Stationary BTZ space-time in Ricci-inverse and f(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(R,A,AμνAμν), with R and A representing the Ricci and anti-curvature scalars, respectively and Aμν 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(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≠0,J≠0, (ii) massless BTZ space-time, M=0 and J=0, and (iii) M=-1,J=0, that is AdS3-type, and analyze the results in modified gravity theories and compare with the general relativity case.