Stationary BTZ space-time in Ricci-inverse and \(f({\mathcal {R}})\) gravity theories

Abstract

In this paper, we explore a stationary BTZ space-time within the framework of modified gravity theory, specifically focusing on Ricci-inverse gravity. It is important to clarify that “Ricci-inverse” refers to the inverse of the Ricci tensor, not the Ricci scalar. We consider a general class of this gravity theory, where the function f is defined by \(f({\mathcal {R}}, {\mathcal {A}}, A^{\mu \nu }\,A_{\mu \nu })\), with \({\mathcal {R}}\) and \({\mathcal {A}}\) representing the Ricci and anti-curvature scalars, respectively and \(A^{\mu \nu }\) is the anti-curvature tensor. We demonstrate that stationary BTZ space-time is a valid solution in this gravity theory, wherein the cosmological constant undergoes modifications due to the coupling constants. Moreover, we study another modified gravity theory known as \(f({\mathcal {R}})\)-gravity and analyze the stationary BTZ space-time. Subsequently, we fully integrate the geodesic equations for BTZ space-time constructed within the Ricci-Inverse gravity, expressing the solutions in terms of elementary functions and compared with the GR result. We classify different types of geodesics, including null and time-like geodesics, under three conditions: (i) nonzero mass and angular momentum, \(M \ne 0, J\ne 0\), (ii) massless BTZ space-time, \(M=0\) and \(J=0\), and (iii) \(M=-1, J=0\), that is \(AdS_3\)-type, and analyze the results in modified gravity theories and compare with the general relativity case.