Einstein gravity with torsion induced by the spinor

Annals of Physics 410 (2019) 167940

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Einstein gravity with torsion induced by the spinor ∗

R. Kaya , H.T. Özçelik Physics Department, Yıldız Technical University, 34220 Davutpaşa, Istanbul, Turkey

article

info

Article history: Received 30 April 2019 Accepted 3 August 2019 Available online 4 September 2019 Keywords: Torsion Spinor field Exact solution

a b s t r a c t We consider spinor fields, minimally coupled to Einstein gravity with torsion in homogeneous and isotropic background in (2+1) dimensions. We set up the total action function of spin 1/2 particles with gravitation. We find the Dirac equation, Einstein and Cartan equations by taking the variation of the action with respect to the spinor field, the dreibein field and the Lorentz connection field. We solve these equations analytically for the spinor fields. We find the product of the spinor field and its conjugate takes a constant value. The circularly symmetric rotating exact solutions of Einstein and Cartan equations are found. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The interest in (2+1) dimensional gravity has grown in recent years due to its role in our attempts to properly understand the intricate structure of the realistic (3+1) dimensional general relativity [1–6]. The (2+1) dimensional Einstein gravity coupled to a scalar field is studied in [7–12]. The geometrical frame of general relativity is a Riemannian space–time. Mielke and Baekler proposed a new, non-Riemannian approach to (2+1) gravity [13]. In this approach, the gravitational dynamics is characterized by both the torsion and the curvature. (2+1) gravity with torsion has been paid much attention and appeared in the work of several authors [14–20]. Einstein gravity in (2+1) dimensions coupled to spinor fields is studied in Refs. [21,22]. One of us has discussed only the static solutions in Ref. [21]. The stationary solutions are found using the algebra of exterior differential forms and the Majorana representation of the Dirac matrices [22]. In this paper, we study the Einstein gravity with torsion in (2+1) dimensions minimally coupled with a Dirac spinor field. We use Dirac matrices to introduce the Lagrangian of the spinor field. ∗ Corresponding author. E-mail addresses: [email protected] (R. Kaya), [email protected] (H.T. Özçelik). https://doi.org/10.1016/j.aop.2019.167940 0003-4916/© 2019 Elsevier Inc. All rights reserved.

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R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

We derive the field equations by varying the total action and obtain circularly symmetric rotating solutions. The paper is organized as follows: In Section 2 we present the geometrical apparatus necessary for the formulation of the Einstein gravity with torsion. We set up the total action function of the spinor field with gravitation. We give explicit forms of the Dirac equation, Einstein and Cartan field equations. In Section 3, we show that these equations can be solved analytically. Finally, the conclusions and perspectives are discussed in Section 4. 2. Formalism of the Einstein gravity with torsion and coupled spinor fields in (2+1) dimensions The Einstein theory of gravity with torsion has been studied since its inception with Cartan, before the discovery of spin. Spin angular momentum is the source of a gravitational field that is coupled to the geometry of space–time. The gravitational interacting of spinning matter can be described correctly in the framework of this theory. Interest in Einstein gravity with torsion has been renewed in recent years [23–26]. More recently, Ivanov and Wellenzohn have claimed that torsion can serve as a geometrical origin for the cosmological constant or dark energy density [27]. Alvarez-Castillo, Cirilo-Lombardo and Zamora-Saa have treated helicity effects of solar neutrinos using a dynamic torsion field [28]. Torsion can also be a source for inflation [29]. Minkevich solves acceleration with torsion instead of dark matter [30] In this section, we review the description of a spinor field minimally coupled to the Einstein gravity with torsion [31–33]. In the (2+1) space–time with the metric tensor gµν , the anti-symmetric part of the affine ρ connection Γµν defines the non-zero torsion tensor ρ ρ − Γνµ ), Tµν ρ = (Γµν

(1)

and the trace of the torsion is defined as Tµ = Tσ µ σ .

(2)

From the metricity condition,

∇λ gµν = ∂λ gµν − gην Γ η

λµ

− gµη Γ η

λν

= 0,

(3)

we have ρ Γµν = {ρµν } − Kµν ρ

(4)

where {ρµν } is the Christoffel symbol

{ρµν } =

1 2

g ρσ (∂µ gνσ + ∂ν gµσ − ∂σ gµν ),

(5)

and Kµν ρ is the contortion tensor Kµν ρ = Rρσ µν

1

(−Tµν ρ + Tµρν − T ρ µν ). 2 is the curvature tensor in the (2+1) space–time with torsion ρ

ρ

ρ ρ λ λ Rρσ µν = ∂µ Γνσ − ∂ν Γµσ + Γµλ Γνσ − Γνλ Γµσ .

