Approximate axially symmetric solution of the Weyl-Dirac theory of ...

Approximate axially symmetric solution of

arXiv:1611.08251v1 [gr-qc] 24 Nov 2016

the Weyl–Dirac theory of gravitation and the spiral galactic rotation problem O. V. Babourova, B. N. Frolov, P. E. Kudlaev Moscow Pedagogical State University, Institute of Physics, Technology and Information Systems, M. Pirogovskaya ul. 29, Moscow 119992, Russian Federation Abstract On the basis of the Poincar´e–Weyl gauge theory of gravitation, a new conformal Weyl–Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan– Weyl spacetime with the Dirac scalar field representing the dark matter model. A static approximate axially symmetric solution of the field equations in vacuum is obtained. On the base of this solution in the Newtonian approximation one considers the problem of rotation velocities in spiral components of galaxies.

Keywords: Dark matter, Dirac scalar field, Weyl–Dirac theory of gravitation, Cartan–Weyl spacetime, axially symmetric solution, rotation curves of spiral galaxies

1.

Introduction

As a consequence of the gauge theory of gravitation of the Poincar´e–Weyl group developed in [1]–[3], it appears that spacetime is allocated by an additional geometrical structure as a Dirac scalar field [4] and geometrical properties of the post-Riemannian Cartan–Weyl space with curvature, torsion, and nonmetricity of the Weyl’s type, in which a Weyl 1-form is defined by the gradient of the Dirac scalar field β . As a result, a theory of gravitation arises, which is pertinent to name the Weyl–Dirac theory of gravitation, since it generalizes the Einstein–Cartan theory of gravitation to include nonmetricity of the Weyl’s type in the Cartan–Weyl space with the Dirac scalar field.

1

2 In papers [5]–[11] it was considered a cosmological aspect of the given theory, and in [12]– [14] the spherically symmetric solution was found, which was conformal to the well-known Yilmas–Rosen metrics [15], [16] of the Majumdar–Papapetrou [17], [18] class of metrics. In the given work an axially symmetric solution of the given theory will be obtained for compact central mass, and also its possible application for the solution of the well-known problem of rotation curves of spiral galaxies will be considered.

2.

Lagrangian density and field equations in the axially symmetric case

We shall use a variational technique in the formalism of exterior forms [8], [19]. We shall take the Lagrangian density LG in vacuum proposed in [8]–[11], in which the term with the effective cosmological constant and terms square-law on curvature are omitted,  1 LG = 2f0 β 2 Ra b ∧ ηa b + ρ1 β 2 T a ∧ ∗Ta + ρ2 β 2 (T a ∧ θb ) ∧ ∗(T b ∧ θa ) + 2 2 ρ3 β (T a ∧ θa ) ∧ ∗(T b ∧ θb ) + ξβ 2Q ∧ ∗Q + ζβ 2Q ∧ θa ∧ ∗Ta +    1 a 4 ab l 1 dβ ∧ ∗dβ + l2 βdβ ∧ θ ∧ ∗Ta + l3 βdβ ∧ ∗Q + β Λ ∧ Qab − gab Q . (1) 4 Here Ra b and T a are the curvature and torsion 2-forms, Qab is the nonmetricity 1-form, ∧ is the exterior multiplication symbol, d is the exterior differentiation symbol, ∗ is Hodge dual conjugation [8]. The first term in LG is the Gilbert–Einstein Lagrangian density generalized to the Cartan–Weyl space (ηa b = ∗(θa ∧θb ), f0 = c4 /16πG). In the last term Λab is the 3-form of undetermined Lagrange multipliers, upon variation of Lagrangian density (1) over which the Weyl condition Qab = 41 gab Q , where Q is the Weyl 1-form (trace of the nonmetricity), arises as the field equation. As the independent variables of the theory, we choose the nonholonomic connection 1form Γa b , the cobasis 1-forms θa , and the Dirac scalar field β. We use the exterior form variational formalism on the base of the Lemma on the commutation rule between variation and Hodge star dualization [19], and obtain three field equations: the Γ-equation, the θequation, and the β-equation, which represent a special case of the field equations received in [8]–[11] (in view of the simplifications accepted at construction of Eq. (1)). As well as in the spherically symmetric case [12]–[14], we adopt that the torsion 2-form has the form, T a = (1/3)T ∧ θa

T = ∗(θa ∧ ∗T a ) ,

(2)

where T is a torsion trace 1-form, and also that a torsion 1-form and a nonmetricity trace

3 1-form (a Weyl’s 1-form) are determined by a gradient of scalar Dirac field T = sdU ,

Q = qdU ,

U = ln β ,

(3)

where s and q are arbitrary numbers, determined by the coupling constants of the Lagrangian density (1). By analogy to the spherically symmetric case [12]–[14], the metrics of the axially-symmetric solution of these equations of the gravitational field for a compact central mass, we search in the form
ds2 = e−2U e−µ dt2 − e2(µ−ν) (dρ2 + dz 2 ) + eµ ρ2 dφ2 ,

(4)

where ρ, θ, φ, z are the axial coordinates. Here µ = µ(r), r 2 = ρ2 + z 2 , ν = ν(ρ, θ) and U = U(ρ, θ). One can obtain the two consequences of the Γ-equation,   2 1 q = 2 − l2 , (ρ1 − 2ρ2 − 1)s + ζ + 3 4     1 3 3 1 1 ζ+ − q+ l3 + = 0. ξq + 2 4 64 2 4 In analogy with the [12]–[14], we choose the unknown function µ(r) as, m m µ(r) = =p . r ρ2 + z 2

(5) (6)

(7)

