Self-interacting scalar field in (2+1) dimensions Einstein gravity with torsion

We study a massless real self-interacting scalar field $\varphi$ non-minimally coupled to Einstein gravity with torsion in (2+1) space-time dimensions in the presence of cosmological constant. The field equations with a self-interaction potential $V(\varphi)$ including $\varphi^{n}$ terms are derived by a variational principle. By numerically solving these field equations with the 4th Runge-Kutta method, the circularly symmetric rotating solutions for (2+1) dimensions Einstein gravity with torsion are obtained. Exact analytical solutions to the field equations are derived for the proposed metric in the absence of both torsion and angular momentum. We find that the self-interacting potential only exists for $n=6$. We also study the motion of massive and massless particles in (2+1) Einstein gravity with torsion coupled to a self-interacting scalar field. The effect of torsion on the behavior of the effective potentials of the particles is analyzed numerically.


I. INTRODUCTION
In recent years there has been much more interest in the self-interacting scalar field, which has played a great role in general relativity, astrophysics, cosmology, and unified theories of elementary particles [1][2][3][4][5][6][7][8][9][10][11].These fields in the framework of general relativity are often used to describe the behavior of the early universe, the inflationary period [12,13], dark energy which could explain the accelerated expansion of the universe [14,15], dark matter [16,17], and other cosmological phenomena.
The self-interacting field theory is more complex than the free scalar field theory.As the self-interacting field theory includes the interaction terms, it is more difficult to find the solution of Einstein and Klein-Gordon field equations.
In the past years, a potential explanation of scalar field theories has been given by a generalization of Einstein gravity with torsion, where the torsion field is generated by a scalar field.Torsion has important consequences from the phenomenological point of view.In this respect, the non-minimally coupled scalar field with curvature and torsion in four and three dimensions has been studied in the literature [27][28][29][30][31][32][33][34][35][36].
In this paper, we consider a massless self-interacting scalar field, non-minimally coupled to Einstein gravity with torsion in (2+1) space-time dimensions, in the presence of a cosmological constant Λ.Firstly, we define the action including the self-interacting potential.Then, we obtain Einstein field equations and the Klein-Gordon equation derived from the variation of the action with respect to the dreibein field and the scalar field.We find the exact solution of them when the torsion and angular momentum vanish.We could not solve field equations in the presence of both torsion and angular momentum.These equations are solved numerically with the 4th Runge-Kutta method.
To discover the space-time structure, the motion of test particles can be used to classify an arbitrary space-time.The stability of circular orbits of particles has been studied in the four-dimensional gravity.Moreover, these studies have been extended to the five-dimensional gravity.The motion of test particles in (2+1) dimensional gravity with and without torsion has been also investigated in Refs.[37][38][39][40][41][42][43][44].
We also investigate torsion effects on massive and massless particles in the (2+1) dimensions of Einstein gravity.We derive the equations of motion for massive and massless particles by using the Euler-Lagrange equation.We give the effective potential for the radial motion for self-interacting scalar fields in this space-time.
The outline of this paper is as follows.In Sec. 2 we introduce the metric and obtain the Einstein field equations and Klein-Gordon equation.Sec. 3 includes a class of solutions to these field equations.In Sec. 4, we present properties of the motion of massive and massless particles in the (2+1) dimensions of Einstein gravity with and without torsion.Finally, our conclusions are given in Sec. 5.

II. EINSTEIN GRAVITY WITH TORSION INDUCED BY THE SELF-INTERACTING SCALAR FIELD
We consider the massless scalar field interacting with itself as the source of torsion.The action for a massless self-interacting scalar field non-minimally conformal coupled to Einstein gravity with torsion in (2+1) space-time dimensions in the presence of a cosmological constant Λ is given by where κ = 8πG/c 4 is the Einstein gravitational constant, R = g µν R µν is the curvature scalar of the Riemann-Cartan space-time, g is the metric tensor determinant, ϕ is the scalar field and V (ϕ) is a potential of self-interaction of the scalar field with the coupling constant λ n describing the strength of the interaction between the scalar fields.
The non-minimally coupling constant ξ will be taken to be equal to 1  8 in (2+1) dimensions, resulting in a conformally invariant field theory [45].
From the dreibein postulate ∇ µ e a ν = 0 one can obtain the Γ-connection as follows with the zero-torsion part and the Lorentz connection field The second term in Eq. ( 5) gives the contribution from the torsion.When α = 0 it means zero-torsion.When α = 1 it means that there is a torsion.We take the following circular stationary and rotational symmetric (2+1) dimensional space-time as Here we assume that the metric components v, w and the angular momentum J will be functions of the radial coordinate r.
An orthonormal base for the metric (8) can be found From the metricity condition ∇ σ g µν = 0 and Γ-connection (5), the non-zero components of the Lorentz connection coefficients can be obtained as follows with Here ′ denotes the derivative with respect to r.

