Topological classes of rotating black holes

In this paper, we investigate the topological numbers for the singly rotating Kerr black holes in all dimensions and the four-dimensional Kerr-Newman black hole. We show that for uncharged black holes, the rotation parameter has a significant effect on the topological number, and for rotating black holes, the dimension of spacetimes has a remarkable effect on the topological number too. In addition, we find that the topological numbers of the four-dimensional Kerr and Kerr-Newman black holes are the same, which seems to indicate that the electric charge parameter has no effect on the topological number of rotating black holes. Our current research provides more evidence that the conjecture put forward in Wei et al. [Phys. Rev. Lett., 129, 191101 (2022)], according to which all black hole solutions should be separated into three different topological classes, is accurate, at least in the pure Einstein-Maxwell gravity theory.

Figure 1

Zero points of the vector ${\varphi }^{r_h}$ are shown on the $r_h-\tau$ plane. To facilitate comparison with previous results in Ref. [10], the blue and red solid lines respond to the Schwarzschild black hole (Sch BH) and Kerr black hole (Kerr BH) with $a/r_0=1$, respectively. The black dot with ${\tau }_c=6\sqrt{3}\pi a$ represents the generation point for the Kerr black hole. There is only one Schwarzschild black hole when $\tau ={\tau }_1$, while there is one Schwarzschild black hole and two Kerr black holes when $\tau ={\tau }_2$.

Figure 2

The red arrows represent the unit vector field $n$ on a portion of the $r_h-\mathrm{\Theta }$ plane. The unit vector field is plotted for the four-dimensional Kerr black hole with $\tau /r_0=34.48$ and $a/r_0=1$. The zero points (ZPs) marked with blue dots are at $(r_h/r_0,\mathrm{\Theta })=(1.46,\pi /2)$, and $(2.15,\pi /2)$, for ${\mathrm{ZP}}_1$, and ${\mathrm{ZP}}_2$, respectively. Due to the two winding numbers $w_1=1$ and $w_2=-1$, the topological number is: $W=w_1+w_2=0$.

Figure 3

Zero points of the vector ${\varphi }^{r_h}$ are shown on the $r_h-\tau$ plane for five- to nine-dimensional singly rotating Kerr black holes. The red, blue, green, brown, and purple solid lines are for five- to nine-dimensional singly rotating Kerr black holes with $a/r_0=1$, respectively. The black dot with ${\tau }_c=4\sqrt{3}\pi a/3$ represents the generation point for the five-dimensional singly rotating Kerr black hole.

Figure 4

The red arrows represent the unit vector field $n$ on a portion of the $r_h-\mathrm{\Theta }$ plane. The zero points (ZPs) marked with black dots are at $(r_h/r_0,\mathrm{\Theta })=(0.28,\pi /2)$, $(3.14,\pi /2)$, $(1,\pi /2)$, and $(1,\pi /2)$, for ${\mathrm{ZP}}_1$, ${\mathrm{ZP}}_2$, ${\mathrm{ZP}}_3$, and ${\mathrm{ZP}}_4$, respectively. The blue contours $C_i$ are closed loops surrounding the zero points.

Figure 5

Zero points of the vector ${\varphi }^{r_h}$ are shown on the $r_h-\tau$ plane for the Kerr-Newman black hole, with $a/r_0=1$ and $Q/r_0=1$. The black dot with ${\tau }_c=6\pi \sqrt{3(a^2+Q^2)}$ denotes the generation point for the Kerr-Newman black hole. At $\tau ={\tau }_c$, there is only one Kerr-Newman black hole, while at $\tau ={\tau }_1$, there are two Kerr-Newman black holes.

Figure 6

The red arrows denote the unit vector field $n$ on a portion of the $r_h-\mathrm{\Theta }$ plane. The unit vector field is plotted for the Kerr-Newman black hole with $\tau /r_0=50$, $a/r_0=1$, and $Q/r_0=1$. The zero points (ZPs) marked with black dots are at $(r_h/r_0,\mathrm{\Theta })=(1.95,\pi /2)$, and $(3,\pi /2)$, for ${\mathrm{ZP}}_1$, and ${\mathrm{ZP}}_2$, respectively.