Study of cylindrically symmetric solutions in an $f(R)$ gravity background

We investigate static cylindrically symmetric solutions of the Weyl and Gödel space–times in the framework of modified $f(R)$ gravity. With this aim, we consider the modified higher-order theory of gravity based on nonconformal invariant gravitational waves. From the modified Einstein equations, we derive two exact solutions of the Weyl space–time and find one exact and one numerical solution of the Gödel space–time. In particular, we obtain a family of exact solutions with a constant scalar curvature $R$ depending on arbitrary constants for both space–times. It is interesting that the second solution for the Weyl metric has a nonconstant Ricci scalar. We find that the result obtained by solving the higher-order theory of gravity is similar to the result for the Einstein field equations with a cosmological constant. Moreover, we graphically study the role of the metric coefficients in both space–times.