Dark Energy in the Theory of Physical Vacuum

G. Shipovshipov@aha.ru, website http://www.shipov.com

Introduction

Einstein believed that one of the main problems in unified field theory is the one ofthe geometrization of the energy-momentum tensor of matter on the right-hand side ofhis equations. This problem can be solved using the geometry of absolute parallelism andCartans structural equations in this geometry [1]:

[keam] eb[kT a|b|m] = 0, (A)

Rabkm + 2[kT a|b|m] + 2T ac[kT c|b|m] = 0, (B)i, j, k... = 0, 1, 2, 3, a, b, c... = 0, 1, 2, 3.

These equations are essentially a matrix form, where the matrices eam, Tabm and R

abkm

appear as the main gauge potential and fields in the theory of physical vacuum.

1 Vacuum equations as an extended set of theEinstein-Yang-Mills equations. Geometrization ofmatter fields

The geometry of absolute parallelism is a space of events in the theory of physicalvacuum. Speaking here of events, we mean the interaction of whole and part in a certainphysical situation.

Consider an event that represents some excitation of physical vacuum and show thatthis excitation is described by Einstein-like equations with geometrized energy-momentumtensor.

Contracting the equations (B) written as

Ri jkm + 2[kT i|j|m] + 2T is[kT s|j|m] = 0 (1)

in indices i and k, we will have

Rjm = 2[iT i|j|m] 2T is[iT s|j|m]. (2)

Contracting further the equations (2) by the metric tensor gjm, we get

R = 2gjm([iT i|j|m] + T is[iT s|j|m]). (3)

1

Forming, using (2) and (3), the Einstein tensor

Gjm = Rjm 12gjmR,

we will get the equations

Rjm 12gjmR = Tjm, (4)

similar to Einsteins equations, but with the geometrized right-hand side, defined as

Tjm = 2{([iT i|j|m] + T is[iT s|j|m])

12gjmg

pn([iT i|p|n] + T is[iT s|p|n])} (5)

Using the notationPjm = ([iT i|j|m] + T is[iT s|j|m])

we will have from (5)

Tjm = 2(Pjm 1

2gjmg

pnPpn). (6)

Tensor (6) can be represented as the sum of parts symmetrical and skew-symmetricalin the indices j and m, and

Rjm 12gjmR = T(jm), (7)

T[jm] =1

(i . . ijm mAj As . . sjm ) = 0, (8)

Aj = Tiji.

Relationship (8) can be taken to be the equations obeyed by the torsion . . ijm of absoluteparallelism geometry 1, which form the energy-momentum tensor (5).

1In the article Riemann-Geometrie mit Aufrechterhaltung des Bergiffes des Fernaparralelismus Sitzungs-ber. preuss. Akad.Wiss., phys.-math. Kl., 1928, 217-221 A.Einstein used the torsion of absolute parallelism inthe formula (10) determining torsion as

=12( ),

where

= eae

a , , , =

x, , , ... = 0, 1, 2, 3, a, b... = 0, 1, 2, 3

connection of absolute parallelism, = . . ijm

anholonomity object in J. Schouten definition. In the same article A. Einstein has specified, that when torsion ( anholonomity object ) is equal to zero the space becomes Minkovski space. Object of anholonomity inthe theory appears when we use the manifold of oriented points and anholonomic tetrad. Tetrad formulationof Einstein equations connects curvature Rijkm with Ricci rotation coefficients T

ijk us

Rijkm = 2[kT i|j|m] 2T ic[kT c|j|m],where

T ijk = ..ijk + gim(gjs..smk + gks..smj)

2

We now decompose the Riemann tensor Rijkm into the irreducible parts

Rijkm = Cijkm + gi[kRm]j + gj[kRm]i +1

3Rgi[mgk]j. (9)

Using equations (4), written as

Rjm = (Tjm 12gjmT ), (10)

we rewrite (9) as

Rijkm = Cijkm + 2g[k(iTj)m] 13Tgi[mgk]j, (11)

where T is the trace of (6).Let us now introduce the tensor current

Jijkm = 2g[k(iTj)m] 13Tgi[mgk]j (12)

and represent the tensor (11) as the sum

Rijkm = Cijkm + Jijkm. (13)

Substituting this relationship into (1), we will obtain

Cijkm + 2[kT|ij|m] + 2Tis[kT s|j|m] = Jijkm. (14)

Equations (14) represent the Yang-Mills equations with a geometrized source given by(3.12). In (3.14) the role of Yang-Mills field is played by the Weyl tensor Cijkm, and that ofpotentials by the Ricci rotation coefficients T ijk.

