Time-machines construct in f(ℛ, 𝒜, Aμν Aμν ) and f(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 { 59}, 084016 (1999)), within the context of modified gravity theories. We consider Ricci-inverse gravity of all Classes of models, { i.e.}, (i) Class-{ I}: f(R, A)=(R+κ\,R²+β\,A), (ii) Class-{ II}: f(R, Aμν\,Aμν)=(R+κ\,R²+γ\,Aμν\,Aμν) model, and (iii) Class-{ III}: f(R, A, Aμν\,Aμν)=(Rκ\,R²+β\,A+δ\,A²+γ\,Aμν\,Aμν) model, where Aμν is the anti-curvature tensor, the reciprocal of the Ricci tensor, Rμν, A=gμν\,Aμν is its scalar, and β, κ, γ, δ are the coupling constants. Moreover, we consider f(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(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.