Geodesic deviation equation in generalized hybrid metric-Palatini gravity

Abstract

Abstract In the context of general relativity, the geodesic deviation equation (GDE) relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In this paper, we consider the GDE for the generalized hybrid metric-Palatini gravity and apply it in this model to investigate the structure of time-like, space-like, and null geodesics in the homogeneous and isotropic universe. We propose a particular case $$f(R,{{\mathcal {R}}})=R+{{\mathcal {R}}}$$ f ( R , R ) = R + R to study the numerical behavior of the deviation vector $$\eta (z)$$ η ( z ) and the observer area–distance $$r_{0}(z)$$ r 0 ( z ) with respect to redshift z. Also, we consider the GDE in the framework of the scalar–tensor representation of the generalized hybrid metric-Palatini gravity, i.e., $$f(R, {{\mathcal {R}}} )$$ f ( R , R ) , in which the model can be considered as dynamically equivalent to a gravitational theory with two scalar fields. Finally, we extend our calculations to obtain the modification of the Mattig relation in this model.