Let us now substitute (13) into the second Bianchi identities in the geometry of absoluteparallelism [20]

[nR |ij|km] +Rsj[kmT |is|n] T sj[nR |is|km] = 0. (15)As a result we have the equations of motion

[nC |ij|km] + Csj[kmT |is|n] T sj[nC |is|km] = Jnijkm (16)

for the Yang-Mills field Cijkm. The source Jnijkm in these equations will then by definedthrough the current (3.12) as follows:

Jnijkm = [nJ |ij|km] + Jsj[kmT |is|n] T sj[nJ |is|km]. (17)

Using the geometrized Einstein equations (4) and the Yang-Mills equations (14), we canrepresent the equations of physical vacuum (A) and (B) in the form of the extended set ofthe Einstein-Yang-Mills equations

(A) [keaj] + T i[kj]eai = 0,Ricci rotation coefficients (torsion field). If in tetrad formulation to put object of anholonomity equal to zeroRiemann curvature addresses in zero. From my point of view anholonomity has new physical sense - we enteras elements of space-time new rotational degrees of freedom - anholonomic rotational coordinates.

3

(B.1) Rjm 12gjmR = Tjm ,

(B.2) Cijkm + 2[kT i|j|m] + 2T is[kT s|j|m] = J ijkm,where the geometrized sources Tjm and Jijkm are given by (5) and (12).

For the case of Einsteins vacuum the equations (A) and (B) are significantly simplifiedand become

(I) [keaj] + T i[kj]eai = 0,

(II) Rjm = 0,

(III) Cijkm + 2[kT i|j|m] + 2T is[kT s|j|m] = 0.

2 Inertial field as a torsion field

The equations of motion for an oriented point in the theory of physical vacuum coincidewith the equations of the geodesics of the space of absolute parallelism

d2xi

ds2+ i jk

dxj

ds

dxk

ds+ T ijk

dxj

ds

dxk

ds= 0, (18)

which differ from the equations of motion in Einsteins theory of gravitation by the additionalterm

T ijkdxj

ds

dxk

ds.

The very name of the quantities

T i jk = eiakeaj (19)

the Ricci rotation coefficients suggests that they describe rotation. It follows from (19)that the quantities T ijk describe the change in the orientation of the tetrad vectors e

aj when

the tetrad shifts by an infinitesimal distance dxi (the covariant derivative k is taken withrespect to the connection i jk, therefore in normal coordinates k = k). Using the Riccirotation coefficients we can form the four-dimensional angular velocity of rotation of thetetrad vector

i j = Tijk

dxk

ds(20)

with the symmetry propertiesij = ji. (21)

Suppose now that the tetrad vectors coincide with the vectors of a four-dimensionalarbitrarily accelerated reference frame, then, by (20), the rotation of the reference frame isfully determined by the torsion field T ijk. Since the field T

ijk transforms following a tensor

4

law relative to the coordinates transformations xi, the rotation of reference frames relativeto the coordinate transformations is absolute. The nontensor transformation law of T ijk isvalid for transformations in the angular coordinates 1, 2, 3, 1,2, 3, therefore rotation is only relative for the group of rotations O(3.1) [1].

Let us now write the nonrelativisitc equations of motion of a mass m under inertiaforces alone, assuming that at a given moment of time it passes through the origin of anaccelerated system

d

dt(mv) = m(W + 2[v]). (22)

These equations can be written in the form

d

dt(mv) = m(Wo + 2 dx

dt), , = 1, 2, 3, (23)

where W = (W1,W2,W3) = (W10,W20,W30), = (1, 2, 3),

= = 0 3 23 0 12 1 0

. (24)On the other hand, equations (18), if we take into account (20), can be represented as

d2xi

ds2+ i j

dxj

ds= 0. (25)

Multiplying these equations by mass m, we will write the nonrelativistic three-dimensional part of these equations in the form

mduds0

= m0dx0

ds0 2m dx

ds0. (26)

Since in a nonrelativistic approximation

ds0 = cdt, u =vc

and dx0 = cdt, then the equations (26) become

mdvdt

= mc20 2mc2 1c

dx

dt. (27)

Comparing (27) with (23) gives

10 =W1c2

, 20 =W2c2

,

30 =W3c2

, 12 = 3c,

13 =2c, 23 = 1

c.

