objects matter, I mean so-called 'realist ic' equations o f state for the interiors of neutron stars and, possibly, quark stars, must guarantee stable hydrostatically equilibrium configurations at condit ion o f strong pressure of upper layers on lo wer ones. In GD the situat ion may turn out to be abso lutely opposite for the object having attained the sphere r = r * . In that case the equation of state of the matter inside the bag must correspond to the total absence o f the pressure o f upper layers on the lower ones o f such matter. In other words, on the surface o f the bag with radius R = GM/c2 the gravitation is already 'turned out'. Thus, if t he description of the co llapsar field is assumed by means o f vacuum potentia l (4) for the particle, 'slipped in' the region of the sphere r = r
*

(of course, with the loss for

13


radiat ion), then near the equilibrium state the gravitat ion as if disappears. There is no attraction on the sphere r = r * , though 00 is large. In that case the situation begins to remind the state of asymptotic freedo m for bound quarks in a hadron in QCD. The analogy wit h QCD may be intensified, if to mean the decrease o f the rest mass o f the test particle (see Sect ion 2), 'immersed' in field (4). In GD also one can consider that the masses of particles bound in the bag are less than the masses o f the 'same' particles measured at distances between them much more than r * . Thus, in GD in the case when the distance between bound particles beco mes one of the order of GM/c2, their masses decrease and the forces of gravitat ion (and we speak for the present only about the gravitation) acting on such particles, can tend to zero. In general, such a picture of bound particles behaviour could be expected at once for the theoretical scheme, which allows consistent ly for the self-action of gravitons. In particular, in QCD one o f the consequences o f essent ial nonlinearit y o f the theory, showing itself in the self-act ion o f gluons, is the property of asymptotic freedo m, i.e., of the decrease o f the force of quarks interact ion at their rapprochement till very small distances. Thus the processes o f the type of

played its role. However, the ment ion of QCD is here only a more or less suitable analogy. The equilibrium o f the scalar force o f repuls io n F(0) and the tensor force of attraction F device which 'turns o ff the gravitat ion on the sphere r = r densit ies
00 (0)
*

(2)

is the

in GD. For all this, the energ y

and

00

(2)

o f both components o f gravitat ion, equal to each other, achieve the

maximum on the surface with R = r * :



00



00 ( 0)



00 ( 2)



1 GM 2 1 GM 2 1 c8 16 r 4 16 r 4 8 G 3 M

2

.

As was done already in [P1], one can present force (12) as an algebraic sum o f forces acting on test particle in field (4), i.e., it will be the sum o f forces: F(2) - the force arising because of the presence of the tensor component of gravitation in (4), and F(0)- the force arising because of purely scalar co mponent of field (4). For the 'tensor' force we have

14


F( 2 )

3 GmM 2 Gm / c 2 (1 )r0 . 2 r2 3r

(13)

And for the purely scalar component of field (4) the repulsio n stays as before because of the linearity of the scalar field at the condit ion mm 0, which guarantees the existence of the classic limit of (4) to be
F( 0 ) 1 GmM r0 . 2 r2

Now it is seen that at r = r * the total force for potential (4) F = F(0) + F(2) beco mes zero, i.e., the gravitat ion (tensor component (4)) is compensated on the sphere r = GM/c2 by the scalar 'ant i-gravitation'. The opportunit y of the usage o f formulae (4) for the vacuum potential, when the bag radius beco mes less and essent ially less than r = r * will be studied especially in the fo llowing sect ions. But to anticipate the end of the paper I say that the vacuum potential (4) and formula (12) are applicable t ill one can speak about their classic limit. Correspondingly, for the potential and the force o f attraction around 'the point' wit h the rest mass equal to M in the space wit h r>> GM/c2 we must have formulae (29) and (35) fro m [PI], i.e., Newton's law. Ult imately, the bottom limitat ion of r fo llows the fact that the total mass of the configurat ion (the collapsar) cannot be the field one only, when the classic limit of (4) is absent. Though here the question is always on the classical theoretic scheme for the gravitational field of the collapsar with vacuum potential (4), but so far as the dynamic treatment of gravitat ion const itutes the base o f such a scheme, I shall have (as in [P2]) to touch more or less profoundly upon the very nature of gravitational interact ion. The nature of dynamic (gauge) fields must be thought about with the usage of quantum notions, quantum elementary processes with the participat ion o f photons, gluons, weak bosons and, lastly, gravitons (of tensor and scalar form). If we are not too dogmat ic about GR, one could make up his mind to express the fo llo wing. It seems to me, that purely classic descript ion of gravitat ion pheno menon in GR by means of curves space-time, now (from the point of view of the theory of dynamic fields) can be assumed good but only pheno meno logically. Here we can recall that up to recent ly for the descript ion o f weak interact ions the scheme o f 4-fermio ns interact ion served well and for a long t ime. Time came, and one had to think about the real theory. Eventually, a new scheme was created, unifying two interactions in the theory by Weinberg-Salam. An attempt to understand more profoundly the nature of weak interaction led to the introduction of weak bosons into the theory - the theory became 15


more general and more exact as a result. By means of this (lyric) digressio n I want to emphasize once more that attempts to understand the gravitation leads inevitably to the usage of the experience of classic GR on one hand, but on the other hand one cannot do so without a resort to all quantum-field theories. How close we approach the understanding of the nature of the gravitat ional field in the proposed scheme here (and in [P1, P2]) with one more, massless scalar graviton? I should like to think that like other field theories and in general as it is in high-energ y physics, the answer to this quest ion will be given in corresponding experiments (observat ions). Anyway, the appearance of fundamental scalar elementary part icles in the theories with spontaneous vio lat ion o f gauge symmetries is considered now (almost) inevitable. As to the massless scalar graviton, which naturally appears in GD and which is a n essent ial co mponent of gravitational interact ion in the scheme, proposed here, then, as it was shown in [P1], such a scalar is naturally connected wit h the total mass o f the who le configuration - the collapsar by means of a source

T Mc 2 (r ) *c

2

1 v2 / c 2 .

