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 permit a $k$-almost yamabe soliton ($g,k,ρ,λ$), then the integral curves of the vector field $ρ$ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field $ρ$. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and $ρ(a)=0$ where a is a scalar, then either the perfect fluid spacetime represents phantom era, or the potential function $Φ$ is invariant under the velocity vector field $ρ$. Finally, we prove that if a perfect fluid spacetime permits a gradient $k$-almost yamabe soliton ($g,k,DΦ,λ$) and $R, λ, k$ are invariant under $ρ$, then the vorticity of the fluid vanishes.

Theorems & Definitions (5)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1