Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mavr.sao.ru/hq/grb/team/vvs/01_Linear_and_non-linear_gravid.pdf Äàòà èçìåíåíèÿ: Tue Sep 21 16:00:52 2010 Äàòà èíäåêñèðîâàíèÿ: Sat Feb 12 05:52:49 2011 Êîäèðîâêà:

Astrophysics and Space Science 191: 231-258, 1992. © 1992 Kluwer Academic Publishers. Printed in Belgium.

LINEAR AND NONLINEAR GRAVIDYNAMICS: STATIC F I E L D OF A COLLAPSAR
VLADIM IR V. SOKOLOV

Department of Relativistic Astrophysics of the Special Astrophysical Observatory of the USSR Academy of Science, Nizhij Arkhyz, U.S.S.R.

(Recei ved 17 July, 1991) Abstract. In the context of a consistent dynamic interpretation of gravitation (gravidynamics) the gravitational field has been divided into two components: scalar and tensor, each one interacting with its source with the same coupling constant. Consequently, the spherically-symmetrical gravitational field generated by a massive object (a source) influences test bodies as an algebraic sum of attraction and repulsion. The field energy in vacuum around the source is also a sum of energies of two components - purely tensor and scalar ones of gravitation. At distances from a gravitating object much greater than its gravitational radius, energies of each separate field component are equal to each other at the same point of space. In the bounds of the gravidynamics based on the so-called Einstein's 'linearized' equation and proceeding from general principles of theory of classical fields a statement (a theorem) has been formulated on the static gravitational field of a collapsar: a spherically-symmetric object generating a static field in vacuum can alwa ys occupy only a finite, nonzero volume.

1. Introduction The quest ion we wish to raise below concerns an outer gravitational field generated by a spherically-symmetric distribut ion o f matter in vacuum fro m the po int of view o f a consistent dynamic interpretation of gravitat ion. We have in mind a certain spherically-symmetric configuration o f the system 'matter + gravitational field' wit h any radius o f the sphere filled by matter down to the smallest dimensions of the sphere (a compact configuration) of the order of GM/c2, where G is the gravitat ional constant and c is speed o f light. The total energy of every one of these configurat ions co incides with the rest energy o f the system and is denoted as Mc2. This paper tries to answer such a concrete quest ion as what are limits where the outer gravitational field under the discussio n can st ill be considered as static. One can formulate the quest ion in another way: to what extent one can compress matter for an obtained configuration to have st ill a static field in vacuum? The paper, however, concerns mainly a stationary stable state of the system 'matter + gravitat ional field' (i.e., a co llapsar), but not the ver y process surely nonstationary o f the co llapse, i.e., of the co mpressio n of matter to dimensio ns of a sphere with a radius o f the order of GM/c2. The collapse as a process of transit ion o f a system to a

1


more bound state is not considered here. For the present the author's aim is to study the stationary spherically-symmetric configuration with the greatest possible binding energy fro m the po int of view o f dynamic field theoretical interpretation o f gravitat ion that we have been adhering to in our papers (Sokolov and Baryshev, 1980; Baryshev and Sokolov, 1984). 2. Energy of Gravitational Field in Gravidynamics and the Static Field of Collapsar As is generally known, the problem of gravitational field energy exists in the General Relativity (GR) nearly since the mo ment of its discovery, and the debate (in particular) on localizabilit y and posit ivit y o f this energy in GR cont inues up to the present. More simply, it is not clear t ill now how to understand the conservation of energy in GR. One of the most known articles on this topic by Zel'dovich and Grishchuk (1986) does not solve, as earlier, all the problems related to field energy. But these authors agree at least that one can try to explain gravitation dynamically, i.e., without ident ifying it with geo metry of space-time by GR. They concluded that, all the same, as a result we shall co me to the same GR. But not all phys icists share such an opinio n (see e.g. Logunov and Mestvirishvily, 1985; Vlasov and Logunov, 1987), in particular, when it concerns the quest ion on the field of a co llapsar. This static and spherically-symmetric fie ld in vacuum around a region filled by matter will be discussed below from the point of view o f gravidynamics (GD). Thereby, since the very beginning we accept as axiom here that the energy of gravitational fie ld is localizable, posit ive and understood in the same sense as it is understood in the classical electrodynamics (ED). What is concretely this energy, or rather energy density, for example, in the case of the co llapsar static field? In ED, the field energy is defined by the second power of electric and magnet ic fields. In GD the gravitat ional field energy densit y is also defined by t he second power of its field strength g2/G, where g is the gravitat ional accelerat ion. As we have shown in our paper (Sokolov and Baryshev, 1980), the field energy densit y near a gravitat ing body can be calculated by the formula

00

1 g2 / G ( N ) 2 . 8 G 8

(1)

Where N

GM is the usual Newtons's potential. Subsequent sect ions o f this paper will r

consider in detail the cho ice of particularly this formula for the static field energy densit y. In this sect ion it is shown to what consequences for the collapsar field the fulfilment of natural requirements of posit iveness and localizabilit y of gravitat ional field energy can lead. In our opinio n, the o ld debate on the field energy is first of all a debate on our understanding (our 'reading' or interpretation) o f equat ions describing the gravitat ional field. One of the aims o f this paper is to look from another point of view at 'old' concept ions by describing where possible in
2


detail the phys ical sense of equat ions, formulae, idealizations known for a long time. I would like to emphasize right now that the base of our point of view on gravitat ion is the consistent, dynamic, field-theoretic interpretation of the same 'o ld' Einstein's equat ions. (But Einstein's equat ions are written fro m the point of view of GR in the so-called 'linearized' form.) In GD it concerns both weak and strong fields eventually. The following shows that the choice of a correct formula for 00 is connected with the fact that the gravitat ional fie ld energy cont inuously distributed in vacuum around the Sun direct ly affects the Mercury perihelion shift. It turns out that for the right explication of such an effect in GD the field energy must be positive only: i.e., the localization of energy (it means that the amount of field energy around the Sun in any volume or in every cubic cm is definite) and its posit iveness (it means that field energy is posit ive like any mass) are directly connected in GD wit h observat ions, wit h experiments. Just this fact (noticed for the first time by Thirring, 1961) sets out the possibilit y of the dynamic interpretation of gravitat ion for elucidation of the meaning and the value of 00. But 00 is only one, temporal co mponent of the energy-mo mentum tensor (EMT) ik of gravitational field. The cho ice of the formula (1) for
00

besides the symmetry ik = ki must take

into account the fact that the gravitation field is massless field. In other words, the field act ion radius r is unlimited and the corresponding field quanta (gravitons) are particles wit h the zero rest mass. All this is available in ED. In ED the fie ld masslessness is connected in particular with the lack o f a trace of the electro-magnet ic field energy-mo mentum tensor (EMT). Correspondingly, in the GD for the gravitational field EMT we proceeded fro m the condit ion ikik 0, (i, k = 0 , 1 , 2 , 3 ) . (2)

Where ik = diag(+ 1, - 1, - 1, - 1) is the diagonal Minkowsky's metric tensor. Thus, for the fo llowing, an assumption that three conditions (or axioms) 00 > 0, ikik 0, ik = ki, which influence the cho ice o f the formula for 00 are satisfied, is of great importance. Since the present work concerns, as a matter of fact, the features of problem statement and this sect ion will show how to answer quest ions posed in the Introduction, we are going to elucidate here in detail the sense of the so-called 'point idealization'. Ult imately, the difference between the linear and nonlinear GD, will be clear at once, a gravitatio nal radius will appear as the main parameter of GD. The fact is that in any classical relat ivist ic field theory (in ED and in GD) one may strictly speaking not ascribe without reserve some finite dimensions both to test particles and to particles which are the sources o f field. In other words, only po int sources or a system of po int sources as a macroscopic gravitat ing body in GD may be on the right side of corresponding field equations. It is