Then the θ-equation yields the following field equations, ′′ νρρ

+

′′ νzz

m2 + 4 − r

 1 1 1 + l 1 + l2 s + l3 q − q + s + 1 = 0 , 2 8 3 2 2 2 1 ′ m (ρ − z ) ν − + ρ ρ r6   1 3 ′2 Uρ l1 + l2 s + l3 q + q − s − 3 − 2 8   1 1 1 ′2 Uz l1 + l2 s + l3 q − q + s + 1 = 0 , 2 8 3 2 2 2 1 ′ m (ρ − z ) ν − + ρ ρ r6   1 1 1 ′2 Uρ l1 + l2 s + l3 q − q + s + 1 − 2 8 3   1 3 ′2 Uz l1 + l2 s + l3 q + q − s − 3 = 0 , 2 8 2 1 ′ 2m ρz ν − + ρ z r6   1 1 1 ′ ′ 2Uρ Uz l1 + l2 s + l3 q − q + s + 1 = 0 , 2 8 3 (Uρ′2

Uz′2 )

(8)

(9)

(10)

(11)

4 and also the β-equation comes to the form, ′′ ′′ νρρ + νzz +

m2 − 4 r

 1 3 2 2 + l1 + l2 s + l3 q − q − s − 3 + ρ2 s (1 − 2ρ1 ) + 2 8 3    1 3 1 ′ ′′ ′′ l 1 + l2 s + l3 q + q − s − 3 = 0 . Uρρ + Uzz + Uρ ρ 2 8

(Uρ′2

Uz′2 )

(12)

The equations (9) and (10) have the consequence, 3 q − s−3 = 0, 8

(13)

which is the same condition as in the spherically symmetric case [12]–[14]. If we introduce the condition, ρ2 s2 (1 − 2ρ1 ) = 0 .

(14)

then, as its consequence, the field equations (8) and (12) yield the equation, 1 ′′ ′′ Uρρ + Uzz + Uρ′ = 0 , ρ

(15)

which allows to determine the unknown function U = U(ρ, θ). ′′ We find the particular solution of this equation obeying the condition, Uzz = 0 [21],   |z| ρ m R m 1− ln , m , R = const , ≪ 1, < 1, (16) U(ρ, θ) = − R h R R ρ

where m , R , h are arbitrary constants.

3.

Approximate axially symmetric solution and the rotation curves of spiral galaxies

Following [20], we apply the axially-symmetric solution (to be found) to solve the well-known problem of the rotation curves of spiral galaxies. Toward this end, we interpret this solution as describing the gravitational field of both the spiral component and the halo of spiral galaxies [21]. Here we interpret the constants R and h as the radii of galactic bulge and halo, accordingly. The constant h is large enough so that the solution will describe the motion of all stars of the galaxy, for which the condition, |z|/h < 1, is fulfilled. We interpret the constant 2m in the metrics (2) as the gravitational radius of the galactic bulge having mass M , 2m = 2GM/c2 = rg ,

2m R

≪1.

5 We consider the Newtonian approximation of the metrics (2). As a consequence of the chosen notations, the quantities U , Uρ′ , Uz′ are small enough in order that quantities (Uρ′ )2 , (Uz′ )2 can be omitted in the equations (8)–(12) in the first approximation. As a consequence of this, the field equations (8)–(11) yield the equations, ′′

′′

νρρ + νzz +

m2 = 0, r4

1 ′ m2 (ρ2 − z 2 ) ν − = 0, ρ ρ r6

1 ′ 2m2 ρz ν − = 0. ρ z r6

(17)

The solution of these equations is m2 ρ2 ν(ρ, θ) = − 4 . 2r

(18)

We see that this quantity is very small. Therefore we have obtain the approximate axially symmetric solution,     2m |z| ρ 2 ds = exp 1− ln ds2GR , R h R   −m2 ρ2 2m 2 2 2 2 2 2 2 − 2m 4 r r r (dρ + dz ) + ρ dφ . c dt − e e dsGR = e

(19) (20)

Here the metrics (20) realizes the known axially symmetric solution1 of General Relativity (GR), which can be found in the monograph [22]. In the Newtonian approximation, the metrics (19) after calculating, g00 ≈ 1 +

2ϕ , c2



ln

(21)

yields the gravitational potential, GM ϕ= R

|z| 1− h



ρ GM − . R r

(22)

Solving the problem of the orbital motion of the stars of the spiral component for this potential, for the square of the velocity of a star we find   GM |z| GM  ρ 2 2 v = 1− + . R h r R

(23)

For stars of the spiral component of the galaxy (with the introduced notation taken into account), the first term in (23) is significantly greater than the second, and therefore, for these stars the equality v 2 ≈ const is valid. For this reason, the magnitude of the velocity of the orbital motion of a star (for any value of z into galaxy arms) does not depend substantially on the distance of the star from the center of the galaxy. After substitution in (23) the data of a typical galaxy, we find, v ≈ 290 1

km , c

In [21] an unfortunate misprint appeared in the analogous formula for this metrics.

(24)

6 that is in agreement with observational data. At a certain distance from the center, the influence of the axial symmetry of the spiral component is decreased and advantage shifts to the influence of the spherical halo, as a consequence of which constancy of the velocities of the stars in the galaxy not to be support. The solution (19) does not completely rule out the hypothesis of the existence dark matter (other then the scalar field in its nature) inside of galaxies, but can substantially reduce the required magnitude of its mass. At the present time, it is assumed that to explain the rotation curves, the mass of dark matter inside galaxies should be an order of magnitude greater than the mass of the luminous (baryonic) components of the galaxies. The results were obtained within the frame of performance of the Task No 3.1968.2014/K.

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