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Astrophysics and Space Science 197: 87-108, 1992. © 1992 Kluwer Academic Publishers. Printed in Belgium.

NONLINEAR GRAVIDYNAMICS: ENERGY-MOMENTUM TENSOR OF COLLAPSAR FIELD

VLADIMIR V. SOKOLOV Department of Relativistic Astrophysics of the Special Astrophysical Observatory of the Russian Academy of Science, Stavropol Territory, Russia

(Recei ved 28 November, 1991) Abstract. Within the scope of a theoretical scheme treating gravitational interaction consistently as a dynamical (gauge) field in flat space-time, an expression was obtained for the density of gravitational field energy-momentum-tensions in vacuum around a collapsed object (collapsar). The case was studied of an interacting static spherically-symmetric field of the collapsar in vacuum with taking into account all possible components (spin states of virtual gravitons) contributions into energy for the symmetric second rank tensor ik. The radius of a sphere filled by matter for the collapsar with mass M can reach values up to GM/c2.

1. Introduction This paper continues the paper (Sokolov, 1992) which began the study of physical properties o f objects with extremely strong gravitational fields on their surfaces, i.e., on co llapsars fro m the standpoint of the consistent dynamical descript ion of gravitational interaction. The main purpose of the paper is the so lut ion of the problem in energy-mo mentum-tensions o f the collapsar field when this gravitational field is strong. In what fo llows, the question is basically on strong fields of objects with masses o f the order of several so lar masses - i.e., on the co llapsars wit h stellar masses. Macroscopic average densit ies o f such objects are at least nuclear ones and achieve supernuclear densit y. Correspondingly, gravitat ional field energy densit ies on such collapsar surface may become equal or even more than
nucl

c2. If we take as an example such object as a neutron star with the mass M = 1.44 and wit h

the radius 10 km and evaluate the gravitational field energy densit y on the neutron star surface by the formula ( N)2/8G, then the energy connected with the field alone turns out to be enormous and approximates to the rest energy of the neutron star matter itself. Can the field energy densit y greater than or of the order of
nucl

c2 be non-localizable? In the

context of a purely geo metrical interpretation of gravitat ion field the answer to this quest ion is
1


known for a long time: the field energy is nonlocalizable even in such a case. The consistent dynamical formulat ion o f the gravitat ional interaction theory proceeds as a matter of fact fro m the notion that every cubic cent imeter of space contains a co mpletely determined quantit y o f gravitat ional field energy-mo mentum-tensio ns. Of course, the final answer in this debate will be obtained as a result o f a space regio ns observatio n wit h the strong field. Just in the context of possible new observational consequences I cont inue to formulate here the collapsar problem in gravidynamics. In part icular, as was noted in the previous paper (Sokolov, 1992), the collapsars can have surfaces always. But for a rigorous proof of that, one must first of all clear up co mpletely the quest ion on the gravitat ional field energy-mo mentum-tensor (EMT). In connect ion wit h the foregoing I shall emphasize through all the paper the characterist ic features of formulat ion o f the co llapsar field problem in direct connect ion wit h the field energy problem. For all this I consistent ly adhere to the theoretical scheme in which the gravitat ional interact ion, equally with other ones, is considered as a dynamic field plunged into flat space-t ime. I note here once more that in such a case one ma y accept at once that energy is localizable, posit ive and is understood in the same sense as in any other field theory, in particular, in the classical electrodynamics. I am not going to prove here especially the justice of such natural demands (axio ms) in the dynamical field theory. It is more interesting to elucidate what observat ional consequences their fulfilment brings to, if the axio ms are really true. We begin here (in the Introduction) with the most important, principle aspects underlying the approach developed by us (Sokolov, 1992; Sokolo v and Baryshev, 1980; Baryshev and Sokolov, 1984) to the descript ion of gravitat ional interact ion. The term 'gravidynamics' (GD) used below (and also frequent ly used by specialists in gravitat ion) seems to me the best one reflect ing the features of our approach. As was shown in detail in the previous paper (Sokolov, 1992), the field energy densit y near a gravitat ing body with the mass M at the distance r fro m its centre, can be given as a matter of fact by 00 = ( N)2/8G, where N = ­GM/r ,

if at its deduction one take into account the fulfillment of three main condit ions for the field EMT:
00

0 ,

ik = ki , ikik = 0 ,

i,k = 0,1,2,3.

Where ik = diag(+1, ­1 , ­1 , ­1 ) is Minkowsky's metric tensor. I emphasize at once that in GD you may use only this always the same constant metric at the consistent dynamic description o f gravitat ional interact ion you may do that in the case of all other interactions. To understand correctly fo llowing sections one should not forget and consistent ly adhere to the concept (which became already a commo n place) that, in the relativist ic field theory, one may not ascribe straight away so me finite dimensio n both to test particles and to particles ('matter') - sources
2


of the field. I.e., at the formulation of a gravitational interact ion relat ivist ic theory (like for all modern field theories) it is more logical at least at the beginning to proceed fro m the fact that the right-hand sides o f corresponding field equat ions can contain a po int source or a system (a ma ) of point sources: i.e., a gravitat ing 'body'. In part icular, every macroscopic regio n, which the real gravitat ing body is formed o f, can be presented as a 'point' with the mass ma. These regio ns are the 'po ints' between which mainly only gravitat ion acts. In GD the fundamental Special Relat ivit y concept about interacting points (usual for local theory) is used as an initial concept of 'gravitat iona l charges'. Of course, a quest ion arises on the just ice of these idealized notions for the macroscopic theory which the gravitation theory is. As we will see from the following, an exhaustive answer to such a question can be obtained ult imately as a full so lut ion of the problem of physical properties of GD collapsars. As it was in the paper by Sokolov (1992), one can begin again to investigate first of all what is a unit 'elementary' po int source with the mass M. In a sense, the co llapsar itself is such an 'elementary' object by analogy with the elementary po int charge - electron - in electrodynamics (ED). In GD around any spherically-symmetric distributed mass there is also the field with the energy densit y
00

- 'a coat' of virtual gravitons. At some distance from the centre of the object in 'vacuum' (i.e., out of the sphere filled by matter) 00 can turn out to become o f the order of the average rest energy densit y of the collapsar - i.e., of the system 'matter + field' 00 = GM2/8r4 Mc2/r3 . Thus, in GD which is a macroscopic theory a question arises, which we meet one way or another in the classical ED, quantum ED (QED), quantum chromodynamics (QCD), etc.: where in this case and in what form is the mass o f such a 'po int' object concentrated? What is the co llapsar rest mass in general in GD? Such problems inevitably arise in GD also when the distance fro m the co llapsar centre become o f order of the gravitat ional radius GM/c2 o f this 'po int' object. Here it turns out that the quant it y GM/c2 is a direct analogue of the classical electron radius. Just as in the classical ED, one can show that unt il the point distribut ion (a ma ) compressed to a dimensio n when distances between the points beco me o f the order of Gma/c2, we deal wit h a theory quite analogous to the classical (and linear) ED. From the foregoing it is clear that in GD the 'po int' source with the mass M is in fact 'so mething ' having a finite dimensio n greater-than or ~ GM/c2. At r >> GM/c2 one can be uninterested at all in the structure of such an 'elementary' po int object. But then the mass of the 'coat' of the virtua l gravitons must be automat ically included in the source mass. Thus, that part of the theory, in whic h the notion o f the po int gravitating object with the mass M is true, can be described by linear equations.
3