In this connect ion it is to the point to recall the source of Higgs's scalars. In the conclusio n of this sect ion I remark that the notion of virtual gravitons (scalar and tensor ones, see [P2]), which led eventually to vacuum potential (4), arises naturally in t hat idea environment, which modern physics o f high energies created: QED, QCD, etc. It are these ideas, const ituting the base o f dynamic interpretation of fields, which lead to the presentation o f gravit y field as a totality of two components - the scalar and the tensor ones. It is from here something arises which looks like an asymptotic freedom of QCD at r r * . 4. The Scale of Forces, a Maximum Acceleration of Gravity Force Near the Collapsar Surface and the Sphere of Maximum Instability for a Given Mass M It is difficult (though it may be appropriate) to keep fro m referring once more to QCD, but an opinio n is expressed already rather often, that apparent ly the same nonlinearit y of the theory leads to the fact, that at distance more or of the order of a neutron dimensio n the force of attraction between quarks beco mes so large that it does not allow to quarks to be free. As is seen fro m Figure 1, the field wit h vacuum potent ial (4) in nonlinear GD gives for something like the confinement of QCD. But here the difference between the potentials
GD GD

for a bound particle (r r * ) and for a free particle of GD in finite ( c2/2), and the attraction 16


force, act ing on the test particle, 'immersed' in field (4), cannot exceed a certain maximu m value. The largest force of attraction acts on particles sit uated on the sphere r = 3/2 r * - in the point of maximum gradient of the potential
Fmax FG m r0 M
GD

in Figure 1. This force is equal to r = rmax = 1.5GM/c2 , (14)

for

where

FG

4 c4 ; 27 G

(*)

and FG can be understood as a limit possible force of attraction act ing between two points with equal masses, situated at the critical distance l.5GM/c2 to each other. The difference between the maximum force of attraction Fm
ax

for a given M and the

limit possible force FG is emphasized the best by reasoning fo llowing hereafter. In any cases in quest ion it is implied that vacuum field (4) is the given field. It means that the test particle of mass m moving in this field disturbs weakly the mo vement of the massive gravitating centre. That is why the gravitating object can be assumed to be at rest with the centre to be in the origin o f the frame o f reference - practically it is this fact whic h defined (see in [ P 1 ] ) the frame o f reference. If we shall assume that in that case the condit io n is fulfilled m << M (and it was always implied before), then the words 'the given field' or 'the given frame of reference' mean in fact (see (14)) that

m Fmax 1 M FG
on the sphere r = rmax.

(15)

In these cases I mean always the given static field with vacuum potential (4), i.e., here we call test particles the particles for which even on the sphere r = rmax = l.5GM/c2 the force of attraction is still sufficient ly small in co mparison with the limit possible force of attractio n FG. Thus, the introduction o f the limit possible force of attraction FG makes absolutely definite such notion as the test particle (the passive attracted mass), the given field (the active attracting mass in the expressio n for rmax = l.5GM/c2) and, ult imately, the quest ion about the special status (or the dist inguishing state) of inertial frames of reference in GD can be considered cleared up definit ively. So, in GD the scale for the force of attraction is introduced quite unambiguously, and the 17


quant it y FG w 1.79 x 1048 dynes plays the role of the fundamental force. However, it is not worthwhile to hurry with the direct comparison of FG wit h other forces exist ing in nature. Here it is very essential to keep in mind the radius o f rapprochement 1.5GM/c2 between two gravitating objects. So far as the force of attraction between two objects cannot exceed the value Fmax = FG(m/M) and it depends on the rat io of masses o f these bodies (with the attracting active mass to be in the deno minator and the attracted passive mass to be in the numerator), then this force of attraction never is more than FG. It is excluded simply because otherwise the attracting body and the attracted one will exchange their roles. Of course, one should not forget that the condit io n m/M << 1 for the active and passive masses corresponds to the given static field (in a fixed frame o f reference). The cases, whe n the ratio m/M becomes of the order of 1, mean as a matter of fact (in condit ions, when the radius of rapproachement ~ r
max)

the pass to situations wit h nonstatic gravitational field

already. Such problems inevitably will demand the allowing for the role of the gravitat iona l radiat ion. (And not only for the tensor one, as it was with the double pulsar PSR 1913 + 16 at rather large st ill dimensio n of the orbit.) As it was emphasized many times before, the movement of particles at relat ive distance to each other of the order of Gma/c1 will demand ult imately the considerat ion of objects with variable masses. At r ~ GM/c2 the movement, for example, of two equal masses take place in the region of nonlinear GD, when their gravitat ional 'coats' essent ially intersect each other. I endeavour here, in this paper, to restrict the range of problems only to stationary ones, and in the limit - to static situat ions wit hout radiatio n. But nevertheless I wish to remark that the limit possible force of attraction FG is equal everywhere both for stellar masses and for masses of elementary particles. The other matter is that the rapprochement, for example, of two nucleons to the crit ica l radius 1.5GM/c2 ~ 10
-52

cm would require 'to turn out' at first all other interactions exist ing in

nature. In fact, the nucleon will cease to be a nucleon much earlier. Thus, the limitat ion arise of really achievable value o f gravitat ional interaction Fmax, connected ult imately with the existence of other interactions. Before going further, I remark that the condition Fmax/FG = m/M << 1 is naturally connected also with the fact that we consider still not too small M, corresponding to nonquantum, macroscopic dimensio ns o f bodies. Otherwise, reducing the mass unlimit-edly, we shall have to abandon the presentation o f the gravitat ional field of the object in the centre as so me cont inuous medium wit h the tensor ik. Ult imately, one will have to consider a n essent ially quantum situat ion, in which the condit ion ikik = 0 can be broken. 18


The essentially quantum situation implies the considerat ion of the processes of type of

with particles equal in rights instead of processes

(static potential). For all that the mo vement of all part icles-sources must be considered already wit hin the bounds o f quantum theory. In our case of large (macroscopic) GM/c2 the movement of test particles in macroscopic vo lumes stay also quite classical. Thus, in what fo llows, the question is only on macroscopic gravitat ing objects and, accordingly, for M the values of the order of stellar masses and more are taken. Then for the maximum acceleration of free fall on the sphere r = rmax = 1.5GM/c2 (the sphere of maximu m instabilit y) in vacuum for the object with the mass M we have

g

max

(r rmax )

4 c4 1 . 27 G M

(16)

In accordance with (11) it is simply the tension of the gravitational field on the surface o f the sphere r = r
ma x

and here I especially draw the reader's attention to the fact that the more

the mass M of the object, the less the tensio n o f the gravitat ional field, maximally possible at this value o f M. Below some est imat ions will be adduced, having for the purpose to emphasize of this feature of vacuum potential (4). But at first we estimate the Newtonian accelerat ion on the surface of the stable neutron star with M = 1.4M and the radius R g
NS NS

= 10 km by usual formula gNS = -GM/R2. It is equal to
NS

= 2 x 1014 cm s-2. The mean densit y o f such an object is equal

= 6.7 x 1014g cm-3.