5

Consequently, the matrix of the four-dimensional angular velocity of rotation of an arbi-trarily accelerated reference frame (matrix of the four-dimensional classical spin) has theform

ij =1

c2

O W1 W2 W3W1 0 c3 c2W2 c3 0 c1W3 c2 c1 0

. (28)It is seen from the matrix that the four-dimensional rotation of a reference frame caused

by the inertial fields T ijk is associated with the rotation

. . ijk = T i [jk] (29)of a space of events in universal relativity theory. Fields determined by the rotation ofspace came to be known as torsion fields. Accordingly, the torsion field T ijk represents theinertial field engendered by the torsion of a space of absolute parallelism.

3 Inertial field in an inertial frame

Most of modern physical theories have been formulated for inertial reference frames. Areference frame is inertial if it moves linearly and uniformly (without rotation) relative toanother one, similar to it. In actual fact, we have here two inertial frames defined througheach other.

There is another definition of the inertial reference frame based on the concepts ofinertial fields and inertia forces. We then have the following definition: a reference frame isinertial if there is no inertia forces in it.

Since inertia forces are engendered by an inertial field, it seems that inertial fields ininertial reference frame must vanish as well. But for the universal theory of relativity thisis not the case inertial fields (or torsion fields) are nonzero even in inertial referenceframes. Indeed, in inertial reference frame the inertia force in (18) becomes zero

F iI = mTijk

dxj

ds

dxk

ds= 0. (30)

In this equation the inertial field T ijk is defined through torsion . . ijk = . . ikj of a space

of absolute parallelism

. . ijk = eiae

a[k,j] =

1

2ei a(e

ak,j eaj,k) (31)

in the following manner [1]:

T ijk = . . ijk + gim(gjs . . imk + gks . . imj ). (32)Substituting this relationship into (30) gives (for m 6= 0)

. . ijkdxj

ds

dxk

ds+ gim(ggs

. . smk + gks

. . smj )

dxj

ds

dxk

ds= 0. (33)

6

Since the relationshipdxj

ds

dxk

ds

is symmetrical in indices j and k, and the rotation ijk

is skew-symmetrical in theseindices, the first term in (33) goes to zero. As a result, it follows from (33) that

gim(gjs. . smk + gks

. . smj ) = 0, (34)

or(gjs

. . smk + gks

. . smj ) = 0, (35)

hencemkj = mjk. (36)

Since mkj is skew-symmetric in indices m and k, then it follows from (36) that ininertial reference frames the torsion of a space of absolute parallelism is skew-symmetricalin all the three indices.

Substituting (35) into (32) gives

Tijk = Tjik = Tikg = ijk. (37)Therefore, in inertial frames the equation (8) becomes simpler in form:

i . . ijm = 0, (38)The energy-momentum tensor (5) is symmetrical in the indices j and m to yield

Tjm =1

( . . ism

. . sji

1

2gjm

. jis

. . sji ). (39)

We can define the auxilliary pseudovector hm as follows

ijk = ijkmhm, ijk = ijkmhm (40)

where ijkn is fully skew-symmetrical Levi-Civita symbol and write the tensor (39) as

Tjm =1

2(hjhm 1

2gjmh

ihi). (41)

Substituting (40) into (38) gives

hm,j hj,m = 0. (42)These equations have two solutions: hm = 0 (a trivial one), and

hm = ,m, (43)

where is pseudoscalar.Writing the energy-momentum tensor (41) through this pseudoscalar, we have

Tjm =1

2(,j,m 1

2gjm

,i,i). (44)

7

In quantum field theory the tensor (44) is the energy-momentum tensor of a masslesspseudoscalar field, where the pseudoscalar plays the role of the wave function in quantumequations of motion.