I.e., in this linear approximat ion GD when gravit ating sources may be quite assumed to be point structureless objects for which the mass conservation is st ill fulfilled to a high precisio n as

dx k 0 , where dt , k


a

ma(r ­ ra),

one can, by using usual rules, put down the Lagrangian o f a symmetric tensor field ik interact ing with its sources. I emphasize that the consistent dynamic interpretation of the field equat ions fitting to this Lagrangian rely on fact that potentials of the field ik (just as in ED) should be understood absolutely independent ly of the chosen metrics ik. In particular, in GD it is senseless to speak about the condition ik << ik. Like the vector 4-potent ial in ED, ik can be o f any value in virtue of indeterminancy o f ik ik + Ai,k + Ak,i + ,ik This transformat ion for ik is the gauge one wit h an arbitrary 4-vector Ai and arbitrary 4-scalar . It can be connected as usual wit h the masslessness of the tensor field ik. Ult imately it is not the Lorentz invariance demand alo ne but the demand of the gauge invariance in the linear approximat io n of GD also which determines both the field Lagrangian and the field equat ions in a unique fashio n. In the linear approximat ion the interaction of the field ik wit h it s sources is described by t he term fikTik of the Lagrangian (f is the coupling constant). If adhering consistent ly to the structurelessness (or the pointness) of particles wit h the rest mass ma 0 interacting with gravitat ion, then to describe the substance which is usually called the 'matter' we must choose as the point of departure the tensor (EMT) of the system of structureless po int objects-particles: Tik = cuiuk(ds/dt), where ds = c dt 1 v 2 / c
2

where ui is the velocity 4-vector and v is the usual velocity of particles. In the linear approximation of GD, for one ('elementary') motionless po int particle wit h mass M in the origin o f coordinates (ra r = 0 and v = 0) this tensor has the simple form Tik = Mc2(r) diag(1,0,0,0). For that massive gravitat ing centre it fixes the reference frame in which we may investigate the field of such a source. In the same linear approximat ion if we base on the interaction f ik Tik = fc(ds/dt) uiukik which one can write down in the symbo l form

4


. Consistently adhering to the dynamical interpretation of the field ik, we can obtain also the equations of motion for particles in a given field ik just as it is done for an electron moving in a given electromagnet ic field (cf. Landau and Lifshit z, 1973). One can understand as the universalit y o f gravitat ional interact ion in GD the fact that f is identical for all fields. In accordance wit h that, the interact ion of electromagnetic field wit h the gravitational field ik must be written in the form

. Ult imately it gives the correct description o f interaction effects between light and given gravitational field (such as light deflect ion and radio signals lag in the field o f the Sun). But for the correct descript ion o f the redshift effect one must add the interaction f ik
(e) ik

wit h the spinor (e-e+) field
ik (el)

constructed by the same rule, i.e., with the same f to the interaction f ik t

(Mosinsky, 1950).

In the theoretical scheme based on a consistent dynamical description of gravitat ional interact ion, the introduction o f nonlinearit ies turns out to be direct ly connected wit h the field energy problem. The unviersalit y of gravitat ional interact ion leads to the fact that any field (including the gravitat ional one) interacts with any gravitational field the stronger the higher is its energy. It is evident that the nonlinear GD is the interact ion when the distance between po int particles with mass ma beco mes of the order of Gma/c2. For the co llapsar it means that the distance fro m its centre can reach the value o f the order of GM/c2. I.e., for the field in vacuum the condition
00

~ Mc2/r3 is already sat isfied. In

accordance wit h the universalit y of the gravitat ional interact ion one can assume the field itself to be the source of gravitat ion (that lacks in ED). It means that in Lagrangian, besides the term f ik Tik, terms arise of the type

. Here it is seen especially well that the gravitation field energy localizabilit y, as well as the 'po intness' of the part icles interact ing wit h the field, can be connected simply wit h the demand of locality o f
5


gravitat ional interact ion. (Such a method of introduction o f nonlinearit ies in GD is analogous in a sense to the transit ion fro m the tree approximat ion to the one-loop approximat ion in QED.) Accordingly, in the right-hand side of the field equations it leads to the including of the gravitat iona l field EMT ik into the sources also. Then near the collapsar gravitat ional radius a kind o f 'splitt ing' of the point source Tik = Mc2(r)diag(l, 0, 0, 0) occurs. I.e., in such a case we can say t hat the degeneration by M is as if it was taken away. A part of the collapsar mass can be now a field mass. It is natural that the account ing o f the ik in vacuum around the region filled by matter influences also the form o f the 4-potential ik of the co llapsar field. That leads in part icular to a complete explicatio n of observat ional effects of minor planets perihelio ns shift, the shift of periastron in a binary syst e m with the radio pulsar PSR 1913 + 16, etc. The cho ice o f the EMT ik of the co llapsar strong field is a part of the so lution of the co llapsar problem in GD. I want to emphasize here that I do not try to describe at once the nonstationary process of co llapse as the passage o f the system into a bound state, i.e., into the co llapsar. It is more simply to state first the problem o f probabilit y itself o f steady stationary states of the system 'part icles + gravitational field' = the system wit h the given rest energy Mc2. I.e., the quest ion may be on a possibilit y (depending on the nearness o f an object to its gravitat ional radius) to speak about such a system as about a 'point' with a definite rest mass M and with the Newton gravitational field (in virtue of the accordance principle) = - GM/r at r >> GM/c2. Thus we can assume that the problems o f the co llapsar (but not collapse) in the GD may be: the problem of probabilit y of existence of a stationary steady state with a given total energy (Mc2), the problem of the regio n dimensio n (a 'bag' filled by particles-matter) ~ GM/c2, the problem of the total mass o f the 'bag' and the particles densit y in it, the problem of the field energy densit y in vacuu m around the 'bag' and the total 'mass' contained in the field surrounding the 'bag'. After this section's remarks we can pass to the substantiat ion of the cho ice o f the co llapsar field EMT. But before, it is necessary at first to elucidate the physical sense of Hilbert-Lorentz gauge condit ion for the 4-potentials ik in connection with particularit ies o f the interacting gravitat iona l field in vacuum near the 'bag' surface (Sokolov, 1992; hereafter referred to as [PI]). 2. The Components of Vector and Tensor Massless Fields and Gauge Conditions Certainly, the main purpose of this section is a refinement of the gauge condition sense for the case of the symmetric tensor field ik describing gravitational interaction. But first I shall try to introduce well-known examples from ED whose ideas underline all modern interaction theories. 2.1. VEC
TOR FIELD

In ED the field Lagrangian is constructed usually from three invariants:

6


I1 = Ai,kAi,k ,

I2 = Ai,kA

k,i

,

I3 = Ai,iA

k

,k

.

