In this paper, we investigate a causality-violating four-dimensional space-time, an extension of the two-dimensional Misner space within the framework of modified gravity theories, specifically in the context of Ricci-inverse gravity. To explore this, we focus on two classes of models denoted as f ( R , A mu nu A mu nu ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathcal {R}}, A<^>{\mu \nu }\,A_{\mu \nu })$$\end{document} and f ( R , A , A mu nu A mu nu ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathcal {R}}, {\mathcal {A}}, A<^>{\mu \nu }\,A_{\mu \nu })$$\end{document} , and derive the modified field equations using the four-dimensional space-time under consideration. Here, A mu nu \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A<^>{\mu \nu }$$\end{document} is the anti-curvature tensor defined by inverse of the Ricci tensor R mu nu \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\mu \nu }$$\end{document} , that is, A mu nu = R mu nu - 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A<^>{\mu \nu }=R<^>{-1}_{\mu \nu }$$\end{document} . Additionally, A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} denotes its scalar, defined as A = g mu nu A mu nu \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}=g_{\mu \nu }\,A<^>{\mu \nu }$$\end{document} . Our analysis shows that in both models, the modified field equations can be solved by considering the matter energy content as a pure radiation field including the cosmological constant. Furthermore, we demonstrate that the energy density of the pure radiation field and the cosmological constant undergo modifications due to the coupling parameters, and these modifications converge to the results of general relativity when these parameters are set to zero. This observation confirms the possibility of causality violation in the form of closed time-like curves within the framework of Ricci-inverse gravity theory.
