Kerr-Anti-De-Sitter/De-Sitter Black Hole In Perfect Fluid Dark Matter Background

We obtain the Kerr-anti-de-sitter (Kerr-AdS) and Kerr-de-sitter (Kerr-dS) black hole (BH) solutions to the Einstein field equation in the perfect fluid dark matter background using the Newman-Janis method and Mathematica package. We discuss in detail the black hole properties and obtain the following main results: (i) From the horizon equation $g_{rr}=0$, we derive the relation between the perfect fluid dark matter parameter $\alpha$ and the cosmological constant $\mathrm{\Lambda }$ when the cosmological horizon $r_{\mathrm{\Lambda }}$ exists. For $\mathrm{\Lambda }=0$, we find that $\alpha$ is in the range $0<\alpha <2M$ for $\alpha >0$ and $-7.18M<\alpha <0$ for $\alpha <0$. For positive cosmological constant $\mathrm{\Lambda }$ (Kerr-AdS BH), ${\alpha }_{max}$ decreases if $\alpha >0$, and ${\alpha }_{min}$ increases if $\alpha <0$. For negative cosmological constant $-\mathrm{\Lambda }$ (Kerr-dS BH), ${\alpha }_{max}$ increases if $\alpha >0$ and ${\alpha }_{min}$ decreases if $\alpha <0$; (ii) An ergosphere exists between the event horizon and the outer static limit surface. The size of the ergosphere evolves oppositely for $\alpha >0$ and $\alpha <0$, while decreasing with the increasing $\mid \alpha \mid$. When there is sufficient dark matter around the black hole, the black hole spacetime changes remarkably; (iii) The singularity of these black holes is the same as that of rotational black holes. In addition, we study the geodesic motion using the Hamilton-Jacobi formalism and find that when $\alpha$ is in the above ranges for $\mathrm{\Lambda }=0$, stable orbits exist. Furthermore, the rotational velocity of the black hole in the equatorial plane has different behaviour for different $\alpha$ and the black hole spin $a$. It is asymptotically flat and independent of $\alpha$ if $\alpha >0$ while is asymptotically flat only when $\alpha$ is close to zero if $\alpha <0$.