Since the two last ones differ from each other only by divergence then ult imately for the vector field Ai, interacting wit h its sources, the Lagrangian consistent only wit h the condit ion of relativist ic invariance will be in the most general form: i.e., Lel = ½ (Ai,k Ai,k ­ dAk,k Ai,i) + jk Ak . (1)

Where d is an arbitrary (auxiliary) constant for the present. Corresponding field equations will be

Ak + dA

,i ik

= ­ jk ,

(2)

2 1 2 2 2 xi xi c t In such a theory there is a 'superfluous' scalar: i.e., the vector Ai is a part of field equat ions (2) both directly ( Ak) and in the form o f a scalar ­ the 4-divergence Ai,i (dA
,i ik

). Corresponding to that, they

say that the vector field can also describe simultaneously the scalar field - the scalar component of the field. But if equat ions (2) describe the electromagnet ic field then the experiment demands the use of the a conserved electromagnet ic current in (2): i.e., j
k ,k

=0,

(3)

Thus, the scalar co mponent is absent in the very source jk of the field Ak. In the special case, when d = 1, the whole theory becomes also a gauge invariant one: namely Ai Ai +,i , (4)

where is an arbitrary 4-scalar. This case (d = 1) combines naturally the absence o f the scalar source (3) and the absence o f the scalar co mponent (Ak,k) in the fie ld Ai . The point is that in the gauge invariant theory the very absence o f the scalar (3) at d = 1 beco mes a compulsory (obligatory) condit ion for the vector source ji. Such a condit ion fo llows the so-called strong law - Noether ident it y (see, for example, the book by Konoplyeva and Popov (1973) on the second Noether theorem). The strong conservat ion law ( j
k ,k

= 0) is a direct consequence of gauge invariance (4) of the Lagrangia n

(1) at d = 1. Conformably they say so met imes that condit ion (3) is necessary for the consistency o f field equations with sources (2) at d = 1. In short, the gauge invariance (4) demands the fulfillment of conservation (3). It is well seen in the vector field case in quest ion. Firstly, if we take the 4-divergence o f the left-hand side of equat ions (2) at d = 1, we obtain the ident it y (Noether identit y) in the form

A

k

,k

+A

,ik ik

A

k

,k

- A

k

,k

0.

This ident ity (at d = 1) is fulfilled independent ly of value of the scalar Ai ,i, and this identit y demands also the absence of the scalar source ji,i = 0. (Although in part icular, it is permissible, that Ai,i 0.) Second, in accordance wit h the fact that ji,i = 0, one may use directly (at d = 1) the gauge invariance of equat ions (2) and exclude the superfluous scalar Ai,i stipulat ing for it the gauge condit ion (Lorentz
7


gauge): Ai,i 0 . field has no scalar component (ji,i = 0 Ai,i = 0). In the case o f the gauge invariance absence (d 1) the excluding of the 'superfluous' scalar Ai,i does not appear so natural as at d = 1. If we take the divergence of the left-hand side of (2) then it does not yield already the identical zero: (5) If there is no scalar source then it is natural to assume that there is no corresponding field, i.e., the Ai

A

k

,k

­ d Ai,i = (1 ­ d) Ai,i 0 .

Now, in general, the scalar source may be nonzero, the field equat ions themselves do not demand direct ly its absence. But if to demand nevertheless once more (usually they refer here to experiment) the fulfilment o f differential conservat ion ji,i = 0 but at d 1 , (3')

then the equalit y o f divergence in the left-hand side of (2) to zero is ensured as a consequence o f the addit ional demand (3') (that does not follow the theory direct ly): (1 ­ d) Ai,i = ­ ji,i 0 . We can say that the scalar part (as the component Ai) is the so lution o f the equation with the scalar source tending to zero. It is usually said also that scalar photons remain in the theory but they cannot be radiated and absorbed. I.e., the equat ion Ai,i = 0 does not mean, generally speaking, that Ai,i 0. In that case the demand that Ai,i = 0 for consistency wit h (3') means the fulfillment of one more additional condit ion lacking in the theory. But, on the other hand, ult imately t hey appeal to the scalar field co mponent Ai requiring the addit ional degrees of freedom for virtual photons. And what if in this case also we try to adhere to the idea that the account for addit ional field co mponent is connected with the vio lat ion of differential law (3') while rigorously observing the integral charge conservation? Then if in the gauge invariant theory (d = 1) the conservat ion ji,i = 0 is fulfilled and the field scalar component Ai,i (virtual quanta) can be naturally excluded from the theory, the case d 1 may be considered in a sense as one corresponding to a 'vio lated' gauge symmetry. In that case it is logical to adopt the fact that the different ial law ji,i = 0 is not fulfilled already (for almo st real and virtual quanta): ji,i 0 at d1 . (6)

Ult imately, as we could see, there is no change by such po int of view in ED but now the arising o f scalars in the theory is a more natural alternat ive. Then the appearance of the scalar is connected with its corresponding scalar source.

8


In particular, at the quant izat ion of the electromagnetic field they use, namely, the gauge noninvariant form of the Lagrangian at d 1. Formally it corresponds to the presence of the scalar Ai,i 0 in the theory, because o f the fact that the condit ion Ai,i = 0 remains fulfilled only on the average. Then as far as we may assume that ji,i = 0 also only on the average, the corresponding scalar quanta cannot exist far out of the region of averaging. Thus, if we assume nevertheless the vio lat ion o f the different ial law ji,i = 0 in the regio n o f averaging, i.e., in a sufficient ly small (< 10
-10

cm) space region, then it is in accordance wit h the absence o f gauge invariance in t hese regio ns and the presence o f the scalar Ai,i is appropriate in them. Indeed, since we may demand the fulfilment of the condit ion Ai,i = 0 only on the average (Bogolyubov and Shirkov, 1973) - i.e., only for average values of A,ii = *Ai,i = 0 , then we could not assume that the current conservation is fulfilled also, generally speaking, only o n the average and in the same permissible states J
i ,i

= *ji,i = 0 .