3


connected with a specific character of Special Relativit y (SR) in which all extent bodies are to be represented as a system of point (structureless) objects interacting with each other. In part icular, the 'po ints' wit h masses ma which real macroscopic gravitat ing bodies consist of (a ma ) are meant to be ent ire regio ns of generally macroscopic dimensio ns wit h masses ma. Certainly, these are not mo lecules, atoms, and electrons. These are large macroscopic regions between which basically gravitation force is act ing only. In the fo llowing I shall do my best to emphasize and to use this fundamental concept ion o f interacting po ints in SR (or in the classica l field theory) as an init ial notion of a 'gravitational charge'. Here (in the GD) a natural questio n arises: to what limits is this idealized concept useful and acceptable? In exposit ion/descript ion o f GD as in ED, it is methodologically convenient, at least at the beginning, to eliminate all other interactions but the gravitat ional one only. As was noted before, the most general form of a field source (a macroscopic gravitat ing body) is a system o f po int sources interacting with each other only gravitationally. (But one must think how to provide the stabilit y o f such a system ­ see Introduction.) At first we invest igate what is here (in the GD) a single, stationary, or an 'elementary' po int source with the mass M and wit h the static fie ld in vacuum. One may use an analogy wit h the elementary point charge (electron) in the classical ED. Thus, every po int generates around itself a static spherically-symmetric field. Let this field po int source resting (for the reference frame definit io n) at the origin o f coordinates. For the field creat ion it is necessary to spend some energy, i.e., the field around the point source contains so me part of the source mass. It means that if the field energy is sufficient ly large we deal wit h a material object distributed continuously (and spherically-symmetric as before) in space around. In that case in ED they say about electron surrounded by a 'fur-coat' of virtual photons. In ED the field around the point the electron will be characterized by the definite energy density 2/8 = e2/8R4. When this densit y beco mes co mparable wit h the mean rest energy densit y mec2/R3 of the particle= electron, the question arises: where is the mass o f the electron concentrated (in a vo lume ~ R3)? This is the old ED problem, still unresolved completely in its quantum generalization. But as is generally known, the classical (and linear) ED is applicable t ill the distance between charges is much larger than the classical electron radius e2/mec2. And this is convent ionally a 'point' (the po int source) in the classical theory o f electro-magnet ic field. In the GD there is also gravitational field with the energy densit y
00

(1) around any mass

distributed in the spherically-symmetric way. As soon as the gravitat ional field energy densit y in vacuum (out of the source, i.e., out of a sphere filled by matter) beco mes o f the order of the mean rest energy of the system 'matter + field'

4


00

GM 2 Mc 3 8 r 4 r

2

for r ~

GM c2

,

(3)

in the GD the same quest ion as in ED arises. Where, in that case, is the mass of such a 'po int' object concentrated? Ult imately, what is the rest mass M of the gravitating body on the whole? It is obvious that these quest ions arise at distances of the order of GM/c2 - i.e., of the order of the gravitat ional radius of the 'po int' source (or gravitating centre) as one can see fro m co mparison (3). The est imat ion of the gravitat ional radius is made here by the same reasons as the est imat ion of the classical electron radius e2/mec2 in ED. Just as in the classical ED we may say that we deal with a theory much alike the classical linear ED unt il some po int sources distribut ion is co mpressed to dimensio ns when distances between them beco me of the order of Gma/c2. For the spherically-symmetric distribut ion o f po ints with the centre at the origin o f coordinates it means that the all system (with the rest energy Mc2) of such an 'elementary' gravitat ing object is not compressed to dimensio ns of the order of GM/c2 or as long as the fie ld is measured at distances much greater than GM/c2. Thus, conventionally a 'po int' in the GD is in fact something with finit e dimensio ns of the order of GM/c2. It means that at r >> GM/c2 the mass M of the point source includes automat ically the 'mass' o f the gravitational field itself generated by the source. This is the sense of 'po int' idealization in the GD. It is now obvious that nonlinear GD is the GD at distances of the order of gravitational radius (~GM/c2) from the centre of any spherically-symmetric configuration. In accordance wit h the universal character of gravitational interaction one must consider the field itself to be the source of gravitational field (there is nothing of the kind in ED). On the right side of the field equations it is accounted for by including the gravitat ional field EMT in sources (a kind o f the source 'splitt ing' occurs) which makes the fie ld equat ions nonlinear. Accordingly, corrections to potentials arise which lead in particular to an ent ire explicat ion of the observed effect - the Mercury perihelio n shift (a 'nonlinear' contribut ion in the effect). Nevertheless, the quest ion about the mass M in the nonlinear GD remains: where is the mass located if the gravitational field energy is posit ive, localizable and condition (3) is satisfied? The outer field o f such a co llapsar remains static as before. More precisely, the quest ion concerns the outer static field o f a co mpact spherically-symmetric configuration wit h dimensio n of the region filled by matter of the order of the gravitat ional radius of the who le system (GM/c2). It is this object which we shall call the collapsar in what fo llows. The fo llo wing sections o f this paper will show in more detail that it turns out that, to answer the questions formulated in the Introduction about static field o f the co llapsar, there is no necessit y to resolve the field equat ions to all approximat ions. The essence of the matter can be cleared quite
5


precisely in the following way. Let us assume that there is such a stationary state: the collapsar wit h the radius o f the sphere filled by part icles (matter) close to the gravitational radius o f the who le system (GM/c2). Let us integrate the gravitational field energy densit y 00 in vacuum the integral going from the surface o f the sphere filled by particles (with a radius rx ) to infinit y and being equated to the total rest energy of the who le configuration


4


rx

GM 2 2 r dr M c 8 r 4

2

.

(4)

This equalit y must be correct at some finite rx ~ GM/c2. But then it should be supposed that the total rest mass of the gravitating object is the 'mass' of the gravitational field only. The latter is difficult to reconcile with assumption (2) about the 'masslessness' of gravitational field - i.e., with the assumption on gravitons as on particles with the zero rest-mass. Strictly speaking, in that case it is difficult to maintain the assumption on static character of field around the collapsar. Thus, proceeding from general principles (axio ms) lying at the base of the relativistic field theory, one succeeds in formulat ing a statement (theorem) of the collapsar static field, the sense of which can be expressed in the fo llowing way: If the gravitational field energy is positive (00 0) and if the field is really 'massless' (ikik 0) then the localization of gravitational field energy means that it is impossible to compress matter (particles) to a sphere with a radius smaller than some finite radius, still static sphericallysymmetric field being in vacuum around this sphere. Our paper is dedicated ent ire to development of a detailed and consistent basis of this statement. But first I return to basic equat ions o f GD, comment ing upon them in detail fro m the fieldtheoretical point of view. Then (and it is very important) the static field will be represented explicit ly as a sum o f two field components with equat ions for each of them: namely, a scalar component and a purely tensor component of the only gravitat ion field. The choice of equations of motion for test particles in a given gravitational field will be based on general principles. The socalled 'linear' contribut ion to the Mercury's perihelio n shift will be regarded. Then the cho ice o f formula (1) for 00 will be based in detail and the gravitat ional field energy will be represented as suitable posit ive contributions of each of gravitatio n co mponents, both scalar and purely tensor ones. Ult imately, on the basis o f the fact that the scala r component of gravitat ion by virtue of (2) is described by the linear equation down to r GM/c2, it will be shown that the radius of the sphere filled by matter with outer static field anyway cannot be less than ¼ GM/c2, the contribut ion o f each field component in 00 being posit ive. In that way the quest ion on the singularit y is reso lved in the GD.
6


3. The Linear Gravidynamics and The Scalar Component of Gravitation A considerable part of what fo llows is based on ideas expressed already in papers by Thirring, Moshinsky, Hoopte, Fock, and others. It is evident that the relat ivist ic theory of classical tensor field pretending to a complete description of gravitation will inevitably use the experience of GR by Einstein and of ED by Maxwell. For all this as we will see from the fo llowing that, developing the field-theoretic interpretation of gravitation, we adhere to the field (dynamic) interpretation of basic principles o f this theory. In particular, we proceed from the fact that gauge transformat ions o f potentials are not connected in any way wit h the known transformat ions o f frame o f references in GR. And the principle of invariance under the gauge transformat ions, rather than the principle o f equivalence, must lie at the basis of the consistent field-theoretic approach to gravitation. When we say about some field theory that, means in the first instance it represents the field equations (for example, Maxwellian equat ions) - just as the Einsteinian GR represents first of all the equations at the basis o f this theory. But it is impossible to understand the usual form o f the Einsteinian equat ions in the non-geo metrical way. In latter there already is a curvature, i.e., the curved space-time. Ult imately it defines the geometrical descript ion of test particle motions in gravitat ional field (i.e., in the curved space-time of GR), it defines famous geodesic equations. Everybody agrees that the dynamic interpretation of Einsteinian equat ions begins in case o f rather weak fields. By the linear GD I mean all cases when gravitatio nal field can be assumed weak. In that case distances between part icles ('gravitat ional charges') must be much greater than their gravitat ional radius, and gravitating po ints may be assumed to be real po int structureless objects for which, in particular, the mass conservat ion law is fulfilled: i.e.,

dx k ( )0 . x k dt

(5)

Where = ma(r - ra) is the gravitat ing points mass densit y. Particles in linear the GD can have any velocit ies up to relat ivistic ones. The natural wish for generalizat ion o f the Newtonian gravitat ion up to relat ivist ic velocit ies leads to this first (rather simple) part of GD, i.e., to the relat ivist ic theory of gravitation in flat space-time (the relat ivist ic gravidynamics). Let us begin with the simplest case of a sourceless tensor field. As is known, a symmetric tensor of the second rank consistent with the Klein-Gordon theory - namely,

ik = 0

(6)

describes a massless field (particles wit h spin 2) which we need, if t he fo llowing condit io ns are
7


satisfied:
ik ,k

=0,

(7) (8)

ikik = 0 ,

Equations (6) together with the 5 invariant conditions (7), (8) define the ik accurate up to the gauge transformation of the potentials ik ik = ik + Ai,k + A
k,i

.