It is necessary to say that the densit y o f the matter of neutron stars is defined by the equation o f state P = P() in corresponding equat ions of hydrostatic equilibrium, allowing, in particular, for relativist ic effects also, usually in the bounds o f GR. For all this the macroscopic densit y o f the matter achieves the values 1014-10 supernuclear densit ies. 19
15

g cm-3 - nuclear and


All modern calculat ions (Shapiro and Teukolsky, 1983) show that at the accelerat ion on the surface o f the order of gNS = 2 x 1014cm s-2 one can choose such a law of changing o f the pressure P(r) and o f the densit y (r) inside the star (P = P()) which secures the needed gradient of pressure in case of hydrostatically-equilibrium neutron star. Here I especially choose some average parameters of the neutron star, which (for example) are o ften used at the interpretation o f observat ional manifestations o f such objects. The main fact is here that the object with the acceleration ~2 x 1014cm s-2 on the surface can provide a quite stable format ion. In GD for the object with the mass 1.4M on the sphere r = r vacuum) the maximum accelerat ion g
mzx max

= 3.24 km (st ill in

= 6.24 x 1014 cm s-2. From the point of view of GR

the object is wit hin its Schwarzschild radius (4.3 km) and it is impossible in principle to secure any equilibrium at any equat ion of state. However, from the po int of view o f GD the acceleration g
ma x

(6.24 x 1014 cm s-2) is only

the maximum acceleration for the mass M = 1.4 M and one can say about the macroscopic densit y of matter inside the bag o f the dimensio n (the radius) of the order and less than R = 3.24 km. Wit h the allowing for the field energy around the bag, such a densit y will be about 15 times more than NS. But the question arises what kind o f matter will be in the bag and what equation of state we can speak about in that case? What forces define it now? Here I have to recall QCD again. According to QCD, at distances between the baryo ns less than 10
-13

cm (it is characterist ic radius o f the nucleo n) and, correspondingly, at densit ies
u cl

exceeding the nuclear one (pn

= 2 2.8 x 1014g cm-3) a few times, the matter of the bag

must undergo the phase transit io n in the state of quark-gluon plasma (QGP). As a result we deal wit h the degenerated Fermi-liquid, to the equation o f state of which a lot of papers are dedicated now (see the literature in Chernavskaya and Chernavsky, 1988; Emel'yanov et al., 1990). It can be assumed that in GD the neutron matter in the who le vo lume of the bag for the object with M = 1.4 M and R 3.24 km must be already in the state of QGP. The further concretizat ion of the bag properties fro m the point of view of GD deserves to be dealt in a special paper and the bag properties will be stated many t imes later. But now I am interested first of all in its gravitational manifestations in vacuum. By use of (16), for the bag of the radius not exceeding 10 km but with the mass M = 4.5 M0, one can obtain the same accelerat ion (2 x 1014 cm s-2) as on the surface of a 'usual' (stable) neutron star. According to GR it is already a black ho le (2GM/c2 13.3 km), i.e., in GD the hydrostatic stabilit y o f the object with M = 4.5 M and R = 10 km can be secured apparent ly by an approximately same equat ion of state as for a usual neutron star with the 20


mass 1.4 M and the radius 10 km. For not to worry for the present over parameters, indefinite in many respects, of the phase transit ion in the state of QGP (what will happen at densit ies ~ 5
n u cl

?), one can

estimate the parameters of the 'superdense' neutron star taking one's cue fro m the mean densit y o f usual neutron star with the mass 1.4M, the radius 10 km and NS = 10I4-1015 g cm3

. If the matter of such an 'average' star undergoes a phase transit ion, it occurs somewhere in a

small regio n near its centre. Thus for the object with a mass M = 5.7 M and the radius R = rmax = 12.7 km the acceleration of free fall on the surface will be not greater than g
max

= 1.6 x 1014 cm s-2 wit h a n

average densit y o f matter in the sphere of the radius R equal to the densit y o f a usual neutron star. Here it has been already taken into account (see Sect ion 6) that approximately a half o f the total mass of an object is distributed around 'the bag' in the form of the energy of gravitational field (the energy of 'the gas' of virtual gravitons). One can think that here also the matter densit y in the bag changes approximately in the same range as for a usual ('average') neutron star. The phase of QGP is not developed yet, the basic part of the bag mass is distributed with the density 1014--1013 g cm-3 and thus in the equat ion o f state the same indefinit io ns remain which remain st ill in case o f an 'average' neutron star also (Shapiro and Teukolsky, 1983). For such a supermassive neutron star the neutron matter will const itute only 2.85 M


(i.e., a litt le more than a half) o f the who le object, the rest of mass will be in a purely gravitat ional 'phase' - the coat of virtual gravitons. Nevertheless (for example, in close binar y systems) at r >> rmax the total mass M = 5.7 M will be measured. The quest ion about properties of such objects and collapsars, having attained the limit small dimensio n (i.e., the sphere of total equilibrium) wit h the bag o f the radius R = r
*

=

GM/c2, is ahead. Here I would like to remark that in GD a stable co mpact object is possible, which looks like the black ho le o f GR by the measured mass M and the radius. But nevertheless it is a static object with a finite value of the gravit y force on the surface and a quite acceptable (finite!) value o f the matter densit y. The equat ion of state P = P() can be almost the same, similar one, as in case of an 'average' neutron star. If to increase more the mass in (16), then for the object with mass M ~ 3 x 106 and the radius ~ 10 R (R/c ~ 0.5 min) which in GR inevitably co llapses in the black ho le, the acceleration on the surface is equal to g
ma x

3 x 108 cm s-2. It corresponds to the accelerat ion

of the gravit y force (GM/R2) on the surface of a white dwarf. The mass densit y (~ M/R3) o f such an object is only < 5 x 103 g cm-3 - i.e., by three orders less than a densit y of whit e dwarfs (~ 106 g cm-3). Certainly, such an object cannot be already called a supermassive whit e 21


dwarf. For a body wit h the mass 7 x 1010M and the radius 1.6 x 1016cm ~ 103AU (R/c ~ 6 days) the accelerat ion on the surface is not greater than the gravit y accelerat ion on the surface of the usual star. The mean densit y in that case turns out to be 1000 times less than the corresponding mean densit y o f stars. Table I contains the summary o f made est imat ions, which shows that vacuum potential (4) allows in principle the possibilit y of existence of stable objects of large and
TABLE 1 Accelerations (g
max)

on the sphere of maximum instability (R = rmax = 1.5GM/c2)

for objects with the given mass M and corresponding mean density (see text)

M (M) 1.4 1.4 4.5 5.7 3 x 10 7 x 10
6 10

Radius

Acceleration (cm s )
-2

Density (g cm-3)
4 14

R

NS

= 10 km

g g

NS

= 2 x 101



14

NS

= 6.7 x 1014

R = rmax = 3.24 km R = rmax = 10 km R = 12.6 km RR


max max max

= 6.24 x 10 = 2 x 10
14

15 NS
ucl

g

15 n

g g

= 1.6 x 10
8


WD

= 15 NS

max~ max

3 x 10 g star

5 x 103 10-3WD <10-3 st
ar

R= 103AU

g

superlarge mass, attaining a crit ical for potential (4) radius r = r

ma x

= 1.5GM/c2. Of

course, the properties of such co mpact objects of GD will be different for the objects of so lar and cosmo logical masses, which is also their difference from the black ho les ofGR. The fact must also be noted that the less and less exotic forms of matter are needed for a spherically-symmetric object with vacuum potential (4) to be made stable on the sphere wit h the maximum gravit y accelerat ion or on the sphere of the maximum instabilit y of the object with the given mass M. Fro m the point of view of GR here it is necessary to do something impossible (absurd, paradoxical): it is necessary to take M in (16) larger and larger. In this limit, ult imately one could take the total mass of the metagalaxy. If we summarize everyt hing said in the last two sections, one can mark here that in GD there is a sphere of the maximum instabilit y r = 1.5GM/c2 for the given M - an analogue o f the sphere o f an abso lute instabilit y, i.e., Schwarzschild's sphere in GR. But in GD after the sphere r = rmax = 1.5 r * , the sphere r = r * of the total equilibrium vanishes which cannot be true in GR in principle. 22


The objects 'breaking through' to the sphere r = r * , will be considered in detail in Sections 6 and 7.