But it can be understood as differentially the charge does not conserve near the electron. Because the analogous reasoning will be used below in GD, now we need a more detailed explanat ion though the question is here only on some different point of view to well-known facts. At distances less or of the order of /mec (3.6 â 10
-11

cm) near the electron the effects of vacuu m

polarizat ion beco me more and more important in ED; e-e+ pairs arise. For all t hat, the total charge o f electron at distances >> /mec is equal to e and is rigorously conserved in accordance with experiment. But also in accordance wit h experiment this charge is not conserved (different ially) in regions o f dimensio ns < /mec. Usually it means an increase of electromagnet ic coupling constant = e2/4c at small distances from electron. Such interpretation o f e-e+ pairs influence allows thinking that it is here, where the scalar source differs from zero ji,i 0, the scalar field-scalar photons (virtual photons) must arise. Thus, generally speaking, the vector field Ai corresponds to particles with two spins: 0 and 1, [ Ai ] = 0 1 . Accordingly, the current ji can in the general case be a source of particles of two spins: [ ji ] = 0 1 . (8) (7)

For all this, the scalar parts Ai,i and ji,i correspond to particles wit h spin 0. Usually their exclusion in the gauge invariant (d = 1) limit wit h the use of corresponding gauge condit io n Ai,i = 0 and at conservation o f current ji,i = 0 remains in the theory only photons with spin 1 ­ the 'purely' vector for real photons. But as we have just seen, near an electron the photons may have any possible spins (0
9


1) which the vector Ai contains. (For example, at count of the scattering amplitude one must take

into account four possible states of po larizat ion for this set of spins.) As we show in the fo llowing sect ion, the analogous situat ion arises also in GD for field near the surface o f the 'bag', i.e., in the strong gravitat ional field o f the co llapsar. But here (in the macroscopic theory) the processes o f field self-act ion play the role of quantum processes of pair production. 2.2. S
YMMETRIC TENSOR FIELD

Let us pass now to a more complicated case o f the symmetric tensor field ik. Here also one can construct in the general case the Lagrangian (fo llo wing only relat ivist ic invariance for the present) of five quadratic invariants, formed fro m derivat ives of ik: I1 = ik,l ik,l , I2 = ik,l il,k , I3 = ,k ,k , I4 = ,l k
l ,k

,

I5 = il,i kl,k .

As far as I2 and I5 are dependent, the requirement of only Lorentz invariance leads to the Lagrangian of the interacting field ik in the form
4


A1

CAI A

f ik T ik , 2 c

(9)

with four arbitrary coefficients CA, instead of one, whit h was in the case of vector field. If to proceed at once, as for the main case, fro m a self-consistent, gauge-invariant scheme, which is fulfilled in the case o f the linear GD [ P 1 ] , then these four coefficient C1, C2, C3, and C4 are defined straight away as we require invariance of the field Lagrangian and field equations with respect to the gauge transformat ion of t ype (4): Ai Ai + ,i ik ik + ,ik , (10')

with an arbitrary 4-scalar . But in that case the symmetric tensor field ik = ki allows more genera l gauge transformat ion wit h an arbitrary vector field Ai in the form ik ik + Ai,k + Ak,i + ,ik . Corresponding relativist ic and gauge invariant field equations will be ('lo ng equat ions') (10)

ik + k

m

,m

i

+ im

,m

k

­ ,ik ­ ik(m

n

,mn

­ ,nn) =

f Tik . 2ac 2

(11)

Notations are here ident ical to those in [ P 1 ] and Tik denotes as usual the energy-mo mentum-tensor (EMT) of point particles. Equations (11) lead automat ically to a requirement of the strong conservat ion law T
ik ,k

=0,
10

(12)


that must, as in ED, correspond in certain situat ions to the absence of a vector source now already. It is natural (just as in ED for d = 1) to assume that if there is no vector source Tik,k, then by the direct use of (10) one may also exclude, in particular, the vector field corresponding to this source and contained in the tensor ik. But here we should understand what the vector field corresponds to the vector Tik,k? If we took the divergence of the left-hand side of equation (11) then it would be identical zero (the Noether ident it y, or the strong law), which is a consequence o f the immediate assumpt ion o f the gauge invariance. On the other hand, I shall be interested here mainly in the 'vio lations' of gauge symmetry which is possible in so me situat ions, for example, in static field of the co llapsar, analogously to what was just described for the electron case. I.e., I need a nonzero vector Bi (in a generally speaking gauge noninvariant theory) disappearing at the 'restoration' of gauge symmetry and disappearance of a corresponding vector source. In other words, I need the vector Bi consistent with the an equation of t ype Bi = ­f Tik,k. The point is that the symmetric tensor ik yields a more wide choice of 'superfluous' field components than the vector field Ai. Here it is easy to pick out a scalar mm, two possible vectors ik,k and ,k and also two scalars ik invariant manner, using the identit y ik ik + ¼ ik , where ii 0 .
,ik

and ,kk. One can separate tensor from scalar by

The vector Bi of our interest is a combinat ion of two possible vectors ik,k and ,i : Bi = ik,k + b,i = ik,k + bik,k consistent with an equat ion of type

( ik,k + bik,k ) = ­fT

ik

,k ik

.
,k

(14') = 0 and at this limit the equalit y Bi = 0

In the gauge invariant limit the vector source disappears, T

can be sat isfied. I.e., in that case one can immediately use the gauge invariance by a requirement of disappearance o f the vector Bi, then the correspondent cho ice b = ­1/2 will be determined unambiguously. Here I have only said in other words what is said usually at the Hilbert-Lorentz gauge condit ion cho ice: i.e., the foregoing means that the gauge is chosen in such a way that if in the theory just this vector is absent, Bi = im,m ­ ½ ,i = 0 , then 'long' equat ions (11) are transformed to the known form
1 f ( ik ik ) T ik . 2 2 2ac
11

(13)

(14)


It points that condit ion (13) for the vector B' (in gauge invariant limit) keeps the correctness o f what we had in the case of the 'lo ng' equat ions: identical zero ing of the left-hand side divergence (the Noether ident it y). Now it is fulfilled also for (14) at Bi = 0. One can say that Hilbert-Lorentz gauge (13) just as condit ion (12) for sources are also the consequence o f the strong conservat ion law in the case o f the tensor field ik. But the main thing is here that condit ion (13) guarantees the excluding of the vector field from the theory in the gauge invariant limit and thus the absence of the vector source (12) is consistent with the absence of its corresponding vector field Bi im,m ­ ½ ,i
im ,m ,i

­¼

.

(15)

Just as in the case of the vector field Ai, the symmetric tensor of the second rank ik can be decomposed into corresponding spin parts. Ten independent components of ik can be grouped into two fields of zero spin (0 0), one field of spin 1 ( 1) and one of spin 2 ( 2): [ ik ] 0 0 1 2 . (16)

For all that, the spin parts, corresponding to 'superfluous' scalar and vector components of the tensor field ik, could be presented in the form [ = mm] 0 and [Bi] 0 1 .

And in general case, the symmetrical tensor Tik itself is the source of these fields with four spin parts also ­ i.e., ten independent components of T
ik

can be grouped into four sources: (17)

[Tik] 0 0 1 2 .

