Cylindrically symmetric and plane-symmetric solutions in $f(R)$ theory via Noether symmetries

TL;DR

This paper explores cylindrically and plane-symmetric solutions in $f(R)$ gravity using Noether symmetries, deriving new solutions and analyzing their relation to General Relativity.

Contribution

It introduces a systematic method to find exact solutions in $f(R)$ gravity with symmetry considerations, extending previous work by including matter and non-matter cases.

Findings

New exact solutions in $f(R)$ gravity for symmetric spacetimes.

Demonstration of the relation between $f(R)$ solutions and GR limit.

Application of Noether symmetry method to classify solutions.

Abstract

The $f(R)$ theory is considered for static cylindrically symmetric and plane-symmetric spacetimes. In order to find solutions to the field equations of these models, the Noether symmetry method is used. First, we examine the GR case for cylindrically symmetrical space-time with the $w=-1$ dark energy state. Then, with the assumption of $f(R) = f_0R^n$, cases with matter and non-matter are examined and general solutions are determined for both space-times. Thus, it is shown that inclusive new solutions are obtained, considering the Noether symmetric conditions. In addition, the GR limit for each cases are examined.