(9)

Where Ai is a vector field consistent with the conditions Ai = 0 and Ai,i = 0 (here ik - ik,ll). One can construct (or postulate, if it is wanted) a more general form o f a linear differentia l equation o f second order for the symmetric tensor ik = ki , demanding direct ly an invariance o f these equat ions under the transformat ion (9). We obtain eventually equations co inciding in form with those usually called in GR the 'linear form' of the Einsteinian equat ions (the left-hand side): ­
ik,l l

+

il,k

l

+

kl,i l

­

,ik

+ ik( ,ll ­

mn

,mn

)=0

(10)

For all this the numerical value of the factor (+1) before the brackets is fixed by the condit ion of ident ical equality of the left-side divergence to zero, which is a consequence of the same invariance under the gauge transformat ions group (9). The consistent dynamic interpretation of these equations is that field potential ik (just as in ED) is understood irrespective o f the metric ik. In particular, there is no such a condit io n like ik << ik. The potentials ik by themselves have as little sense as 4-potentials in ED. They may have any value because of their uncertaint y according to (9). Generally speaking, one can write down the gauge (gradient) transformat ion group under whic h equation (10) must be invariant, in more general form (but which does not coincide outwardly wit h the well known group of infinitesimal transformat ion in GR): ik ik = ik + Ai,k + Ak,i +
,ik

,

(10a)

with 5 arbitrary functions Ai (4-vector) and (scalar). Then to obtain equat ions like (6) from the general equat ions (10) (for particles/gravitons wit h spin 2 only) it is necessary to require the satisfact ion of 5 gauge condit io ns like (7), (8) at once, for the 'gauging/calibrat ing ' scalar and vector Ai fields sat isfying the equations

Ai + Al,il = ji ;

= + 2A

l ,l

.

(11)

Where ji il,l ­ ,i ( ikik) and ji,i = 0 according to field equat ions (10). The field Lagrangian, which the equations (10) are obtained fro m, is o f the form:

8


Z = a(FlmnFnml ­ FmnmFlnl) + b( where F
lmn

lm,n



nm,l

­

mn

,m

ln,l) ,

(12)

=

ln,m

­

lm,n

and the value o f a depends on the cho ice o f the measurement units for the

potentials ik (see below). The expressio n in the second bracket is reduced to the divergence and does not influence the field equat ions for any b. (By variat ion o f the values o f a and b one can obtain all Lagrangians met in literature which the same equations (10) follow from.) The interaction between the field ik and its source Tik is described by the Lagrangian
f ik T c2
ik

,

(13)

where f plays the ro le o f an interaction constant or coupling constant and is defined specifically at the cho ice o f the measurement units for ik , and Tik corresponds in the considered case o f the linear GD to an EMT of the system o f point (structureless) objects-particles. That is here Tik = cuiu
k

ds , dt

ds = c dt 1 v 2 / c

2

,

where ui is the 4-velocit y and v is the usual velocity of the particles. I emphasize once more that we consistent ly adhere to point idealizat ion for gravitational field sources ( = ma (r - ra)) as a n init ial notion of the theory. One can obtain a general form of field equat ions with sources in the linear GD by virtue of the action
1 f ik ( Z c2 ikT )d c

(the sources motion is assumed to be specified), which leads to the field equat ions in the form of: ­
ik,l l

+

il,k

l

+

kl,i l

­

,ik

+ ik( ,ll ­

mn

,mn)

=­

f T 2ac 2

ik

.

(14)

Because of general character of these equations they cannot, however, to be applicable direct ly to solve so me concrete problem; for example, to find the fie ld of only one po int source. Both the symmetric tensor ik by itself and equat ions (14) describe a who le 'mixture' o f fields - this is simultaneously scalar, vector and tensor fields. In that case they say about 'the mixture' of fields with different spins ([ik] = 0 0 1 2), see, for example, Sexl (1967). This property must also be attributed to the tensor source Tik on the right-hand side of (14). But the identical equalit y o f divergence o f the left-hand side of these equat ions to zero, or, in other words, gauge invariance (10a), demands the fulfilment of the conservat ion law for the tensor source (so-called 'strong' law)
9


T

ik

,k

= 0,

(15)

as in ED the fulfilment of the conservat ion law for the 4-current ji,i = 0 is a consequence o f the gauge (gradient) invariance o f the Maxwell's equations. Equalit y (15) means that the vector component of the field ik may be regarded as dependent since the corresponding vector source is absent (or, a vector field may be only virtual...). Of course, in the linear GD approximat ion, when gravitat ional interact ion between particles may be assumed weak, the equalit y (15) is fulfilled only approximately. An exact conservation law (Tik + ik),k = 0 will be true only for the sum of the source EMT (Tik + ik). That is just a consequence o f relat ivist ic invariance of the field equat ion (14), but not the gauge one. But as was noticed in the previous sect ion, in the linear GD the strong field as a source is automat ically included in the notion of a 'po int' source with so me mass M (there is no 'splitt ing'). Then the field ik in (14) is the field at distances much greater than GM/c2 for each source in the right side (for the weak field), and equalit y (15) must be regarded as a consequence of gauge invariance of equat ions (14). Thus, if we approximately regard the sources in the right-hand side of field equations (14) as the point sources with given masses, then, allowing for the gauge invariance o f these equat ions, one can bind the vector component of the field ik by the gauge condit io n (the Hilbert-Lorentz gauge condit ion):
im ,m

=½

,i

,

(16)

and below we regard the vector field in ik to be dependent (so we 'exclude' the vector component of ik). In that case, the field equations (14) are transformed to the form

ik = ­
where T = mn
mn

f (Tik ­ ½ ikT) , 2ac 2

(17)

is a scalar present ing the trace of the sources EMT. It is impossible, in particular,

to demand the sat isfact ion for ik of the 5 invariant conditions (7), (8) at once ('to exclude' also the scalar ) for equations (14) because o f the fact that T 0. [Just the nonzero trace of the sources EMT (the scalar) permits an assuming that the scalar cannot be always only a virtual field.] The system o f equat ions (17) still describes 'the mixture' o f fields but the amount of co mponents of this mixture is now less because o f condit io n (16). And now we separate explicit ly these components present ing equat ions (17) in the fo rm o f an equivalent equation system for each component separately - scalar and purely tensor ones. Let us present the potential ik in such an invariant form dist inguishing explicit ly between scalar and tensor components

10


ik ik + ¼ ik

,

(18)

where ikik 0 and ik describes now the tensor component in the field ik, as well as invariant tensor ¼ ik describes only scalar component. In exact ly the same way one can manipulate with Tik : Tik T where T
ik (2) ik (2)

+¼T

,

(19)

ik 0 for the sources T

ik

(2)

of the tensor component ik in the field ik .

Convo lut ion o f equat ions (17) by indices yields an equat ion only for the scalar part of the field ik. If one now substitutes expressions (18) and (19) in (17) and uses the equation for the scalar part then one can obtain an equation only for the tensor component of gravitat ion also. As a result we obtain
f T, 2ac 2 f Tik( 2ac 2
2)

(20)*

ik

.
mn

(21)* = ¼ ,n, excludes

The Hilbert-Lorentz gauge condit ion (16), written down now in the form as before the vector component of gravitat ion.