But in the gauge invariant limit (in the linear GD) the condit ion Bi = 0 at the conservat ion of the current T
ik ,k

retains in the theory the gravitons wit h two spins. Accordingly, for real gravitons [ik] 0 2 , [Tik] 0 2 . (18)

Where the source of purely scalar gravitons is a nonzero trace of the point particles EMT is (T = Tmm) ( = mm) . Thus, one 'superfluous' co mponent () is, however, possible in the GD (see in detail in [P1]), i.e., in GD there is, in principle, a possibilit y o f real zero-spin gravitons or massless scalar bosons emissio n. 3. Virtual Vector Field of the Collapsar Here as the init ial notion of the co llapsar (an 'elementary' object in GD) I shall mean the fo llowing: this is a bound spherically-symmetric object (an analogue o f Schwarzschild's black ho le) ­ a 'bag' filled as before by point particles with m*a 0 and by fields together with the gravitat ion field surrounding this bag on the outside. Inside the bag the field cannot be only a gravitat ional field, it

12


depends on the particles densit y in the bag. The condit ion of spherical symmetry is connected with the fact that the 'elementary' object - massive po int - yields by definit io n a spherically-symmetric field around it. It is natural to study first the case of such symmetry of the 'elementary' object down to its centre. Unlike free (or almost free, as in the linear GD) particles with the rest mass ma we use as in [P1] the designat ion m*a ma for particles bound inside the bag. The bound particles wit h masses m*a 0 move in this bag in such a way that it is natural to imagine the bag itself and the fie ld around as a certain stationary state in which there is a cont inuous exchange between the bag and it s surrounding field, plus gravitat ional field self-act ion processes. The latter are the most essent ial at small and extremely small dimensio n of the bag. For all this, if energy o f the who le configurat ion (the bag + field = the co llapsar) is constant and equal to Mc2, then at a constant energy contained in the bag such an exchange leads ult imately to the reaching o f the steady-state values of energy-mo mentum-tensio ns outside the bag also, i.e., in vacuum surrounding the bag. For macroscopic objects (the collapsars) with mass M of the order of stellar mass or more the above-ment ioned exchange means the existence of a static spherically-symmetric (at sphericallysymmetric distribut ion of matter in the bag) gravit ational field in vacuum around the bag. As in [P1] I shall speak here mainly about this external static vacuum so lut ion o f field equat ions. As to the bag itself, it is for the present sufficient to suppose its spherical symmetry o f the distribution and the mot ion of po int particles with the rest masses m*a 0 bound in the bag (the more specificat ion of the bag features will be in the next paper). Here the quest ion is mainly on the co llapsar gravitat ional field EMT, i.e., on the choice of an expressio n for the EMT when the fie ld interacts with its sources. Generally speaking, we must now keep in mind also the processes of the field self-actio n which can be pictured as

for tensor (spin 2) and scalar (spin 0) gravitons separately. Such processes beco me determining at a large densit y
00

of gravitat ional field energy at distances from the co llapsar centre of the order of

GM/c2. In the next section I shall try to elucidate how the field energy is co mputed near the bag with the dimensio n o f order of the co llapsar gravitat ional radius GM/c2. The designations are here mainly the same as in [P1] though somet imes so me more accurate definit ions are needed. Since the co llapsar field interacts continuously wit h the bag (and the bag with the field) and self13


acts near the bag in accordance wit h (19), it is not surprisingly that EMT of this field is not conserved (strict ly speaking), i.e.,
ik ,k

0

.

(20)

On account of the spherical symmetry and static (stationary) character of the field under consideration in vacuum, there remain only t hree identical nonzero components of the 4-vector (20):
00 ,0

= 0 , 1

1

,1

=

22

,2

= 3

3

,3

0,

(20')

(see formula (48) in [ P 1 ] ) . Formally these co mponents can be ident ified with the presence o f a pressure gradient for an 'environment' around the bag. (Of course, the equalit ies (20') is fulfilled here only approximately, at the characterist ic times >> GM/c3. This will be said in more detail in the next paper.) The nonzero vector
ik ,k

must be the source of the corresponding vector field. In general there is

no innovat ion in it, we should only take the 4-divergence o f both left-hand and right-hand sides of vacuum equation (49) in [P1]:

1 d2 f k [r ,ik (r )] 2 r dr 2ac 2

,k ik

,
ik (p)

(21) of the 'po int' source in
ik (p)

where the equat ion is written only for the addit ion ik to the potential [P1] for ik
ik (p)

+ ik (see formula (28') in [ P 1 ] ) . H e r e o n l y the new designation



(p)

ik

is introduced here for the 4-potential (28) in [ P 1 ] :

( p) ik



3fM 111 diag (1, , , ) . 2 16 a r 333

It is obtained in [ P 1 ] without account for ik in the right-hand side of the field equations, i.e., at r >> GM/c2. For the structureless 'po int' source the Hilbert-Lorentz gauge condit io n
(p) ik ,k

= ¼ ,i is

still sat isfied at this approximation. I.e., the relevant vector field is absent, which means that
(p) ik ,k

­ ¼ ,i = 0 .

(22)

Certainly, at more logical reasoning one should keep in mind that equation (21) is a consequence o f the more general equat ion
1 f ik ( ik ik ), k (T(*) ik ) 2 2 2ac

,k

(23)

with the nonzero (different ially) right-hand side already. This equat ion must be written down excluding the bag (i.e., the region wit h T
ik (* )

0 for bound part icles) and taking into account the
14


spherical symmetry

1 d2 1 f [r ( ik ik ),k ] 2 r dr 4 2ac 2

ik ,k

.
ik ,k

Now condit ion (22) is not satisfied already for the vector field Bi = Lorentz gauge condit ion for the 4-potential ik =
(p) ik

­ ¼ ,i. I.e., the Hilbert-

+ ik (see formula (28') in [ P 1 ]) obtained
ik

with the account of field self-action processes (19) is not satisfied. By use of the fact that gauge condit ion (22) is sat isfied for the 'po int' potential have for the left-hand side of (21):
(p)

we shall really

1 d2 [ r ( r dr 2

ik ( p)

1 ik 1 d2 k ),k ] [r ,ik ] . 2 4 r dr
ik

Thus the vector Bi which arises in the nonlinear approximation of GD or at the refusal of the gauge invariance is equal to Bi =
ik ,k

­ ¼ ,i =

ik

,k

0 ,

(24)
(p) ik

i.e., this vector is reduced to vector 4-divergence of nonlinear addit io n ik to 'point' potential
i

.

