Twisting shadows: Light rings, lensing, and shadows of black holes in swirling universes

Using the Ernst formalism, a novel solution of vacuum general relativity was recently obtained [Phys. Rev. D 106, 064014 (2022)], describing a Schwarzschild black hole (BH) immersed in a nonasymptotically flat rotating background, dubbed swirling universe, with the peculiar property that north and south hemispheres spin in opposite directions. We investigate the null geodesic flow and, in particular, the existence of light rings in this vacuum geometry. By evaluating the total topological charge $w$, we show that there exists one unstable light ring ($w=-1$) for each rotation sense of the background. We observe that the swirling background drives the Schwarzschild BH light rings outside the equatorial plane, displaying counterrotating motion with respect to each other, while (both) corotating with respect to the swirling universe. Using backward ray-tracing, we obtain the shadow and gravitational lensing effects, revealing a novel feature for observers on the equatorial plane: the BH shadow displays an odd ${\mathbb{Z}}_2$ (north-south) symmetry, inherited from the same type of symmetry of the spacetime itself: a twisted shadow.

Figure 1

Schematic representation of the metrics generated from Schwarzschild spacetime using conjugation and Kinnersley transformations.

Figure 2

Kretschmann scalar of the SU (gray) and SBHSU (black) spacetimes, for $\theta =\pi /2$ and $j=0.05$, 0.075, 0.1. For the BH cases we have set $M=0.1$ and $\theta =\pi /2$.

Figure 3

Representation of the surface $g_{tt}(r,\theta )=0$ in the section $y=0$ of the 3-dimensional Euclidean space, for $j{M}^2=0.01$, 0.05, 0.1. As the parameter of the swirling background increases, the ergosurface gets closer to the horizon surface.

Figure 4

Isometric embedding of the horizon into the 3-dimensional Euclidean space, for $j{M}^2=0,1/4,1/2$. As the parameter of the swirling background increases, the horizon gets more prolate and thinner along the equator.

Figure 5

3D plot representations showing the potentials $H_{+}$ (left panel) and $H_{-}$ (right panel), as a function of the $(r,\theta )$ coordinates for $M^{2}j=0.05$. The critical points for both $H_{\pm }$ are highlighted in red, while the Schwarzschild LR locations are denoted by black points.

Figure 6

LR position on the $(r,\theta )$-plane for various values of the rotation parameter $j$. We considered a sequence of values of $j$ defined by $j=0.025\times n$ for $n\in \mathbb{N}$. The sequence starts at $j=0$, for which the LR is located at $(3M,\pi /2)$, and for $j\to \infty$ the LR positions go to $(2M,0)$ and $(2M,\pi )$ in the used coordinate system, which, albeit not geometrically invariant, clearly shows the off-equatorial displacement of the LRs with increasing $j$.

Figure 7

Vector field ${\mathbf{v}}_{+}=(v_{\rho }^{+},v_z^{+})$ in the $(\rho /M,z/M)$-plane for the Schwarzschild BH (top), Schwarzschild-Melvin BH with $BM=0.1$ (middle) and SBHSU with $j{M}^2=0.025$ (bottom). The red points represent the location of the LRs of the corresponding spacetime.

Figure 8

Schematic representation of the contour $C$, enclosing a compact region outside the horizon in the $(\rho ,z)$-plane. The region inside the curve $C$ is designed to encompass all of the spacetime outside the horizon as $\xi \to \infty$.

Figure 9

Plot illustrating the vector field ${\mathbf{v}}_{+}=(v_{\rho }^{+},v_z^{+})$ in the $(\rho /M,z/M)$-plane for the SU with $j=0.025$. Since Eq. (85) does not depend on the coordinate $z$, the pattern of the vector field displayed in the plot repeats itself infinitely upward and downward.

Figure 10

Plot illustrating the backward ray tracing method for the SBHSU spacetime. We evolved, numerically, 50 null geodesics with initial conditions given by Eqs. (86)–(89), $\theta =\pi /2$ and $r=10M$. We have chosen the observation angles at random.

Figure 11

Shadow and gravitational lensing of the SBHSU, starting with the Schwarzschild case and increasing the swirling parameter $j$ in equal steps of $\delta j=0.0001$. In all the images, the observer is positioned at the equatorial plane ($\theta =\pi /2$) and at the radius $r=15M$.

Figure 12

Shadow and gravitational lensing of the SBHSU, for $j{M}^2=0.0005$, for several observer positions. We start with $\theta =0$ (on axis observer seeing the BH from the top) and we increase the angle $\theta$ in equal steps of $\delta \theta =\pi /8$, until we reach $\theta =\pi$ (on axis observer seeing the BH from the bottom). We kept the radius fixed at $r=15M$.

Figure 13

Shadow edge of the SBHSU for $j{M}^2=0.0005$, for observers positioned at $\theta =0$ (top left), $\theta =\pi /4$ (top right) and $\theta =\pi /2$ (bottom). We kept the radius fixed at $r=15M$. The points $P_1$, $Q_1$ and $R_1$ are randomly chosen points on the shadow’s edge. The points $P_2$, $Q_2$ and $R_2$ are obtained by reflecting $P_1$, $Q_1$ and $R_1$ with respect to the horizontal axis at the middle of each plot. The points $P_3$, $Q_3$, and $R_3$ are obtained by reflecting $P_2$, $Q_2$ and $R_2$ with respect to the vertical axis.

Figure 14

Deviation of geodesics from the equatorial plane. We evolved, numerically, 50 null geodesics with initial conditions given by Eqs. (86)–(89), setting $\alpha =0$, while $\beta$ varies randomly.