Existence and Physical Properties of Gradient Ricci–Yamabe Solitons

Abstract

We first prove the existence of the gradient Ricci–Yamabe soliton (briefly GRYS) by constructing an explicit example endowed with the Robertson–Walker metric. Then we focus on the physical properties of the gradient Ricci–Yamabe solitons satisying Einstein’s field equations, under the assumptions of different subspaces of Gray’s decompositions. For instance, we prove that if a GRYS space-time satisfying Einstein’s field equations, in which the gradient of the potential function \(\psi\) is a unit-timelike torse-forming vector field, belongs to the subspaces \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\), then it is a Robertson–Walker space-time with vanishing shear and vorticity. Moreover, its possible local cosmological structures are of Petrov types I, D, or O. Finally, we obtain the equations of state of a perfect-fluid space-time admitting the GRYS whose velocity field is a unit-timelike Killing vector field.