Shadow of rotating non-Kerr black hole

The shadow of a rotating non-Kerr black hole has been studied, and it was shown that in addition to the specific angular momentum $a$, the deformation parameter of non-Kerr spacetime essentially deforms the shape of the black hole shadow. For a given value of the black hole spin parameter $a$, the presence of a deformation parameter $ϵ$ reduces the shadow and enlarges its deformation with respect to the one in the Kerr spacetime. Optical features of the rotating non-Kerr black hole in terms of rotation of the polarization vector along null congruences have been studied. A comparison of the obtained theoretical results on the polarization angle with the observational data on Faraday rotation measurements provides the upper limit for the dimensionless deformation parameter as $ϵ\le 19$.

Figure 1

The radial dependence of the effective potential of radial motion of the massless particles for the different values of the $ϵ$ parameter: solid line for $ϵ=0$, dashed line for $ϵ=-5$, and dotted-dashed line for $ϵ=-10$ (left panel) and solid line for $ϵ=0$, dashed line for $ϵ=5$, and dotted-dashed line for $ϵ=10$ (right panel).

Figure 2

The schematic geometry of the gravitational lens. An observer far away from the black hole can set up a reference coordinate system (x, y, z) with the black hole at the origin. The Boyer-Lidquist coordinates coincide with this system only at infinity. The tangent vector to an incoming light ray defines a straight line which intersects the $\alpha \mathrm{\text{−}}\beta$ plane at the point $({\alpha }_i,{\beta }_i)$.

Figure 3

The observables for the apparent shape of a rotating black hole are the radius $R_s$ and the distortion parameter ${\delta }_s=D_{cs}/R_s$. Here $D_{cs}$ is the difference between the left endpoints of the circle and of the shadow.

Figure 4

Silhouette of the shadow cast by a non-Kerr black hole situated at the origin of coordinates with inclination angle $\theta =\pi /2$, having a rotation parameter $a$ and a deformation parameter $ϵ$. Upper row, left: $a/M=0.35$, $ϵ=0$ (solid line), $ϵ=-5$ (dashed line), and $ϵ=-10$ (dashed-dotted line). Upper row, right: $a/M=0.4$, $ϵ=0$ (solid line), $ϵ=-5$ (dashed line), and $ϵ=-10$ (dashed-dotted line). Lower row, left: $a/M=0.45$, $ϵ=0$ (solid line), $ϵ=-5$ (dashed line), and $ϵ=-10$ (dashed-dotted line). Lower row, right: $a/M=0.5$, $ϵ=0$ (solid line), $ϵ=-5$ (dashed line), and $ϵ=-10$ (dashed-dotted line). The shadow corresponds to each curve and the region inside it.

Figure 5

Silhouette of the shadow cast by a non-Kerr black hole situated at the origin of coordinates with inclination angle $\theta =\pi /2$, having a rotation parameter $a$ and a deformation parameter $ϵ$. Upper row, left: $a/M=0.4$, $ϵ=0$ (solid line), $ϵ=5$ (dashed line), and $ϵ=10$ (dashed-dotted line). Upper row, right: $a/M=0.5$, $ϵ=0$ (solid line), $ϵ=5$ (dashed line), and $ϵ=10$ (dashed-dotted line). Lower row, left: $a/M=0.55$, $ϵ=0$ (solid line), $ϵ=5$ (dashed line), and $ϵ=10$ (dashed-dotted line). Lower row, right: $a/M=0.6$, $ϵ=0$ (solid line), $ϵ=5$ (dashed line), and $ϵ=10$ (dashed-dotted line). The shadow corresponds to each curve and the region inside it.

Figure 6

Observables $R_s$ as functions of the $ϵ<0$ and $ϵ>0$ parameters, corresponding to the shadow of a black hole situated at the origin of coordinates with inclination angle $\theta =\pi /2$ and spin parameters $a=0.5$.

Figure 7

Observables ${\delta }_s$ as functions of the $ϵ<0$ and $ϵ>0$ parameters, corresponding to the shadow of a black hole situated at the origin of coordinates with inclination angle $\theta =\pi /2$ and spin parameters $a=0.5$.

Figure 8

The radial dependence of the polarization vector for the different values of the deformation parameter $ϵ$ : solid line for $ϵ=0$, dashed line for $ϵ=-1$, dotted-dashed line for $ϵ=5$, and dotted line for $ϵ=10$. $a=0.5$.