(6)

(7)

The curvature scalar of the Riemann–Cartan space–time R is defined as R = g µν Rµν

(8)

where Rµν = Rρµρν is the Ricci tensor. Greek indices = 1, 2, 3 denote (2+1) space–time coordinates. We consider the Dirac spinor field Ψ and the conjugate of the spinor field Ψ¯

Ψ =

(

ψ11 ψ12

)

,

Ψ¯ =

(

ψ21

ψ22

)

.

(9)

R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

3

The covariant derivatives of the spinor field are defined to be

∇µ Ψ = ∂ µ Ψ + Γµ Ψ ,

∇µ Ψ¯ = ∂µ Ψ¯ − Γµ Ψ¯ ,

(10)

where Γµ is the spinor connection

Γµ =

1 8

[ ] τ eτ b . [γ a , γ b ]ea ρ eρ b,µ − Γµρ

(11)

Here γ a = γ 1 , γ 2 , γ 3 are Dirac matrices

γ1 =

(

1 0

0 −1

)

,

γ2 =

(

0 i

i 0

)

,

γ3 =

(

0 −1

1 0

)

.

(12)

and the dreibein field eµ a is defined as gµν = eµ a eν b ηab ,

(13)

where ηab = (−, +, +) is the Minkowski metric tensor. Roman indices flat coordinates. The dreibein field satisfies the dreibein postulate

∇µ eν a = ∂µ eν a + eν b ωµ a b − eρ a Γ ρ

µν

= 1, 2, 3 denote locally

= 0,

(14)

where ωµ a b is the Lorentz connection field. From the above postulate we can obtain the Γ -connection as follows

Γ ρ µν = ea ρ (∂µ eν a + eν b ωµ a b )

(15)

ωµ a b = ηac ωµ ac .

(16)

with

We consider a Dirac field coupled to Einstein gravity with torsion in (2+1) space–time dimensions in the presence of a cosmological constant. Here we use the convention that a positive value for Λ will denote the AdS space. Its total action reads

∫ S=

√ −gLd3 x =

∫

√ −g(LG + LD )d3 x.

(17)

Here g is the metric tensor determinant and LG is the gravitational Lagrangian 1

(R − 2Λ), (18) 2κ with the Einstein gravitational constant κ and LD is the minimally coupled Dirac Lagrangian LG =

LD =

) 1 ( i Ψ¯ γ µ ∇µ Ψ − (∇µ Ψ¯ )γ µ Ψ − mΨ¯ Ψ , 2

(19)

where Ψ¯ = Ψ † γ 1 , γ µ = ea µ γ a and the mass of the Dirac field is m. We assumed our solutions in (2+1) dimensions for the Einstein and Cartan equations will be functions of only the radial coordinate r, a circularly symmetric solution. We consider the Einstein gravity with torsion in a homogeneous and isotropic universe described by the metric

(

ds2 = − v (r) +

J 2 (r) r2

)

dt 2 + w 2 (r)dr 2 +

(

rdφ +

J(r) r

)2 dt

(20)

in plane polar coordinates (t , r , φ ). J(r) is the angular momentum. This metric describes an AdS black hole with gt φ = 0. The black hole horizon is determined by taking roots of the equation g rr (r) = w −2 (rh ) = 0.

(21)

The metric with the co-tetrad fields is given by ds2 = −(e1 )2 + (e2 )2 + (e3 )2

(22)

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R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

with

√ 1

e =

J 2 (r) + r 2 v (r) r

dt ,

e = w (r)dr , J(r) e3 = rdφ + dt . r 2

(23)

2.1. Explicit forms of the Dirac equation and Cartan equations The Dirac equation in (2+1) dimensions to Einstein gravity with torsion is considered. By varying the total action with respect to the spinor field Ψ and the conjugate of the spinor field Ψ¯

δL = 0, δΨ

δL = 0, δ Ψ¯

(24)

the Dirac equations are obtained as : 4J

√

J 2 + r 2 v (U − ψ12 ψ22 ) − r w (J 2 + r 2 v )(U − ψ12 ψ22 )(3κ U + 8m)