If the pseudovector hm is time-like, it can conveniently be represented as

hm = ,m = (xi)um, (45)

whereumu

m = 1 (46)

and (xi) is a scalar quantity.Substitution of (45) into the tensor (41) yields the energy-momentum tensor of the form

Tjm =1

2(ujum 1

2gjm). (47)

The tensor (47) in its structure looks rather like the energy-momentum tensor of anideal liquid.

Let us show that matter described in inertial reference frames by energy-momentumtensors of the form (47) is relative, i.e., it obeys the principle of universal relativity.

Defining the density of matter as

= T/c2, (48)

whereT = gjmTjm, (49)

we find from (37), (39) and (41)

= T/c2 = 12c2

hjhj = 1c2

.jis . . sji =

1

c2T jis T

sji . (50)

Suppose now that the density (50) describes the mass of a vacuum purely field particlewith the energy-momentum tensor (39), then the mass can be written as

m0 =(g)1/2dV, (51)

whereg = det gjm, dV = dx

1dx2dx3,

and the density is given by (50).Density (50), and hence the mass (51), behave as absolute magnitudes relative to coor-

dinate transformations xi, since the inertial field Tijk is a tensor relative to these transfor-mations. Using the tetrad eai we can pass over from the base indices i, j, k... to the fiberindices a, b, c.... For example,

T abk = eaiT

ijke

jb.

Using this property and the orthogonality conditions ei aeak =

ik, we can write the

density (50) in the form

= 1c2

T abkTb ka . (52)

8

Applying now transformations defined on the rotation group O(3.1) to T abk we can turnthese quantities to zero. Consequently, the density (52), and hence the mass (51), arerelative in nature, as follows from the principle of universal relativity.

The relativity of mass has been found experimentally in the theory of physical vacuumproposed by Dirac who created quantum electrodynamics. In quantum electrodynamics, ex-periments on the production of electron-positron pairs from vacuum when vacuum absorbed-quanta with energy E 2moc2, where mo is the electron rest mass, have shown that therest mass mo is a relative quantity. Till the pair production, the rest mass of the systemwas zero, since there existed only -quanta with zero rest mass. After the production wehave an electron and a positron, both with a non-zero rest mass m0, so that in the systemwe now have a rest mass 2m0.

4 Field model of a point particle

Let us examine the solution of the vacuum set of the Einstein-Yang-Mills equations(A), (B.1) and (B.2), which describes the spherically symmetrical formation produced fromvacuum. This enables us to establish the correspondence of the equations of physicalvacuum to the fundamental equations of field theory, which yield a point, spherically sym-metrical model of a particle. Indeed, the only model of a particle that follows from thefundamental equations of modern field theory is the model of a point particle with thematter density proportional to the Dirac -function

(r). (53)

Static solutions of field equations in Newtons gravitation theory and the Maxwell-Lorentz electrodynamics with a point source on their right-hand side yield the Coulomb-Newton interaction potential

r. (54)

We will consider the spherically symmetrical solution of the vacuum equations (A) and(B), which describe the vacuum excitation with a variable Coulomb-Newton potential andfor which the Energy-momentum tensor (5) is different from zero. This solution has thefollowing characteristics [1]:

Solution with a variable Coulomb-Newton potential

(55)

1. Coordinates x0 = u, x1 = r, x2 = , x3 = .2. Components of the Newman-Penrose symbols

i00 = (0, 1, 0, 0), i11 = (1, U, 0, 0),

i01 = (0, 0, P, iP ),

00i = (1, 0, 0, 0), 11i = (U, 1, 0, 0), 01i =

1

2P(0, 0, 1, i)

9

U(u) = 1/2 + 0(u)/r, P = (2)1/2(1 + /4), = x2 + ix3,0 = 0(u).

3. Spinor components of the torsion field

= 1/r, = = 0/r, = 0(u)/2r2,

= 1/2r +0(u)/r2, 0 = /4.4. Spinor components of the Riemann tensor

2 = = 0(u)/r3, 22 = = 0(u)/r2 = 0

u

1

r2.

