Einstein gravity with torsion induced by the scalar field

Abstract

We couple a conformal scalar field in (2+1) dimensions to Einstein gravity with torsion. The field equations are obtained by a variational principle. We could not solve the Einstein and Cartan equations analytically. These equations are solved numerically with 4th order Runge-Kutta method. From the numerical solution, we make an ansatz for the rotation parameter in the proposed metric, which gives an analytical solution for the scalar field for asymptotic regions. I. INTRODUCTIONIn a series of three recent papers [1-3] dark energy is explained in terms of metric-scalar couplings with torsion. In these papers, Sur and Bhatia discuss "the replacement of the cosmological constant Λ with a scalar field nonminimally coupled to curvature and torsion to overcome the problems of the cosmological constant due to the "fine tuning" problem [4-10] to explain the driving of the late time cosmic acceleration [3,11,12]". Sur and Bhatia state that [3] "although observations greatly favor the ΛCDM model (cold dark matter with Λ) [13][14][15][16] there is still some room for models which replace the cosmological constant with scalar field coupled to torsion". This fact motivates researchers to investigate models which couple fields with torsion as well as curvature.The Einstein gravity with torsion is the simplest generalization of Einstein's general relativity theory, allowing the possibility of relating space-time with torsion. It reduces to Einstein's original theory when torsion vanishes. The Einstein's general relativity is in agreement with all experimental facts in the domain of macrophysics. It has been argued, however, in the microscopic level space-time must have a non-vanishing torsion, and so, microscopic gravitational interactions should be described by the Einstein gravity with torsion [17]. It has been also shown that torsion is required for a complete theory of gravitation [18]. The spin of matter, as well as its mass plays a dynamical role this theory. All the available theoretical evidence that argues for admitting spin and torsion into a gravitational theory is summarized in .The spin-gravity coupling has been paid much attention and appeared in the work of several authors, who have been mainly interested in the study of the matter fields, namely, scalar, gauge, and spinor fields [22,23].Among the other recent papers on the arXiv for Einstein gravity with torsion, one can cite the paper by Ivanov and Wellenzohn [24] where the torsion field acts as the origin of the cosmological constant or dark energy density. Still another paper treats helicity effects of solar neutrinos using a dynamic torsion field [25]. Torsion is also necessary for the stability of self-accelerating universe [26,27]. Minkevich solves acceleration with torsion instead of dark matter [28]. Torsion can also be a source for inflation [29]. Alencar finds that torsion is necessary to localize the fermion field in the Randall-Sundrum 2 model [30].The non-minimally coupled scalar field is of interest for general relativistic gravitational theories,...

Scale invariant Einstein–Cartan theory in three dimensions

“…Correspondingly in three-dimensions the bare Einstein's general relativity is not a dynamical theory. Therefore there is a wide literature on modified general relativity in three dimensions [1][2][3][4][5][6][7][8][9].…”