,m

Thus we have formulated the basic equat ions in the linear GD which will be naturally generalized below for strong fields, i.e., when sources cannot be already represented as systems of only 'po int' structureless objects and 'a splitting' o f them into a 'field' part and a 'po int' one will be needed. But the main conclusio n fro m equat ions (20)* and (21)* is the fo llowing: gravitat ion in the GD has two components: scalar and tensor ones each interact ing with its source with the same coupling constant f. In our opinio n it is this condit ion which is the essence o f the dynamic interpretation of equat ions of type (10) or (14), and this is a radical difference of the approach developed by us fro m different versio ns of alternat ive theories (the scalar-tensor one, bimetric formalis m, etc.). 4. The Outer Field of a Massive Gravitating Centre at Distances of r >> GM/c2. Motion in Given Field Section 2 has mentioned that in the consistent GD description it is convenient, at least at the beginning, to eliminate all interact ions except the gravitat ional one. A gravitat ing 'body' in the righthand side o f the equations (*) (20* and 21*) is in fact a system of po int sources interacting only gravitat ionally. All the more, in the linear GD one can obtain the field o f any system o f po int sources (both inside the system and outside it) as a superposit ion of fields o f such 'elementary' sources. Furthermore, the gravitat ional potentials of the point objects coincide wit h the potential o f
11


real spherically-symmetric bodies. But we begin wit h gaining an understanding o f what this spherically-symmetric and static field of only one 'elementary' motionless po int source in the origin o f coordinates is. One can obtain the EMT of the mass M point particle located in the origin o f coordinates from the general expressio n for T ik c 2u i u of coordinates c2 = c2(r - r) = c2(r). Hence, for the point source in the centre we have Tik = Mc2 (r) diag(l, 0,0,0). (22)
k

1 v 2 c 2 , in which one must take for the particle wit h the mass M in the origin

Here the frame of reference is fixed at last: i.e., this is the frame of reference (inertial, of course) of a body o f reference + generally speaking any frame of coordinates in which the source-particle (the body of reference) rests in the origin of coordinates (r = 0 and v = 0). Since the source is at rest, the field must be static, centrally symmetric, and the system of the equations (*) will be rewritten as

1 d2 f [ r ( r )] T, 2 r dr 2ac 2 1 d2 f [r ik (r )] T( 2 2 ik r dr 2ac
For the scalar source T from (22) we have T = ikTik = Mc2 (r) , and equat ion (23) for the scalar field (r) has the solut ion
(r ) fM 1 , 8 a r
2)

(23)

,

(24)

(25)

(26)

consistent with the addit ional boundary condit io n: the scalar potential must be zero at infinit y. For the purely tensor source T
(2) ik

from (19) and (22) we have (in case when r >> GM/c2 or when

the total mass of the source is assumed to be in the centre): T
(2) ik

= ¾ Mc2 (r) diag (1, 1/3, 1/3, 1/3) .

(27)

In that case the so lut ion o f equation (24) with the same boundary condit io n (the potential at infinit y

12


must be zero) will be the tensor field ik(r) = 00(r) diag(1, 1/3, 1/3, 1/3) ,
00 ( r ) 3 fM 1 , 2 16 a r

(28)

r >> GM/c2 .

Thus the static field of a massive point in the origin o f coordinates is described by two potentials (which are functions of r), namely,
(r ) C r

and

00 ( r )

3C , 2r

if

C

fM . 16 a

And one can unite it, if desired, in one single tensor field:
ik 3C 1 1 1 1C diag (1, , , ) diag (1, 1, 1, 1) = 2r 3 3 3 2r fM 1 diag (1,1,1,1) . 16 a r

(29)

And there is already no ambiguit y or arbitrariness connected with the gauge invariance. The static potential (29) is determined quite uniquely (of course without allowing for radiation incident fro m outside) by gauge condit io n (16) and by the addit ional condit ion: ik must be zero at infinit y. Evidently, the so lut ion (29), so as (26) and (28), can be quite applicable as descript ion of outer field outside the sphere of the radius r0 filled by matter with any centrally symmetric distribut ion of mass (r). In that case one can take right away the EMT of a macroscopic body (as a cont inuous medium) in the form c2uiuk as the source in the right side of field equat ions. Here is now the body mass densit y and ui is so me element average 4-velocit y of (macroscopic) 'elementary' volumes of averaging. For the static case or if macroscopic motions are slow, the EMT of a gravitat ing body will be of the form c2 diag(l, 0, 0, 0). Solving the field equat ions one can find both an inside so lution and an outside one. The outside solution for any r r0 will be determined only by the total mass inside the sphere r = r0 and will co incide with the so lution (29) simply because of the Poisson's equat ion property for centrally symmetric (r):
r0

4 (r )r 2 dr M M (r )dV ,
0 V0

where V0 is a vo lume of the sphere wit h the centre in the origin o f coordinates. In that case (as in the Newtonian gravitat ion) the dimension of the gravitat ing body is of no importance, of course, till
13


the configuration is negligible in comparison with the total rest energy Mc2. In accordance with the more consistent discrete conception o f gravitating bodies accepted here (in the GD), it means that 'points' wit h masses ma inside the body are located at distances fro m each other much greater than Gma/c2. Hence, t ill t his configurat ion is quite far fro m the dimension of the order of GM/c2 and the masses ma of the 'points' are the masses of rather large (macroscopic) regions indeed, then an ant ipressure needed to secure the stabilit y of such a sphere is simply an elasticit y of matter which these regions consist of. In other words, here so me nongravitat iona l interact ion is meant which defines the equation of matter state. Let us now consider the motion in a given field. As was noticed before, the interaction of the field ik with particles is described by the scalar (f/c2)ikTk (13) and one can obtain the particles mot ion in field fro m the action

S

1 f ik ik (ikT c 2 ikT )d c

,

(30)

which gives for test point particle wit h the mass m
S ( mcds fm ik u i dx k ) . c

(31)

If we compare this wit h the act ion for a charged particle in a given electromagnet ic field and introducing the 4-vector A
(g) k

ikui ,

(32)

we can write the action for a particle in a given gravitational field in the 'electrodynamic' form
e S (mcds Ak( g ) dx k ) c

,

(31)'

where e fm. A variat ion of (31) or (31)' (as in Landau and Lifshitz, 1973; and see detail in paper by Baryshev and Sokolov, 1983) gives equations of motion that are convenient also to be written down by the vector Ak(
g)

in the same 'electrodynamic' form

mc

du i e ( g ) g ( Ak ,i Ai(,k ) )u ds c

k

.

(33)

To emphasize that in GD one can use explicit ly the conception of force, as in ED, one can even introduce the 3-dimensio nal vectors E(g)

1 A ( g ) c t

(g)

,

H

(g)

A(g) ,

14


where

(g)

­A0, A(g) = ­A ( = 1, 2, 3). Then 3-dimensio nal equat ions of mot ion fro m (33)

coincide by form wit h the Lorentz force and with the equat ion for energy in ED
dp 1 = e ( E(g) + [v · H(g)] ) ; dt c dEk dt
in

= e(E(g) · v) .

(34)

The fact that in (30) Tik enters both the 'inert ial' part of act ion and the 'gravitat ional' one, is a direct consequence (or rather a generalization) of equalit y of the inertial mass min the gravitational one mg
ra v e rt

of particles and

( minert = mg

ra v

). Thus, if fro m the very beginning we proceed direct ly

fro m the experimentally tested law of equalit y of the inertial mass and the gravitational one for test bodies understood in GD as point (structureless) objects but not from the 'equivalence principle', then equations (33), introduced for the first time in GD by Birkho ff (1944), can be a perfect ly relativist ic generalization of the Newton's equations of motion v m
N grav ert

min

,

where mg

ra v

= minert m .

One can now write an expressio n for a force acting on a motionless test particle located in the field (29), explicit ly presenting this force as the sum o f forces: F(2) is the force connected with the tensor component of field (28) and F(0) is the force connected with the scalar co mponent (26). To do that it is convenient to use the 4-vector A
(g) k

(32) and the equat ion of mot ion fro m (34). If the

particle is at rest or, generally speaking, moving but very slowly (v/c ~ 0), then for each term in (29) only 0-co mponent of correspondent 4-vectors will be nonzero. I.e. (further the symbo l g is omitted for (g))



( 2)



3C 2r

and

( 0)



1C . 2r

Correspondingly, for the forces we have

F( 2 ) e

d

r dr r
(2)

and

F( 0) e

d

r . dr r
( 0)

And if we connect the constants f and a with the Newtonian gravit y constant G like this: G f 2/16a, one can write for each of forces separately
F( 2 ) 3 GmM r 2 r2 r

and

F( 0 )

1 GmM r 2 r2 r

.

