Shadow Geometry of Kerr Naked Singularities

In our study, we use geometric units $c=G=M=1$. The Kerr metric is an axially symmetric, stationary vacuum solution to the Einstein field equations, which describes uncharged rotating compact objects, whose mathematical formulation is derived in Kerr (1963). The line element of the Kerr metric in Boyer-Lindquist coordinates ($t,r,\theta,\phi$) is (Boyer & Lindquist, 1967):

where

Here, $a=J/M^{2}$ is a dimensionless spin for a Kerr compact object with mass $M$ and angular momentum $J$. For $|a|<1$, the metric describes a KBH; and for $|a|>1$, the metric describes a KNS. We further consider only $a>0$ without loss of generality.

Because the Kerr metric is stationary and axially symmetric, there are two Killing vectors $K^{\mu}=(1,0,0,0)$ and $R^{\mu}=(0,0,0,1)$ representing the time and azimuthal translation. Conserved quantities can be derived from the symmetries of the physical laws. By projecting the Killing vectors along the covariant four-momentum vector $p_{\mu}$, we obtain energy $E$ and angular momentum in the $\phi$ direction $L_{z}$:

To analyze the spherical photon orbits and the shadow of KNSs, we employ the Hamilton-Jacobi equation:

Here, $S$ is the action as a function of the affine parameter $\lambda$ and coordinates $x^{\mu}$. The solution to equation (4) can be separated into different components that only depend on each of the Boyer-Lindquist coordinates (Carter, 1968):

Equations (4) and (5) yield the following equations of motion for null geodesics for each of the coordinates (Sharp, 1979):

where the overdots represent derivatives with respect to the affine parameter $\lambda$ along the geodesics. The radial equation of motion is written in terms of the radial effective potential $R(r)$, which is of major interest in this study (Stewart & Walker, 1973):

Here, we define impact parameters $\Phi=L_{z}/E$ and $Q=C/E^{2}$, where $C$ is the Carter’s constant, a third conserved quantity discovered from the separability of the Hamilton-Jacobi equation (Carter, 1968). $C$ is relevant to geodesics in the latitudinal direction. We solve $R(r)=dR/dr=0$ for spherical photon orbits with constant radius $r=r_{\mathrm{p}}$ and obtain two sets of solutions, but only one set of solution is physical and thus relevant to our study (Teo, 2003; Charbulák & Stuchlík, 2018):

These parameters are related to the image plane at infinity with orthogonal coordinates $\alpha$ and $\beta$, which observes the KNS at a polar inclination angle $i$ (Bardeen et al., 1972):

Unstable photon orbits occur when $d^{2}R/dr^{2}<0$. On the equatorial plane of KBHs, the Carter’s constant vanishes and there are two unstable solutions outside the event horizon (Bardeen et al., 1972):

where the inner solution $r_{\mathrm{ph-}}$ and outer solution $r_{\mathrm{ph+}}$ correspond to the prograde and retrograde equatorial orbit, respectively. For non-equatorial orbits, unstable photon orbits with radii $r_{\mathrm{p}}$ exist in the range $r_{\mathrm{ph-}}<r_{\mathrm{p}}<r_{\mathrm{ph+}}$, which distinguish orbits that get captured by the event horizon and those that can return back to infinity. For KNSs, the prograde equatorial orbit does not exist. The retrograde equatorial orbit separates between orbits that terminate at the singularity and those that recede to infinity. Off the equatorial plane, all orbits that approach the KNS can escape. Unstable photon orbits surrounding KNSs exist in the range $r_{\mathrm{ms}}<r_{\mathrm{p}}<r_{\mathrm{ph}}$, where $r_{\mathrm{ms}}$ is the marginally stable radius which satisfies $d^{2}R/dr^{2}=0$ and $r_{\mathrm{ph}}$ is the equatorial retrograde circular radius (Charbulák & Stuchlík, 2018):

While stable photon orbits with $r<r_{\mathrm{ms}}$ are normally hidden inside the event horizon for KBHs, they have physical significance for KNSs due to the lack of an event horizon. Nevertheless, bounded photon orbits cannot be seen by distant observers, so they are irrelevant to our study on the observational features of KNSs.