The vector B arises here as a consequence the very gravitat ional field as the sources, in other words, because o f the account ing (next approximat ion in the Lagrangian o f interaction (9)) for the process of type (19). Of course, for all that the gauge invariance is already vio lated. I want to emphasize once more that there is no absolutely new vector field here. If we assume that near the bag (at r ~ GM/c2) the different ial conservation is not fulfilled for sources (in this connect ion it is appropriate to recall different ial and integral conservat ion in GR): T
ik (*),k

+

ik

,k

0 at r GM/c2 ,

(25)

then the vector component of the tensor field which was a purely tensor component at r >> GM/c2, is sure 'to start operating' (see [P1]). (For all that (20) is a particular case o f (25) for vacuum.) This vector component was excluded then because of an absence of the appreciable input of the vector source at r >> GM/c2. But at r ~ GM/c2 everything that the symmetrical tensor of the second rank (16) yields is used. Thus fro m the foregoing the conclusio n is that near the bag (at r ~ GM/c2) the tensor component ik of the collapsar field ik gives raise an addit ional vector component. For all that we can say that an addit ional degree o f freedo m is 'unfrozen' - the vector component ik of the fie ld (and after it the scalar
ik ,ik

0 also). Now we may not say about ik as about a purely tensor field, we may say about

tensor field without one scalar. I.e., the field ik contains at r ~ GM/c2 three spin parts: i.e., [ik] 0 1 2 ,
m m

0 .
15

(26)


What occurs with tensor field ik at r GM/c2 is natural fro m the quantum-field point of view. The nearer to the bag the more virtual this field becomes. Accordingly (and analogously to what was in QED), in that case the co mponents of spin 1 (and 0) must appear, lacking in real gravitons o f spin 2. But at 'unfreezing' o f addit ional co mponents of the field ik the question arises on correct account for contribut ion of these components into the field energy. In case o f the purely scalar field = mm there is nothing to 'unfreeze'. There remains also only one component even near the bag. For this field (with always well-determined spin) both real and virtual gravitons have spin 0. The last is in good agreement with the fact that energy dependence on r for the purely scalar field remains invariable up to the distance o f the order of ~ GM/c2 fro m the bag centre. (Of course, it is fulfilled t ill one may speak about gravitation only, see [ P 1 ] . ) For the collapsar massless gravitat ional field the EMT of the -component at any distance fro m the bag centre up to r GM/c2 is equal [P1] to



ik (0 )



1 GM 2 111 diag (1, , , ) . 4 16 r 333

(27)

It is this fact which allows me insist ing confidently in [PI] on absence o f the singularit y in GD. In what fo llows, also at comput ing of the co llapsar tensor field energy, I will do my best to use the possibilit y o f divisio n of the field ik into two components ik and , when the scalar co mponent features are clear in many respects.

4. The Energy-Momentum Tensor of the Interacting Gravitational Field of the Collapsar Now I pass directly to the apparent ly most co mplicated mo ment in the who le scheme of GD. I will try to understand how the EMT of the tensor component ik of the field ik behaves close to the bag surface or at distances fro m its centre of the order of GM/c2, i.e., where all possible co mponents o f the tensor field co me into play ([ik] 0 1 2), or where tensor gravitons depart from their mass surface and beco me essent ially virtual part icles, for example, where they exist only a short time o f the order of (GM/c2)/c ~ 10-4-10-5 s for stellar mass objects and correspondingly huge (> 1014 g cm-3) densit y of the gravitational field on surface o f such bags. Let us take once more the example of the vector field in ED. In this theory, in principle the same formula for the electromagnet ic field energy densit y may be applied both in the case o f a free fie ld (corresponding to real photons) and in the case of a field around o f a source (electron), i.e., in the case o f a static fie ld in vacuum (corresponding to almo st real and virtual photons) interacting wit h the source. It is possible, at least in principle, because the obtaining o f the field EMT in ED is done without use of the Lorentz gauge Ai,i = 0 (see Landau and Lifshitz, 1973). Consequently, the scalar component of the vector field Ai is not excluded and this addit io nal degree of freedo m may, in principle, yield its contribut ion into the EMT of electromagnet ic field.
16


Of course, here the quest ion is only on possibilit y in principle o f application of ident ical formulae for the field energy in the classical ED in two situations of 'charge absence'. Strict ly speaking, the same formula for energy is applicable both in case of the free field and in the case o f the field interact ing wit h it s sources only unt il photons near the charge (electron) depart considerably fro m their mass surface, i.e., photons must nevertheless be sufficiently 'almost real' for one may neglect the quantum effects of the e - e + pair creation. Coming back to GD in the scheme, suggested in [P1], the obtaining o f the field EMT is substant iated only for every co mponent and ik of the free gravitat ional field, i.e., of the fie ld in the wave zone as a matter of fact. In this case one manages to secure confidently the positive definiteness of the field energy also. Therefore, strictly speaking, in GD (unlike what is at least in principle permitted in ED) one may not use directly, wit hout reservat ion, the 'purely tensor' EMT formula (see formula (43) in [ P 1 ] for of the field with definite spin 2) when the tensor ik has a vector part also. This can be particularly important in the case o f a 'strongly' interact ing (self-act ing, more exactly here) gravitational field near the bag surface at r GM/c2, when the vector ik,k differs considerably fro m zero: i.e., this is the case o f essent ially virtual gravitons - the vector Bi is 'unfrozen' at least. But here one should keep in mind that apparently there is no method (like the analogous situat ion in QED) of Lorentz-covariant divisio n o f the field ik into components for such virtual gravitons unlike what was made in [P1] for real gravitons. I emphasize once more that in [ P 1 ] the EMT formula was obtained for the free fie ld rea l gravitons. Then it was applied (see equat ion (48) etc. in [ P 1 ] ) for the case o f the co llapsar field, i.e., for virtual gravitons. Now it requires elucidations. Apparent ly, the first step which was made in [ P 1 ] , i.e., the subst itution
(p) ik



ik

(2)

,

proves its value by the fact that at r >> GM/c2 we still deal wit h almost real gravitons o f spin 2. But the next step like all the other 'approximations' ik =
(p) ik

+ ik

ik

(2)

,
ik (2)

will be of a small sense at the use of the same 'pure' EMT formula definite spin 2, see (43) in [ P 1 ] ) for the tensor component of the field.

(for gravitons with the

An approach to the bag means that gravitons become more and more virtual, they depart more and more from their mass surface (pipi 0). One can assume that in that case the applicat ion of formula (43) from [ P 1 ] selects ('cuts out') the energy only for the spin 2 particles, though the vector Bi =
ik ,k

is nonzero already and conformably the particles of spin 0 and 1 appear. For all that this part of

the total EMT (for spin 2 only) remains diagonal and traceless as before. If here also, as in the case wit h virtual photon, there is no a covariant select ing method for
17


