S L O W L Y R O T A T I N G S U P E R D E N S E C O N F I G U R A T I O N S I N

R O S E N ' S B I M E T R I C T H E O R Y OF G R A V I T A T I O N

EDWARD V. C H U B A R I A N andARMEN V. SARKISSIAN

Department of Theoretical Physics, Yerevan State University, Armenian SSR, U.S.S.R.

(Received 3 August, 1983)

Abstract. In the limits of the Bimetric Theory of Gravitation the equations of axial-symmetric gravitational field, describing the field of rotating configurations, are obtained and examined. The analytical solutions of these equations out of configurations are found. The equations are solved numerically in the first approxi- mation in the angular velocity for the configurations consisting of the real baryon gas (cf. S ahakian, 1972). The moments of inertia of these configurations in BTG are calculated. It is shown that the equations and the appropriate solution, obtained by Falik and Opher (1979), are incorrect.

1. Introduction

The bimetric theory suggested by Rosen (1973, 1974) is a modified variant of the General

Theory o f Gravitation, which satifies the principles of covariance and equivalence. As

is well-known, the Schwarzschild singularity puts on a specific limitation on the class

of stationary solutions of the Einstein equations. The mass of configurations consisting

of degenerate barion matter cannot significantly exceed 1.55M o. 'Stars ' with

M > 1.55M o masses must surely collapse and form so-called 'black holes'. However,

there are not certain observational facts confirming the existence of the latter.

On the other hand, the solutions of equations of the Bimetric Theory of Gravitation (BTG)

N~,~ 1 - ~g~,vN : - 8~tkT~,,, (1.1)

for the spherical-symmetric distribution of masses do not possess such singularities and

the masses of the degenerate matter are not exposed to such strong limitations (S ahakian

et al., 1978, 1979). The main difference in the formalism of both theories is the use of

the tensor ?~v, rather than the metrical tensor g ~ , in the BTG. Principally it allows us

to separate the pure gravitational effects from those connected within the non-inertial nature o f the frame. Essentially, the concrete form of the flat metric fixes the system of

the frame (cf. Avakian et al., 1980). Thus, the choice of the flat metric in the form (in units of Oppenheimer-Volkoff, G = c = 1)

da 2 = dt 2 - dr 2 - r 2 (dO 2 + sin 2 0 dq~ 2) (1.2)

means that we deal with an inertial system in this frame. In this case the form of the

metric describing the curved space formed by the static spherical-symmetric distribution of masses is represented by

( t3"2 = e 2q~(r) dt 2 - e 20(r) dr 2 - r 2 e 2x(r) (dO 2 + sin 2 0 dcp2). (1.3)

Astrophysics and Space Science 98 (1984) 1-14. 0004-640X/84/0981-0001 $02.10. �9 1984 by D. Reidel Publishing Company.

2 E . V . C H U B A R I A N A N D A. V. S A R K I S S I A N

If the metric of the curved space is chosen in the Schwarzschild form

ds 2 = e za(R) dt 2 - e 2ram dR 2 - R z (d02 + sin 2 0 dq92) , (1.4)

then do -2 - and, consequently - the inertial (flat) frame will be unique and given by

dcr 2 = dt 2 - e 2;~(R) dR 2 - R 2 e 2~(R) (d02 + sin 2 0 d~02). (1.5)

Thus, we see that the form of the metric of the curved space is hardly connected with the form of the flat metric; i.e., with the chosen frame. In effect (cf. Avakian et aL, 1980) arbitrariness in the choice of the frame allows us to put some conditions on the coefficients g,~, and the metrical coefficients 7,~ are connected with correlations expressing the flatness of do z . These conditions not only enable us to solve the field equations, but also define the form of the flat metric in a single way in every separate case (frame).

2. Axial-Symmetric Gravitational Field

The spherical-symmetric gravitational fields have been examined by N. Rosen and also other authors (cf. Sahakian e taL , 1978; Sarkissian e taL , 1978, 1979; Rosen and Rosen, 1977). Stationary, axially-symmetric fields have been examined by us; we have obtained field equations and external solutions of these equations in the quadratic approximation to the angular velocity (approximation fi = f~2/8 rcp(0)) (Papoyan et al.,

1979). It is interesting also to solve the internal problem - that concerning the rotating configurations which consist of the degenerate matter in the limits of Rosen's BTG - and to compare the results with those similar in the GRT. The problem in the present paper is solved to the first approximation in the angular velocity f~ that corresponds to the examination of the rotating sphere, taking into account the Coriolis-type forces.