5. The Motion in the Given Field (4) and the Possibility of the Periodic Pulsation of the Sphere with R = r
*

Now, unlike that was said in Sect ion 3,1 shall consider some cases of the allowing for the dependence on v2/c2 for the force acting on a test particle in field (4), i.e., I shall endeavour to study such situat ions when a test particle (or particles) performs rather slow (for the radiation could yet be not taken into consideration) radial mot ions in the same vacuu m potential. As before, m <<, that guarantees in fact a small value o f the force F
max

at r =

l.5 r * in comparison wit h FG. Here the case of accretion on the central object under the act ion of the force Fmax can be included, i.e., on the sphere of the maximum instabilit y r = 1.5 r * , 'the rope is broken' and the fall of test particles is possible to the centre or, more exact ly, to the sphere r = r * of the total equilibrium, where the force of gravit y beco mes zero. Thus, let us assume that the motion of test particles in field (4) occurs in such a way that
v v r0 v( r ) r r

and rot v = 0 ,

(17)

i.e., v - the value o f the velocit y of every part icle - depends only o n r, and the velo cit y vector is always directed along the radius-vector r. Beside the fall to the centre, here one could also imagine, in principle, some spherically-symmetrical pulsat ions of the system o f test particles near the sphere r = r * of the stable equilibrium (see Figure 1). In such a case of whirlwindless motion of test bodies the vector field A is also spiral-free,

H rotA rot (

1 11 (r ) v(r ) r) 0 ; c 1 v2 / c 2 r

and here, as in Section 3, the second addend (the 'magnet ic' term) in force (10) wit h 'unco mfortable' vector H vanishes. Thus for the force, producing the work, we obtain
F dP fm A fm dt c t

(18)

where 23


A 1 11 (r ) 11 (r ) v (v ) 2 2 t c t 1 v / c 1 v 2 / c 2 t
As far as we exclude the explicit dependence of the particle velocit y on t, i.e., there is no 'detached' force setting the function v(t) in advance, then v/t 0 (we keep in mind that m << and there is no radiat ion). But then everything becomes again to look like the case o f 'statics' considered in Sect io n 3. As before, for the force act ing on every test particles, only 00-component of potential (4) is of importance: i.e., of

F fm fm

d 00 ( r ) r0 . dr 1 v 2 / c 2

(19)

In co mparison wit h formula (12) the difference is here in the root 1 v2 / c 2 under the sign of derivat ive d/dr:

F m

GD d r0 . dr 1 v 2 / c 2

I repeat once more that all measurements are made here in the frame of reference in which the central attracting body rests (m << M). It concerns also the acceleration, wit h which test particles mo ve and which can be obtained from the general relat ivist ic correlat ion
dv 1 v2 / c 2 v {F 2 ( v F )} , dt mi c

where F is any force act ing on the part icle wit h the inertial mass mi. (Here I use the notation mi along wit h the notation of gravitat ional mass m.) So far as in that case the velocit y o f test particles changes only in quant ity and the force is directed along the velocit y, we have

00 (r ) dv f m v 2 d 00 ( r ) 2 2d 1 v / c ( ) dt mi dr 1 v 2 / c 2 c 2 dr 1 v 2 / c 2
Thus we obtain that the centrally-symmetric mot ions in field (4) occur with the acceleration dv 1 v2 (1 2 ) dt mi c
3/ 2

fm

d 00 (r ) dr 1 v 2 / c

2

.

(20)

Of course, the accelerat ion, which the particles move with, does not depend on part icles 24


mass, but using the equalit y mi/m = 1 in (20) it is necessary to keep in mind that this equat ion for dv/dt is obtained at the assumption mi = m << M. By use of (20) the notations (r) (1 ­ v2/c2) and
GD

= - f00 for the acceleration, whic h

changes the particles velocit y only in quant it y, one can write down in more compact form:
d dv 1 ( r ) GD dt dr 2 d ( r ) . dr

GD

Now we have an equation which describes the time change of the velocity field of test particles performing centrally-symmetrical movements in a given gravitational field wit h potential (4). As before, it is meant that the sum of the particle masses is much less than the total mass of the who le configuration. Now I shall endeavor to study the possible 'regimes o f work' of formulae (19) and (20). Rewrite once more formula (20) as

dv r*c 2 r* 1 r*c 2 d 1r 2 (1 ) (1 * ) . 2 dt r r 2 r dr 2r
For the force F in the same notations we obtain
F mc
2

(20')

r* r2

1 / 2

r r d 1 (1 * ) mc 2 *2 r 2 r dr

3 / 2

(1

1 r* ) 2r

or F GmM (1 r* / r ) 1 GmM (1 r* / 2r ) d(1 v 2 / c 2 ) . 2 2 3/ 2 r2 dr 1 v2 / c 2 2 r (1 v / c ) (18')

At v = 0, = 1, and d/dr = 0 the force and the accelerat ion (of the free fall) correspond to case (12) considered already. The novelt y of formulae (18') and (20') is the presence of the gradient of particles velocit y, i.e., of the quant it y d(l - v2/c2)/dr, and o f 'dangerous' deno minators (1 - v2/c2) in the formula for the force. Let us assume for the mo ment that in case in questio n of radial mot ions of test particles, their energy is, nevertheless, conserved in so me way down to the 'depth' o f the order of r = r
*

in field (4). Let a single part icle falls to attracting centre from infinit y wit h so me init ia l

velocit y v0, which it acquired wit h respect to the centre in any way. So far as the energy is conserved, if we use (5) one can write the equation
mc
2 2

1 v2 / c

{1

r* 1r mc 2 (1 * )} r 2r 1 v2 / c2 |

const ,
r

then 25


2 1 v0 / c 2

1 v2 / c 2 (1 GD / c 2 )

It follows fro m this that the particle velocit y (if its energy is conserved or almost conserved) will be maximum irrespect ive of the value o f its init ial velocit y 0 in the moment when
>

will be minimum, i.e., at r = r * , where v
2 ma x

GD

= - c2/2 (see Figure 1): at m << M. (21)

/c2 = 1 ­ ј (1 ­ v20/c2)

