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Rotating black strings in $f(R)$-Maxwell theory

In general, the field equations of $f(R)$ theory coupled to a matter field are very complicated and hence it is not easy to find exact analytical solutions. However, if one considers traceless energy-momentum tensor for the matter source as well as constant scalar curvature, one can derive some exact analytical solutions from $f(R)$ theory coupled to a matter field. In this paper, by assuming constant curvature scalar, we construct a class of charged rotating black string solutions in $f(R)$-Maxwell theory. We study the physical properties and obtain the conserved quantities of the solutions. The conserved and thermodynamic quantities computed here depend on function $f'(R_{0})$ and differ completely from those of Einstein theory in AdS spaces. Besides, unlike Einstein gravity, the entropy does not obey the area law. We also investigate the validity of the first law of thermodynamics as well as the stability analysis in the canonical ensemble, and show that the black string solutions are always thermodynamically stable in $f(R)$-Maxwell theory with constant curvature scalar. Finally, we extend the study to the case where the Ricci scalar is not a constant and in particular $R=R(r)$. In this case, by using the Lagrangian multipliers method, we derive an analytical black string solution from $f(R)$ gravity and reconstructed the function $R(r)$. We find that this class of solutions has an additional logaritmic term in the metric function which incorporates the effect of the $f(R)$ theory in the solutions.