Perfect fluid spacetimes and k-almost Yamabe solitons

In this article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study k-almost Yamabe solitons and gradient k-almost Yamabe solitons. First, two examples are constructed to ensure the existence of gradient k-almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a k-almost Yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permits a k-almost Yamabe soliton (g, k, rho, lambda), then the integral curves of the vector field rho are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field rho. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and rho(a) = 0 where a is a scalar, then either the perfect fluid spacetime represents a phantom era, or the potential function phi is invariant under the velocity vector field rho. Finally, we prove that if a perfect fluid spacetime permits a gradient k-almost Yamabe soliton (g, k, D phi, lambda) and R, lambda, k are invariant under rho, then the vorticity of the fluid vanishes.