Let us write the metrics of the curved and flat spaces as follows

d s 2 = e2~ d t 2 - e2~Odr 2 _ r 2 e2Z (d02 + sin2 0 d(o 2) -

- 2cor 2 e 2z sin 2 0 d~o dt ,

do -2 = dt 2 - dr 2 - r 2 (d02 + s i n 2 0 d q ) 2 ) , (2.1)

where co is an odd function of f~. v v 1 v.~. Furthermore, after calculating the components of N~ tensor by N~ = ~(g g~zl~)l~

formula, it turns out that

N 3 = r 2 s i n 2 0 e 2 r 1 j_ + •1 - - ~bl "]- ( '~ + + r 2r z

[ ~ 01 - c~ sin2 0e2~'- 2e + sin20e 2~ + 3co 2 cotg 2 (2.2)

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION 3

will adjoin to the components of N~, known from the static-spherical problem. In this formula the indices 1 and 2 correspond to the derivatives r and 0.

In turn, the number of non-trivial field equations will be equal three: i.e.,

2 (~11 q- -- q~l = 4zck(p + 3P), k = e ~+3~, (2.3)

r

2 Oi l + -- O1 = -- 4zk(p - P), (2.4)

r

(011 " t - I ~ q - 2 ( 0 1 - q} l ) ] (01 " 1 - I ! (01 -- t~l) - - 16r&(p + P)](0 +

1 + r 2 [(022 "{- 3(0 2 cotgO] = 16=k(p + P) f l , (2.5)

where we take into account that 0 = Z (cf. Sahakian et al., 1978), and that the non- zero components of the four-dimensional velocity and the energy-impulse tensor in the examining approximation are equal to u ~ 1 6 2 u3=f~u~ T~ T l = TEE = r 3= -P;

T ~ = - ( P + p)r 2 sin20e2q'-2r ((0 + f~).

It is necessary to add to these equations also the equation of hydrodynamic equi- librium

e 1 ~b 1 - (2.6)

P + p

Inserting P = p = 0 in (2.3)-(2.5) we shall obtain the field equations out of the mass distribution, the solutions of which are (cf. Papoyan et al., 1979)

M M' ~b- , O = - ; (r > R~)

r 1"

( 0 = ~ t=o ~ r t+2 ct F ( I + I 2 l+2, , 2 (M~M') )P~ 1), (2.7)

where

Rs Rs

M = + 3 )r2 dr, M, : f )r2 dr, 0 0

and c t are constants of integration, while F(e, fi, x) stands for a confluent hypergeometric function.

4 E. V. CHUBARIAN AND A. V. SARKISSIAN

Before we start numerical calculations, it is necessary to modify Equation (2.5). We shall seek its solution in the form co(r, 0) = x/~ q = x/~ s o Q,(r)P~ ~) (cos 0). By taking into account the fact that P~)(y) satisfies the equation

d 2/9(/l) dp~l) - - + 3 cotg 0

d0 2 dO + ( l - t ) ( [ + 2 ) P ~ 1)= O,

we shall obtain for (2.5)

d2 Q~ + + 2(01 - q51) dr dr 2

+

when I = 1, and

d2 QI dr 2

[2 ] "1- 7 (I//1 -- (Pl) -- 16zrk(p + P) Q1 = 16~zk(p + P) 8 ~ ,

- - + [ ~ + 2(~1 - q } l ) l ~ S +

(2.8)

[! 1 Q, (29, + (~1 - q~l) - 16r&(p + P) Q, = (l - 1) (I + 2) ~S '

when l > 1. In the homogeneous Equation (2.9) we represent Ql in the form Qt = bzqz. The values

of the constants cz and b t are defined from the condition of continuity of co and its first derivative by r on the surface of the configuration: Le., by

c, ( + M ' ~ = O , btqt(R~) + - F l+ 1 , 2 l + 2 , - 2 M Rls +2 R~ ]

d d [ + ( 7:0 b, drr q,(R~) + c~ dr F l+ 1 ,2 l+2 , - 2 + M'

