f(R,A) gravity theory in Einstein space background and causality violation

In this paper, we exploration a Petrov type-N vacuum solution to Einstein’s field equations, while incorporating a negative cosmological constant ($\Lambda <0$) within the framework of modified gravity theories. This solution intriguingly accommodates closed time-like curves at a particular moment in time, effectively violates the causality condition, thus acts as a time-machine model. A key observation is that the determinant of the Ricci tensor $R^{\mu \nu }$ for this particular Einstein space metric differs from zero. This noteworthy finding suggests to the existence of an anti-curvature tensor defined $A^{\mu \nu }=R_{\mu \nu }^{-1}$ and hence, an anti-curvature scalar $A=g_{\mu \nu }{A}^{\mu \nu }$, which is introduced with the Lagrangian of the system, thereby giving rise to as Ricci-inverse gravity theory. We consider class-I models of Ricci-inverse gravity, where the function $f=f(R,A)=(R+\kappa A)$ with $\kappa$ is the coupling constant. We demonstrate that this Einstein space metric serves as a vacuum solution with a negative modified cosmological constant within the framework of Ricci-inverse gravity. Consequently, the violation of causality persists within this new gravity theory as well. Moreover, we solve the modified field equations by considering matter content other than vacuum and demonstrate that the energy-density and isotropic pressure satisfies the equation $\rho =-p=\frac{3\kappa }{\Lambda }$.