Optical caustics of Kerr spacetime: The full structure

I present an exhaustive numerical investigation of the optical caustics in gravitational lensing by a spinning black hole for an observer at infinity. Besides the primary caustic, I examine higher order caustics, formed by photons performing one or several loops around the black hole. My investigation covers the whole parameter space, including the black hole spin, its inclination with respect to the line of sight, the source distance, and the caustic order. By comparing my results with the available analytical approximations, I find perfect agreement in their respective domains of validity. I then prove that all caustics maintain their shape (a tube with astroidal cross section) in the entire parameter space without suffering any transitions to different caustic shapes. For nearly extremal spin, however, higher order caustics grow so large that their cross sections at fixed radii wind several times around the black hole. As a consequence, for each caustic order, the number of images ranges from 2 to $2(n+1)$, where $n$ is the number of loops spanned by the caustic. As for the critical curves, I note that for high values of the spin they develop a small dip on the side corresponding to prograde orbits.

Figure 1

The shadow border as seen by an equatorial observer for different values of the black hole spin. ${\theta }_1$ and ${\theta }_2$ are the angular coordinates in the observer’s sky in units of $2M/r_o$. The black hole is in the origin and the spin axis points in the positive ${\theta }_2$ direction.

Figure 2

Cross sections of the primary caustic for $a=0.9998M$ and ${\mu }_o=0$. From left to right, the radial coordinate is $r_s=5M$, $4.5M$, $4M$, $3M$, $2.3M$, $1.82M$, $1.42M$, $1.22M$.

Figure 3

The primary caustic tube for $a=0.9998M$ and ${\mu }_o=0$. In building this 3-dimensional representation, I have chosen coordinates such that $x=r\sin \vartheta \cos \varphi$, $y=r\sin \vartheta \sin \varphi$, $z=r\cos \vartheta$, following Ref. 15. The spin axis is directed toward the top. The straight solid line indicates the direction towards the observer, whereas the dashed line points in the opposite direction. The primary caustic for a Schwarzschild black hole ($a=0$) coincides with this dashed line.

Figure 4

A schematic picture of the images of a point source at $r_s=12M$ generated by a black hole with spin $a=0.2M$ as seen by an equatorial observer. (a) The primary caustic cross section and four possible positions of the source, indicated by the cross, the filled circle, the empty circle, and the box. (b) The corresponding images in the observer’s sky together with the critical curve (solid line) and the shadow of the black hole (dashed line).

Figure 5

(a) Azimuthal coordinate of the retrograde cusp of the primary caustic as a function of the radial coordinate. (b) Size of the primary caustic as a function of the radial coordinate. In both figures, from bottom to top, the curves are drawn for $a=0.2M$, $0.4M$, $0.6M$, $0.8M$, $0.9998M$.

Figure 6

Comparison between numerical calculations (solid lines) and the analytical approximation by Sereno and De Luca [16] (dashed lines). (a) Position of the retrograde cusp as a function of the radial coordinate. (b) Size of the caustic as a function of the radial coordinate. From bottom to top, the curves are for $a=0.2M$, $0.4M$, $0.6M$, $0.8M$, $0.9998M$.

Figure 7

Second order (a) and third order asymptotic caustics (b) for an equatorial observer ${\mu }_o=0$ and different values of the spin. From left to right, $a=0.02M$, $0.2M$, $0.4M$, $0.6M$, $0.8M$, $0.9998M$.

Figure 8

Second order (a) and third order asymptotic caustics (b) generated by a Kerr black hole with spin $a=0.6M$ for different values of the inclination ${\mu }_o=\cos {\vartheta }_o$. From the center to the periphery, ${\mu }_o=0$, 0.3, 0.6, 0.9, 0.99.

Figure 9

Azimuthal position of the prograde and retrograde cusps of the asymptotic caustic as a function of the black hole spin. (a) Second order caustic; (b) third order caustic. The solid lines are from numerical results in this paper, the dotted lines are the SDL approximation of Ref. 26, the dashed lines are the perturbative results of Ref. 19.

Figure 10

The second order caustic surface for $a=0.2M$ and ${\mu }_o=0$. The straight line indicates the direction towards the observer.

Figure 11

A representation of the caustics on the equatorial plane in which the radial coordinate has been replaced by $\log (r_s-r_h)/2M+11$. The outer circle represents the surface at $\log (r_s-r_h)/2M=4$. Inner circles are in steps of 1 in such a logarithmic coordinate. The observer is in the direction labeled by O on the equatorial plane. The number at the end of each caustic represents the respective caustic order. Here the spin is $a=0.2M$ with the black hole rotating counterclockwise.

Figure 12

Position of the retrograde cusp for the second order (a) and the third order (b) caustics, as a function of the radial coordinate. In both figures, from bottom to top, the curves are for $a=0.2M$, $0.4M$, $0.6M$, $0.8M$, $0.9998M$.

Figure 13

Size of the second order (a) and the third order (b) caustics, as a function of the radial coordinate. In both figures, from bottom to top, the curves are for $a=0.2M$, $0.4M$, $0.6M$, $0.8M$.

Figure 14

Relative error of the perturbative approximation given by Eqs. (13, 14) as a function of the radial coordinate. (a) Error in the shift of the caustic. (b) Error in the size of the caustic. Curves are for caustic orders $m=2$, 3, 4 from top to bottom. $a=0.02M$, ${\mu }_o=0$.

Figure 15

Asymptotic caustics of an extremal Kerr black hole for an equatorial observer. From left to right the caustic order is $m=2$, 3, 4, 5.

Figure 16

Third order asymptotic caustic of an extremal Kerr black hole for an equatorial observer in the $\varphi \mathrm{\text{−}}\mu$ plane. In this plot, I have taken into account the fact that the azimuthal coordinate is periodic with period $2\pi$.

Figure 17

Second order (a) and third order (b) asymptotic caustics of an extremal Kerr black hole for different positions of the observer. From the center to the periphery the caustics are drawn for ${\mu }_o=0$, 0.3, 0.6, 0.9, 0.99.

Figure 18

Three-dimensional pictures of the second order caustic surface of an extremal Kerr black hole for an observer on the equatorial plane. The surface has been plotted for radial distances in the range $[2.2M,20M]$. (a) View from the north pole. (b) View from the observer side, lifted by 10° above the equator. (c) View from the left side. The straight line in (a) and (c) points towards the observer.

Figure 19

Azimuthal position of the retrograde cusp (a) and prograde cusp (b) as a function of the logarithm of the distance from the event horizon. From bottom to top, the curves are for caustic order $m=2$, 3, 4, 5.

Figure 20

Critical curves for a Kerr black hole with spin $a=0.9998M$ and an equatorial observer. The coordinates $({\theta }_1,{\theta }_2)$ in the observer’s sky are in units of $2M/r_o$. (a) Primary critical curve for various source distances: starting from the outside, $r_s=5M$, $3M$, $1.8M$, $1.1M$. (b) Second order critical curve for various source distances: starting from the outside, $r_s=\infty$, $4.2M$, $1.6M$, $1.1M$. (c) Critical curves of order $m=2$, 3, 4 for a source at infinity. (d) Critical curves of order $m=1$, 2, 3 for a source at $r_s=1.1M$. The shadow border is represented by the dashed curve.

Figure 21

Relation between $\psi$ and $ϵ$ for $a=0.2M$, ${\mu }_o=0$, $\eta =\pi /4$. From left to right, the curves are drawn for $r_s=2M$, $3M$, $4M$, $5M$, $6M$, respectively. Note that for these values $r_m=3.16M$.