Stationary BTZ space-time in Ricci-inverse and $$f({\mathcal {R}})$$ f ( R ) gravity theories - Faizuddin Ahmed; Abdelmalek Bouzenada (European Physical Journal C: Particles and Fields)
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 $$f({\mathcal {R}}, {\mathcal {A}}, A^{μ ν }\,A_{μ ν })$$ f ( R , A , A μ ν A μ ν ) , with $${\mathcal {R}}$$ R and $${\mathcal {A}}$$ A representing the Ricci and anti-curvature scalars, respectively and $$A^{μ ν }$$ A μ ν 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 $$f({\mathcal {R}})$$ 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, $$M \ne 0, J\ne 0$$ M ≠ 0 , J ≠ 0 , (ii) massless BTZ space-time, $$M=0$$ M = 0 and $$J=0$$ J = 0 , and (iii) $$M=-1, J=0$$ M = - 1 , J = 0 , that is $$AdS_3$$ A d S 3 -type, and analyze the results in modified gravity theories and compare with the general relativity case.