′ + 2r ψ21 (ψ12 (2JJ ′ + r(r v ′ + 2v )) − 4(J 2 + r 2 v )ψ12 ) √ − 2 J 2 + r 2 v rJ ′ (ψ12 ψ22 + U) = 0,

(25)

8r(U − ψ12 ψ22 )ψ21 (J + r v ) + 2r ψ21 (4(J + r v )(−U + ψ22 ψ12 ) ′

2

2

2

2

′

′

′ + U(2JJ ′ + r 2 v ′ + 2r v ) + ψ12 (4ψ22 (J 2 + r 2 v ) − ψ22 (2JJ ′ + r 2 v ′ + 2r v ))) √ 2 − ψ12 ψ21 (r w (J 2 + r 2 v )(3κ U + 8m) + 2 J 2 + r 2 v (rJ ′ − 2J)) = 0, √ ψ21 ((2J ′ − 4J /r)/ J 2 + r 2 v + 3κ U w + 8mw)

(26)

′ + 2ψ22 (2JJ ′ + r 2 v ′ + 2r v )/(J 2 + r 2 v ) − 8ψ22 = 0, √ ′ ′ 2 2 2 J + r vψ22 (−2J + rJ ) + 2r ψ21 (2JJ + r(r v ′ + 2v ))

(27)

′ + r(J 2 + r 2 v )(wψ22 (3κ U + 8m) − 8ψ21 ) = 0,

(28)

where U = ψ11 ψ21 + ψ12 ψ22 .

(29)

′

Here denotes the derivative with respect to r. From the metricity condition (3) and the Γ -connection (15) obtained from the dreibein postulate with Eq. (16), we can obtain the following Lorentz connection coefficients as follows:

ω2

11

ω2 22 ω1 23 ω3 21 ω3 11

√ 2J J ′ − r − J 2 + r 2 vω2 31 −2J 2 + 2JrJ ′ + r 3 v ′ 13 ( ) √ , ω2 = , = 2r J 2 + r 2 v J 2 + r 2v w′ 1 = − , ω2 33 = − , ω1 13 = −ω1 31 , ω1 21 = −ω1 12 , w r = −ω1 32 , ω2 21 = −ω2 12 , ω2 23 = −ω2 32 , ω3 13 = −ω3 31 , = −ω3 12 , ω3 23 = −ω3 32 , ω1 11 = 0, ω1 22 = 0, ω1 33 = 0, = 0, ω3 22 = 0, ω3 33 = 0.

Variation with respect to the Lorentz connection field ωµ

√ √ ∂ ( −gL) ∂ ( −gL) − ∂ρ = 0, ∂ωµ ab ∂ (∂ρ ωµ ab )

(30)

ab

(31)

R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

5

gives Cartan field equations. From the above equation and the Lorentz connection coefficients (30), we can get following Cartan field equations

√

(J 2 + r 2 v ) κ rU − 2ω312 − 2 J 2 + r 2 v r 2 ω132 = 0,

)

(

(

r J′ −

√

ω332 = 0,

)

J 2 + r 2 vω231 + r wω132 − 2J = 0,

ω331 = 0,

ω232 = 0,

ω131 = 0,

J κ U − 4r ω112 = 0,

√

κ J 2 + r 2 v U − 4r ω132 = 0, ω212 = 0.

(32)

The field equations (32) can be solved and remaining non-zero Lorentz connection coefficients can be found as follows

ω1

12

ω2

11

ω2 31

= −ω1

21

=

UJ κ 4r

,

ω1

23

= −ω1

32

=−

Uκ

√

J 2 + r 2v 4r

1

,

ω2 33 = − , r

−2J 2 + 2JrJ ′ + r 3 v ′ U κw w′ = , ω2 13 = − , ω2 22 = − , 2 2 2r(J + r v ) 4 w Uκr rJ ′ − 2J 21 12 13 = √ − ω2 , ω3 = −ω3 = . 4 r J 2 + r 2v

(33)

From Eqs. (4) and (15) with (16), we can obtain the non-zero components of the contortion tensor UJ κw , K12 1 = −K21 1 = − √ 4 J 2 + r 2v 1