The Riemann metric of the solution (55) in (quasi) spherical coordinates has the form

ds2 =

(1 2

0(t)

r

)c2dt2

(1 2

0(t)

r

)1dr2

r2(d2 + sin2 d2). (56)

Using the solution (55), we can determine the explicit form of the energy-momentumtensor (6). Calculations will show that the tensor is

Tjm = c2ljlm, (57)

where is the matter density of a vacuum excitation given by

= 2o(u)

c2r2, o(u) < 0 ; (58)

lmlm = 0 is the light like vector, which is the spinor basis of the solution (55).We now consider the limiting process o(u) o = const of the matter density in the

solution (55). We introduce the auxiliary parameter with the dimensionality of length

=pi|o|r22o

. (59)

Through the parameter the density module (58) can be represented as

= + =8pio

c21

2pir2

r2=

8pio

c21

2pir2

(r2 + 2)

(1 +

2

r2

), (60)

where the + sign implies that the density + defines right-hand matter with a positivedensity and positive mass. Taking the limit in (60) for 0, i.e., for o(u) o = const,and using the well-known formula

1

2pir21

pilimx0

(x

x2 + r2

)=

1

2pir2(r) = (r),

10

where (r) is the three-dimensional Dirac function, we will get

+ =8pio

c21

2pir2(r) =

8pio

c2(r). (61)

It is seen from this relationship that when a vacuum excitation becomes stationarythe matter density distributed over space coincides with the matter density for a pointparticle (Diracs -function describes the distribution of a point source).

The result obtained substantiates Einsteins assumption that in a purely field theory apoint particle must appear as a limiting case and not be introduced into the theory in anartificial manner, since: ... a combination of the idea of a continuous field with concepts ofmaterial points located discretely in space turns out to be contradictory. A consistent fieldtheory requires that all the elements be continuous not only in time but also in space, andat all its points at that. Consequently, the material point is out of place in field theory [2].

The fact that a material point appears in a purely field theory as a limiting stationarycase is one of the most important results of the theory of physical vacuum.

5 Real point massive particles produced from vacuum

We will now consider the correspondence of the equations of physical vacuum (A) and(B) to those fundamental equations of modern physics in which particles are point andstable particles. If we take into account the above results, the equation (B.1) in the set (??)for a point stationary source can be written as

Rjm 12gjmR = Tjm, (62)

where

Tjm =8pi0

c2(r)ljlm. (63)

Now we compare equations (62) with Einsteins equations describing a point source. Wenote that these equations coincide when in the relationship (63) before the -function standsthe mass of the point source, i.e.,

M =8pi0

c2. (64)

On the other hand, as the source goes stationary, the metric (56) becomes the Schwarzshieldmetric (i.e., the solution of Einsteins equations for a point source) provided that

0 =MG

c2. (65)

Substituting (65) into the equality (64), we will obtain the value of initially arbitraryfactor in the vacuum equations (62)

=8piG

c4. (66)

11

In that case the equations (62) completely coincide with Einsteins equations that describethe gravitational field of a point source with constant mass. Accordingly, the metric (56)becomes the Schwarzshield metric

ds2 = (1 2MGrc2

)c2dt2 (1 2MGrc2

)1dr2 r2(d2 + sin2 d2). (67)

The equations of motion for a test particle in the theory of physical vacuum coincidewith the equations of geodesics of the space of absolute parallelism

d2xi

ds2+ i jk

dxj

ds

dxk

ds+ T ijk

dxj

ds

dxk

ds= 0. (68)

To clarify the correspondence of the equations of physical vacuum and Einsteins theorywe will use the equations (68). These equations coincide with the equations of motion inEinsteins gravitation theory on the condition that

F iI = mTijk

dxj

ds

dxk

ds= 0,

i.e., if the inertia forces are zero. This means that the equations of motion in Einsteinstheory

d2xi

ds2+ i jk

dxj

ds

dxk

ds= 0 (69)

are written either in inertial or locally inertial reference frames.To show this we will write for simplicity the solution (55) with a constant source func-

tion 0 = MG/c2 = const in conventional quasi-Cartesian coordinates. In these coordinatesthe tetrad eai becomes

e(0)

0 = (1 +2

c2)1/2, e

(1)1 = e

(2)2 = e

(3)3 = (1

2

c2)1/2, (70)

where in parentheses we have the tetrad indices and = MG/r.The Riemannian metric for the tetrad (70) can be derived using the relationships

gik = abeaie

bk, ab =

ab = diag(1 1 1 1).

