A note on constant curvature solutions in cylindrically symmetric metric $f(R)$ Gravity

TL;DR

This paper derives a new family of exact constant curvature solutions in cylindrically symmetric $f(R)$ gravity, extending previous work and applicable to cosmic strings, with analysis of stability and relation to General Relativity solutions.

Contribution

It introduces a two-parameter family of exact solutions with constant Ricci scalar in cylindrically symmetric $f(R)$ gravity, expanding the set of known solutions.

Findings

Derived a 2-parameter family of solutions including cosmological constant and a new parameter.

Showed the solution corresponds to a constant Ricci scalar and relates to the Tian family in GR.

Discussed the stability of solutions under initial conditions.

Abstract

In the previous work we introduced a new static cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric f(R) theories of gravity\cite{1}. Now we obtain a 2-parameter family of exact solutions which contains cosmological constant and a new parameter as $\beta$. This solution corresponds to a constant Ricci scalar. We proved that in $f(R)$ gravity, the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in GR. We show that our constant curvature exact solution is applicable to the exterior of a string. Sensibility of stability under initial conditions is discussed.