The unstable photon orbits with radii $r_{\mathrm{ms}}<r_{\mathrm{p}}<r_{\mathrm{ph}}$ can be projected to the image plane at infinity using equations (8) and (9). We define this projection to be the shadow of KNSs. There is a one-to-one correspondence between $r_{\mathrm{p}}$ and the ($\alpha,\beta$) coordinates on the image plane. The shadow of a KBH always has a closed geometry, but the shadow of a KNS might have a gap due to the non-existence of the prograde equatorial circular orbit. The shadow is symmetric with respect to the $\alpha$-axis on the image plane. Thus, we can determine the geometry of the shadow for different values of $a$ and $i$ by solving for $\beta(r_{\mathrm{p}})=0$ and count the number of roots in terms of $r_{\mathrm{p}}$, where $\beta$ is expressed as a function of $r_{\mathrm{p}}$ by substituting equations (8a) and (8b) into equation 9b, given fixed values of $a$ and $i$. If there are two roots, the shadow is closed. If there is one root, the shadow is open with a gap. If there is no root, the shadow vanishes. Since $\beta(r_{\mathrm{p}})=0$ is a sextic polynomial, it does not have an analytical closed-form solution, so we solve this equation numerically and obtain figure 1.

In figure 1, we translate the root(s) of $\beta(r_{\mathrm{p}})=0$ downward by $r_{\mathrm{ms}}$ to illustrate the range of relevant unstable photon orbit radii above the marginally stable radius for different values of $a$ and $i$. As $a$ increases, both $r_{\mathrm{p}}$ and $r_{\mathrm{ms}}$ increase; because $r_{\mathrm{p}}$ increases at a slower pace, the $r_{\mathrm{p}}-r_{\mathrm{ms}}$ curve shifts downward, so it appears to shift rightward as shown in figure 1. This shifting pattern of $r_{\mathrm{p}}-r_{\mathrm{ms}}$ results in the topological change of the KNS shadow. For $a\lesssim 1.18$, there are unstable roots for a range of $i$ from negative values with small magnitude to $90^{\circ}$, which guarantees the existence of an unstable root. Using the axial symmetry of the Kerr metric, the negative roots can be reflected over the $i=0^{\circ}$ axis to represent second roots for small $i$ (denoted as dotted lines in figure 1). This means that for $a\lesssim 1.18$, a shadow can be closed for small $i$ or open for larger $i$. For $a\approx 1.18$, the $r_{\mathrm{p}}-r_{\mathrm{ms}}$ curve shifts rightward such that the unstable root of $\beta(r_{\mathrm{p}})=0$ only occurs for non-negative $i$, which means that there is only one root, representing an open shadow with a gap. When $a\gtrsim 1.18$, the $r_{\mathrm{p}}-r_{\mathrm{ms}}$ curve continues to shift rightward, so there is no unstable root for small $i$, which corresponds to the vanishing of the shadow. As $a$ increases, the minimum $i$ where one unstable root occurs shifts rightward, so the minimum $i$ for the shadow to exist increases. $a\approx 1.18$ is an important critical parameter as it marks the transition between two, one, or zero unstable roots.

The physical implication of the above analysis is summarized in figure 2, which shows the parameters of $a$ and $i$ where the shadow is closed (region A), open (region B), or vanishing (region C), corresponding to two, one, or zero roots of $\beta(r_{\mathrm{p}})=0$, respectively. As $a$ decreases towards $1$ (maximally spinning KBH), the maximum $i$ for a closed shadow asymptotically approaches $90^{\circ}$. This is consistent with the fact that KBHs have closed shadows. For $1<a\lesssim 1.18$, the shadow can be closed for smaller $i$. As $a$ increases towards $1.18$, the maximum $i$ for a closed shadow decreases to zero. For $a\gtrsim 1.18$, the shadow is no longer closed due to the significant frame dragging (Lense-Thirring) effect for larger spins, and the shadow vanishes for small $i$. As $a$ increases to infinity, the minimum $i$ for the shadow to exist increases. The location ($a\approx 1.18,\,i=0^{\circ}$) marks the triple point among region A, B, and C on the $a$–$i$ phase space where the topological change in the KNS shadow occurs. Figures 3 and 4 depict the analytical KNS shadow by projecting the unstable photon orbits to an image plane at infinity with orthogonal coordinates $\alpha$ and $\beta$ for different values of $a$ and $i$, according to equations (8a) to (9b). These figures are consistent with the analytical results in figure 2.

The effective angular momentum of unstable photon orbits $\Phi=L_{z}/E$, as defined by equation 8a in section 2.1 can provide further physical insights into the gap opening and vanishing behaviors at different spins and inclinations. For a $<$ 1 (KBH), d$\Phi$/d$r_{p}$ $<$ 0 outside the event horizon, so the maximum $\Phi$ for unstable photon orbits $\Phi_{max}$ occur at $r_{p}=r_{ph-}$, the inner equatorial orbit. For a $\geq$ 1 (maximal KBH and KNS), by setting d$\Phi$/d$r_{p}$ = 0 and d${}^{2}\Phi$/d$r_{p}^{2}$ $<$ 0, the $\Phi_{max}$ occur at $r_{p}=r_{ms}$, the marginally stable orbit. Thus, by substituting $r_{p}=r_{ph-}$ for $a<1$ and $r_{p}=r_{ms}$ for $a\geq 1$ into equation 8a, we can find a function of $\Phi_{max}$ as a function of spin $a$, as plotted in figure 5.

