Photon regions and shadows of Kerr-Newman-NUT black holes with a cosmological constant

We consider the Plebański class of electrovacuum solutions to the Einstein equations with a cosmological constant. These space-times, which are also known as the Kerr-Newman-NUT–(anti–)de Sitter space-times, are characterized by a mass $m$, a spin $a$, a parameter $\beta$ that comprises electric and magnetic charge, a NUT parameter $\ell$ and a cosmological constant $\mathrm{\Lambda }$. Based on a detailed discussion of the photon regions in these space-times (i.e., of the regions in which spherical lightlike geodesics exist), we derive an analytical formula for the shadow of a Kerr-Newman-NUT–(anti–)de Sitter black hole for an observer at given Boyer-Lindquist coordinates $(r_O,{\vartheta }_O)$ in the domain of outer communication. We visualize the photon regions and the shadows for various values of the parameters.

Figure 1

Angular radius $\alpha$ of the shadow of a Schwarzschild black hole, given by Synge’s formula, Eq. (1).

Figure 2

Legend for Figs. 3, 4, 5, and 6.

Figure 3

Photon regions in Kerr space-time for spins $a=\lambda a_{\max }$, where $a_{\max }=m$. The plots on the right show a magnified inner part.

Figure 4

Photon regions in Kerr-NUT space-time with $\ell =\frac{3}{4}m$, $C=0$ for spins $a=\lambda a_{\max }$, where $a_{\max }=\sqrt{m^2+{\ell }^2}=\frac{5}{4}m$. The plots on the right show a magnified inner part.

Figure 5

Photon regions in Kerr-Newman-NUT space-time ($\beta =\frac{5}{9}{m}^2$, $\ell =\frac{4}{3}m$, $C=0$) with a cosmological constant ($\mathrm{\Lambda }={10}^{-2}{m}^{-2}$) for spins $a=\lambda a_{\max }$, where $a_{\max }\approx 1.51m$. The plots on the right show a magnified inner part.

Figure 6

Photon regions for varying singularity parameter $C$ with fixed $a=\frac{4}{5}{a}_{\max }$, $\beta =\frac{5}{9}{m}^2$, $\ell =\frac{4}{3}m$, and $\mathrm{\Lambda }=\{\begin{array}{ll}{10}^{-2}{m}^{-2}& \text{for}C\le 0\\ 0& \text{for}C>0\end{array}$, where $a_{\max }=\{\begin{array}{ll}1.51m& \text{for}C\le 0\\ 2\sqrt{5}m/3& \text{for}C>0\end{array}$. If existent, the cosmological horizon restricts the region where the causality is violated. If $C=1$ or $C=-1$, one of the two half-axes is regular and it is not surrounded by a causality violating region.

Figure 7

At an observation event with Boyer-Lindquist coordinates $(r_O,{\vartheta }_O)$ we choose an orthonormal tetrad $(e_0,e_1,e_2,e_3)$ according to Eqs. (19). For each light ray that is sent from the observation event into the past the tangent vector can be written as a linear combination of $e_0$, $e_1$, $e_2$ and $e_3$. In this way we can assign celestial coordinates to the direction of the tangent vector (see Fig. 8).

Figure 8

To each light ray at the observation event we assign celestial coordinates $\theta$ and $\psi$ with the help of Eq. (21) (see figure on the left). The figure on the right shows the stereographic projection (red ball) of the point $(\theta ,\psi )$ on the celestial sphere (black ball). The dotted (red) circles indicate the celestial equator $\theta =\pi /2$ and its projection.

Figure 9

Shadow of a black hole for different parameters $a$, $\beta$, $\ell$ and $\mathrm{\Lambda }$, seen by an observer at $r_O=5m$ and ${\vartheta }_O=\pi /2$. The cross hairs indicate the spatial direction towards the black hole, i.e., the spatial direction of the principal null congruences with respect to our observer with four-velocity $e_0$. The dashed (red) circle indicates the celestial equator, cf. Fig. 8.

Figure 10

Shadow of a black hole for an observer at $r_O=5m$ and different inclination angles ${\vartheta }_O$, with fixed $\beta =\frac{5}{9}{m}^2$, $\ell =\frac{4}{3}m$, $\mathrm{\Lambda }={10}^{-2}{m}^{-2}$ and $a=a_{\max }\approx 1.51m$. As in Fig. 9, the cross hairs indicate the spatial direction towards the black hole and the dashed (red) circle indicates the celestial equator.