(35) '

15


It produces as a result the usual good old Newtonian law :-)
F = F(2) + F( 0 ) GmM r , r2 r

(35)

i.e., the attraction in the sum. Thus, here I would like to emphasize that even at r >> GM/c2 the static, spherically-symmetric gravitat ional field in vacuum, generated by a massive object in the centre (in particular, by a quasistatic, centrally-symmetric distribution o f mass with the densit y (r)) as the physical act ion on test bodies is an algebraic sum o f the attraction F(2) (the gravitat ional tensor co mponent proper) and the repulsio n F(0) (the scalar co mponent of gravitation). The foregoing has shown that this property fo llo ws the most general principles lying in the base of the dynamic interpretation of the gravitat ional field. Or rather (35) must be considered as a direct consequence o f the conclusio n formulated at the end of the previous sect ion for the case of weak field. A descript ion of interaction between a part icle o f a given rest mass m moving wit h so me velocit y v, and centrally-symmetric field (29) of a motionless massive centre, can be obtained fro m act ion (31) in which the quantit y L mc
2

1 v2 / c 2 fm

(1 v 2 / c 2 )
00

1 v2 / c

2

(36)

is an analogue of the Lagrange's function wit h 00 C/r. An explicit separat ion of the test particle interact ion with both components of the field allo ws writing down L in the form L mc
2

1 v 2 / c 2 fm

(1 v 2 / 3c 2 )
00

1 v / c

2

2



1 fm 1 v 2 / c 2 . 4

(36)'

The equat ions o f mot ion o f the particle in the centrally-symmetric field accurate up to terms of the order of v2/c2 << 1 fo llow direct ly the Lagrange's funct ion

L

(

mv 2 1 mv 2 3 v2 m N ) ( m N ) 2 , 2 42 2 c

(37)

L

(

mv 2 1 1 mv 2 5 1 v2 fm 00 fm) ( fm 00 fm ) 2 , 2 4 42 6 8 c

the expressio n for which can be obtained fro m (36) by expansio n into a power of (v/c) series up to terms o f the second order inclusive. And it was taken into account here that (v2/c2) and (N/c2) are terms of an equal infinitesimal, when the part icle moves by act ion of gravitat ional field only. (A small addit io n in (37) to the classical Lagrange's funct ion only depends ult imately on r, if v is

16


expressed by the velocit y o f the undisturbed Kepler problem.) Here we co me clo se to the explication o f the Mercury perihelio n shift effect in GD connected, in particular, with the cho ice o f ik - the EMT gravitat ional field. In GD this effect is in fact an algebraic sum of two effects: (1) The first (linear) effect arises because relativistic corrections of the order of v2/c2 are allowed at the test particle mot ion in the static field of a massive centre. In point of fact this effect is connected a relat ivist ic lag of gravitat ional interaction between the Sun and the Mercury at motion of the latter by its orbit. (2) The second (nonlinear proper) effect is connected with corrections of the tensor potentials (ik) of a massive centre (corrections of the temporal co mponent 00) which arises when allowing gravitational field continuously distributing positive energy around the Sun. It is this part of the effect that depends on the correct choice of the formula for 00, which will be the quest ion in the next section. Here we only explain the first effect arising due to the test particle interaction wit h the field (29) (with the 2 components). Considering the Keplerian motion (m<< M) with a small addit io n

(

mv 2 3 m 8 2

N

)

v2 c2

to the classical Lagrange's funct ion we obtain the value of the perihelio n shift equal to

1

7 GM / c (1 e 2 ) a

2

.

(38)

In this formula (for the perihelio n shift only) a is the usual semi-major axis designation o f the orbit and e ­ orbit eccentricit y. Other effects (the light deflection in the so lar field, the lag of a radio signal in the same field, the gravitat ional drift of atom frequencies = the gravitational red shift) are considered in detail in the paper by Mosinsky (1950) and in the paper by Baryshev and Sokolov (1983) using the same no menclature that is accepted here. The light deflect ion and the signal lag (i.e., the interactio n between electromagnet ic field and a given gravitational field) is understood in GD as interactio n between light and a non-ho mogeneous matter 'medium' wit h the refract ion index

n 1 2

GM / c r

2

and, correspondingly, wit h the velocit y of light propagation in this 'medium' cg = c/n. Both this effects arise because o f the interact ion only with the tensor component of gravitation since the corresponding scalar in

17


f ik T( c2

ik el )



f f1 ik T(ikl ) 2 ik T( e 2 c c4

ik el )



f ik T( c2

ik el )

is identically equal to zero (ik electromagnet ic field.

ik (el)

0) to account of the property of

(el)

ik

- the EMT o f

A rigorous descript ion of the gravitat ional frequency shift demands considerat ion of the atoms behaviour in gravitational field. The cause of the frequency change (the gravitat ional red shift) of a photon radiated by an atom is the shift of the atom quantum levels as a result of electromagnet ic and spinor fields interactions with a given gravitational field (Mosinsky, 1950). The result of a rigorous analys is gives the effect value equal to v/v = ­ GM/c2r which co incides wit h the experimentally measured one. 5. Energy of the Central Source Static Gravitational Field This sect ion shows in detail why formula (1) for 00 was chosen as the temporal co mponent of the spherically-symmetric field EMT of any massive gravitat ing centre. The energy o f each co mponent of gravitat ion ( and ik) will be found separately. The EMT of gravitational field (without gravitational charges in it) was found in the GD in the paper by Sokolov and Baryshev (1980) where we used the fo llowing limitat ions as addit ional condit ions for its cho ice: l) the obtained EMT must be symmetric (ik = ki); 2) ik must have the trace (ikik 0) ident ically equal to zero, whith is connected with the 'masslessness' o f gravitat iona l field; 3) it must always give the positively determined gravitat ional field energy densit y (00 0); 4) the last condition concerns each of its components separately (i.e.,
00 (0)

0 and

00 (2)

0).

But general principles alo ne are apparent ly not sufficient for the gravitational field energy densit y value final cho ice. To the opinio n of investigators who tried to look at this quest ion (Thirring, 1961; Sexl, 1967), the account of the posit ive gravitat ional fie ld energy continuously distributed in space around a central source gives an appreciable contribution co mparable by order of magnitude wit h observat ions of planets orbit perihelio n shifts in the Sun gravitat ional field. It is the effect which cannot be described by allowing for only the relativist ic lag of gravitat ional interact ion. In other words, as was noted in quoted papers, the effect of the Mercury perihelio n shift demands consideration o f the so-called gravitational potent ial 'nonlinear' correction, arising because of account of the gravitat ional field EMT itself in the equations (20*, 21*) right-hand side. Naturally, we used this circumstance and chose in a sense the simplest expressio n for the EMT of the possible ones sat isfying 4 conditions ment ioned above, and which leads (as it turned out) to explicat ion o f the observed Mercury perihelio n shift for 100 years. I.e., we directly used the fact that in GD the cho ice o f ik is limited by experiment too.
18


To obtain the EMT (in particular, in the so-called 'canonical' form) one must proceed fro m the gravitat ional field Lagrangian (12) alo ne. And one must take into account that the vector component of the field ik is absent in it, i.e., the Hilbert-Lorentz condit ion
im ,m ,i

=½

(16) is

satisfied. Besides, to obtain the canonical EMT in a symmetric form at once, one must choose the constant b in the field Lagrangian (12) in front of a negligible divergence addit io n like this: b a. Thus, the Lagrangian (12) is reduced to the form Z = a
mn,l mn,l

­ a ½ ,m

,m

;
ik

(12)'

or, if to separate explicit ly the purely tensor ik and scalar components of the field (if
ik ,k

= ¼ ,i) we obtain Z = a
mn,l mn,l

­ a ¼ ,m

,m

,

(39)

Hence, we have for the symmetrical (and canonical) EMT tik = 2a(t where t
ik ik

­ ¼ tik)

,

(40)

= t
ik 00

mn,i



mn

,k

­ ½ ik

mn,l ,m

mn,l

,

= ,i,k ­ ½ ik,m

.

If now one calculates the value o f t ( N)2/8G.

for the gravitational field around a massive centre with the
00

use of (26) and (28), we obtain already the result coinciding wit h the formula (1): t

=

However, the point is that the definit ion of the canonical EMT as a conserved quantity is still ambiguous: why exact ly this EMT and not any other one? In part icular, by means o f a certain arbitrariness at the cho ice o f the EMT one can redefine so (see, for example, Medvedev, 1977, p. 206) that a new tensor should have the trace identically equal to zero, which corresponds more to properties o f the field under discussio n. Thus, a hope appears to choose the unique EMT fro m a n infinite set of various ones wit h the help of another general condit io n. And however, if to redefine now at once the EMT into form (40) wit h Lagrangian (39) so that a new tensor should have no trace, then as a result we shall obtain the field energy densit y wit h (generally speaking) an uncertain sign. In particula r, for the static field we co me to an absurd result: energy of this field is equal to zero everywhere. (The fields condition
ik ,k ik

and are still connected by the

= ¼ ,i , see below... )

If we calculate the field EMT, it is more logical to proceed from the notion of free field, e.g. away fro m its source. But for an interacting field (with its source) one may not speak by definit io n
19


about any conserved quant ities, including the conserved complex of the field energy-mo mentum. Only for the pure case of free field one can introduce dynamic (or conserved) quantit ies characterizing the system. These are like quantum numbers. Hence, it fo llows fro m the notion o f the free field of a definite spin meaning that we deal wit h the field ik which is a mixture of the purely tensor and scalar free fields (but interacting with matter by the same coupling constant in general). Besides, the energy densit y o f static gravitational field around a massive object which is ult imately to be calculated by means o f the field EMT obtained expressio n, is most likely to be unchangeable and equal to ( N)2/8G. In spite of ambiguit y of the field EMT canonical definit ion, its temporal co mponent (in accordance with its physical sense) must uniquely determine some certain part of the system 'matter + field' total energy (Mc2), but outside the sphere filled by matter. (Here, in the GD, we can proceed from the notion o f field energy accepted in ED...) As it will be shown below, the corresponding addit io n to the Mercury perihelio n shift is just connected with the fact that a certain part of system energy (energy outside the sphere of the radius r around the Sun) is as if 'excluded', and at the distance r from the Sun (in vacuum) an effect ive decrease of the massive central body mass occurs by the factor of (1 ­ ½ GM/rc2). So, the equations o f free gravitat ional field away fro m its sources are obtained fro m equations (20*, 21*):
0 ik 0

