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

TL;DR

This paper derives Kerr-AdS and Kerr-dS black hole solutions in a perfect fluid dark matter background, analyzing their properties, horizons, ergospheres, singularities, geodesics, and rotational velocities, revealing how dark matter influences black hole characteristics.

Contribution

It introduces new Kerr-(A)dS black hole solutions incorporating perfect fluid dark matter and thoroughly examines their physical and geometric properties.

Findings

Horizon structure depends on dark matter parameter and cosmological constant.

Ergosphere size varies oppositely for positive and negative dark matter parameters.

Stable orbits exist for certain dark matter parameter ranges.

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 $\alpha$ and the cosmological constant $\Lambda$ when the cosmological horizon $r_{\Lambda}$ exists. For $\Lambda=0$, 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 $\Lambda$ (Kerr-AdS BH), $\alpha_{max}$ decreases if $\alpha>0$, and $\alpha_{min}$ increases if $\alpha<0$. For negative cosmological constant $-\Lambda$ (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 $\Lambda=0$, 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 $\alpha<0$.