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

Abstract

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, $f(\mathcal{R})=(\mathcal{R}+α_1\,\mathcal{R}^2+α_2\,\mathcal{R}^3+α_3\,\mathcal{R}^4+α_4\,\mathcal{R}^5)$ and $f(\mathcal{R})=\mathcal{R}+α_k\,\mathcal{R}^{k+1},\quad (k=1,2,...n)$. We solve the modified field equations incorporating zero energy-momentum tensor, $\mathcal{T}^{μν}=0$ 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: $f(\mathcal{R}, \mathcal{A})=(\mathcal{R}+β\,\mathcal{A})$, (ii) Class-\textbf{II} model: $f(\mathcal{R}, A^{μν}\,A_{μν})=(\mathcal{R}+γ\,A^{μν}\,A_{μν})$, and (iii) Class-\textbf{III} model: $f(\mathcal{R}, \mathcal{A}, A^{μν}\,A_{μν})=(\mathcal{R}+α_1\,\mathcal{R}^2+α_2\,\mathcal{R}^3+β_1\,\mathcal{A}+β_2\,\mathcal{A}^2+γ\,A^{μν}\,A_{μν})$, where $\mathcal{A}=g_{μν}\,A^{μν}$ is the anti-curvature scalar, $A^{μν}$ is the anti-curvature tensor, the reciprocal of the Ricci tensor $R_{μν}$. 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 $Λ$, and thus, shifted the result in comparison to the general relativity case.