Russian Physics Journal, Vol. 51, No. 12, 2008

ON SOME COSMOLOGICAL PROBLEMS О. N. Radchenkov

UDC 514–539

A regular solution describing a periodic expansion – contraction process of the Universe is derived. Keywords: cosmology, Universe, dark matter.

Recent astronomical data suggest that our Universe appears to be spatially flat, its expansion accelerating. These findings can be reconciled with theory, assuming predominance of special kinds of matter, referred to as dark matter and dark energy, in narure [1]. The experiment, however, does not confirm the presence of such matter. In this work, a possible explanation of these phenomena is proposed on the basis of a compensatory gravitation model [2]. Geometrically, the model assumes that the common ideas of curved basic and flat tangent spaces change places. The gravitational field acts as a physical field being described by the mixed tensor Gμν in the Minkowski space. It is its gage specificity of interaction with matter that causes the motion of matter to be locally perceived as occurring mechanically in a certain non-flat spacetime with the metric

g μν = ηλσ e(μλ ) e(νσ) ,

e(νμ) ≡ δμν − Gμν ,

(1)

where ηλσ is the Minkowski space metric. This process resembles propagation of light beams through an optically inhomogeneous medium in some way: this medium can be considered as a certain three-dimensional analog of the space, abandoning the concept of interaction with light. Field equations reproduce a tetrad form of the Einstein equations [3], provided the tensors e(νμ) and eν(μ ) are treated as reciprocal tetrads

1 ∂ λ Cμ λν + ∂ λ C ν λμ + ∂ ν C λ μλ + ∂ μ Cλ νλ + Cμ λσ C ν λσ 2 πk ⎛ 16 1 ⎞ − Cτνσ + C σν τ C τμσ − Cμ τν + C ντμ C σ τσ = 4 ⎜ Tμν − δμν Tλλ ⎟ . 2 ⎠ c ⎝

(

)

(

)

(2)

Here raising and lowering of the indices is performed by means of the metric ηλσ .

(

)

(

)

C λμν = ∂ μ e(σν ) − ∂ ν e(σμ) eσ(λ ) ≡ e(σμ) e(τν ) − e(σν ) e(τμ) ∇ τ eσ(λ ) ,

(3)

where ∂μ ≡ e(νμ) ∇ν is the operator of an extended covariant derivative in the Minkowski space (do not confuse with partial derivation!). In addition to these equations, the following important equalities result in theory:

Institute of Applied Mathematics of the Far East Branch of the Russian Academy of Sciences, Vladivostok, Russia, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 46–51, December, 2008. Original article submitted February 29, 2008. 1294

1064-8887/08/5112-1294 ©2008 Springer Science+Business Media, Inc.

∂μ C τνσ + ∂ ν C τσμ + ∂ σC τνμ = C τλμ C λ νσ + C τλν C λ σμ + C τλσC λ νμ ,

(4)

these are corollaries from the Jacobi identities for the commutator of the operators ∂ μ . According to their role, Eqs. (2) and (4) prove to be similar to the pair of four-dimensional Maxwell equations. Equation (3), however, relates the strength C τμν with the potential e(νμ) . Let us go to the statement of the problem. In the Universe, one can generally observe homogeneously isotropic distribution of matter. In order to determine how its parameters are varied, we will try to find a solution to the corresponding averaged equations (2)–(4) similar to electrodynamics of continua. The solution will be sought in the Galileian coordinate system for a certain spatial region Λ of volume V , so large-sized that it can be subdivided into a number of subregions where distribution of matter can still be assumed homogeneous and isotropic. Let us take the averaged 3-D tensor field, say, of the second rank, to mean time functions invariant on spatial rotation of the coordinates

aki =

1 i ∫ ak (ct , x, y, z )dxdydz , VΛ

(5)

where the Latin indices run over the values 1, 2,3 , and disregard fluctuations, that is, assume that

aij alk = aij

alk .

(6)

In fact, the latter assumption can, naturally, be fulfilled only approximately. It will be considered to do the better the larger the region volume V . Let us separate out the field components which can be invariant to the rotation group e0(0) , ek(i ) , C i 0 k ,

1 ijk n ε C jk , T00 , Tki , 2

(7)

1 ijk 0 ε C jk , T0i , Tk0 2

(8)

where εijk is the discriminant tensor. The rest terms e0(i ) , ek(0) , C 00 k ,

are of no interest. Upon averaging over directions, they will disappear. Let us consider e0(0) . One can see that tensor (3) has no time derivatives of this function, that is, in the problem, it acts as a peculiar integrating factor at the variable ct . Therefore, for example, redefining time or merely fixing the corresponding gauge, we can write

e0(0) = 1 .

