Characterizations of η–ρ-Einstein solitons in spacetimes and f(ℛ)-gravity

Abstract

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then the integral curves generated by the velocity vector field [Formula: see text] are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient [Formula: see text]–[Formula: see text]-Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient [Formula: see text]–[Formula: see text]-Einstein soliton in [Formula: see text]-gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function [Formula: see text] remains invariant under the velocity vector [Formula: see text]. Also, for two models [Formula: see text] and [Formula: see text] ([Formula: see text] = constant and [Formula: see text] is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition.