Does the shape of the shadow of a black hole depend on motional status of an observer?

In a recent work on rotating black hole shadows [Z. Chang and Q.-H. Zhu, Phys. Rev. D 101, 084029 (2020)], we proposed a new approach for calculating size and shape of the shadows in terms of astrometrical observables with respect to finite-distance observers. In this paper, we introduce a distortion parameter for the shadow shapes and discuss the appearance of the shadows of static spherical black holes and Kerr black holes in a uniform framework. We show that the shape of the shadow of a spherical black hole is circular in the view of arbitrary observers, and the size of the shadows tends to be shrunk in the view of a moving observer. The diameter of the shadows is contracted even in the direction perpendicular to the observers’ motion. This seems not to be understood as length contraction effect in special relativity. The shape of Kerr black holes is dependent on motional status of the observers located at finite distance. In spite of this, it is found that there is not a surrounding observer who could view the shape of the Kerr black hole shadows as circularity. These results could be helpful for observation of the Sagittarius A* in the center of the Milky Way, as our Solar System is moving around the center of the black hole.

Figure 1

Schematic diagram of black hole shadows in terms of astrometrical observables in the celestial sphere, where the angles between light rays from photon region are $\alpha \equiv \angle \mathrm{BOC}$, $\beta \equiv \angle \mathrm{AOC}$ and $\gamma \equiv \angle \mathrm{AOB}$, and the celestial coordinates of point $\mathrm{C}$ are $\mathrm{\Psi }\equiv \pi /2-\angle \mathrm{COD}$ and $\mathrm{\Phi }\equiv \angle \mathrm{BOD}$.

Figure 2

Angular diameters as a function of the observers’ radial coordinate $r$ in Schwarzschild space-time for selected observers.

Figure 3

Ratio $\sin ({\gamma }^{(A)}/2)/\sin ({\gamma }^{(B)}/2)$ as a function of observers’ radial coordinate $r$ in Schwarzschild space-time for observers A and B.

Figure 4

Left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for observers $u_{\mathrm{geo},-}$ at selected position $r$. Right panel: the corresponding shape of shadows in the left panel. Here, the spin parameter $a=0.999$.

Figure 5

Left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for observers $u_{\mathrm{geo},+}$ at selected position $r$. Right panel: the corresponding shape of shadows in the left panel. Here, the spin parameter $a=0.999$.

Figure 6

Left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for observers $u_{\mathrm{ZAMO}}$ at selected position $r$. Right panel: the corresponding shape of shadows in the left panel. Here, the spin parameter $a=0.999$.

Figure 7

Left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for observers $u_{\mathrm{Car}}$ at selected position $r$. Right panel: the corresponding shape of shadows in the left panel. Here, the spin parameter $a=0.999$.

Figure 8

Top-left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for selected observers at position $r=4\mathrm{M}$. Top-right panel: the corresponding shape of shadows in the top-left panel. Bottom-left panel: distortion parameter $\mathrm{\Xi }$ of the shadows as a function of $\frac{\mathrm{\Phi }}{\gamma }$ for selected observers at position $r=14\mathrm{M}$. Bottom-right panel: the corresponding shape of shadows in the bottom-left panel. Here, the spin parameter $a=0.999$.

Figure 9

Distortion parameter $\delta$ of the Kerr black hole shadows as a function of ZAMOs’ radial coordinate $r$ using Bardeen’s approach and the astrometrical observable approach. Here, we choose the spin parameter $a=0.9999$.

Figure 10

Distortion parameter $\delta$ of the Kerr black hole shadows as a function of ZAMOs’ radial coordinate $r$ using the approach by Grenzabach et al. and the astrometrical observable approach. Here, we choose the spin parameter $a=0.9999$.