(9)

For the rest field values we get ek(i ) = aδik ,

T00 = ε,

Tki = − pδik ,

C i 0 k = Dδik ,

1 ijk n ε C jk = H ηin , 2

(10)

where a, ε, p, D, and H are some unknown time functions. Let us find the relation between a, D , and H . Averaging Eqs. (3) and (4), we derive

1295

H = DH ,

D = − a a ,

(11)

where the dot means partial differentiation with respect to the variable ct . Hence, H =Ω a,

(12)

where Ω is a pseudo-scalar integration constant. The averaged equations (2) in terms of functions (10) are written as 4πk D − D 2 = 4 (ε + 3 p ), 3c

1 4πk D − 3D 2 − H 2 = 4 ( p − ε) . 2 c

(13)

The first equation corresponds to similar time indices, and the second – to spatial ones. According to Eqs. (11) and (12), they can be written as the first-order equations on the functions a and ε using superposition and differentiation, and we have

a 2 a

2

+

1 Ω2 8πk = ε, 4 a 2 3c 4

a ε = −3 ( ε + p ) . a

(14)

To close the system, one needs the equation of state p (ε ) . The observations show that matter in the Universe consists mostly of hydrogen (~75%) and helium (~25%). The matter is largely concentrated in galaxies. In the context of scale of the problem, this part of the Universe is conveniently considered as a certain specific gas where galaxies are particles. However, matter can be modeled as a two-component thermodynamic system “gas–radiation”, taking into account stellar and background radiation in space. The equation of state for such a system cannot be readily derived. To this end, one must average, at least, energy-momentum tensors of the continuous isotropic medium and radiation field. They are given by formulas (3.6) in [2]. For radiation this averaging is obvious: its energy-momentum tensor spur is zero. However, for the medium one must define the function of intrinsic specific internal energy U as well as other conditions. For example, stellar radiation must be considered as a result of gravitational contraction of matter, that is, a decrease in U. Nonetheless, one can easily derive an approximate equation of state for the system. On account of prodigious galaxy masses this component will not be much different from dust. The equation of state for radiation is known. Integrating the second equation (14) for the case of dust, we obtain

pg = 0, ε g = ε0 a −3 ,

(15)

where ε 0 is the integration constant. Designating the energy density as εr = ε − ε g ,

(16)

where ε is the total energy density of the system, we derive 3 p = ε − ε 0 a −3 .

(17)

a a ε + 4 ε = ε0 4 . a a

(18)

The right equation (14) becomes

Its general solution 1296

⎞ ε ε −ε 1⎛a ε = ⎜ ∫ I ε0 a −4 da + ε1 ⎟ = 03 + 1 4 0 , I ⎝1 a ⎠ a

⎛ a ⎞ I = exp ⎜ 4 ∫ a −1da ⎟ = a 4 , ⎝ 1 ⎠

(19)

where I is the integrating factor and ε1 = const . The following difference between two integration constants plays the key role in the foregoing solution: Δε = ε0 − ε1 .

(20)

In essence, ε 0 is the energy density of dust with highly uniform distribution in space. Generally speaking, under such distribution there must be neither gravitational field nor radiation which should be treated as a result of gravitational forces acting upon dust. However,

ε ( a ) a =1 = ε1 .

(21)

Therefore ε1 can be treated as energy density of dust already grouped in certain starless island formations – protogalaxies. These formations will be safely stable, provided ε1 < ε 0 . In this case, the difference Δε should be identified with background radiation. It follows from Eq. (17) that this radiation exerts a negative pressure on dust, thereby interfering with its highly uniform distribution. Let us show how to derive dependence (15) using averaging. In a continuous medium, the “density” μ formally acts as a source of gravitational field (see Eq. (2.15) in [2]). However, the real intrinsic density of medium mass is ρ from Eq. (2.12)

(

μ = F ( a1 , a 2 , a 3 ) ημν eτ(μ ) eσ( ν ) b0τ b0σ

)

12

(

det eλ(α ) bβλ

)

−1

(

, ρ = F ( a1 , a 2 , a 3 ) ημν b0μ b0ν

)

12

( )

det bβα

−1

.

(22)

It is the function ρ that identically satisfies the law of conservation of mass

∇μ (ρu μ ) = 0 .

(23)

It appears that the difference ( μ − ρ ) can be associated with the density of notorious dark matter. In our problem, matter is considered to be uniformly distributed. Hence it follows from Eqs. (9) and (10) that μ = ρ a −3 .