addit ional co mponents of virtual graviton, then correspondingly there is no a general method of the deduction of EMT of such gravitons (as a conserved quant it y of a close system). There is a litt le sense in such 'deduction' because virtual particles are essent ially interact ing part icles. We may speak about EMT of such particles only as about a stationary set in (for example, as a result of an exchange with the bag + the self-act ion processes) values o f energy-mo mentum-tensions of gravitat ional 'coat' around the bag. Of course, one may hope that the total theoretical solution of the problem o f energy of interact ing gravitons will be at last carried out in a totally quant ized GD. Here, like QED, the essent ial will be the question on a graviton propagator choice in the theoretic scheme under considerat ion wit h two types ( and ik) of free fields. But it seems to me that preliminarily one can understand much in GD by the careful study of the sense o f gauge condit io ns, conservat ion laws, etc., it is this that is made in this paper. One must always keep in mind that the quest ion is mainly on the macroscopic theory (GD) and macroscopic objects. Remarks on these features which differ GD fro m microscopic quantum theories, such as QED and QCD, will be made at the end of the paper. Now I attempt to move forward going by the way of natural phys ical supposit ions and cont inuing the use of the quantum field theory notions. For all that I shall always mean macroscopic situations, when there are so many gravitons that one may speak to (still wit h a high precisio n) about some 'environment' around the bag, properties o f which are given by the tensor ik. Ult imately observable effects predicted in this direction can reso lve the gravitons energy problem by experiment as it was as a matter of fact in QED. Below we shall use here the signs
ik ( ) ik

and

()

for EMTs of the and ik co mponents
ik

correspondingly for the interact ing fields near the bag. And the interact ing scalar field EMT coincides with EMT (27) of real gravitons of spin 0: Let us assume that: 1) Since the spherical symmetry remains valid also up to r GM/c2 it is natural to suppose that the 'unfreezing' of addit io nal co mponents of the field tensor ik does not 'spoil' the diagonalit y of it s EMT
ik () ik ( )

=

(0)

.

.

2) If the field ik remains for r GM/c2 also massless as before (and quantum corrections/effects are still small), then such an 'unfreezing' of spins 0 1 must not 'spo il' also the tracelessness of the EMT
ik ()

, taking into account all spins.
00 ()

3) At last, the assumption remains valid about the energy posit ivit y

for the interacting field

ik (of integer spin) even at the 'unfreezing' addit ional spins; and what is more, I continue to consider that the energy o f these addit ional co mponents remains posit ive (at least in sum) for spins 1 and 0 in ik. Thus, near the bag for the case of the interacting field ik the EMT
18
ik ()

, taking account of all


spins, must remain as before diagonal, traceless and always wit h a posit ively determined component. However, now let us make the subst itution ik = we obtain the diagonal tensor
(p) ik ik

00

()

-

+ ik
(2)

ik

(2)

,

in equat ion (43) from [P1] for the EMT

which takes into account only spin 2 in ik. As a result

(020) (1 4
where
00 (2)

re r2 111 4 e2 )diag (1, , , ) , r r 333

(28)

= (l/l6)(GM2/r4) and the re 1/3 GM/c2 is introduced here for the notationa l

convenience. I would remind once more that the question is always on the space regions in vacuu m around the bag at r > GM/c2 or r GM/c2, i.e., the bag itself is excluded fro m considerat ion. In co mparison wit h what was at the subst itution in the formula for in [P1] of the 'po int' potential energy for every r by the factor
ik (p) ik (2)

accounting only for spin 2

(the case o f almost real gravitons), we note now the decrease of

(1 4

re r2 4 e2 ) 1 r r

at

r re ,

and as a consequence, the condit io n o f the equalit y o f energies in every po int of space for both components and ik, which was fulfilled in the case of almost real gravitons (see formula (47) in [ P 1 ] ) is now broken
00 ( )



00

(0)



00

(2)

(1 4

re r2 4 e2 ) . r r

(29)

But now (at r GM/c2) both types of gravitons ( and ik) can already be essent ially virtual particles. If one considers that the application of formula (43) from [P1] selects ('cuts out') the energy of only spin 2 gravitons in ik, then the decrease of the right-hand side o f (29) is explainable: simply in the right-hand side of inequalit y (29) it is not everything that is accounted for. Suppose that a contribution o f 'unfrozen' addit io nal virtual co mponents of the tensor field ik in accordance with the above-ment ioned assumpt ions 1), 2), 3) is given by the tensor



00 ( 2)

(

4

re r2 111 4 e2 )diag (1, , , ) . r r 333

(30)

Then if we add it to (28) we obtain for EMT of the field ik the 'o ld' expressio n



ik ( )

111 (020) diag (1, , , ) 333

ik ( 2)

,

(31)

in which the decrease of energy in (29) is compensated by the posit ive energy o f the 'unfrozen' virtual co mponents from diagonal and traceless tensor (30). As a matter of fact, a statement is formulated here which may serve as a basic, experimentally
19


verifiable assumption for the gravitational field around the bag: energy of the tensor field ik with all its possible spin parts ([ik] 0 1 2) is equal as before to the energy o f purely scalar field in every po int, i.e. at any r > GM/c2 or r GM/c2. It means that in every point around the bag the condit ion
ik ()

=

ik

()

(*)

is fulfilled (see condit io n (47) in [ P 1 ] ) which is true also for virtual gravitons of the co llapsar static or stationary (more exact ly) field. Thus equation (27) (
ik (0) ik



( )

) together with the condit ion (*) are proposed here as a so lut ion

of the energy problem for the interacting co llapsar field, if a bag radius R GM/c2 and M of order of solar masses and greater. From the foregoing it follows that the energy-mo mentum-tensio n for the 'environment' around the bag o f such a (macroscopic) dimensio n can be as before given by the tensor

ik

ik ( )



ik ( )



ik ( 0)



ik ( 2)



1 GM 2 111 diag (1, , , 4 8 r 333

)

.

(32)

In that case the tensor potential ik for the outer field o f the bag will have in all 'approximat ions' the ident ical appearance, co inciding wit h formula (28') in [P1]. I.e., nonzero components of the potential ik for the collapsar field in vacuum will be, as before,

00

2 GM / f (1 1 GM / c ) , r 2 r 2 GM / f (1 1 GM / c ) . r 6 r

(33)

11

22

33

5. Conclusions It should be said that the theoretical scheme developed in [P1] and in this paper differs radically fro m different variants of theoretical alternat ives to GR by the fact that here gravitat ions are two types: scalar and tensor ones interacting with the matter by the ident ical coupling constant. I.e., there is a scalar field ­ the unremovable 'superfluous' field co mponent which corresponds to the scalar source T. Thus we develop the variant of the theory in which side by side wit h real spin 2 massless gravitons there may exist real massless particles of spin 0 ­ massless scalar bosons. Unlike what was made in the paper by Sexl (1967), our scheme contains massless scalar field which is included in st ill completely gauge invariant theory wit h 5 gauge functions (10). What was called in the paper by Sexl the expulsio n of the scalar , in our case is only the separation of the scalar part from the tensor one in free field with 5 gauge condit ions for the tensor component (see in [P1]).
20


Furthermore, it is essent ial to dist inguish between two approximat ions in the theory: (1) The linear GD is a gauge-invariant theory where sources may be only po ints or a system o f massive po ints bound by massless fields (gauge fields) as it is usual in a lo cal theory o f field. It is in this approximat ion where the Lorentz covariant divisio n is made into scalar and purely tensor gravitat ional fields. The law T particles. (2) The nonlinear GD is a gauge-noninvariant theory. I.e., the introduction o f non-linearit ies into GD is connected first and foremost with the 'vio latio n' of the gauge invariance. In this approximat ion the mass of interact ing particles is not conserved; the law T
ik ,k ik ,k

