Symmetric Teleparallel Gravity: Some Exact Solutions and Spinor Couplings

Abstract

In this paper we elaborate on the symmetric teleparallel gravity (STPG) written in a nonRiemannian spacetime with nonzero nonmetricity, but zero torsion and zero curvature.Firstly we give a prescription for obtaining the nonmetricity from the metric in a peculiar gauge. Then we state that under a novel prescription of parallel transportation of a tangent vector in this non-Riemannian geometry the autoparallel curves coincides with those of the Riemannian spacetimes. Subsequently we represent the symmetric teleparallel theory of gravity by the most general quadratic and parity conserving lagrangian with lagrange multipliers for vanishing torsion and curvature. We show that our lagrangian is equivalent to the Einstein-Hilbert lagrangian for certain values of coupling coefficients. Thus we arrive at calculating the field equations via independent variations. Then we obtain in turn conformal, spherically symmetric static, cosmological and pp-wave solutions exactly. Finally we discuss a minimal coupling of a spin-1/2 field to STPG.

Parity-violating gravity and GW170817 in non-Riemannian cosmology

“…is of precisely the same form as the propagation speed found in parity-violating Riemannian theories [17]. By noting that |λ A | = 1, we may express the above as 21) in order to compare with the propagation speed observed from the coincident detections GW170817/GRB170817A. The LIGO experiment tightly constrains the propagation speed of GW within the bounds −7 × 10 −16 < −c A T + 1 < 3 × 10 −15 , [1, 40,41] and, as a result we find that parity-violating gravity in the STG geometry is constrained by 4 |β 1 − β 3 |k < 6 × 10 −15 .…”

Scale Transformations in Metric-Affine Geometry

“…[64,72]. It was further shown there that by demanding vanishing curvature and torsion and taking a 1 = −a 3 = 1/4, a 2 = −a 5 = −1/2, a 4 = 0, b i = 0 = c i one obtains the symmetric teleparallel equivalent of GR [73,74] from the above action. Furthermore if we pick b 1 = 1, b 2 = −1, b 3 = −4 , a 1 = −a 3 = 1/4, a 2 = −a 5 = −1/2, a 4 = 0, c 1 = −c 2 = c 3 = 2 and impose only the vanishing of curvature, we may expect to reproduce a generalized equivalent to GR that admits both torsion and non-metricity.…”

Family of scalar-nonmetricity theories of gravity

“…Much of the literature on symmetric teleparallelism is phrased in terms of differential forms (see, e.g., [3,[21][22][23][24]), and only recently coordinate basis and explicit formulation in terms of tensor components have gained more attention (see [4,12,[25][26][27][28][29]31, 32 1 / 3 ]). Thus, for the benefit of the reader, we included some foreknowledge in the Section II.…”