ρ = const,

(24)

Averaging the energy-momentum tensor of the continuous medium (3.6) from [2], where U = const , we derive Eq. (15). Taking into account Eq. (19), we rewrite the left equation (14) as follows: 4a 2 a 2 =

32πk 3c 4

( ε0 a − Δε ) − Ω 2 a 2 .

(25)

Its right-hand side cannot be negative. Thus in the model, there occurs the field domain a . Let us take a unity as such limiting value – the situation where no gravitational field is present. Hence Ω2 =

32πk 3c 4

ε1 .

(26)

1297

Yet, such relationship is not obvious. In addition to ε1 , there is another constant value similar in meaning. This is the total energy density of matter ε 0 (in our approach, the Universe is not expanding). However, in this case, one should make allowance for another circumstance. The fact of presence of pseudoscalar Ω implies violation of spatial even parity in nature, which is really observed only in weak and strong interactions. Therefore Eq. (26) actually relating Ω with the rest energy density of matter will be more substantiated. The aforesaid can be illustrated in a different way. The desired solution is a result of averaging the field generated by many island sources. Hence P-odd parity should be inherent in the field of one island source. It will be assumed centro-symmetrical. In the spherical coordinates x1 = r , x 2 = θ , and x 3 = ϕ , we write the following relation general for the tensor e(νμ) :

e(νμ )

φ+χ 0 0 ⎞ ⎛ ν ⎜φ − χ μ 0 0 ⎟⎟ ⎜ = , ⎜ 0 λ ωλ sin θ ⎟ 0 ⎜⎜ ⎟ −ωλ sin −1 θ λ ⎟⎠ 0 ⎝ 0

(27)

where the functions φ(t , r ) and ω(t , r ) belong to the anti-symmetrical part. This configuration is not invariant to spatial coordinate inversion due to the presence of ω . Since the solution must be СРТ-symmetrical, the sign of ω appears to be dependent on whether the field is generated by matter or antimatter. Outside the source this function will be zero. It is ω that is responsible for the components C132 , C 213 , and C 321 . However, it follows from Eq. (10) that it is their averaging that results in the field H . On this basis, let us rewrite Eq. (25) as

2aa = Ω [ (a − Δε ε1 )(1 − a)]

12

.

(28)

Due to the presence of a radical, the equation can have two partial solutions. The first solution is written as Ωc ( t − t1 ) =

ε0 2a − ε0 ε1 arcsin ε1 ε0 2 ε12 − 4 Δε ε1

(

12

)

12

ε ⎛ Δε ⎞ − 2 ⎜ −a 2 + 0 a − ⎟ ε1 ε1 ⎠ ⎝

,

(29)

where the integration constant t1 , taking advantage of random choice of time origin, can be assumed zero. The second solution differs from the first one by the sign of t and nonzero integration constant t2 . Selecting t2 , these two multivalued solutions must be smoothly joined at the points where the derivative a vanishes. Thus we obtain a general continuous solution. In terms of metric (1), it describes the periodic process of “expansion – contraction” of a spatially flat Universe with the interval ds 2 = c 2 dt 2 − a 2 (t )(dx 2 + dy 2 + dz 2 ) .

(30)

This process is regular (!) and includes the stages of both “deceleration” and “acceleration”. In connection with the possible cyclic evolution of the Universe, we shall touch briefly on thermodynamic aspects of the problem. The two-component system “gas – radiation” is not closed: it interacts with the “external” gravitational field. Therefore its state should not by any means be equilibrium but should also depend on a certain “relaxation” process. In view of the uniform distribution of matter in space, this process should obviously be considered isochoric on the whole. Taking into account the fact that gravitational interaction is universal, in addition, it can be assumed reversible! Indeed, dissipation will be caused only by conversion of one form of energy into another. Yet, gravitation is generated by any forms of energy and, in turn, it can redistribute them. It is this unique property that allows one to determine the process. It is a mere exchange of energy between the field, gas, and radiation. Taken 1298

together they already form a closed system. The energy of the field will be spent on nonequilibrium “heating” of the components, and, vice versa, their transition to equilibrium will be accompanied by return of the energy to the field. In this approach, the gas component is conveniently considered as a certain equilibrium mixture of real gases in different aggregate states (relaxation processes in galaxies proceed relatively fast). As a result, additional, the so-called “correction terms” must appear under the radical sign in Eq. (28). Nonetheless, the new solution will most likely retain regularity. Finally, let us discuss the propagation of light. We are interested in the interpretation of a cosmological red shift. Its character can be easily determined from what has been said. The radiation energy and frequency will change due to interaction with a variable gravitational field. The change will be the higher the longer it takes the light to propagate, that is, it will appear to be in proportion to the distance covered. Conceivably, no stars might be observed at fairly large distances in this case. Upon establishment of equilibrium by matter, the energy radiated is likely to convert into the energy of the gravitational field. Only background radiation will remain. In this case, only a limited region of the Universe will be accessible to observation from a fixed point of space. The Lagrangian extremals (2.24) adopted in [2] are initial here, and we have 1 ∂ μ F μν + C ν μλ F μλ − C λ μλ F μν = 0, 2 Fμν = ∂ μ Aν − ∂ ν Aμ − C σμν Aσ .