Rs ,/A

The determinant of the system differs from zero; consequently, b z = c z = 0 (l > 1) and bl r 0, c~ r 0. This means that the external solution in the f~-approximation is defined only by the constant c~, so that

cl ( 2(M + M ' ) ) (2.10) c o ( r ) = ~ 7 F 2, 4, -r '

And within it - by representing q(r) in the form of the sum of the solution of the homogeneous and non-homogeneous equations q(r) = Q(r) + Bj(r) - we shall obtain, instead of ( 2 . 5 ) ,

j , l + I ! - 2 ( ~ , - q51)]J1+ I~ ( ~ q - ~ 1 ) - 1 6 n k ( P + P ) I J = O , (2.11)

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION

Qlt + [~ + 2(~ - ~bt)]Qi +

+ [~(~//1- ~ b l ) - 1 6 r c k ( p + P ) l Q = 16rck(p+P) 8 ~ / ~ . (2.12)

Thus Equations (2.3), (2.4), (2.6), (2.11), and (2.12) form the desired complete system of equations. The values of the constants B and c 1 are specified by the continuity of the go3 component of the metric tensor and its first derivative on the surface of the configuration as

B - QIU- Qfl , r = Rs,

Jfl - J l f

where

JQ1 - Jl Q el - , r = R , ;

Jfl - J l f

sh y) M + M' f = y e y c h y - , y -

y / r

The most important characteristics of the rotating configurations is the moment of inertia, defined from the expression of the total angular momentum of the star as

J = Igt = f T ~ x / / ~ d3x,

R~

[ = - - (P+o) I ~ + 1 r4eS~ (2.13) 3

0

Let us point out that the equation and the solution for slowly rotating configurations in the BTG, obtained by Falik and Opher (1979) are incorrect (see Papoyan et al., 1979; and the present paper).

First of all, while getting their equation for the unknown function N (see Equation (2.10) in Falik and Opher, 1979), the authors used a wrong method, writing it down from a formal comparison of intervals ds 2 written in their and Hartle's (1967) articles. They have not considered the fact that besides d s 2 there exists also the flat metric do- 2 in the BTG. When choosing the curved metric in the form (see Equation (2.6) in Falik and Opher, 1979)

ds 2 = H 2 dt 2 _ Q2 dR 2 _ R2k 2 (dO e + sin 2 0 (dq~ - L dt) 2)

one is obliged to write down do "z, not in the form (1.2), but in the form (1.5), which completely changes the values of the components of the tensor N~ (depending on the covariant derivatives with respect to ~',v)-

6 E. V. CHUBARIAN AND A, V. SARKISSIAN

Secondly, Falik's method is inadmissible, because otherwise it would be possible to obtain all equations ofRosen's BTG (including also those for the spherically-symmetric field) from a simple comparison with the Einstein equations, which is completely ruled out.

Lastly, Equation (2.5) and its solution (2.10) obtained by us from direct calculations (making use of the formalism of the BTG) do not coincide with the corresponding ones from Falik's paper. They have only the same asymptotic behaviour when r ,> Rs, which is a natural result of the fact that the BTG coincides with the GRT in the first approximation (far from gravitating bodies, weak fields). And Falik's solution is no more than a solution of the GRT written down in bimetric notations.

3. Numerical Calculations

Equations (2.3), (2.4), (2.6), (2.11), and (2.12) have been integrated numerically on the IBM computer for a number of values of the energy's density in the centre. The internal solution on the surface must pass continuously over into the external one. It follows that, for each configuration defined by the parameter Pc = p(0), the values q~(0) and ~(0) were found by the method of repeated integrations from the centre up to the surface P(Rs) = 0. The search was carried out up to the fulfillment of the boundary conditions

RA I(Rs) + (Rs) = 0 ,

Rs@a(Rs) -1,- ~l(Rs) = O. ( 3 . 1 )

We have inserted new variables q~= ~(r)-q~(0), t~= ~ ( r ) -~ (0 ) , r = = x e-(e(o)+ 3q,(o))/2 not to have to carry out test integrations, which allows us to write down the field equations with quite determined initial conditions

p(O) = Pc, ~(0) = if(O) = Q(O) = Q,(O) = j,(O) = O, j(O) = const. (3.2)

From the boundary conditions (3.1) we find that

d e(o) = -Xo 8(Xo)- 8(Xo),

d 0(o) = - X o -

dx (3.3)

where

Xo = Rs e l/z(e~(o) + 30(0))

Calculations have been carried out for real baryon stars with the equation of state from Sahakian (1972) (equation of state 1); and with the equation of state accounting for the existence of the re--condensate in nuclear matter (Sarkissian et aL, 1978): (equation of state 2).

