Multiple shadows from distorted static black holes

We study the local shadow of the Schwarzschild black hole with a quadrupole distortion and the influence of the external gravitational field on the photon dynamics. The external matter sources modify the light ring structure and lead to the appearance of multiple shadow images. In the case of negative quadrupole moments we identify the most prominent mechanism causing multiple shadow formation. Furthermore, we obtain a condition under which this mechanism can be realized. This condition depends on the quadrupole moment, but also on the position of the observer and the celestial sphere.

Figure 1

Illustration of the spherical observer sky for a given spacetime (with partial cutout). The observer is located at the center of this sphere.

Figure 2

Illustration of the equirectangular projection for lens and shadow maps for the observer sky shown in Fig. 1.

Figure 3

The effective potential $h^2(x,0)$ in the equatorial plane for different values of the quadrupole moment: (a) For $a_2<a_2^{\mathrm{crit}}\approx -0.021$ no stationary points exist, and consequently no light rings. At $a_2^{\mathrm{crit}}$ an inflection point appears, which corresponds to a marginally stable light ring. (b) For $a_2^{\mathrm{crit}}<a_2<0$ a maximum and a minimum exist, representing a stable and an unstable light ring with respect to radial perturbations. (c) For $a_2\ge 0$ a minimum exists corresponding to an unstable light ring.

Figure 4

Existence of circular null geodesics. Circular orbits are located on the red curves defined by (20) and (25), respectively, for equatorial/off-equatorial light rings. The blue curve determines the regions of stability of the light rings with respect to radial and vertical perturbations (see the Appendix). Light rings below the blue curve are stable with respect to radial perturbations, and unstable with respect to vertical ones, while in the region above the opposite is true. The two curves intersect at the location of the ISCO.

Figure 5

Contour plots of the potential $h(x,y)$ for negative quadrupole moments. In the range $a_2\in (-\infty ,-0.021)$ two saddle points are observed, located symmetrically with respect to the equatorial plane [(a) and (b)]. Decreasing the quadrupole moment, the light rings move away from the equatorial plane, and approach the horizon. In the range $a_2\in (-0.021,0)$ the saddle points lie in the equatorial plane [(c) and (b)]. Decreasing $a_2$, the outer light ring moves closer to the horizon, and approaches the inner one.

Figure 6

Contour plots of the potential $h(x,y)$ for positive quadrupole moments. A single saddle point is observed lying in the equatorial plane. Increasing the quadrupole moment $a_2$, the light ring approaches the horizon.

Figure 7

Effective potentials for quadrupole moment $a_2=-0.015>a_2^{\mathrm{crit}}$ and various values of impact parameter $\eta$ illustrating the effect of increasing $|\eta |$.

Figure 8

Effective potentials for quadrupole moment $a_2=-0.03<a_2^{\mathrm{crit}}$ and various values of impact parameter $\eta$ illustrating the effect of increasing $|\eta |$.

Figure 9

The effective potential at impact parameter ${\eta }_o$ for an illustrative choice of observer (dot) and celestial sphere (vertical line). At this value of the impact parameter the observer is on the boundary of a forbidden region.

Figure 10

Qualitative features of the potential allowing multiple shadows (left), and not allowing multiple shadows (right). The observer position is marked by the dot, and the celestial sphere by the vertical line.

Figure 11

Lensing image for quadrupole moment $a_2=-0.015$, an equatorial observer at $x_o=4.5$ and a celestial sphere at $x_{cs}=14.4$. Lines of constant impact parameter $\eta$ are shown for $\pm {\eta }_1\approx \pm 4.784$ (small circles) and $\pm {\eta }_{cs}\approx \pm 0.725$ (large circles).

Figure 12

Shadow maps for quadrupole moment $a_2=-0.015$, an equatorial observer at $x_o=4.5$, and a celestial sphere at $x_{cs}=14.4$. Lines of constant impact parameter $\eta$ are shown for $\pm {\eta }_1\approx \pm 4.784$ (small circles) and $\pm {\eta }_{cs}\approx \pm 0.725$ (large circles).

Figure 13

Photon trajectories for quadrupole moment $a_2=-0.015$, an equatorial observer (dot) at $x_o=4.5$ and a celestial sphere (vertical line) at $x_{cs}=14.4$ for different initial conditions, points 1–4, in Fig. 12. The position of the inner light ring is indicated by a red triangle, that of the outer light ring by a green triangle (behind the observer).

Figure 14

Photon trajectory for quadrupole moment $a_2=-0.015$, an equatorial observer (dot) at $x_o=4.5$ and a celestial sphere (vertical line) at $x_{cs}=14.4$ for a point in the shadow just left of the impact parameter ${\eta }_{cs}$ boundary in Fig. 12 12. The inset shows a zoomed image demonstrating an interaction with the external region without actually bouncing off of it.