(41**)

Far from the sources one may completely assume the field and ik to be independent and free, then one can obtain each o f the equations noted by (41**) by independent variat ion o f corresponding parts in Lagrangian (39). But if the fields and ik are really independent then, generally speaking, they must not alread y be scalar and tensor parts of some unique tensor ik = ik + ¼ ik ( Hilbert-Lorentz condition
ik ,k m m

0) consistent with the
ik ,k

= ½ ,i , which connects two fields and ik (

= ¼ ,i). Such a

connection is natural in (12)' or in (39), when the fields and ik: interact with its sources. The third sect ion has ment ioned already that when considering the free field in vacuum one can, even on the level of Lagrangian (12) with field equations (10) by means of gauge transformat ion (10a), separate himself fro m scalars and vectors, independent ly o f anyt hing, demanding the satisfact ion o f 5 gauge condit io ns (7) and (8) (wit h the help o f the arbitrary scalar and vector Ai) to obtain equat ion ik = 0 at once. That is to say, in vacuum for the free field (i.e. without direct interact ion wit h sources of field) we have the right to demand independently t he sat isfact ion o f the

20


5 conditions for the tensor field ik only right away in form
ik ,k

=0

and

ikik = 0.

(41)

The first (vector) condit ion is true now not because of the equat ion = 0 for the scalar field (as if the connect ion
ik ,k

= ¼ ,i exist as before) but directly as the gauge condit ion excluding the vector

field. In conformit y wit h this pure case, one can obtain both the field equat ions and the EMT of the tensor component from Lagrangian a
mn,l mn,l

,

(42)

which fo llows from (12) and fulfilment of condit ions (41). Hence, the EMT of such a free field (of the determined spin) can be obtained fro m the corresponding part of (40) if to redefine the canonical EMT (see details in Medvedev, 1977; Sokolov and Baryshev, 1980) so that it should have the trace ident ically equal to zero (and owing to ik = 0 too): where t
ik (2) ik (2)

2at

ik

(2)

,

(43)

= 2/3

mn

,i



mn,k

­ 1/6 ik

mn,l

mn,l

­ 1/3 mn

mn,ik

The independent scalar field is by no means excluded because of gauge condit io ns (41) for the purely tensor field. As fo llows from the equat ion (20*) in the system (20*, 21*), in principle the scalar field can be radiated by the corresponding part (the coupling constant f is the same) of the same source and afterwards become the free field independently of ik (in the linear GD). That cannot be said o f the vector field which is not radiated at all because o f conservat ion law (15), i.e., because of the vector source absence. The possibilit y of radiat ion o f scalar (longitudinal) waves is a separate question, it will be ment ioned below. It is now important that the equation for a free scalar field = 0 in (41**) is obtained fro m the second term in (39), both the value and the constant sign to be not essent ial. It is necessary only t hat the scalar and tensor in (39) should be measured ult imately in ident ical unit s. This is connected once more with the genet ic connection essence of these fields as both components of gravitat ion field interacting with matter by the same coupling constant f. But for obtaining the EMT of the independent and free scalar field the negat ive constant before its Lagrangian in (39) does not fit. First of all, it leads to negative energy o f the -field and, second (as it was said above), in sum wit h energy o f the tensor field ik it gives an uncertain sign o f gravitat ion field energy in who le. Hence, for the posit ive definiteness of energy one must take the free scalar fie ld Lagrangian in the

21


form + a ¾ ,m ,m , (44)

with the same equation for free scalar field = 0 (without a direct interaction with sources). The select ion of the constant ¾ is connected here with the requirements (the axio m also) ment ioned above, i.e., that the field energy densit y in vacuum or around a massive gravitat ing centre must coincide wit h one which the canonical (also symmetrical) EMT (40) gives in that case. Hence, one can obtain for the EMT of the scalar field with the trace identically equal to zero where t
ik (0) ik (0)

2 at

ik

(0)

,

(45)

=1/2 ,i ,k ­ 1/8 ik ,m

,m

­ 1/4

,ik

.

Thus, the EMT of gravitational field is a sum o f tensors concerning the scalar and tensor components of the field ik =
ik (2)

+

ik

(0)

2a(t

ik

(2)

+t

ik(0))

.
m m

(46) 0, 00 0,

This sum is consistent with the above ment ioned requirements (axio ms): ik = ki,
00 (0)

0 and

00 (2)

0 (one can become sure of the latter by direct calculat ion). But besides,

everywhere we adhere to the idea that it is necessary to proceed not only fro m Lagrangians whose cho ice is always ambiguous by definit io n (as is seen in (12)), but from the field equat ions which fo llo w them and which are tested afterwards in experiments. From (43) and (45) one can now obtain for the static field tensor and scalar components (28), (26) for a central source:

t

00 (2)

1 6

mn , l



mn , l



1 C2 ,t 2 r4

00 ( 0)

1 1 C2 fM ,m , m , with C . 4 8 2r 16 a

One more property must, therefore, be added for the found field EMT to the properties ment ioned above. It turns out that (at least when r >> GM/c2) field energies o f each co mponent separately are equal to each other in the same space point



00 (0 )



00 ( 2)

1 ( N ) 2 1 GM 2 8 G 16 r 4

2

,

(47)

if the connect ion G f 2/16a for the constants G, f and a fo llows fro m Newtonian law (35). Property (47) for fields (26) and (28) is connected to a certain extent with the cho ice o f the gravitational field EMT. The fundamental character property
22
00 (0)

0

0

(2)

(47) is to be tested in


future. But of principle is the fact that in that case the planet orbit perihelio n shift s is co mpletely accounted for. Let us return again to basic equations (20*, 21*) with sources in connection with the main aims o f this article: the static co llapsar field in GD, the energy problem for gravitational field, and the nonlinear GD. (What is the rest mass of a gravitating object at all and where is it concentrated?) Evidently, when approaching the gravitating centre down to r GM/c2 one must take into account that the total energy (Mc2) of the system 'matter + gravitational field' consists o f fie ld energy in vacuum and energy inside the sphere filled by matter (particles). Here a simple 'po int' notion of the central gravitat ing object ceases to be applicable and one must already account for a kind of 'a splitt ing' of sources in equations (24). First let us consider a static centrally symmetric field (r). Scalar source (25) in equat ion (23) remains always the same (i.e., does not 'split ') when approaching the centre fro m r >> GM/c2 down to r GM/c2. At least the gravitation alo ne in vacuum around a source cannot change this source because identical equalit y to zero of the gravitational field EMT trace (
m m 2

0 for massless field).

Hence, it fo llows that equation (23) is linear down to the limit r GM/c . It means that both the potential (r) and the force (of repulsion) F(0) and the energy densit y 00(0)(r) are found exactly for all r down to r GM/c2. Let us now consider a static, centrally symmetric tensor field ik(r). When mo ving from regions with r >> GM/c2 closer and closer toward a central object with r GM/c2, a simple po int approximat ion for purely tensor source T
ik (2)

(27) must be sat isfied fro m bad to worse (the source is

splitted, i.e., one may not already use the same M in (27), the mass of the central 'po int' as if reduced when approaching r GM/c2). When approaching the centre, the traceless tensor ik (46) or the energy-tensio n o f gravitational field it self continuously distributed in vacuum around a central source (body) must play an ever increasing role in vacuum as an addit io nal source. Hence, equations (24) for purely tensor field ik(r) become nonlinear, which is different from linear equation (23) for the scalar component (r). If we use the results of this sect ion one can calculate the EMT for field (29) in so me point r for some given direct ion of Cartesian axes X, Y, Z:

C2 ik 2a 4 r0 0

10 0 (


0 x x r
2

0
,

2

)

where , = 1, 2, 3; = 0 ( ), = 1 ( = ). But since the cho ice of Cartesian axes directio n
23


in the same frame o f reference for every po int r is essent ially arbitrary because of central symmetry of the problem under considerat ion, then an averaging over all equiprobable (having equa l probabilit y) direct ions o f axes X, Y, Z, drawn from the centre, gives x x = 0, if and x = r2/3. Thus, the energy-tension of gravitational field o f the centrally-symmetric problem in every po int located at a distance r fro m the origin o f coordinates is given independent ly o f axes direct ion by the tensor ik = 00 diag(1, 1/3, 1/3, 1/3),
m m
2

0.