,r

x

Fig. 2.

O[ - - ,' -1.5 -1

~ GRT

- 0 . 5 0 0.5 1 1.5

Fig. 1. D e p e n d e n c e o f the re la t ivis t ic m o m e n t o f iner t i a I(pc) f rom the cen t r a l densi ty . O n the axis o f the i

a b s c i s s a the va lues o f the p a r a m e t e r r/p = a r c t g lgpc/pn are pu t , w h e r e p , is the nuc lqa r densi ty .

1

o

0 . 5

0

',d"

o ,p.

x o')

O3

(I X 10 3) 2

3

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION 7

0 0.5 1 r/R~

Dis t r i bu t ion o f the re la t iv is t ic m o m e n t o f iner t i a I(r) a n d the dens i ty a long the rad ius :

Pc = 5.64 • 1014g c m - 3 - 1, Pc = 2.12 x 1015g c m - 3 - 2, Pc = 3.44 • 1018 g c m ~ 3 - 3,

Pc = 3.44 • 10t5 g c m - 3 - 4, Pc = 4.22 • 1016 g c m - 3 - 5, Pc = 4.63 • 1018 g c m ~ 3 - 6.

8 E. V. CHUBARIAN AND A. V. SARKISSIAN

Fig. 3.

150

100

50

0

2 -- ((.0 X 102)

1 - ( ~ X 1 0 2 )

0.5 1 r/R~

Dependence of co(r) for the configurations with Pc = 5.46 x 10X4g c m - 3 _ 1,

Pc = 2.12 x 101s g c m -3 - 2, Pc = 4.63 x 1018 g cm -3 - 3.

TABLE I

pc M R 0 I ~max Erot N

g c m - 3 M o km 1045 g cm 2 103 cm - 1 1050 erg 1057

BTG 2.45 • 1011 0.99 594 438 0.03 1.37

2.64 x 1013 0.89 1400 2300 0.007 0.49 5.64 x 1014 0.67 12.28 0.55 6.16 105

2.12 x 1015 1.89 9.38 2.15 11.8 1500 8,5 x 1015 3,7 8.41 8.63 13.3 7660 4.22 x 1016 7.31 9.62 73.72 9.55 3.4 x 104

6.27 x 1017 14.59 12.77 1085 5.16 1.4 x 105 4.63 x 1018 16.29 13.35 1395 4.58 1.5 x 105

1.26 • 1019 15,74 13.04 380 4.8 4.4 x 104

GRT 2.45 • 1011 0.99 594 438 0.3 1.37

2.64 • 1013 0.89 1400 2300 0.007 0.49 5.97 • 1013 0.73 3800 1.9 • 104 0.001 0.16 5.51 • 1014 0.64 1.3 0.49 6.22 94.7

8.14 • 1015 1.18 11.8 0.99 9.76 472 1.65 • 10 ]5 1.39 10.9 1.12 11.9 797 2.44 x 1015 1.51 10.2 1.11 13.7 1048 5.58 x 1015 1.53 8.31 0.85 18.8 1509 5.53 • 1016 1.17 6.51 0.33 23.7 928

1.18

1.06 0.83 2.6

6.13 16.9

53.6 65.1 61.2

1.18

1.06 1.01

0.81 0.81 0.82 0.85 0.88 0.68

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION

TABLE II

Pc M R 0 I ~max Erot N g cm- 3 M o km 1045 gcm 2 103 cm- 1 10 s~ erg 1057

BTG 1.2 x 1033 0.02 262 102 0.12 7.7 0.03 6.7 x 1033 0.24 8.6I 0.l I 6.63 23.8 0.29 2.2 x 1034 0.44 8.99 0.25 8.04 81.8 0.54 4.5 • 1034 0.46 3.53 0.23 8.83 90.1 0.57 2.2 x 1035 0.45 8.09 0.12 11.43 78.5 0.52 4.7 x 1037 3.2 3.74 6.19 22.20 1.5 • 104 8.72 6.5 x 1038 10.74 7.23 379 6.08 7 • 104 65.25 3.9 x 1039 12.49 7.87 389 5.05 2 X 104 85.92 6.5 • 104o 11.17 7.29 379 5.93 2.6 x 105 69.51

