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

TL;DR

This paper investigates cylindrical black hole solutions within various modified gravity theories, deriving new solutions and analyzing test particle motion, revealing how coupling constants affect the effective cosmological constant compared to general relativity.

Contribution

It provides new cylindrical black hole solutions in $f( ext{R})$ and Ricci-Inverse gravity theories, and examines how these modifications influence particle trajectories and the effective cosmological constant.

Findings

Modified gravity coupling constants alter the effective cosmological constant.

New solutions for cylindrical black holes in $f( ext{R})$ and Ricci-Inverse gravity.

Test particle geodesics are affected by the modified gravity parameters.

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 $f(\mathcal{R})=\mathcal{R}+\alpha_k\,\mathcal{R}^{k+1},\quad (k=1,2,...n)$. 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.