(48)

In such a form the gravitat ional field EMT around a gravitating centre corresponds to some medium as if consist ing of a relativist ic gas (of virtual gravitons). Since here the quest ion is all over on field in vacuum around a sphere o f r GM/c2, then we exclude for the present this small regio n r ~ GM/c2 in the centre. Then the equat ions for the potential ik(r) in vacuum will be

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

ik

.

(49)

Consequent ly, for 00-component of this equat ion we have

1 d2 f C2 [r 00 (r )] 0 2 4 , r dr 2 cr

(49)'

outside the sphere r ~ GM/c2. A sum o f the integral of this equat ion wit hout the right side plus a solut ion o f equat ion (49) with the right side will be a centrally symmetric, t ime-independent solut ion of this equat ion. Then we have

00 ( r ) a

b CGM / c 2 ( ). r 2r 2

Evidently, at r >> GM/c2 the obtained potential must pass into potential (28), which defines (together with the (r)) the densit y
00

in (49)'. On the other hand, we require, as before, the

satisfact ion of the old condit ion about the potential to be zero at r . As a result, for the constants a and b we have a 0, b C3/2, and for the corrected potential ik(r) we obtain ult imately

3 fM 1 1 GM / c 00 ( r ) (1 2 16 a r 3 r

2

)

;

(28)'

ik(r) = 00(r) diag(1, 1/3, 1/3, 1/3) .
24


Here we can return to the unique tensor field with the fo llowing non-zero components

C 1 GM / c 00 (r ) (1 r 2 r

2

)

GM (1 GM / c 2 / 2r ) ; fr

11 22 33

C 1 GM / c 2 (1 ). r 6 r

(29)'

At calculat ion o f a corresponding contribut ion 2 into the perihelio n shift only the correction for 00-component of the potential ik gives ult imately an essent ial addition to the classical Lagrange's funct ion. Other small corrections (to 11, 22, 33) give an addit io n to the Lagrange's functio n proportional to v2/c2, which leads to a negligible contribut ion into the effect. Thus, a nonlinear contribution to the Mercury perihelio n shift turns out to be equal to

2

GM / c (1 e 2 ) a

2

,

(50)

which in sum wit h the linear effect of gravitational interaction lag (38) gives the well-known GR result

1 2

6 GM / c (1 e2 ) a

2

.

(In this formulas for the perihelio n shift a is the semi-major axis o f the orbit and e its eccentricit y, as was in (38).) The sign of the nonlinear perihelio n shift 2 is evident ly connected direct ly wit h the sign o f the gravitat ional field energy 00, as was seen from the foregoing. And the nonlinear contribution into the effect of Mercury perihelio n shift is completely connected only wit h the corrections to the tensor component of gravitat ion. 6. Nonlinear Gravidynamics and the Theorem on the Collapsar Static Field Wit hin the bounds o f the linear GD one can also predict new (respect ive to GR) effects in the weak field of the Earth. The descript ion of such effects in the experiment with a gyroscope on orbit of the Earth, which could differ dynamic interpretation of gravitation (GD) fro m the geo metrical one (GR), is accounted in the paper by Baryshev and Sokolov (1983). But the most essential difference from GR ment ioned above can be the existence in GD o f the socalled scalar co mponent of gravitat ion. In particular, the possibilit y o f scalar radiation or lo ngitudinal gravitat ional waves emerging, for example, at spherically-symmetric pulsat ions and spherically-symmetric co llapse o f a gravit y field source, follows equat ion (20*) in the system (20*, 21*). The possibilit y in principle o f such a radiat ion in GD allows approaching abso lutely
25


otherwise (than in GR it was) to description o f the very process o f a relativist ic, sphericallysymmetric gravitat ional collapse. Such a co llapse should be understood in GD as a process (catastrophic may be) o f the system 'part icles + field' transit io n into a more and more bound state at which the system loses a part of its energy - rest mass - in the form o f longitudinal (scalar) gravitat ional waves. At the collapse o f spherically-symmetrical distributed matter (or a particles system) and at format ion o f a maximum bound body wit h the dimension o f order of the gravitat ional radius (GM/c2), the particle rest mass and the rest mass o f a co llapsing object on the who le can really change. As a matter of fact this is just the statement of the paper by Baryshev and Sokolov (1984). The decrease of the rest mass o f a gravitat ing body in GD follows just the conservation of energy at such a spherically-symmetric collapse, since now (unlike GR) 'an evacuat ion' o f energy is permissible in principle in the form o f scalar lo ngitudinal waves. Thus, the possibilit y in principle of the change o f the rest mass of gravitationally interact ing bodies

( dxk/dt),k 0 can be a basic feature of the x k

nonlinear GD, dist inguishing it fro m the linear approximation, where the law o f mass conservat ion (5) was fulfilled and gravitat ing 'po ints' were assumed to be really po int structureless objects. The descript ion of the co llapse nonstationary process itself as the transit ion of the system in a bound state is however a nontrivial problem. In the total extent this problem requires calculat ing effects of the falling matter brake by radiation arising at the collapse. In accordance wit h the foregoing, the particle rest mass in equat ion (33) cannot already remain constant when falling to the centre, i.e., in general case in GD we deal with the mot ion o f particles wit h the changing mass in principle. Here, using the previous we invest igate the posing of problem about results of such a co llapse. First, one can attempt to study stationary stable states of the system 'particles + gravitat ional field' in the mo ments when the system does not radiate. Analogously to that, as in quantum mechanics o f atom, the system is considered in stationary states or between transit io ns fro m one given energy level into another one. Correspondingly, the collapsar is understood here as some stationary stable state when the fo llo wing is given: 1) the total energy, 2) the dimension of a regio n filled by matter, 3) the total matter mass in this region, 4) the energy densit y of field in vacuum and its 'mass'. The circumstance which was already ment ioned in the previous sect ion and which is to be begun with (in our opinio n) is the fact that the scalar potent ial, i.e., the potential of repulsio n found in the linear approximat ion o f GD for the field in vacuum around a po int with the mass M, does not change, the most probably, in the nonlinear approximation also down to r GM/c2. It fo llows the

26


general requirement, which will remain valid for GD also, namely that the EMT trace T of the system of interact ing part icles comes to the EMT trace of particles only (Landau and Lifshitz, 1973). Such a requirement is satisfied indeed precisely for the gravitat ional field itself and for electromagnet ic classical ('massless') fields also. In any case we use the noted circumstance and in the fo llowing we shall proceed as far as possible fro m the requirement: for spherically-symmetric gravitational field in vacuum to be static down to a distance o f order of GM/c2 fro m the centre it is necessary that two condit ions for the EMT trace of the system 'part icles + field' should be sat isfied. On the one hand (the 'external' integral),
V0



TdV Mc 2 (r )dV Mc 2 ,
V0

(51a)

where V0 > (GM/c2)3 4/3 4/3 rx And on the other hand (the 'internal' integral),
3

.

rx GM / c

2

V0



TdV 4


0

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

(51b)

From the spherical symmetry it fo llows that the funct ions * and v2 depend on r only. The discrete descript ion accepted in GD ho lds that the densit y o f po int particles wit h rest masses ma* bound in a sphere o f radius rx can be represented in the form * = ma* (r - ra). Then condit ions (51a,b) must be understood in the fo llo wing way: a 'po int' source of scalar field located in the origin of coordinates (i.e., somewhere in the centre of a large sphere of volume V0 (51a)) is determined in po int of fact by the trace of the interacting po int particles EMT located in the small sphere of radius of order of GM/c2 (in the 'internal' integral 51b). In a sense here we answered the quest ion what is this co llapsar mass. In po int of fact, conditions (51a,b) can serve as a definit io n o f the rest mass of supposed spherically symmetric co mpact configuration wit h the dimensio n of the region filled by matter of order of gravitat ional radius. However, here the parameter Mc2 describes here the system 'matter + field' on the who le, defining by the condit io ns (51a,b) the static character of the outer gravitational field o f the configurat ion. The mass of the compact object in the centre (the collapsar itself) will be discussed more below. One can suppose that condit ions (51a,b) will be broken in so me way at the decreasing of the dimensio n of the sphere filled by matter down to r 0, without excluding in principle its quantum dimensio n. But we shall do our best below to adhere consistently to the idea that one can always
27