U κ r 2w

1

K23 = −K32 = K23 1 = −K32 1 =

√

4 J 2 + r 2v U κ r 2w

√

4

J2

r2

+

v

U κvw K12 3 = −K21 3 = − √ , 4 J 2 + r 2v Uκ

√

J 2 + r 2v

,

K13 = −K31 =

,

UκJw K23 3 = −K32 3 = − √ . 4 J 2 + r 2v

2

2

4w

, (34)

Using Eqs. (1), (15) with (16), the non-zero components of the torsion tensor are defined as T12 1 = −T21 1 =

2

T13 2 = −T31 2 = − T23 3 = −T32 3 =

U κvw

UJ κw

√

J2

Uκ

+ √

r2

v

, T12 3 = −T21 3 = √

2 J 2 + r 2v

J 2 + r 2v

2w UκJw

√

4 J 2 + r 2v

,

U κ r 2w

, T23 1 = −T32 1 = − √

2 J 2 + r 2v

.

, (35)

Only the totally antisymmetric component of the torsion tensor couples to spinor fields, whereas the other components do not couple. The torsion trace vector (2) can be obtained as Tµ = 0. The torsion cannot propagate through the vacuum as a torsion wave [23]. From the torsion tensor (1), the affine connection (4), the contortion tensor (6) and the line element (20), the Cartan field equations are obtained as in the Ph.D. dissertation prepared by Hasan Tuncay Özçelik (YTU 2016) [34]. ρ As a check on our calculations, by varying the total action with respect to the contortion Kµν

√ √ ∂ ( −gL) ∂ ( −gL) − ∂σ = 0, ρ ρ ∂ Kµν ∂ (∂σ Kµν )

(36)

we can obtain the Cartan field equations. Solving these equations, we can arrive the same result as in Eq. (35) [34].

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R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

2.2. Explicit form of the Einstein field equations Varying the total action with respect to the dreibein field ea µ

√ √ √ ∂ ( −gL) ∂ ( −gL) ∂ ( −gL) − ∂ + ∂ ∂ = 0, ρ σ ρ ∂ ea µ ∂ (∂ρ ea µ ) ∂ (∂σ ∂ρ ea µ )

(37)

leads to Einstein field equations

√ 2 −4κ J 2 + r 2 vw2 ψ21 (2J 3 U + 2Jr 2 U v + J 3 rU ′ + Jr 3 v U ′ ) ′ ′ 2 ′ + 8κ (J 2 + r 2 v )r w 2 (ψ22 (ψ12 ψ22 − U)ψ21 − ψ21 (−ψ22 U ′ + U ψ22 + ψ22 ψ12 )) ( ) 2 4 2 ′ + r ψ21 (J w w(κ U(3κ U + 16m) + 16Λ) − 8κψ12 ψ21 ′ + r 2 v (4wJ ′2 + r v (r w2 (w(κ U(3kU + 16m) + 16Λ) − 8κψ12 ψ21 ) − 16w ′ )) ′ + 2J 2 (v (r 2 w3 (κ U(3κ U + 16m) + 16Λ) − 8κ r 2 w 2 ψ12 ψ21 − 8r w′ + 8w)

− 2w (J ′2 − 2r v ′ )) + 4Jr(2r vwJ ′′ − J ′ (r wv ′ + 2v (r w′ + 3w))) 3 ′ + J 3 (8wJ ′′ − 8J ′ w′ )) + 8κ (J 2 + r 2 v )r w2 ψ21 ψ12 = 0, √ 4J ′2 + J 2 + r 2 vw 2 (κ U(3κ U + 16m) + 16Λ) + 8r v ′ = 0, √ 2 (−2J 3 r − 2Jr 3 v + J 2 r 2 J ′ + r 4 v J ′ ) 8κ J 2 + r 2 v U w 2 ψ21

(38) (39)

3 ′ ′′ + 8κ (J 2 + r 2 v )r 2 w2 (ψ21 ψ12 + ψ22 (ψ12 ψ22 − U)ψ21

′ 2 ′ 2 ′ − ψ21 (−ψ22 U ′ + U ψ22 + ψ22 ψ12 )) + ψ21 (−8κ r 2 w 2 ψ12 ψ21 + 16r w′ − 16w