As the spin increases, the frame dragging effects become stronger so photons that have too much angular momentum would scatter instead of sustaining a spherical orbit around the KBH or KNS. Hence, $\Phi_{max}$ is a decreasing function of spin $a$, which can also be demonstrated mathematically by showing d$\Phi_{max}$/da $<$ 0 in addition to the physical argument above. It is physically significant to examine when $\Phi_{max}$ decreases to 0 where all prograde photon orbits no longer occur. By substituting $r_{p}=r_{ms}$ into equation 8a and setting $\Phi=0$:

This results in the critical spin $a_{crit}=\sqrt{6\sqrt{3}-9}\approx 1.18$, meaning that prograde photon orbits vanish for $a>a_{crit}$. Here, we reproduce the same critical spin where the polar photon orbits disappear as shown in Charbulák & Stuchlík (2018) using a different mathematical approach, and we will take a step further to draw physical connections between photon orbits and KNS shadow geometry. The unstable photon orbits projected to the left and right part of the shadows on the image plane (according to our direction convention) are prograde ($\Phi>0$) and retrograde ($\Phi<0$), respectively. The physical picture is that the left side of the shadow consists of prograde photons that travel along the rotational orientation of the KNS, and vice versa. Because $\alpha$ and $\beta$, the orthogonal coordinates of the image plane as defined in equation 9a and 9b in section 2.1, are continuous functions of $\Phi$, as $\Phi$ changes from positive to negative values, it smoothly traces out the shadow from left to right. When $a>a_{crit}$, the KNS spins too rapidly such that the prograde photons can no longer sustain spherical orbits, so the left side of the shadow vanishes, and the shadow opens up a gap.

Besides the critical spin, the effective angular momentum of photon orbits can also physically explain the different trends we find in our spin-inclination phase space (figure 2). For a fixed spin, as the inclination increases from face-on to edge-on, the photons have greater (in magnitude) effective angular momentum because it gains motion in the $\phi$ direction. For a fixed spin $1<a<a_{crit}$, as the inclination increases, the prograde photon trajectories become more prograde. At some higher inclination angle, their effective angular momentum exceeds $\Phi_{max}$ so they can no longer sustain spherical orbits around the KNS. Thus, the shadow opens up its gap at higher inclinations. For a fixed spin $a>a_{crit}$, as $\Phi_{max}<0$, retrograde photon orbits at lower inclination might exceed the $\Phi_{max}$, so the shadow vanishes. As the inclination increases, the retrograde photon orbits become more retrograde, so they can resist the frame dragging effects and fall below $\Phi_{max}$, allowing them to orbit the KNS and the shadow to re-emerge again. The connections between the effective angular momentum of photons and the projected shadows are visualized in figures 6 and 7 for different spins and inclinations.

To confirm our analytical study and obtain better physical insights, we numerically integrate null geodesics in Kerr spacetimes and study their deflection angles. We place a KNS centered at the origin. The KNS has mass $M$ and dimensionless spin $a$. We set up an image plane located at a distance 10,000$\,M$ away from the KNS at a polar inclination angle $i$ with respect to the $z$-axis, defined to be perpendicular to the ring singularity. The center of the image plane is at the intersection of this plane with the radial vector originating from the singularity. The image plane is effectively at infinity with respect to the KNS, so its orthogonal coordinate $\alpha$ and $\beta$ are related to conserved quantities of the photons—energy $E$, angular momentum $L_{z}$, and Carter’s constant $C$—by equation (9). We arrange the photons in a square grid of $128\times 160$ light rays in the domain $(\alpha,\beta)\in(-8\,M,8\,M)\times(-8\,M,10\,M)$ with a spacing of $0.125\,M$. The setup of the image plane is visualized in figure 8.

