Spherical and cylindrical solutions in f(T) gravity by Noether symmetry approach

Ali Nur Nurbaki,Cemsinan Deliduman,Salvatore Capozziello

doi:10.1140/epjc/s10052-020-7666-7

Abstract

We find exact solutions for f(T) teleparallel gravity for the cases of spherically and cylindrically symmetric tetrads. The adopted method is based on the search for Noether symmetries of point-like Lagrangians defined in Jordan and Einstein frames. Constants of motion are used to reduce the dynamical system. We first consider the Lagrangian defined in the Jordan frame for a spherically symmetric tetrad and, by the help of two constants of motion, we eliminate a tetrad potential and integrate the other. The more complicated structure in the Einstein frame is also overcome by the same method. After that we obtain the Jordan frame Lagrangian for a general cylindrically symmetric tetrad. Following the same procedure adopted in the spherically symmetric case, we again obtain the tetrad potentials and then the exact solutions.

Highlights

General Relativity is accepted to be the most successful theory of gravity, due, for example, to the recent discovery of gravitational waves confirming the validity of the Einstein approach, it is worth noticing that this success is valid observationally below the solar system scale and theoretically far away from the Planck scale
We find exact solutions for f (T ) teleparallel gravity for the cases of spherically and cylindrically symmetric tetrads
The adopted method is based on the search for Noether symmetries of point-like Lagrangians defined in Jordan and Einstein frames

Summary

Introduction

General Relativity is accepted to be the most successful theory of gravity, due, for example, to the recent discovery of gravitational waves confirming the validity of the Einstein approach, it is worth noticing that this success is valid observationally below the solar system scale and theoretically far away from the Planck scale. Independent of renormalization concerns, mostly for observational motivations, higher-order terms were introduced as general functional forms f (R) in the Lagrangian This theories are called Extended Theories of Gravity if General Relativity belongs to the group or Modified Theories of Gravity if the Einstein approach is not retained. As pointed out in [19], a conformal formulation can always be adopted in a pure and extended TEGR, that is for an action containing just T or its extension f (T ) These authors propose conformal scalar and gauge theories in TEGR and study the conformal torsion gravity. They demonstrate the existence of cosmological solutions like power-law acceleration and de Sitter expansion, realized in the framework of f (T ) gravity where a conformal scalar field and a conformal torsion are present Following these works, it is possible to write the conformal transformation for the torsion scalar as. 108 Page 4 of 9 we obtain the point-like Lagrangian in the Einstein frame given as

Ar Mr M B

A Noether symmetry leads to the existence of a constant of motion given by

Jordan frame

Einstein frame

A B M Mr F

A M ωφ FB