Time-Machines Construct in $f(\mathcal{R},\mathcal{A},A^{\mu\nu}\,A_{\mu\nu})$ and $f(\mathcal{R})$ Modified Gravity Theories

In this paper, our objective is to explore a time-machine space-time formulated in general relativity, as introduced by Li (Phys. Rev. D {\bf 59}, 084016 (1999)), within the context of modified gravity theories. We consider Ricci-inverse gravity of all Classes of models, {\it i.e.}, (i) Class-{\bf I}: $f(\mathcal{R}, \mathcal{A})=(\mathcal{R}+{\kappa\,\mathcal{R}^2}+\beta\,\mathcal{A})$, (ii) Class-{\bf II}: $f(\mathcal{R}, A^{\mu\nu}\,A_{\mu\nu})=(\mathcal{R}+{\kappa\,\mathcal{R}^2}+\gamma\,A^{\mu\nu}\,A_{\mu\nu})$ model, and (iii) Class-{\bf III}: $f(\mathcal{R}, \mathcal{A}, A^{\mu\nu}\,A_{\mu\nu})=(\mathcal{R}{\kappa\,\mathcal{R}^2}+\beta\,\mathcal{A}+\delta\,\mathcal{A}^2+\gamma\,A^{\mu\nu}\,A_{\mu\nu})$ model, where $A^{\mu\nu}$ is the anti-curvature tensor, the reciprocal of the Ricci tensor, $R_{\mu\nu}$, $\mathcal{A}=g_{\mu\nu}\,A^{\mu\nu}$ is its scalar, and $\beta, {\kappa}, \gamma, \delta$ are the coupling constants. Moreover, we consider $f(\mathcal{R})$ modified gravity theory and investigate the same time-machine space-time. In fact, we show that Li time-machine space-time serve as valid solutions both in Ricci-inverse and $f(\mathcal{R})$ modified gravity theories. Thus, both theory allows the formation of closed time-like curves analogue to general relativity, thereby representing a possible time-machine model in these gravity theories theoretically.