Thus, it turns out that even if we assumed the total conservat ion of the energy at the fa ll to the centre (which is not correct, generally speaking), then in that case also the maximu m attainable velocit y o f the fall in potential (4) would be bounded fro m above by a value less than c for the particle with m 0. The real velocit y v
ma x

must be still less than the value fro m

(21) in consequence o f the account for the influence of energy loss (braking) for the radiation. By this I want to emphasize the fact that the gravity force (18) is always a regular function of/and v. This force becomes zero on the sphere r = r
*

irrespective of the values of finite

quantities -1 and d/dr, i.e., irrespect ive of the value of the particle velocit y. In the same (extreme) assumption that the energy is conserved in case of radial mot ions in quest ion, one can obtain the dependences (r) and d(r)/dr at = const.:

(r )

m2c 4 r 1 r2 (1 * *2 ) 2 , E2 r 2r

d (r ) m2c 4 r 1r rr 2 2 (1 * * )(1 * ) *2 . dr E r 2r rr
Thus for the acceleration and the force at the given = const, we shall have

dv m2c6 r r r 1 r2 2 *2 (1 * )(1 * *2 ) , at m << M ! dt r r r 2r
F r* (1 r* / r ) 2 r (1 r / r 1 r 2 / r 2 ) * * 2
,
2

(22)

fro m that it is seen that as a fact the deno minator does not become zero at any real r. From (21) at v0 = 0 we obtain the known result (6) for the limit velocit y in case of the parabo lic fall to a massive centre. On the who le, a natural result is obtained also in case o f the fall of an ultrarelat ivist ic part icle, i.e., in the case when at infinit y the particle was flying already with the velocity almost equal to c. Then from (21) at v0/c 1 it fo llows that v 26
max

c.


It can be understood also in such a way that in that case the field (4) changes weakly the
2 init ial velocit y and, consequent ly, the energy mc 2 / 1 v0 / c 2

(which is very big for v0/c

1 in co mparison with mc2) of such part icles. So far as on this sect ion the question is only o n the force producing the work and changing the value o f part icle accelerat ion, then fro m formulae (20') and (22) for dv/dt it is seen that the cases v c (0, d/dr0) and m0 can also be understood in a sense as the 'switching off gravitation in this limit ('chiral' one for GD). In particular, it must be true also for photons leaving the sphere r = r * . Formulae (22) can be used in all cases of centrally-symmetric mot ions, when the particle energy can be considered constant with a high precisio n. Let us suppose the test particles to be already (in so me way) in bound, stable state with a definite (conserved) nonzero total energy on the sphere r = r * (see Figure 1). As it is seen from (22), the force and the accelerat ion beco me zero on this sphere. It is natural to consider small (wit h small small amplitude and velocit y - wit hout the radiat ion) deviat ions fro m the equilibrium state for such particles. For example, we can imagine a thin spherical layer o f the bag matter, being situated near the sphere r = r * . Thus, we can consider harmonic oscillat ions o f the bag surface which can arise due to the action o f any (including nongravitat ional) disturbances near the surface r = r
*

of the

sphere of the total equilibrium. Let us assume that the bag has here the radius almost coinciding with r * . I consider here the simplest case of rather slow motions when there is no radiat ion at all (i.e., v2/c2 << 1) and the oscillat ions are not damped. From formulae (20) at = (1 - v2/c2) 1 and, correspondingly d/dr 0, we have

dv r c2 r rr * 2 (1 * ) c 2 (1 * ) *2 dt r r rr
For oscillat ions with small amplitude near the sphere r = r
*

we write

r = r * + r = r * (1 + r/ r * ) = r * (1 + x) , where x r/ r * << 1. Then we can write for dv/dt: dv 1 1 / r* c2 x . c 2 (1 ) 2 dt 1 x (1 x) r* (1 x)3 Just the first term of the expansion over small deviation x << 1 from the equilibrium 27


state gives dv c2 x , dt r* r c2 r , r* ( ) 2 r 0 , r

or

where

c 2 / r*2 c / r* c /

GM . c2

(23)

In other words, for the case of rather slow motions in potential (4) on the who le a n obvious result is obtained: a system o f particles with minimum energy (on the bag surface with R = r * ), i.e., near the sphere of the total equilibrium r = r * , can perform harmo nic (r/ r * << 1 and v2/c2 << 1) oscillat ions, pulsations with the period r * /c. I emphasize once more that this paper deals only with stationary states of the co llapsar. The case o f strong deviat ions (r/ r
*

~ 1) fro m the equilibrium must lead ult imately to the

pulsations at considerable decrease o f the total energy Mc2 owing to the loss for the gravitat ional radiat ion, i.e., it can be already an essent ially nonstationary situation.

6. Utmost Compact Objects - Collapsars and Some Properties of the Bag with the Radius R = r
*

In fact, at the end o f the previous sect ion the questio n is on the object 'breaking through' to the sphere of total equilibrium r = r * , i.e., this is the object which found itself already for some reasons under its sphere o f maximum instabilit y r = 1.5 r * . It is this object, strictly speaking, which may be called the co llapsar, it is, apparent ly, the stage of relat ivist ic co llapse - the shrinking under the sphere r = 1.5 r * . As it will be seen fro m the fo llowing, such an object can really be an analogue of what is called Schwarzschild black ho le in GR in the sense that the co llapsar o f GD also possesses so me universal properties which are determined ultimately by its mass M. But there are here considerable particularit ies also. First of all I emphasize once more that tensor potent ial (4), one component of which is pictured in Figure 1, is the potential of the gravitat ional field in vacuum, created by an 'elementary' source, which at distances r >> GM/c2 is perceived as a point object with the mass M. The field of the point is spherically symmetric by definit ion and potential (4) is diagonal. But if we begin to approach the central po int, then for the same central symmetry o f potentia l (4) at distances r ~ GM/c2 fro m the centre such a 'point' or such an object turns out to consist 28


of so me more elementary, more 'fundamental' po int objects wit h masses ma*. This situat ion reminds evident ly o f high-energy physics - the collisio ns o f particles with large transferred mo mentum. Like all consistent relat ivist ic field theories (QED, QCD, electroweak theory) the theory of gravitat ional interact ion also must contain the notion of the po intness o f the sources of (gauge) field. Thus, in accordance wit h general principles of the theory of gauge fields, the bag is a system of bound points with masses ma* generat ing the gravitat ional field with potential (4) in vacuum surrounding the bag. The main purpose of the paper is the descript ion o f vacuu m field properties o f such an 'elementary' object up to r ~ GM/c2. For this I departed from the fact that such an object exists, i.e., it is stable and its total energy is equal to Mc2. In other words, the existence of the co llapsar was supposed in advance (see [P1, P2]), and then, proceeding from so me general axio ms o f theory, an attempt was made to prove this existence. Essential was to assume the existence o f static solut ions for the collapsar and, in part icular, the existence of Newtonian limit for the field of the point with the mass M was important. For vacuum potential (4) and in Figure 1 it is implied that if a po int (a sphere) is chosen on finite distance from the centre, then it means either the point on the bag surface or one above the bag, i.e., in 'vacuum' - in the region where the energy densit y is not zero and there is the pressure of the gravitat ional field only. The most important is to remember that at r GM/c2, where the repulsive force begins to act on a matter point or on the points of the bag surface. At it is seen fro m Figure 1, this force or these forces can be rather big for the values of M of the order of (for example) one solar mass. For at r GM/c2 the field cannot be already described by the static potential (4). Somewhere a situat ion arises when it is more and more difficult to secure the condition o f the static character of the who le configuration and ult imately it beco mes impossible to speak about the object with the rest mass M and Newtonian field of gravit y at r >> GM/c2. Let us take such an integral of the energy densit y o f the gravitat ional field 00:


4


r

GM 2 2 r dr Mc 8 r 4

2

and make it equal (at some r) to the total energy of the whole configuration (the bag + the field = the collapsar). At r = rf r * /2 = (GM/c2)/2 it turns out that for field (4) the total energ y of such a spherically-symmetric object is the energy o f massless gravitational field alone. But lastly, it is difficult to coordinate the condit ions (2) and (3) of the static character of the who le configurat ion (see formulae (51a) and (51b) in [PI]), which we used at the grounding of vacuum so lut ion (4). Strict ly speaking, such an object with the bag radius equa l 29


to rf , will not have the rest mass because the gravitat ional field itself does not have it. Here, for macroscopic objects the same difficult ies with totally field mass arise which were in electrodynamics with the electromagnet ic mass o f electron. And here, in GD, only a part of the total mass o f the who le configuration can be the fie ld one. Otherwise, 4-potential of suc h an object has not already its classic limit N = - GM/r and it can be said that simply at r >> GM/c2 such an object does not exists. The impossibilit y itself o f the existence o f the stable bound object with the bag radius equal to /yean be understood also in such a way that the forces o f repulsio n in the limit whe n r rf beco me so large that they do the bag to scatter in the form o f gravitat ional waves wit h the total energy equal to Mc2, i.e., this is ut most unstable situat ion which does not realize in nature for sure. Absolute opposite case to such an utmost unstable situation is the case when we have the bag of the radius R = r * = GM/c2. As before we assume the fields and matter (particles) inside the bag to be distributed and mo ving spherical symmetry. In that case the total energy of the who le configuration wit h the bag radius R = r
*

is divided into equal parts: a half of the total energy o f the collapsar is the

energy o f the field alo ne ('the coat' of virtual gravit ons), and the other half o f the energy is the total energy o f the bag. Thus, for the co llapsar - the object (more precise, for the bag) which found itself under its sphere r = 1.5 r * of maximum instabilit y, an equation is true in the form Ѕ Mc2 (the bag with R = GM/c2) + Ѕ Mc2 (the field) = Mc2 . (24)

How the energy (Mc2/2) is divided inside the bag between bound particles wit h masses ma* 0 and the fields, I do not know at present. But it is clear already that the bag with radius R = r * must be in equilibrium wit h its own gravitatio nal fie ld in vacuum if this field is give n by potential (4). Indeed, the forces acting on surface part icles o f such a bag are equal to zero. It means that the field outside the bag is as if 'switched off, upper layers of the bag do not press alread y on lower layers, that is quite unusual in Newtonian physics and abso lutely inadmissible in GR. Thus we can assume that in GD the physics inside the bag wit h R = r
*

is co mpletely

determined by the fields o f interact ion between bound ('half-naked') particles wit h the masses ma*. Everywhere above I avo ided the concretizat ion o f the bag properties. For the obtaining of the outer solut ion (4) I needed only spherical symmetry of the distribut ion and the motion 30


of matter - the particles with masses ma* in the bag. Now, when the properties of the vacuu m field o f the co llapsar became so mewhat more clear, I shall endeavor carefully to elucidate what is inside? Of course, inward properties o f utmost co mpact objects wit h R = r * deserve a special invest igat ion. Here I limit myself for the present to the most commo n concepts and semi-qualitat ive est imat ions. If the radius o f a region filled by point particles (the bag radius) is close to R = r
*

or

even the bag radius is equal to r * , then the integral of the energy densit y of only part icles in the bag is anyhow less than the total energy, and for R = r * we have an inequalit y
R r*

4


0

*c
2

2 2

r 2 dr

1 v / c

1 Mc 2 . 2

(25)

Even this strict inequalit y alo ne forbids all part icles bound in the bag to be in rest (v2 0) in the who le volume o f the bag. Only free (remo ved to infinit y) particles can be in rest everywhere. Really, at v2 0 we should have fro m (25) an inequalit y
r*

4 * c 2 r 2 dr
0

1 Mc 2

2

.

But here we enter again in conflict with ment ioned condit ions (2) and (3) of the static character (and in point of fact, of the existence and the stabilit y!) o f the who le configurat ion. According to conditions (2) and (3) for the case in quest ion the scalar source 'is conserved', i.e., the integrat ion alo ng vo lume for T gives always the same result (Mc2) for any surfaces in vacuum, embracing the bag ent irely. Thus we can write the integral
r r*



TdV 4


0

* c 2 1 v 2 / c 2 r 2dr Mc

2

,

(26)

where the integrat ion extends only alo ng the bag vo lume, where *(r) 0. And if to assume that in this vo lume everywhere v2 0, then we obtain an equalit y contradict ing the inequalit y written above for *2. The equalit y


4 c 2 r 2 dr Mc
0

2

at v2 0 can take place, but only in limit case of particles infinitely removed from each other. Of course, on the bag surface the particles velo city can become zero (v2(r) is equal to 31


zero at r = r * ). Thus, we can really speak about some stationary (but no statical) state of the 'gas' of particles inside the bag. The particles in the bag cannot be absolutely 'cold'. They have to move, but in such a way that second powers of their velocit ies ('the temperature') v2(r) would depend only on r. Of course, everything that was said is true till we do not take into account possible quantum corrections in (3) for (nongravitational) massless fields inside the bag. It is not excluded that the contribut ion of such effects in T should beco me determining at v c in the whole volume of the bag. But close to the centre, as will be seen from the following, the particles really mo ve wit h ultra-relat ivist ic velocit ies the most probable. If is understood (as in the manual of Landau and Lifshitz, 1973) as a general energ y densit y of particles and fields inside the bag, then we can learn much about this quantit y meaning the spherical symmetry o f the problem with potential (4) for the case of the bag wit h R = r *. It turns out that the total energy densit y inside the bag wit h R = r according to a law as
*

must change wit h r

1 c4 1 (r ) 8 G r 2

(27)

which is determined only by fundamental constants of the relativist ic theory of gravitation. Indeed, for (27) we have: (1) the integral of e(r) along the sphere with R = r * gives
r*

4


0

1 c4 1 2 1 r dr Mc 2 8 G r 2

2

as it must be in accordance with Equation (24), (2) the quant it y e(r) turns continuously into the energy densit y o f the gravitational field around the bag

00 ( r )

1 GM 2 1 c8 8 r 4 |r r* 8 G 3 M

2

}

(28)