+ J 4 (r 2 w3 (κ U(3κ U + 16m) + 16Λ)) + 4v (r 4 w(J ′2 + 2r 2 v ′′ ) − 2r 6 v ′ w′ ) + 2J 2 r 2 (w(−6J ′2 + 4r 2 v ′′ + 8r v ′ ) + r v (r w 2 (w(κ U(3κ U + 16m) + 16Λ) ′ − 8κψ12 ψ21 ) + 8w ′ ) − 4r 2 v ′ w ′ ) + 16Jr 3 (r vw J ′′ − J ′ (r wv ′ + v (r w ′ + w )))

+ 16J 3 r(r wJ ′′ + J ′ (w − r w′ )) + r 6 v 2 w2 (w(κ U(3κ U + 16m) + 16Λ) ′ − 8κψ12 ψ21 ) − 4r 6 wv ′2 ) = 0,

(40)

√

−4J (w − r w ) + 2Jr (w (r v − J ) + 2r vw ) − r (2κ J 2 + r 2 v U vw √ + v (−2r wJ ′′ + 2J ′ (r w′ + w) + κ J 2 + r 2 v r w 2 U ′ ) + r wJ ′ v ′ ) √ − J 2 r(2rJ ′ w′ − 2w (rJ ′′ + 2J ′ ) + κ J 2 + r 2 vw 2 (rU ′ + 2U)) = 0, √ 2 −4κ J 2 + r 2 vw2 ψ21 (J 2 (2J 2 U + 2Ur 2 v + J 2 rU ′ + r 3 (2v U ′ + U v ′ )) 3

2

′

′

′2

′

3

2

(41)

3 ′ ′ + r 5 (v 2 U ′ + U vv ′ )) + 8J κ (J 2 + r 2 v )r w2 (ψ21 ψ12 + ψ22 (ψ12 ψ22 − U)ψ21

′ 2 − ψ22 (ψ12 ψ22 − U)ψ21 ) + r ψ21 (−12J 2 r w J ′ (r v ′ + 2v ) ′ + J 5 w2 (w(κ U(3κ U + 16m) + 16Λ) − 8κψ12 ψ21 ) + J 4 (8w J ′′ − 8J ′ w ′ )

+ 4r 3 v (J ′ (r wv ′ + 2v (r w′ + w)) − 2r vwJ ′′ ) + 2J 3 (w(4r(r v ′′ + v ′ ) − 2J ′2 ) ′ + v (r 2 w3 (κ U(3κ U + 16m) + 16Λ) − 8κ r 2 w2 ψ12 ψ21 − 8r w′ + 8w)

− 4r 2 v ′ w′ ) + r 2 J(−4r 2 wv ′2 + r v 2 (r w2 (w(κ U(3κ U + 16m) + 16Λ) ′ − 8κψ12 ψ21 ) − 16w ′ ) + 4v (w (3J ′2 + 2r 2 v ′′ − 2r v ′ ) − 2r 2 v ′ w ′ ))) = 0,

(42)

ψ21 ψ11 − ψ22 ψ12 − ψ11 ψ21 + ψ12 ψ22 = 0,

(43)

ψ22 ψ11 + ψ21 ψ12 − ψ12 ψ21 + ψ11 ψ22 = 0.

(44)

′ ′

′ ′

′ ′

′ ′

R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

7

3. Exact solutions In this section exact solutions of Einstein and Cartan equations in (2+1) dimensional space–time are given by considering spinor fields as external source for torsion of space–time. From the Dirac equations (25)–(28) we obtain the following equations:

√ ′ (r) = (−r wψ12 (J 2 + r 2 v )(3κψ12 ψ22 + 8m) + 4J ψ12 J 2 + r 2 v ψ11 + r ψ11 (−3J 2 κwψ12 ψ21 + 4JJ ′ + r v (4 − 3κ r wψ12 ψ21 ) + 2r 2 v ′ ) √ − 2r ψ12 J ′ J 2 + r 2 v )/8r(J 2 + r 2 v ),

(45)

ψ12 (r) = −(J r wψ11 (3κψ11 ψ21 + 3κψ12 ψ22 + 8m) + 2r ψ11 J P 2

′

′

+ r 2 v (r wψ11 (3κψ11 ψ21 + 3κψ12 ψ22 + 8m) − 4r ψ12 ) − 2r 3 ψ12 v ′ − 4J(ψ11 P + r ψ12 J ′ ))/8r(J 2 + r 2 v ), ψ21 (r) = (r wψ22 (J + r v )(3κψ12 ψ22 + 8m) − 4J ψ22 2

′

2

(46)