It can be written as

ds2 = (1 2MGrc2

)c2dt2 (1 + 2MGrc2

)(dx2 + dy2 + dz2). (71)

If now we consider the nonrelativistic approximation and take gravitational fields to beweak, i.e., assume that

2

c2 1, gik ' ik, ds ' ds0 = cdt(1 v

2

c2)1/2, (72)

Ri jkm 'o

Rijkm = 0,

v2

c2 1, ds ' ds0 ' cdt,

12

we will derive from (69) the following equations of motion of a mass m

md2x

dt2= mc200 = m

MG

r3x. (73)

It is easily seen that the equations derived are the equations of motion of the Newtoniantheory of gravitation, which are known to be written in inertial reference frames. Note thatin passing from Einsteins equations of motion (69) to the Newtonian equations of motion(73) we have never used transformations of the coordinates xi. Therefore, transitionsfrom accelerated reference frames to inertial ones are absolutely out of the question. Thissuggests that both (73) and (69) are written for inertial reference frames, although theyallow coordinate transformations that correspond to a transition to a locally inertial referenceframe. In transition to an accelerated locally Lorentzian reference frame the equations (69)take the form d2x/ds2 = 0. These are the equations of free motion.

A consistent description of the transition to a locally Lorentzian reference frame is onlypossible using the equations (68). Indeed, in passing to an accelerated reference frame inNewtons equations (73) an inertia force should appear. It is this situation that is describedby the equations (68). For the conditions (72) the equations of motion (68) for a mass willbe written as

md2x

dt2= m

MG

r3x mMG

r3x = 0. (74)

Here FI = mc2T00 = mMGx/r3 is the inertia force that compensates for the lo-cally gravitational force FG = mMGx

/r3. It is owing to this compensation that a localweightlessness condition in an accelerated locally Lorentzian frame is produced.

To conclude, we can say that the equations of physical vacuum describe stable pointgravitating particles produced from vacuum and satisfying the equations in Einsteins theoryof gravitation provided that:

6 -term and dark energy

When spherically symmetrical massive matter becomes stationary, the field equations(B.1) and the energy-momentum tensor (47) in an inertial reference frame become

Rjm 12gjmR =

8piG

4Tjm, (75)

whereTjm = Mc2(r)(ujum 1

2gjm). (76)

Equations (75) with the matter tensor (76) can be rewritten as

Rjm 12gjmR + gjm = 8piG

c4T

(d)jm , (77)

whereT

(d)jm = Mc

2(r)ujum (78)

13

is the dust tensor and

= 4piGMc2

(r) (79)

is sam kind of -term.Equations (77) are interesting in that they enable a point particle to be modeled as a

microscopic black hole and the approximate solutions of the vacuum equations to be soughtwithin the framework of Einsteins theory beyond the source (when 2MG/rc2 1), nearthe horizon of the black hole (when 2MG/rc2 1) and inside the source (2MG/rc2 > 1).Corresponding to these three cases are the following equations:

Rjm = 0, 2MG/rc2 1, (80)

Rjm 12gjmR = 8piG

c4T

(d)jm , 2MG/rc

2 1, (81)

Rjm 12gjmR + gjm = 8piG

c4T

(d)jm , 2MG/rc

2 > 1. (82)

Equations (81), which in Einsteins theory are fundamental and describing a point source,in the theory of physical vacuum are approximate equations that are valid near the horizon.Exact correspondence of the equations of physical vacuum to the field equations in Einsteinstheory of gravitation only holds for the vacuum equations (80).

Considering that gjmgjm = 4, we can rewrite the tensor (47) as

Tjm =1

2(xi)(

1

4gjmg

jmujum 12gjm) = (83)

=1

2(xi)(

1

4gjm 1

2gjm) = 1

42(xi)gjm,

orTjm = (x

i)gjm, (84)

where

(xi) = 14

2(xi) =c2

4.

In this case -term is generated by a scalar field which has torsional nature and also is asource of a dark matter.

References

[1] Shipov G.I. A Theory of Physical Vacuum. A New Paradigm M.: ZAO GART,1998,p.312.(in English)

[2] Einstein A. The meaning of relativity. Princeton Univ. Press, N.Y., 1921.

14