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

Abstract

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 $α$ and the cosmological constant $Λ$ when the cosmological horizon $r_Λ$ exists. For $Λ=0$, we find that $α$ is in the range $0<α<2M$ for $α>0$ and $-7.18M<α<0$ for $α<0$. For positive cosmological constant $Λ$ (Kerr-AdS BH), $α_{max}$ decreases if $α>0$, and $α_{min}$ increases if $α<0$. For negative cosmological constant $-Λ$ (Kerr-dS BH), $α_{max}$ increases if $α>0$ and $α_{min}$ decreases if $α<0$; (ii) An ergosphere exists between the event horizon and the outer static limit surface. The size of the ergosphere evolves oppositely for $α>0$ and $α<0$, while decreasing with the increasing $\midα\mid$. 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 $α$ is in the above ranges for $Λ=0$, stable orbits exist. Furthermore, the rotational velocity of the black hole in the equatorial plane has different behaviour for different $α$ and the black hole spin $a$. It is asymptotically flat and independent of $α$ if $α>0$ while is asymptotically flat only when $α$ is close to zero if $α<0$.