√

J2

+

r2

v

+ r ψ21 (3J κwψ11 ψ22 + 4JJ + r v (3κ r wψ11 ψ22 + 4) + 2r v ) √ + 2r ψ22 J ′ J 2 + r 2 v )/8r(J 2 + r 2 v ), √ ′ ψ22 (r) = (r wψ21 (J 2 + r 2 v )(3κψ11 ψ21 + 8m) − 4J ψ21 J 2 + r 2 v 2

2 ′

′

(47)

+ r ψ22 (3J 2 κwψ12 ψ21 + 4JJ ′ + r v (3κ r wψ12 ψ21 + 4) + 2r 2 v ′ ) √ + 2r ψ21 J ′ J 2 + r 2 v )/8r(J 2 + r 2 v ).

(48)

From the Einstein field equations (38), (39), (41) and (42) we find the following equations J ′′ (r) = (8J 2 w (2κ

√

J 2 + r 2 v U w J ′ + 6J ′2 + κ (J 2 + r 2 v )U w 2 (3κ U + 8m)

√ − 8v ) + 4(J 2 + r 2 v )(4w J ′ (κ J 2 + r 2 v U w − J ′ ) + r v (r w3 (3κ 2 U 2 − 16Λ) √ + 16w′ )) + J(4κ J 2 + r 2 v rU w2 (4v − J ′2 ) − 8r wJ ′ (J ′2 − 12v ) − κ (J 2 + r 2 v )3/2 rU w4 (κ U(3κ U + 16m) + 16Λ) + 2(J 2 + r 2 v ) J ′ (16w ′ − r w 3 (κ U(3κ U + 16m) + 16Λ))))/32J(J 2 + r 2 v )w,

v ′ (r) = −

4J ′2 + (J 2 + r 2 v )w 2 (κ U(3κ U + 16m) + 16Λ) 8r

(49)

,

(50)

√

w ′ (r) = w(r 2 (4J ′ (J ′ − κ J 2 + r 2 v U w) + r 2 vw2 (16Λ − 3κ 2 U 2 )) + J 2 (r 2 w2 (16Λ − 3κ 2 U 2 ) + 16) √ + 8Jr(κ J 2 + r 2 v U w − 2J ′ ))/16(J 2 + r 2 v )r , √ √ 4J + s1 J 2 + r 2 v 16 − r 2 w 2 (κ U(3κ U + 16m) + 16Λ) ′ J (r) = ,

(51) (52)

2r

where s1 is ±1. From Eqs. (49) and (52), we get U =−

∂ J ′ (r) ∂r

− J ′′ (r) = 0. Using this equation we can obtain U

2Λ . κm

(53)

Substituting U into v ′ (r) (50), w ′ (r) (51) and J ′ (r) (52) we can rearrange these equations as follows

√

v ′ (r) = −

2(J(r) J(r)2 + r 2 v

√

Λr 2 w 2 (4 − r3

3Λ ) m2

+ 4 + 2J(r)2 + r 2 v )

,

(54)

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R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

√

w (r) =

w( 2 √

′

3Λ ) m2

√ m J(r)2 + r 2 v Λr 2 w 2 (4 −

s1

′

s1 Λw Λr 2 w 2 (4 −

1

J (r) =

+4 3Λ ) m2

+ Λr w2 (4 −

+ 4 + 2J(r)

r Using Eqs. (54) and (56), we can obtain

v (r) =

C1 − J 2 (r)

3Λ m2

)+

2 r

),

.

.

r2 If we solve Eq. (55), the solution can be written as follows

w(r) = √

2mr 2C2 Λr 2 + C22 + 4Λr 4 (Λ − m2 )

.

(55)

(56)

(57)

(58)

From J ′ (r) (56) we can get J(r) as

√ 2

J(r) = C3 r +

2s1 m C1 r C2 w (r)

,

(59)

where C1 and C2 are constants. C3 can be chosen as follows 2s1

C3 = −

√ √ C1

Λ(Λ − m2 )

(60)

C1

to obtain a finite value of the angular momentum J(r) when r is going to infinity lim J(r) =

r →∞

1 2

s1

√

√

C1

Λ , Λ − m2

(61)

2

where C1 = Λ−Λm . ′ ′ ′ ′ Substituting v (r) (57), w (r) (58) and J(r) (59) into ψ11 (r) (45), ψ12 (r) (46), ψ21 (r) (47) and ψ22 (r) (48) we obtain the following Dirac equations ′ ψ12 (r)Q ′ (r) + ψ11 (r) = 0, ′ ′ ψ11 (r)Q (r) + ψ12 (r) = 0, ′ ψ21 (r) − ψ22 (r)Q ′ (r) = 0, ′ ψ22 (r) − ψ21 (r)Q ′ (r) = 0.