(r ) |r r*

1 c8 8 G 3 M

2

(3) the functions 00(r) and (r) do not connect smoothly on the boundary r = r * :

d 00 4 c10 (r ) |r r* dr 8 G 4 M

3

,

d (r ) 2 c10 |r r* dr 8 G 4 M

3

,

32


as it must be. It means to cross the boundary o f two media: the gravitat ion (virtual gravitons) the bag matter (with 'half-naked' particles with mass ma*). Now, let us assume that the total energy densit y on the surface of the bag wit h the radius R = GM/c2 attains several (n = 2-10?) nuclear ones ( of such a bag is determined fro m (27): n x 2.8 x 1014 g cm-3 c2 1 c4 1 , 2 8 G Rn Rn 13.84 km /n . (29)
n u cl

2.8 x 1014g cm-3). Then the radius

Accordingly for the mass of the collapsar (the bag + the field) in that case we have Mn = c2Rn/G = 9.39M/n. For example, in the case when the surface energy densit y o f the bag is equal to two nuclear ones, we have, R2 9.79 km and M2 6.64M at the mean density of the bag matter 5.75
n u cl

(of course, with the account for (24)!). If

the surface densit y achieves four nuclear ones, then we have for the bag radius and the collapsar mass, R4 6.92 km and M4 4.69 M ,
n u cl

which will take place at the mean densit y of the bag matter 13 again.

taking account of (24)

If (24) is not taken into account, the mean densit ies increase twice. Fro m the point of view of GR such an object must collapse in the black hole long ago, and, consequently, no equation of state can be said here in principle. From the point of view o f GD, in all cases we deal wit h an object with finite densit y at nonzero mass and, as it was mentioned in Sect ion 4, according to QCD at densit ies exceeding the nuclear densit y in several t imes (n = ?), the bag matter must undergo the phase transit ion into the quark-gluon plasma (QGP). It can be supposed that the objects with the parameters ment ioned above (n = 2-4) are the gigant ic macroscopic quark bags, in which even on the surface neighbouring quarks (and gluons) are compressed in such a way that they are o n distances to each other less than 1 Fermi = 10
-13

cm. I.e., even on the surface of such a bag

there are no neutrons in which quarks could exist on distances greater or even equal to 1 Fermi = 1 F. As it is seen fro m (27), the total densit y only increases deep into the bag, and, consequent ly, at the approach to its centre the neighboring quarks turn out to be co mpressed 33


more and more narrowly, i.e., any two neighboring particles o f QGP turn out to be at less and less (in co mparison with 1 Fermi) mean distance from each other at r 0. Thus the rule o f the change of densit y in (27) does not contradict at least to the QCD notion of asymptotically free quarks and gluons in such a macroscopic QCD-bag. And really, for the 'running' interact ion constant of QCD in case o f 6 variet ies (flavours) of quarks one can write

s

12 , (33 2k ) ln(1F 2 / l 2 )

(30)

where k is the number of 'unfrozen' flavours of quarks. Here the dimensio n o f 1 Fermi is chosen as a value which is called the radius of the confinement of co lour in QCD. For the present it is st ill a free parameter of the theory which must be determined fro m experiment. (For example, fro m experiments on co llisio ns o f ultrarelat ivist ic ions.) In formula (30) the quant it y l can be understood as the distance on which the quarks approach. Famous 'running' (decrease) of the constant s is obtained at unlimit ed approach of quarks (and gluons), i.e., at l 0. But to regard the quantity as purely pragmatically, retaining for as only the requirement that the co lour confinement in so me vo lume, the QCD does not forbid apparent ly such a view to objects also. The same 'running' (decrease) of the constant of the strong (colour) interaction can be obtained by taking the value 10 km as a radius of confinement, i.e., the value o f radius of macroscopic QGP-bag for the collapsar with the mass M ~ 6.7 M and the radius of the bag R ~ 10 km. In that case the constant of colour interaction



sm



12 (33 2k ) ln(10 2 km 2 / r 2 )

(31)

remains exactly the same (large), as in (30), but at r 10 km. Here already r can be understood as everywhere in this paper and, in particular, as in formula (27), i.e., r being the distance to the center of the bag. As it was mentioned above, it fo llows from (27) that at r0 the quarks (and the gluons) turn out to be pressed more and more t ight, aiming to beco me ult imately asymptotically free (as it fo llo ws fro m QCD in that case). Thus, for such (hypothetic) model o f the macroscopic QGP-bag besides formula (31) the increase o f the energy densit y (27) towards its centre must be essent ial. Specifically, in the centre of such a bag wit h R = 10 km (i.e., at r = 1 Fermi) the 34


'macroscopic' constant of co lour forces will be only about 3 constants of electromagnetic interaction (Q
ED

0.0073). For all that the density (r)/c2 will be of the order of 5.4 x 1052 g

cm-3, and the total energy (mass) in a so small sphere (r = 1 F) will be 7 x 1014g ~ 10-19M at M~6.7M. Consequent ly, on one hand a
sm

really provides the confinement of co lour in the bag

volume with the radius ~ 10 km (i.e., as a result the bag remains white), and on the other hand in spheres (relatively to the bag centre) with decreasing radii the quarks (and the gluons) beco mes more and more 'weakly interact ing' particles. Of course, for the time being the adduced estimat ions o f co lor (chromodynamic) properties of the bag must be regarded only as qualit y estimates. I do not insist on the abso lute correctness (from the standpo int of mo dern problems o f QCD also) of the who le ident ificat ion with quarks o f po int particles with masses ma* 0 bound in the bag and at densit ies /c2 >
nuci.

But even in this paper so me coincidences can be marked which allo w to

think that the study o f the co llapsar properties in GD is just 'the cross-road', where QCD and GD meet suddenly. Specifically, it is known that the quarks masses are the smaller the t ighter they are connected in a baryo n. From the results o f Sect ion 2 it fo llows that in GD also any particle (a nucleon, for example) finding itself in a bound state on bottom of the potential well at r = GM/c2, i.e., on the surface of the bag wit h R = r * , must decrease its mass by half exact ly, i.e., to change its structure! After that the gravitat ion is 'turned off on the surface of such a bag, it is natural to assume that inside the bag 'so me' fields of interact ion between bound ('halfnaked') particles wit h masses ma* 0 and at (r)/c2 more and much more than p
nucl,

can

reduce these masses further at r-> 0, down to the chiral limit with ma* 0 and c. 7. Conclusions Through all the paper the strong gravitat ional field of the co llapsar means concretely t he case when the energy densit y o f the gravitational field on the surface o f it s bag (28) co incides with the value (r) and turns out to be equal to several nuclear densit ies (>
2 nuclc