GRT 6.5 x 1025 1.05 3739 49700 0.002 0.66 1.04 6.5 x 1028 0.87 809 1200 0.015 1.31 1.11 6.5 x 1032 0.86 1380 2700 0.007 0.59 1.14 1.2 x 1033 0.02 262 102 0.12 23.7 1.12 6.5 • ]033 0.22 8.82 0.097 6.52 20.63 0.31 5.6 X 1034 0.45 8.89 0.216 9.22 91.77 1.65 6.5 X 1035 0.35 6.78 0.072 12.21 53.63 1.21 6.5 X 1036 0.29 3.09 0.012 36.11 78.24 0.9 6.5 • 1037 0.54 2.27 0.031 78.26 949 0.91

Numerical integration was carried out for a number of values of central densities,

covering the region of the white dwarfs and baryon stars. The results of the calculations

are represented in the Tables I and II and Figures 1-3.

The results of the numerical integration with the equation of state 1 and equation of

state 2 in the B T G and G R T are correspondingly represented in the Tables I and II. The

values of the mass densities, masses of the configurations and their radii are given in

the first three colunms.

The maximal value of dense configurations is removed on 3 orders towards larger

densities. So, if the maximum of the mass is obtained by the Einstein theory when

Pmax = 3.58 X 1015 g c m - 3

then in the B T G

p(O) 3.6 x l0 is m a x = g c m - 3

e (M~a • = 1.55214o) ,

B (MmB~x = 16 .08Mo) .

Taking into considerat ion that it is possible to use the same criterion of stability (see

Baleck, 1980) for 'bimetric ' configurations, the displacement of the region of the growth

of the M(pc) curve in the B T G indicates that the class of stationary neutron stars in the

B T G considerably exceeds the 'Einstein ' class. If we compare the values for M w i t h the

results of earlier papers (obtained on the basis of 'softer' equations of state) and take

account of the existence of re- -mesons (MmSax = 12.71Mo) in the nucleus and equation

10 ~E. V, CHUBARIAN AND A. V. SARKISSIAN

of state for ideal baryon gas (MmBax = 0.1797MG) (cf. Sarkissian, 1978), it is possible to convince ourselves once more that the harder the equation of state, the larger is the maximal possible value of the mass.

In the last column of the tables the values of the full number of particles are written. If n is the density of the number of baryons, it is possible to calculate the full number N by the formula

N = f nu ~ x / ~ d3x. (3.4)

Taking into account that, in the f~-approximation u ~ = e - ,(r), we obtain for the total number of baryons of the bimetric configurations the value

Rs t ~

N = 4re I e3~n(r)r2 dr. (3.5) , d

0

The maximum value of the angular velocity of rotation is defined by the absence of outflow of matter during the rotation. This condition can be found from the equations of hydrodynamic equilibrium

Tv~. = 0, (3.6)

which are independent and do not exist in the field equations in the bimetric theory, The covariant derivative of the tensor T~ is equal to

1 ~ ( ~ - - g r~) - 1TUp Og, p (3.7)

Furthermore, we observe that

�89 ?g.p _ 8 In x f Z g ~X v ~X v

(3.8)

and

8gP'~ dg~P (3.9) g~p - g~p 8x" 8x ~

If so, then by resort to Equations (3.7)-(3.9), the equation of hydrodynamical equilibrium for the matter can be written in the form

1 8 P _ 1 1 8 { . , / ~ u . u ~ , ( p + p ) _ ~ u . u p a g ~ , p ' ] ,

P + p 3x ~ ~ (P + P) ~x ~ (3.1o)

where u ~ is the four-vector of velocity. The first member on the right side of (3.10) disappears in the case of stationary fields (independent of time) and in the presence of

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION 11

axial-symmetry (no dependence on the angle q~). Using the condition

8 8x ~ (g~ou~u ~ = O, (3.11)

we obtain for the Equation (3.10) of hydrodynamical equilibrium

1 81' 8u~ - - U u

P + p 8x v ~3x v

which gives

1 8P 8 In u ~

P + p 8r 8r

1 OP O In u ~

P + p 80 80

(3.12)

- - ; (3.13)

from which it following that

dP - d l n u ~ . (3 .14)

