Symmetric Teleparallel Gravity: Some exact solutions and spinor couplings

TL;DR

This paper explores symmetric teleparallel gravity (STPG) in a non-Riemannian spacetime, deriving exact solutions and analyzing spinor couplings, revealing equivalences with Einstein-Hilbert gravity and novel geometric properties.

Contribution

It introduces a new prescription for nonmetricity, derives exact solutions in STPG, and discusses spinor couplings, expanding the understanding of non-Riemannian gravity theories.

Findings

Exact conformal, spherically symmetric, cosmological, and pp-wave solutions obtained.

The Lagrangian is shown to be equivalent to Einstein-Hilbert for specific parameters.

A minimal coupling of spin-1/2 fields to STPG is discussed.

Abstract

In this paper we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian 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.