We initialize the momentum vectors $\mathbf{k}$ of the photons to be perpendicular to the image plane and satisfy the condition for null geodesics, $k^{\mu}k_{\mu}=0$ (Chan et al., 2013). We integrate null geodesics backwards in affine parameter starting on the image plane in Cartesian Kerr-Schild coordinates ($t_{\mathrm{KS}}$, $x$, $y$, $z$), whose Kerr metric is written as (Kerr, 1963):

where $\eta=\mathrm{diag}(-1,1,1,1)$ is the Minkowski metric representing flat spacetime,

and the radial component $r$ of the Boyer-Lindquist coordinates can be implicitly defined in Cartesian Kerr-Schild coordinates:

We employ the differential geometry software package Fadge ¹¹1https://github.com/adxsrc/fadge and the ordinary differential equations solver XAJ ²²2https://github.com/adxsrc/xaj. Building on top of Google’s GPU-accelerated, composible, and automatic differentiation package JAX (Bradbury et al., 2018), fadge automatically derives the geodesic equations from arbitrary metric according to Chan et al. (2018)’s formulation. XAJ implements the Runge-Kutta Dormand-Prince 4(5) method with an adaptive stepsize control and interpolated dense output (Press et al., 2002). Along the photon trajectory, XAJ controls the numerical error by monitoring $k^{\mu}k_{\mu}$ and ensures the photons remain essentially massless for accurate ray tracing calculations.

Although it is expected that quantum gravity effects emerge as photons travel very close to a gravitational singularity, our numerical study is purely general relativistic. To avoid any unphysical result that arises from the mathematical limitations of GR and demands correction from a theory of quantum gravity, we terminate the integration if a photon gets too close to the singularity and the numerical instability becomes significant. Because we are interested in the behaviors of unstable spherical photon orbits sufficiently distant from the singularity, this do not affect our analysis of the shadow of KNSs. Also, we do not integrate the radiative transfer equation nor model the emitting plasma around KNSs in this study. We focus on how null geodesics affect the observational features for gravitationally lensed images of KNSs.

For different spins $a$ and observational inclination angles $i$, we compute the deflection angle of every photon on the observer’s grid. The deflection angle of a light ray measures the angular difference between the incoming photon that travels towards the KNS and the outgoing photon that travels away from the KNS. For an initial momentum vector $\mathbf{k}_{i}$ and a final momentum vector $\mathbf{k}_{f}$, the deflection angle $\theta_{d}$ is defined by:

From the deflection angle calculations, we compare the numerical ray tracing results with the analytical predictions of the KNS shadow as presented in section 2.

From the numerical ray tracing calculations, we visualize deflection angles of each light ray on the image plane grid for different spins $a$ and observational inclination angles $i$. Figure 9 compares between the analytical shadow computed by equations (8a)–(9b) in section 2.1 and the numerical shadow as an illustration of how to recognize the shadow from the deflection angle visualizations. Given that the outgoing direction of a photon near an unstable spherical photon orbit is very sensitive to the impact parameter, its deflection angle changes rapidly as function of the impact parameter. As the shadow is the projection of unstable spherical photon orbits, it corresponds to the region where light rapidly oscillates between getting reflected back to the image plane (high deflection angle) and traveling to the other side of the KNS (low deflection angle). Figure 10 shows the deflection angle plots on the full $a$–$i$ phase space, which reproduces many analytical characteristics of the KNS shadow in section 2.2.

Figure 10 demonstrates that for $1<a\lesssim 1.18$, the shadow is closed for lower $i$. For the critical parameter $a\approx 1.18$, the shadow opens its gap for all non-zero inclination angles due to the significant frame dragging effect. For $a\gtrsim 1.18$, the shadow vanishes for lower $i$. As the spin increases from $a\approx 1.18$, the minimum $i$ for the shadow to exist increases and the shadow’s arc shrinks because frame dragging becomes stronger such that more photons get scattered off instead of orbiting the KNS and returning to the image plane. The topological transition between open, closed, and vanishing of the shadow is consistent with the analytical result shown in figure 2. Especially, the triple point ($a\approx 1.18,\,i=0^{\circ}$) is a notable prediction from the analytical calculations in section 2.2 reproduced by our numerical ray-tracing calculations. Generally, as $a$ and $i$ increase, the shadow shifts rightwards similarly to KBHs with $a<1$, except for lower $i$ where the shadow vanishes faster than its rightwards shifting and appears to shift leftwards.

Besides the topological change in the shadow, the absence of an event horizon also results in other distinctive observational features for KNSs. For KBHs, the shadow refers to the apparent boundary that separates between photon orbits that are captured by the event horizon and those that can escape to infinity (Falcke et al., 2000). However, because all photons orbits around KNSs can return to infinity apart for a subset of retrograde equatorial orbits that terminate at the singularity, within the KNS shadow, there are interior structures that behave like mirrors (high deflection angle) where light is deflected back to the image plane and lens (low deflection angle) where the region appears transparent for a distant observer. As $a$ and $i$ increases, these mirror-like and lens-like structure appears to shift continuously. A more thorough study on the topological features of these mirror and lens effect can further provide constraints on observations of SMBHs.