Constraining the deformation of a rotating black hole mimicker from its shadow

We consider a black hole mimicker given by an exact solution of the stationary and axially-symmetric field equations in vacuum known as the $\delta$-Kerr metric. We study its optical properties based on a ray-tracing code for photon motion and characterize the apparent shape of the shadow of the compact object and compare it with the Kerr black hole. For the purpose of obtaining qualitative estimates related to the observed shadow of the supermassive compact object in the galaxy M87 we focus on values of the object’s spin $a$ and inclination angle of observation ${\theta }_0$ close to the measured values. We then apply the model to the shadow of the $\delta$-Kerr metric to obtain constraints on the allowed values of the deformation parameter. We show that based uniquely on one set of observations of the shadow’s boundary it is not possible to exclude the $\delta$-Kerr solution as a viable source for the geometry in the exterior of the compact object.

Figure 1

Schematic illustration of the celestial coordinates used for the ray tracing code in the $\delta$-Kerr space-time.

Figure 2

The boundary of the shadow in the $\delta$-Kerr space-time with different values of spin $a$ and deformation $\delta$ for different inclination angle of observation ${\theta }_0$. Each panel shows the shadow’s boundary for ${\theta }_0=15°,45°,75°,90°$. For smaller values of $a$ the effect of $\delta$ on the shadow’s boundary is more noticeable.

Figure 3

The boundary of the shadow in the Kerr metric within experimental bounds from the EHT observations. The two boundaries are obtained for the limiting values of inclination angle and spin of ${\theta }_0=19°$, $a=0.85$, and ${\theta }_0=15°$, $a=0.95$.

Figure 4

The boundary of the shadow in the $\delta$-Kerr metric with fixed inclination angle ${\theta }_0=17°$ and different values for spin $a$ and deformation $\delta$. The two boundaries are obtained for the limiting values of $a=0.80$, $\delta =0.8$, and $a=0.99$, $\delta =1.1$.

Figure 5

The average radius of the shadow $⟨R⟩$ for different values of spin $a$ and quadrupole parameter $\delta$ with fixed value of observation angle ${\theta }_0=17°$. The solid lines correspond to $⟨R⟩$ for the minimum and maximum values obtained in Fig. 3. The purple vertical line corresponds the case of Kerr metric.

Figure 6

The radius of the innermost stable circular orbit (ISCO) in units of $M_0$ as a function of the deformation parameter $\delta$ for different values of the spin parameter $a$. The solid line corresponds to the Zipoy-Voorhees metric. The vertical line for $\delta =1$ corresponds to the Kerr case.

Figure 7

The thin accretion disk in the $\delta$-Kerr space-time with the observation angle ${\theta }_0=15°$, for the different values of $a$ and $\delta$. Image sizes are normalized with $M_0$ while the energy flux of each rows (fixed $a$) is normalized with the maximum value in the Kerr case ($\delta =1$). The top row corresponds to the Zipoy-Voorhees metric, while the middle column corresponds to the Kerr case.

Figure 8

The thin accretion disk in the $\delta$-Kerr space-time with the observation angle ${\theta }_0=15°$, for the different values of $a$ and $\delta$. Image sizes are normalized with $M_0$ while the energy flux of each rows (fixed $a$) is normalized with the maximum value in the Kerr case ($\delta =1$). The top row corresponds to the Zipoy-Voorhees metric, while the middle column corresponds to the Kerr case.

Figure 9

The thin accretion disk in the $\delta$-Kerr space-time with the observation angle ${\theta }_0=15°$, for the different values of $a$ and $\delta$. Image sizes are normalized with $M_0$ while the energy flux of each rows (fixed $a$) is normalized with the maximum value in the Kerr case ($\delta =1$). The top row corresponds to the Zipoy-Voorhees metric, while the middle column corresponds to the Kerr case.

Figure 10

The thin accretion disk in the $\delta$-Kerr space-time with the observation angle ${\theta }_0=15°$, for the different values of $a$ and $\delta$. Image sizes are normalized with $M_0$ while the energy flux of each rows (fixed $a$) is normalized with the maximum value in the Kerr case ($\delta =1$). The top row corresponds to the Zipoy-Voorhees metric, while the middle column corresponds to the Kerr case.