speak about interacting po int particles inside the sphere rx GM/c2. We shall always assume that inside the sphere of r = rx there are particles with the rest masses ma* 0, interacting by means o f some massless fields, or inside the sphere r = rx there is the matter with characterist ic equations o f state with the rigidit y less than p = /3 (cf. Landau and Lifshitz, 1973). Ult imately, the quest ion(s) formulated in Introduction co mes to the quest ion of what limits the condit io ns (51a,b) could be assumed consistent with the requirement that the spherically symmetric field in vacuum should remain static. Now everything is ready for a rigorous proof of the article basic statement, i.e., of the theorem on the collapsar static field. But before go ing direct ly to the formulat ion of this statement I note here that I shall not touch upon uncertainties connected with the notion of an interacting po int of the quantum theory of field. That is why the characterist ic dimensio n under considerat ion GM/c2 is assumed to be quite macroscopic non-quantum dimension for a while. Accordingly, for M values everywhere, far and wide I mean masses of the order of stellar one and more, up to cosmo logica l masses. Besides, one must always keep in mind that within the bounds of GD we deal in result with an idealized situat ion. Surely, real 'po ints' interact not only gravitationally. In particular, at a small distance between these points non-gravitat ional forces can arise which one could neglect at the distances >> Gma/c2. It can be especially important at small GM/c2 ~/>1 km. The properties of the points at the distances of the order of Gma/c2 apart can co mpletely change if to recall that the rest mass can change (ma ma*) at the 'strong' gravitational interaction or at the compressio n of the system down to the r~ GM/c2. Thus, we chose condit io ns (51a,b) as the basic condit ions determining the static character of field in vacuum. It means that scalar source (25) in the right-hand side o f equation (23) remains always the same when approaching the centre from r >> GM/c2 down to r GM/c2. Now it turns out that to answer the quest ion on the static field o f the collapsar - a compact configurat ion (see the Introduction) - one does not need at all to solve equations (49) in all their approximat ions. For it is sufficient to know that the potential (r) and the energy densit y down to rx GM/c2. Let (just for a while) dimensio ns o f the region filled up by po int particles to be unlimited fro m below and let for the present the energy, distributed in space out of the region, to be only the posit ive energy o f gravitat ional field in vacuum - i.e., 00 =
(0) 00 00 (0)

(r) are found already exactly

+

(2) 00

0,



(2)

00

0 .

(52)

How close can one approach the centre for the gravitat ional field in vacuum to remain static? For static scalar field in vacuum we have (in all approximat ions) the 'scalar' energy densit y equal to
28




(0) 00



1 GM 2 . 16 r 4

Consequent ly, one may write direct ly the integral


( vacuum )





(0) 00

4


1 / 4 GM / c

2

1 GM 2 2 r dr Mc 2 . 16 r 4

(53)

It is evident that this integral must also account for the energy densit y o f the gravitational field tensor component
(2) 00

which in all next approximat ions at r GM/c2 can differ, generally

speaking, in so me way from (47). And above all, this energy must be posit ive (non-negat ive) right along. So then the integral for the sum


( vacuum )



(

(0 ) 00



(2) 00

)dV 4 (
rx

1 GM 2 16 r 4

(2) 00

) r 2 dr

is equal to the same value Mc2 at some rx which is anyway greater than¼ GM/c2. Out of dependence upon the concrete form of
(2) 00

0, if the energy o f both gravitational fie ld

and each of its co mponents separately is posit ive, then before the sphere r =¼ GM/c2 will be reached (fro m infinit y to ¼ GM/c2), the total energy Mc2 of the who le configurat ion must already be the energy o f gravitational fie ld so lely in vacuum (
m m

0). It means that even over the sphere r =

¼ GM/c2 the scalar source (the EMT trace of the whole system) must already go to zero:

T *c
(i.e., T=0 everywhere if
m m

2

1 v2 / c 2 0

0). But then either velocities of all particles must become equal to the

velocity of light, or the rest masses in * must become zero, which is all the same. In other words, then the equations of field can be only wave equations all over the place. And the scalar source vanishes everywhere. Consequently, in that case one ma y not speak about any static field at all. (Of course, then there is no any newtonian limit for the gravitational field.) Now one may say that if stable configurat ions wit h the static field outside the regio n filled up by particles are possible, then the dimensio ns o f the region are anyway greater than 2 ¼ GM/c2. And consequent ly, the stationary object in the centre generat ing the static field may occupy only a finite vo lume - the sphere o f a diameter greater than ½ GM/c2. Hence, the impossibilit y fo llows of an infinite mean densit y inside this sphere, also as in principle a possibilit y is excluded exact ly o f approach (the compressio n) to the centre of the gravitat ing object down to r = 0. Thus it was shown that fro m the fact of the positive gravitational field energy ( 'masslessness' (
m m 00

0) and its

0) it follows that it is impossible to compress particles (matter) into the sphere
29


of the volume less than 4/3 (¼ GM/c2)3 having spherically-symmetric static field in vacuum (the tensor source trace T in (20*) not to be equal to zero). Condit ions (51a,b) can be assumed consistent with the static character of the spherically symmetric field in vacuum, if an upper limit in integral (51b) is anyhow greater than ¼ GM/c2. It is natural that the energy so lely o f static gravitational field generated by a central object in vacuum can anyhow only be less than the total energy Mc2 of the who le spherically symmetric configuration:


4 (
rx

(0) 00



(2) 00

)r 2 dr Mc

2

(in vacuum, for rx > ¼ GM/c2) .

(54)

This inequalit y can only be strict. Correspondingly, the co mpact object in the centre (somewhere inside the sphere of the radius r~ GM/c2) must have a finite 'rest mass' M*. And the rest mass M* of the object near the centre (the collapsar proper) cannot be equal to M and must be less than M. It is possible that some part of the configurat ion total energy (mass) must be purely gravitational. Now one can write the statement (the theorem) expressed above about the collapsar static spherically-symmetric field in the form of the integral condit ion 1 GM 2 4 ( 16 r 4 rx where
00 (2)
00 ( 2)

*00 )r 2 dr Mc 2 , rx > ¼ GM/c2 ,

(55)

0,

00 *

0· Here we allow a possible contribution of the positive energy of so me non00 *.

gravitat ional (but massless) field

At fulfillment of condit ion (51a,b), condit ion (55) completely

solves the singularit y problem in GD. 7. Conclusions The previous has shown that in GD co llapsar properties must differ fro m the properties of 'black ho les' (BH) in GR. Though it should be said that the situat ion wit h the co llapsar in GD so mewhat resembles the situation wit h BH in GR. In a sense, the field o f GR may not be assumed static near the Schwarzschild radius. It becomes particularly clear when it is necessary 'to sew' together outside and inside so lutions for the field... In GR they say about a 'so lidifying', infinitely last ing collapse in the frame of reference of a remote observer. In this frame of reference the field in vacuum (Schwarzschild fie ld) is nevertheless assumed to be static. The 'remote observer' frame of reference in GR coincides in po int of fact wit h one determined in Sect ion 4, in which the gravitating body is at rest in the origin o f coordinates. But, it is in this frame o f reference that we have shown absence of any singularit y in GD wit hin the
30


bounds o f the dynamic interpretation of 'o ld' equat ions (14). The general statement expressed in the paper on the spherically symmetric static field following the axio ms lying at the base of the relat ivist ic field theory must greatly help concret izat ion o f the collapsar (the co mpact and stationary, bound object) properties in GD. In part icular, it is possible that the collapsar has nevertheless a surface and its properties do not coincide co mpletely wit h properties of BH described by the known Schwarzschild-To lmen so lut ion. In this connect ion it might be more correct to say not 'BH' meaning the solidifying co llapse, but more carefully to call the corresponding state 'collapsar' and to call basic parameter determining such an object not Schwarzschild radius, but according to its energetic definit io n in the GD to call the value GM/c2 a gravitational radius. References
Baryshev, Y. V. and Sokolov, V. V.: 1983, Trudy A0 LGU 38, 36 (in Russian). Baryshev, Y. V. and Sokolov, V. V.: 1984, Astrofizika 21, 361. Birkhoff, G. D.: 1944, Proc. U.S. Nat. Acad. Sci. 30, 324. Logunov, A. A. and Mesvirishvily, M. A.: 1984, Teor. Mat. Fiz. 61, 323. Landau, L. D. and Lifshitz, E. M.: 1973, Theory of the Field, Nauka, Moscow, p. 504. Medvedev, V. V.: 1977, Nachala teoretichtskoy fiziki, Nauka, Moscow, p. 494 (in Russian). Mosinsky, M.: 1950, Phys. Rev. 80, 514. Sexl, R. U.: 1967, Fortschritte der Phys. 15, 269. Sokolov, V. V. and Baryshev, Y. V.: 1980, in Gravitation and Relativistic Theory 17, 34 (in Russian). Thirring, W. E.: 1961, Ann. der Phys. 16, 96. Vlasov, A. A. and Logunov, A. A.: 1987, Teor. Mat. Fiz. 71, 323. Zel'dovich, Ya. B. and Grishchuk, L. P.: 1976, Usp. Fiz. Nauk 149, 695.

31