P + p

For the rotating configurations with f~ = const., only two components of four-vector velocity differ from zero: namely,

u 3 = f~u o ' u o = [e2~ _ e2~, r2(co + f~)z sin 2 0] - 1/2 (3.15)

As is pointed out by the numerical calculations, the value of the co function is maximum at the centre of the star - always smaller than f~. With increasing of r, the value of co decreases monotonously becoming zero at infinity. That is why we leave out co in (3.15), in order to simplify the further calculations. Then Equation (3.14) on the equator of configurations (0 = 7z/2) will be written as

1 d P d ~ 1 d

P + p dr dr r dr In (1 - e z~- 2~b r 2 ~ 2 ) . (3.16)

Here the 'spherical' values of the functions P, p, (p, and ~ are used. On the surface of configurations the left part of (3.16) is equal to zero because of the

absence of the pressure gradient - i.e., we obtain the condition of equality of the gravitational and centrifugal forces on the equatorial surface of the star (condition of the absence of the outflow of the matter). For the maximum value of the angular velocity of rotation in the examining approximation we get

M 2 e2~p(gs) - 2~0(Rs) 2

~"~max = ( 3 . 1 7 ) R~(1 - r

For the same suppositions from (3.14) in GRT we obtain

2 (goo)l ~'~max -- ( 3 . 1 8 )

2R~

12 E.V. CHUBARIAN AND A. V. SARKISSIAN

Besides the values (3.17) and (3.18) for different configurations also the values of the energy of rotation

2 I f ~ m a x

E r o t = - - ( 3 . 1 9 ) 2

are given in Tables I and II. As can be seen from these tables, in the region of maximum mass and moment of

inertia of the baryon stars, the energy of rotation is ~ 50 ~o (10 ~o for the configrations with re- -condensate) of the total energy. For the 'Einstein' configurations it is only 5.5 ~o (1.1 ~/o) of McZ. Apparently, this circumstance is connected with the compactness of the bimetric configurations, which allow larger values of f~max. The difference noted in the values of the energy of rotation is a relativistic effect (in the Newtonian limit, of course, there is no such difference).

Figure t represents the dependence of the complete relativistic moment of inertia from the central density. In the region of metastable stars (up to the values Pc ~ 1014 g cm- 3) the difference between the values of I in the BTG (continuous curve) and in the GRT (dotted curve) is very small. A further increase of the central density makes this difference more noticeable. In the region of neutron stars it reaches the value IBmax/IEmax ~ 103. Lastly, at super-high densities I(pc) oscillates both in the BTG and GRT. This oscillation of the moment of inertia (also of the mass, radius and of the total number of baryons, apparently as in GRT (cf. Dimitriev and Holin, 1963)), is condi- tioned by a special form of equation of state P --- const, p in the central region of the configuration at relativistic densities . . . .

Figure 2 represents the relativistic moments of inertia I(r) along the radius, at central densities Pc = 5.64 x 10X4gcm -3 (curve 1), Pc = 2.12 x 1015gcm -3 (curve 2), and Pc = 4.63 x 1018 g cm -3 (curve 3).

It is easy to see, that for less dense configurations I(r) attains its maximum value correspondingly in deeper regions of the star. When Pc = oo - as is evident from the calculations - I(r) increases up to the surface.

On the same figure, curves, defining the distribution of density within the configuration are represented, when Pc = 3.44 x 1015gcm -3 (curve4), Pc = 3.2 x 1016gcm -3 (curve 5), and Pc = 4.63 x 1018 g cm- 3 (curve 6). Compact configurations, as can be seen on the figure, confine the main mass to deeper regions: the more compact the star, the smaller the volume in which the total mass is concentrated.

As in the Newtonian theory (and also in GRT) the value of centrifugal forces, acting on the isolated element, depends on the velocity of rotation in the local frame. However, it is known from GRT, that the local system is 'dragged' by the rotating body. This phenomenon is known as the effect of Lense-Thirring (cf. Landau and Lifshitz, 1973). Let us examine the process of dragging of the local inertial system by the rotation of a star in the limits of BTG. The problem of finding the gravitational field, formed by a uniformly rotating body, was solved in a fixed (inertial) frame. Then the flat metric was chosen in the form (2.1). However, when we examine the movement of the testing body in such a field, it is more convenient to solve the problem in a free-falling frame.