)· As a matter

of fact, apparently in that case one should speak already about the utmost strong gravitationa l field of the object with the (bag) radius < 10 km and the (collapsar) mass ~ 4-6.7 M. The precise values of the mass and the radius o f such an uttermost compact object (of co llapsar) depend most probably on precise values of parameters of phase QGP-transit io n. So, the collapsar with the mass 4-6.7 M and the bag radius <10km can be imagined as a two-phase system. The first phase is a bag itself- the region filled by matter (apparently, in the 35


phase o f QGP). The second phase is the gravitat ional field in 'vacuum' around the bag, where it (the field) interacts with itself only

and does not interact already wit h the bag (at least in the sense in which it was said in previous sect ions). The basic properties of such an ut most compact object are presented by the fo llowing parameters: (1) a half o f the co llapsar mass (Mc2/2) is contained in the bag and another half in surrounding gravitat ional field; (2) the potential o f the bag surface achieves its limit value equal to
GD

= - c2/2;
GD

(3) the mass of the particle finding itself on the bag surface in bound state (at c2/2) is two times less than the mass of the same particle in free state (i.e., at r );

=-

(4) the energy densit y (of fields and particles) inside the bag is given by the dependence

(r )

1 c4 1 . 8 G r 2

Of course, all these and other properties o f the co llapsar are ult imately the consequences of an assumpt ion, grounded in previous papers [P1] and [P2]:
ik (0)

=

ik

(2)

I.e., in every po int of 'vacuum' around the bag the energy-tensio n densit y o f scalar and tensor components of gravitat ion are equal to each other. That is why the gravitat ional field of such a limit object is presented by 4-potential (4). It is seen fro m Table I that it is not every object which can be in ut most bound state, i.e., under its sphere of maximum instabilit y r = 1.5GM/c2. Though the exact values of the mass and the radius o f the utmost compact gravitational configuration (the co llapsar) are not known yet, but one can suppose now already that the objects wit h the mass <6M and the acceleration on the surface o f the order of 1014cm s-2 (see Table 1) are 'supermassive' neutron stars, and this are just these objects which are ma ximally instable relatively to the transit ion (the collapse) in the ut most bound state. It can be said already now that such a collapse occurs at the cost of the decrease o f the total energy (Mc2) of such a 'supermassive' object. It is just this co llapse of the objects with masses <6 and radii < 13 km which realizes apparent ly the phase transit io n when a gigant ic bag is formed wit h quarks 'squeezed out' of nucleons (in the 36


process of transit io n through the sphere r = 1.5GM/c2). Such a phase transit io n sizes now the who le vo lume of the bag. It is in this (utmost) situation that the word 'bag' approaches the sense o f the notion of the bag used in QCD. In the rest of cases, and in case of big masses (especially up to cosmological ones) there is no phase transit ion. According to the est imates o f Table I the densit ies in such cases can be arbitrary, moderate and small, though as earlier such objects are the most compact for their masses M. In that case they can be in a stationary state with the region dimensio n filled by matter greater than 2 r * . But here one can, nevertheless, use the word 'the bag' meaning the co mpactness (00/c2 c2) and the gravitat ional connect ion o f such bodies. Of course here there is no total cessation o f gravitat ion action as in the case o f macroscopic QCP-bag wit h R = GM/c2. The outer gravitation at R > r
*

for such bags compresses the object as in Newtonia n

theory, and the farther the bag dimensio n fro m the dimensio n o f limit sphere 2 r * , the more the deviat ions fro m dependence (27) must be for the densit y (r), and there is no jo int (28) on the surface o f such bags already. At the increase of M the states of matter in such bags will be less exotic. At the increase o f the object mass (at M >> 6 ) there is no reason to make the object to 'break through' in the limit bound state at the given mass M. All other interaction determining the equat ion of state (the temperature, the pressure, the rotation mo mentum in hierarchical systems) in such a bag can compete wit h the gravitat ion. These objects are hardly suited to be called collapsars. Though the energy densit y of gravitational field near the surface of their bags can be co mparable here also with the matter densit y in the bag (the compactness!), but an abso lute value 00/c2 is here far less than
2 nuclc

at M >> 6M.

Thus the distribut ion e(r) of the total energy (27) inside the bag with the radius R = r * (< 10 km) can be a distribution at which 'naked' and 'half-naked' point particles inside the bag are packed most closely in it. Then R = r * = GM/c2 is a radius of the most dense packing o f the body rest mass wit h the total energy Mc2, with the body mass approaching the value M at distances r >> GM/c2. The states with smaller dimensio n o f the bags (R < r * , but not less than r * /2 !) are states with the engaged 'ant i-gravitation' - the repulsio n. In such a state (at Figure 1 to the left fro m the point r = r * ) the bag seeks to expand. At masses M < 6 M* the 'ant i-gravitation' forces beco me huge, of the same order of forces which caused the collapse in the utmost bound state at R = GM/c2. The bag wit hdrawn in that way fro m equilibrium must begin to 37


pulse (with the period GM/c3) radiat ing the surplus o f energy in the form o f scalar gravitat ional waves. Summing up the who le aforesaid it is necessary to emphasize that potential (4) is a tensor potential written for arbitrary values of mass M for the field out of the bag. The notions of the mass, the radius and the properties of the utmost bound object (the collapsar proper) demand ult imately t he exit beyo nd the limit s o f gravidynamics. Certainly it is necessary to account here for the informat ion obtained at accelerators when the properties o f matter in the state of QGP are studied. There are a lot of papers in astrophysics now whose authors endeavour to use QGP properties at the grounding of the possibilit y o f the existence of quark stars in nature (within the limit s of tradit ional approach on the basis o f the OppenheimerVolkov equation) (Krivoruchenko, 1987; Haensel, 1987). So, the proposed paper means the mot ion toward almost the same aims in the investigation o f the utmost compact objects properties. In particular, in gravidynamics also (like GR) the experimental (observational) investigat ions stay actual of the same objects - the candidates for black ho les of GR. But as it was said many t imes in [P1, P2], the collapsar physics is abso lutely different. Ult imately the possibilit y o f periodic oscillat ions (pulsations) of the QGP-bag wit h the period (GM/c2)/c < 10 km s-1 c-1 3 x 10-5 s near the posit ion o f its total equilibrium must lead to a definite observational test of the aforesaid. References
Baryshev, Yu. V. and Sokolov, V. V.: 1983, Trudy AO LGU 38, 36. Chernavskaya, O. D. and Chernavsky, D. S.: 1988, Usp. Fiz. Nauk 154, 497. Emel'yanov, V. M., Nikitin, Yu. P., and Vanyashin, A. V.: 1990, Fortschritte der Physik 38, 1. Haensel, P.: 1987, Prog. Theor. Phys. Suppl. 91, 268. Krivoruchenko, M. I.: 1987, Pisma v ZhETF 46, 5. Landau, L. D. and Lifshitz, E. M.: 1973, Field Theory, Nauka, Moscow, p. 504. Shapiro, S. and Teukolsky, S.: 1983, Black Holes, White Dwarfs and Neutron Stars, Wiley Inc., New York. Sokolov, V. V.: 1992a, Astrophys. Space Set. 191, 231. Sokolov, V. V.: 1992b, Astrophys. Space Sci. 197, 87.

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