Cylindrical black hole solutions in $f(\mathcal{R})$ and $f(\mathcal{R},\mathcal{A},A^{\mu\nu}A_{\mu\nu})$ modified gravity

We explore a cylindrical black hole (BH) space-time introduced by Lemos, in the context of modified gravity theories. Specifically, we focus on $f(\mathcal{R})$-gravity framework, where we choose two form functions, $f(\mathcal{R})=(\mathcal{R}+\alpha_1\,\mathcal{R}^2+\alpha_2\,\mathcal{R}^3+\alpha_3\,\mathcal{R}^4+\alpha_4\,\mathcal{R}^5)$ and $f(\mathcal{R})=\mathcal{R}+\alpha_k\,\mathcal{R}^{k+1},\quad (k=1,2,...n)$. We solve the modified field equations incorporating zero energy-momentum tensor, $\mathcal{T}^{\mu\nu}=0$ and obtain the result. Moreover, we study another well-known modified gravity theory called Ricci-Inverse ($\mathcal{RI}$) gravity and investigate this Lemos black hole (LBH) space-time. To achieve this, we consider different classes of models defined as follows: (i) Class-\textbf{I} model: $f(\mathcal{R}, \mathcal{A})=(\mathcal{R}+\beta\,\mathcal{A})$, (ii) Class-\textbf{II} model: $f(\mathcal{R}, A^{\mu\nu}\,A_{\mu\nu})=(\mathcal{R}+\gamma\,A^{\mu\nu}\,A_{\mu\nu})$, and (iii) Class-\textbf{III} model: $f(\mathcal{R}, \mathcal{A}, A^{\mu\nu}\,A_{\mu\nu})=(\mathcal{R}+\alpha_1\,\mathcal{R}^2+\alpha_2\,\mathcal{R}^3+\beta_1\,\mathcal{A}+\beta_2\,\mathcal{A}^2+\gamma\,A^{\mu\nu}\,A_{\mu\nu})$, where $\mathcal{A}=g_{\mu\nu}\,A^{\mu\nu}$ is the anti-curvature scalar, $A^{\mu\nu}$ is the anti-curvature tensor, the reciprocal of the Ricci tensor $R_{\mu\nu}$. We solve the modified field equations under the same aforementioned scenario of energy-momentum tensor, and obtain the result. Subsequently, we study the geodesic motions of test particles around this LBH within the Ricci-Inverse and $f(\mathcal{R})$-gravity theories and analyze the outcomes. We demonstrate that different coupling constants chosen in these modified gravity theories influences the usual cosmological constant $\Lambda$, and thus, shifted the result in comparison to the general relativity case.