Cylindrical black hole solutions in $f(\mathcal{R})$ and $f(\mathcal{R},\mathcal{A},A^{\mu \nu }{A}_{\mu \nu })$ 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, $f(\mathcal{R})=(\mathcal{R}+{\alpha }_1{\mathcal{R}}^2+{\alpha }_2{\mathcal{R}}^3+{\alpha }_3{\mathcal{R}}^4+{\alpha }_4{\mathcal{R}}^5)$ and $f(\mathcal{R})=\mathcal{R}+{\alpha }_k{\mathcal{R}}^{k+1},(k=1,2,...n)$. 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 ($\mathcal{R}\mathcal{I}$) 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.

28 pages, 7 figures, 1 table, accepted in Physics of the Dark Universe (https://doi.org/10.1016/j.dark.2025.101883), mathematics overlap with arXiv:2407.11513, arXiv:2410.11922

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