Solitons and Gradient Solitons on $$GRW$$ Spacetimes

A perfect fluid spacetime is not a generalized Robertson–Walker spacetime, and also the converse does not hold in general. The purpose of this study is to determine the conditions under which a generalized Robertson–Walker spacetime is a perfect fluid spacetime. It is established that if the metric of a generalized Robertson–Walker spacetime is a $$\rho$$ -Einstein soliton, then the spacetime turns into a perfect fluid spacetime. Conversely, a perfect fluid spacetime with divergence-free velocity vector obeying $$\rho$$ -Einstein solitons represents a generalized Robertson–Walker spacetime. Also, we obtain the same result if the metric is either a gradient $$\rho$$ -Einstein soliton or a gradient $$m$$ -quasi Einstein soliton. As a consequence of the above studies, we acquire some fascinating results.