Axially symmetric solutions in Ricci-inverse modified gravity

We investigate a family of axial symmetry solution constructed in general relativity (GR) within the framework of Ricci-inverse (RI) gravity theory. In GR, these solutions admitted closed time-like curves at an instant of time from an initial spacelike hypersurface in a causally well-behaved manner, thus, violates the causality condition. Our aim is to examine these axial symmetry solutions within the context of Ricci-inverse gravity theory to determine whether closed time-like curves still appear in this new gravity theory. We consider two Classes of RI-gravity models: (i) Class-II models defined by a function f=f(R, Aμ ν\,Aμ ν) f = f ( R , A μ ν A μ ν ) gravity and (ii) Class-III models defined by f=f(R,A, Aμ ν\,Aμ ν) f = f ( R , A , A μ ν A μ ν ) , where Aμ ν A μ ν is the anti-curvature tensor, A=gμ ν\,Aμ ν A = g μ ν A μ ν as its scalar, and Rμ ν R μ ν is the Ricci tensor. We are able solved the modified field equations considering these axial symmetry solutions as background in RI-gravity with null radiation as the matter content and the cosmological constant. This confirms that the chosen family of axial symmetry solutions are valid solutions in RI-gravity theory and, consequently, closed time-like curves is still form, analogous to their formation in GR.