(31)

These are written with allowance for unitary symmetry of electromagnetic theory Aμ → Aμ + ∂ μ φ .

(32)

They can be considered as a tetrad representation of the Maxwell homogeneous equations in the gravitational field, but the concepts of basic and tangent spaces change places. The wave solutions to these equations are found in the gauge Aμ = ( 0,− A ) .

(33)

Let us introduce 3D vectors of electrical and magnetic strengths

E = ( F01 , F02 , F03 ) , B = ( F32 , F13 , F21 ) .

(34)

Then in the gravitational field (9)–(12), they can be written as a −1 div E = 0 , a −1 ( rot B + ΩB ) = E +2 ( a a ) E ,

(35)

E = − A − ( a a ) A , B = a −1 (rot A + ΩA) .

(36)

where

A remark needs to be made as to the foregoing solutions. It can be assumed that no sources of the field H exist outside the galaxies. Therefore, for near-field radiation, the constant Ω can be taken zero. For distant objects, however, this is not the case. Let us introduce the following auxiliary vector: a = A α, α ≡1 a .

(37)

The set (35) can be written, with allowance for Eq. (36), as follows:

1299

α 2 div a = 0 , α 2 ( Δa −2Ω rot a −Ω 2 a ) = a − ( α α ) a ,

(38)

where Δ is the 3D Laplace operator. Any wave can be considered flat at relatively small parts of space. Therefore we restrict ourselves to straightforward solutions describing flat waves of the following form: a = R ( r ) T (ct ) ,

(39)

where R is a certain vector function of the radius-vector r and T is the scalar function of ct . Substituting Eq. (39) into Eq. (38) and separating the functions R and T , we derive for the near-field radiation α 2T div R = 0 , ΔR + k0 2 R = 0 , T − ( α α ) T + k0 2 α 2T = 0 ,

(40)

where k0 is the separation constant. The partial complex solutions to this system are written as R = a0 exp (ik0 n ⋅ r ) , T = exp ( −ik0 ∫ αcdt ) ,

(41)

where n is the unit vector in the direction of wave propagation and a0 = const is a certain real vector normal to n . Hence A = A0 exp {i k0 ( n ⋅ r − ∫ αcdt )} , A0 = a0 α .

(42)

Calculating, according to Eq. (36), the strengths, we get

E = iαk0 A , B = iαk0 [n, A] , n ⋅ E = 0, B = [ n, E ] .

(43)

The real values of these force fields describe an ordinary transverse electromagnetic wave. Its cyclic frequency ω and wave vector k change with time as ω = αk0 c, k = α k 0 n .

(44)

Now let us consider the case of distant objects. Let the wave propagate in the direction of the axis x . Substitution of R ( x) = ( X , Y , Z ) into Eq. (38) results in a new set of equations for this vector

(

)

(

)

X ′ = 0, Y ′′ + 2ΩZ ′ + k02 − Ω2 Y = 0, Z ′′ − 2ΩY ′ + k02 − Ω2 Z = 0,

(45)

where prime means a derivative with respect to the coordinate x . Its partial solution is written as follows: X = 0, Y = a0 sin ( Ωx ) exp ( ik0 x ) , Z = − a0 cos ( Ωx ) exp ( ik0 x ) .

(46)

Hence, the wave can still be represented in the form of Eq. (42). However, its amplitude A0 = ( A0 y , A0 z ) = ( a0 α sin(Ωx), −a0 α cos(Ωx) ) , a0 = const ,

(47)

will slowly change its orientation in space. Calculating strengths (36), we again derive relations (43) and (44). Thus, the cosmological red shift can actually be considered as a consequence of a gravitational effect. However, the presence of the scalar field Ω in the Universe will show itself as a change in the polarization of light. The change is not related to the carrier frequency. Therefore, at large distances, a certain degree of polarization must be 1300

observed even for incoherent radiation. It was discovered in the WMAP experiment for background radiation in [4]. Conceivably, appropriate processing of the data of this experiment could obviously support the cosmological model developed in this work.

REFERENCES

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