(62)

′

Here we can define Q (r) as follows: ′

Q (r) =

√ w( (C2 + Λr 2 )2 + r 2 (4m2 − 3Λ))

.

4mr 2 We can solve Eqs. (63) and then obtain the spinor fields

ψ11 (r) = d1 eQ (r) + d2 e−Q (r) , ψ12 (r) = d2 e−Q (r) − d1 eQ (r) , ψ21 (r) = d3 eQ (r) + d4 e−Q (r) , ψ22 (r) = d3 eQ (r) − d4 e−Q (r) .

(63)

(64)

The product of Ψ and Ψ¯ has a constant value as follows:

Ψ Ψ¯ = 2(d1 d4 + d2 d3 ). †

(65)

Using Ψ¯ = Ψ γ and Eq. (65) we can obtain d1 and d2 real constants, and d3 and d4 are imaginary constants d3 = d1 ∗ ,

1

d4 = d2 ∗ .

(66)

R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

9

We can integrate Q ′ (r) (63) and find Q (r) = C4 + LogS(r)

(67)

where C4 is a constant and

√

( r

√

C2 Λ + 4Λr 2 Λ − m2 +

(

S(r) =

)

√ 4

w (r)

Λ(Λ−m2 ) mr

C2 + Λr 2 +

) √2m2 −Λ 4 Λ(Λ−m2 ) .

w (r)

(68)

2mr

Since Q(r) is a real function, S(r) must also be a real function. In order for S(r) to be a real function,

Λ must be greater than m2 (Λ > m2 ).

Substituting Q (r) into Eq. (64) we can get the components of the Dirac spinor as

ψ11 (r) = d2 eC4 S −1 (r) + d1 eC4 S(r), ψ12 (r) = d2 eC4 S −1 (r) − d1 eC4 S(r).

(69)

We can break it up into a left-chiral and a right-chiral part [35],

Ψ =

ΨR + ΨL ΨR − ΨL

(

) (70)

where RΨR = ΨR and LΨR = 0 and a similar set of equations for ΨL [36]. The curvature scalar of space–time with torsion R is calculated R = 2Λ.

(71)

The non-zero components of the Ricci tensor are given [34]. In the absence of torsion, the curvature scalar reduced to R = 6Λ [20]. Einstein’s general relativity including matter with spin, naturally explains why the Universe is spatially flat, homogeneous and isotropic. The torsion of space–time generates the gravitational repulsion in the early universe [37]. The roots of Eq. (21) give the radii of the AdS black hole event horizons. We can obtain metric singularities from the following equations

√

√ r1 =

2

C2

, √ −Λ 3Λ − 4m2

√

√ r2 =

−Λ +

√

2 −

Λ+

√

(72)

C2

. √ −Λ 3Λ − 4m2

(73)

These equations imply that there exist the inner and the outer horizons 1 r1 = √

Λ

√ −

2C2 1+α

,

1 r2 = √

Λ

√ −

2C2 1−α

,

(74)

under the following conditions: Λ > m2 > 43 Λ, m2 = Λ (3 + α 2 ) with 1 > α > 0 and C2 < 0. 4 The black hole in (2+1) Einstein gravity minimally coupled spinor fields has the thermodynamical properties to those found in (2+1) dimensions [3,9]. We are currently working on the entropy, Hawking temperature and the emission probabilities of this AdS black hole. The possible results are going to be presented elsewhere [38]. 4. Conclusions We studied in (2+1) dimensions Einstein gravity with torsion minimally coupled to a Dirac spinor field Ψ . By considering variations with respect to the spinor field, the dreibein field and the Lorentz connection field, the Dirac equation, Einstein and Cartan field equations are obtained. We give circularly symmetric rotating exact solutions of the field equations with a negative cosmological constant in (2+1) dimensional space–time. We show that the torsion has a significant effect on the spinor field in (2+1) dimensions.

10

R. Kaya and H.T. Özçelik / Annals of Physics 410 (2019) 167940

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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