ROTATING SUPERDENSE CONFIGURATIONS IN BIMETRIC TRANSITION 13

In the field of the rotating body, such a frame (and also the testing body), falls with an acceleration along the radius on the central body and rotates around it as a result of dragging. The form of the fiat metric in such a frame allows to define the inertial forces and the angular velocity of dragging.

Writing down the metrics da 2 and ds 2 in the form

ds2 = g~v dxS' dx v, d~ = ?,,v dx~* dx ~, (3.20)

where g,~ and 7,v are the functions of ,, R, and 0. With the help of the transformation of the type

t = t(z, R, 0), r = r(,, R, 0), O' = O, qr = q~ + f ( , , R, O)

one can impose g0o = 1, goi = go3 = 0 conditions, meaning the use of the free-falling frame. The component ?~3 = 733 = - r2 sin2 0 remains unchanged. In this case the ten equations of the field, together with the six conditions, imposed on the fiat metric (P~,~,p = 0), allow to define the 16 remained functions guy and 7,,~. In the free-falling frame the problem of finding these functions was solved in the quadratic approximation to the angular velocity fL Then the flat metric, being far from the rotating body, will be (cf. Avakian et al., 1980)

( o ) d o "2 1 + rg coar 2 = Sin2 C2 dz 2 _ rg dR 2 r C 2 r

d '~ _ r 2 "--(.CIU2 + s i n 2 { / d f p 2 ) - - - - _ 2 0 9 2 r 2 sin2 0 d R - - +

C

+ 2cor 2 sin 2 0 dq~ d~ - 209r 2 sin 2 0 --dR dq~ + C

+ 6ilb I sinOcosOdOc d z - 6ilbrg sin0eos0 ~ dO. (3.21) r

The values r and co are defined by the expressions

Cl ( , 0 m - - ,

r 3

r = rg~3(Rl_- -o~--c~)12/3 { 1 + Coil( 2rg .~1/3p2}2/3 , (3.22) j \ 3 ( R -

where b = c2/~r; bl = c3/w/r; P2 is the Legendre polynomial of second degree. The constants Co, c2, c3 can be found by a match with the internal solution.

As it follows from the form of 7o0, the chosen frame falls with acceleration g ,~ GM/r 2 in a radial direction to the centre and rotates with angular velocity co around the body, which brings to the appearance of the potential rg/r and the potential of centrifugal forces

- - ( ( . 0 2 r 2 / c 2 ) sin 2/9 in the fiat metric

14 E. V. CHUBARIAN AND A. V. SARKISSIAN

So the dependence of co from r is an important characteristics of the star, because it is related with the angular velocity of rotation of the particle falling from infinity on the gravitational field of the rotating body. Outside of this body co has the (2.10) form. The dependence of co from r, along the radius of the configuration, when Pc = 5.64 x 1014gcm -3 (curve 1), Pc = 2.12 x 1015gcm -3 (curve2), Pc = 4.63 x 10 is g c m - 3 (curve 3), is indicated on the Figure 3. As can be seen, co is maximum in the centre and monotonously diminishes with increasing r.

References

Avakian, R. M., Chubarian, E. V., Khachatrian, B. V., and Sarkissian, A. V.: 1980, Astrophys. Space Sci. 68, 347.

Baleck, V.: 1980, Abstracts of the 9th International Conf. GRG, Jena 2, 457. Dimitriev, N. A. and Holin, S. A.: 1963, Problems of Cosmology 9, 254. Falik, D. and Opher, R.: 1979, Gen. Rel. Gray. 10, 343. Hartle, J. B.: 1967, Astrophys. Y. 150, 1005. Landau, L. D. and Lifshitz, E. M.: 1973, Field Theory, Moscow, 504 pp. Papoyan, V. V., Sarkissian, A. V., and Chubarian, E. V.: 1979, Astrophys. Space Sci. 64, 65. Rosen, N.: 1973, Gen. Rel. Gray. 4, 435. Rosen, N.: 1974, Ann. Phys. N.Y. 84, 455. Rosen, N. and Rosen, J.: 1977, Astrophys. J. 212, 605. Sahakian, G. S.: 1972, Stable Configurations of Degenerate Gas Masses, Nauka, Moscow. Sarkissian, A. V. and Khachatrian, B. V.: 1978, Scientific Notes of YSU 2, 55. Sarkissian, A. V., Khachatrian, B. V., and Chubarian, E. V.: 1979, Astrophysics 15, 503.