A. The field equations
The variation of the total action (1) with respect to the dreibein field e a µ ∂( yields the Einstein field equations as follows Varying the total action (1) with respect to the scalar field ϕ leads to the Klein-Gordon equation

III. SOLUTIONS
In this section solutions of the Einstein field equations and Klein-Gordon equation in (2+1) dimensional space-time are given by considering self-interacting scalar fields as an external source for torsion of space-time.
The field equations ( 13)-( 17) and Klein-Gordon equation ( 19) are a set of coupled differential equations.We shall examine a class of solutions to these equations.
A. Solution with J = 0 and α = 0 The line element (2+1) dimensional space-time in a homogeneous and isotropic universe (8) with J(r) = 0 is considered as In this non-rotating case, the metric component w(r) is given by w .
From the Einstein field equations ( 13) and ( 14), for the zero-torsion, α = 0, and the non-rotating case ,J(r) = 0, we can obtain the scalar field ϕ(r) and the metric component v(r) respectively as where A, B and C are arbitrary constants.The scalar field ϕ(r) is in agreement with those derived in Refs.[18][19][20]36].The scalar field ϕ(r) goes to zero as r goes to infinity.From Eq. ( 15) and Eq. ( 19), when we set α = 0 and J(r) = 0, we can find From the above equation, we can obtain The ϕ(r) 6 potential in (2 + 1)-dimensional gravity was previously known [46][47][48] to have good behavior in yielding exact black hole solutions.
The scalar curvature of the (2+1) space-time (20) with α = 0 and J = 0 is obtained as follows We can see that the Ricci scalar R (28) goes to 6Λ in the limits r going to infinity.
B. Solution with J = 0 and α = 0 From the Einstein field equations ( 13)-( 17) and the Klein-Gordon equation ( 19) with α = 0 and J = 0, we can obtain The scalar curvature of the (2+1) space-time with α = 0 and J(r) = 0 is obtained as follows The search for exact solutions of the coupled system of differential equations v ′ (r) (29), w ′ (r) (30), ϕ ′′ (r) (31), and J ′′ (r) (32) is a very hard job.We can not solve these equations in analytical forms.These equations are solved numerically with the 4th order Runge-Kutta method.
The plots of the scalar field ϕ(r), the angular momentum J(r), the metric components v(r) and w(r), and the Ricci scalar R with α = 0 and the angular momentum J(r) = 0 are given in respectively Fig. 1 We consider the metric (8) for a massless self-interacting scalar field non-minimally coupled to Einstein gravity with torsion.The Einstein field equations ( 13)-( 17) and the Klein-Gordon equation ( 19) are reduced to a system of first and second-order differential equations with torsion α = 1 and angular momentum J = 0 as follows The curvature scalar R with α = 1 and the angular momentum J(r) = 0 can be present as In the absence of the coupling constant λ 6 , the curvature scalar of space-time with torsion is reduced to R(r) (Eq.( 50)) in Ref. [36].
The plot of the scalar field ϕ(r), the angular momentum J(r), the metric components v(r) and w(r), and the Ricci scalar with torsion are given respectively in Fig. 6, Fig. 7 We see that the Figs.6-10 for J = 0, α = 1 case are distinctly different from the J = 0, α = 0 case, given in Figs.1-5.We can deduce that the torsion has an effect on the scalar field ϕ(r), the angular momentum J(r), the metric components v(r), w(r), and the curvature scalar R(r).

IV. THE EQUATIONS OF MOTION
To explore the (2+1) dimensions of Einstein gravity with torsion induced by the self-interacting scalar fields further, let us consider the motion of freely moving both massive and massless particles.
Test particles move along geodesics of (2+1) space-time; the geodesics equations being derivable from the Lagrangian where ε = 1 corresponds to time-like geodesics, ε = 0 corresponds to null geodesics.τ is the proper time for massive particles ε = 1 and affine parameter for massless particles ε = 0.Here an overdot denotes the partial derivative with respect to an affine parameter τ .The Euler-Lagrangian equations are Here D 1 , D 2 and D 3 are constants.
We show that there exist stable circular orbits in (2+1) Einstein gravity induced by the self-interacting scalar field.The radius of stable orbits increases with increasing L for all positive values of L. We can see that the depth of the effective potential well decreases with increasing angular momentum L. FIG.24: The plot of the effective potential V(r) with J(r) = 0, α = 0 for massive particles ε = 1 is plotted by Runge-Kutta method with respect to r.The figure shows the four different curves correspond to the angular momentum L = 10, 50, 100 and 150 from the bottom up

V. CONCLUSIONS
In this work, we have studied self-interacting scalar fields in (2+1) dimensions Einstein gravity with torsion α = 1 in the presence of a cosmological constant.We have obtained the field equations with a self-interaction potential by a variational principle.We have then investigated non-rotating J = 0 and rotating J = 0, circularly symmetric solutions.We have obtained analytical solutions of field equations with J(r) = 0 and α = 0. Using the Runge-Kutta method, we have given the numerical solutions of these equations with J(r) = 0 for the case of α = 0 and the case of α = 1 respectively.In the case of J = 0, α = 1, the scalar field ϕ(r) goes to zero faster than in the case of J = 0, α = 0.
We can conclude that the torsion has an effect on the scalar field ϕ(r), the metric components v(r), w(r), the angular momentum J(r), and the Ricci scalar R.
The investigation of the motion of test particles may be a very useful tool to study the nature of the gravitational properties of the corresponding space-time metrics.We perform the numerical calculations with the help of the 4thorder Runge-Kutta method.We have examined the orbits for the effective potential of particles for different cases of J, α, and different values of L and compared the results to these orbits.
We have concluded that the orbits given by ǫ = 0 are not stable but the orbits are stable for ǫ = 1.The depths of the effective potential wells in Fig. 22 are less than those in Fig. 21.
We also have derived first-order equations for particles in the non-rotating J = 0 and torsionless α = 0 backgrounds and analyzed the properties of some special trajectories.An explicit numerical study of the effective potential for specific values of the parameters could lead to interesting results.

FIG. 22 :FIG. 23 :
FIG.22:The plots of the effective potential V(r) with J(r) = 0, α = 0 for massive particles ε = 1 are plotted by Runge-Kutta method with respect to r.The figure shows the effective potential for the four different curves correspond to the angular momentum L = 10, 50, 100 and 150 from the bottom up