= 0 means first of all the mass conservat ion law for interacting

= 0 (in connection wit h the vio lat ion of
ik

the gauge symmetry (10)) is not already fulfilled in the different ial sense. In the nonlinear approximation o f GD the vector divergence o f the sum (T
(*)

+ ik),k is not equal to

zero (25) in the collapsar strong field (near the bag) when elementary vo lumes of averaging are less than or of the order of (GM/c2)3. At distances r >> GM/c2, where the linear approximat ion o f GD is correct, the elementary vo lumes by which the averaging is made in the different ial law, may be much greater than (GM/c2)3. It results in the transit ion (T
ik (*)

+ ik),k T

ik

,k

=0 ,

which can be understood as the conservat ion law for particles together with their gravitational 'coats' (i.e., at r >> GM/c2 the 'degenerat ion' by M appears). We connect the introduction of non-linearit ies into GD first of all wit h the accounting in the interact ion Lagrangian for the terms of type: fikik = fik
ik (0)

+ fik

ik

(2)

= fik

ik

(0)

+ fik

ik

(2)

,

which we can correlate with corresponding elementary processes (19) in accordance with genera l laws. In a sense (see Sect ion 1) it corresponds to transit ion fro m the tree approximat ion to the oneloop approximat ion in QED. But there it is true that the terms fik
ik

violate the gauge invariance

(10) of the theory. (There are self-act ion processes of t ype (19) in QCD, but there they are included primordially in the gauge invariant scheme because o f the non-Abelian character of the gauge group.) In this paper the quantum-field notions are often used and even elementary processes are ment ioned that is more appropriate in a totally quant ized theory. Indeed, after that I tried in [P1] and in this paper to elucidate the sense o f gauge conditions one can try to approach to the procedure of quant izat ion o f the proposed variant of theoretical scheme. But in co llapsar macroscopic cases under consideration quantum-field and classical notions coexist simultaneously. We speak about virtual and almost real gravitons. For the collapsars with the mass 10 the virtual gravitons exist as rea l particles about 10-4s ­ these are rather big times in comparison with the muon lifet ime (e.g.). The
21


almost real gravitons exist more than 3 min in case of the Sun - the Mercury exchange. The virtual gravitons in the case o f cosmo logical objects wit h dimensio ns o f their gravitational radius o f the order of several astronomical units exist already during the hours as real particles. On the other hand, since for the description of the co llapsar properties wit h the mass ~ one may use the tensor ik for the same 'environment' around the bag. It means that there are many gravitons in every (cm)3 for the bag around with the radius greater than or of order of severa l kilo meters. First of all here the 'weight' of every such (cm)3 of 'vacuum' around the bag is important. It becomes comparable with the 'weight' of (cm)3 inside the bag itself. Then the account ing for the gravitat ion of such 'vacuum' beco mes important before the caring about quantum effects. The above remarks have for an object to emphasize the difference between macroscopic theory which GD is, and microscopic ones ­ the quantum theories QED and QCD with massless gauge fields. But here we emphasize also the co mmo n ideo logical base of GD and these theories. In particular, in the consistent dynamical theory o f gravitat ional interact ion one can and must seriously speak about virtual gravitons, not only about real ones. The very problem o f quantization of gravitat ion inevitably poses the quest ion about virtual gravitons (exist ing 'for a lo ng macroscopic time' for the macroscopic co llapsars), the main property of which, like of photons in QED and gluons in QCD is the presence of all possible polarization states for the symmetric tensor field (16). One can be occupied with the cho ice o f a gravit on propagator in this case. But at least it is clear already that the ('lost' in the linear GD) vector Bi = im,m ­ ½ ,i must be restored in the scheme with all the ensuing consequences for field energy and potential ik near the bag. Before the fulfilment of a total quant izat ion procedure it is questionable whether the equalit y
ik ()

=

ik

()

(*),

based still o n co mmo n reasonings, is sufficient ly well-grounded. But my purpose in this paper would be half achieved if I managed to convince the reader that the account for all co mponents of the field ik is as necessary in GD (in the case of the strong co llapsar field) as in QED and QCD in analogous cases demanding the account ing for all possible spin states of virtual photon and gluon. Indeed, since the condit ion ii = 0 causes the linearit y o f the purely scalar co mponent o f gravitat ion, i.e., there is no vertex

then sat isfying once (in gauge invariant scheme st ill) the condit ion of the vector absence (
ik p) ,k

­ ½ ,i =

ik (p) ,k

­ ¼ ,i = 0 ,
ik ,k

one can never do that at r ~ GM/c2. The vector field Bi =
22

0 will counteract. Certainly one


may doubt the cho ice o f field EMT (32), but the taking into account of addit ional spins is necessary nevertheless. Thus the quantum properties of the field ik ­ the presence of addit ional spins in virtual part icles ­ become apparent already in the half-classic, macroscopic situation wit h the collapsar. The tensor potential o f the static field o f the bag (33) obtained by allowing for the equalit y (*) must lead to certain experimental (observable) consequences. It could be testified in part icular by the periastron shift effect in relativist ic close binary systems. The shift effect must be described by the same 'o ld' formula [P1] as

1 2

6 GM / c (1 e2 ) a

2

fulfilled because of (*) at all r >/~ GM/c2. Certainly, for all that one must not forget to take into account the more and more increasing ro le o f gravitat ional emissio n fro m the system influencing the secular effects what is particularly important for small values of r ~ GM/c2. A special paper will be dedicated to the study of observat ional consequences for potential (33). Ult imately, fro m (*) one must obtain the picture of properties o f the bag, of its surface and the field around, consistent with all other physics. These properties must lead to absolutely definite experimental (observational) tests allowing distinguishing the collapsars in GD from black ho les in GR.

References
Baryshev, Y. V. and Sokolov, V. V.: 1984, Astrofizika 21, 361. Bogol yubov, N. N. and Shirkov, D. V.: 1973, Introduction in Gauge Field Theory, Nauka, Moscow, p. 480. Konopleva, N. P. and Popov, V. N.: 1973, Gauge Fields, Atomizdat, Moscow (in Russian). Landau, L. D. and Lifshitz, E. M.: 1973, Theory of Field, Nauka, Moscow, p. 504. Mosinsky, M.: 1950, Phys. Rev. 80, 514. Sexl, R. U.: 1967, Fortschritte Phys. 15, 269. Sokolov, V. V.: 1992, Astrophys. Space Sci. 191, 231. Sokolov, V. V. and Baryshev, Y. V.: 1980, in Gravitation and General Relativity, KGU, Kazan', Vol. 17, p. 34 (in Russian).

23