Perfect fluid spacetimes and $k$-almost Yamabe solitons

Abstract

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.