Looking through the Kerr disk

TL;DR

This work investigates vortical null geodesics that thread Kerr’s maximal analytic extension, crossing both horizons and the ring-disk to connect $r>0$ sources with $r<0$ observers. By recasting constants of motion into impact parameters, the authors identify an inner throat region where the radial potential has no real roots, restricting observable paths to geodesics with no radial turning points. They derive and validate analytic solutions in Eddington-Finkelstein–like coordinates via elliptic integrals and confirm them with numerical integration, correcting prior formulae and enabling precise visualizations of how an observer in the negative-$r$ domain would view sources at $r>0$, including strong distortions and image inversions. The results also apply to Kerr white-hole analogues, offering distinctive observational signatures for interior-geodesic light propagation and clarifying the causal and optical structure of the Kerr interior.

Abstract

We study null geodesics that connect the two asymptotically flat regions of the maximally extended Kerr spacetime. These vortical geodesics traverse both horizons and pass through the ring singularity, linking the positive-$r$ exterior to the negative-$r$ asymptotic side. Using impact parameters, we identify a closed subset of parameter space, the inner throat, where the radial potential has no real roots, and photons exhibit no radial turning points. In this region, at most two constant-latitude geodesics exist, one of which is aligned with the principal null direction. We also identify the forbidden polar-angle band that limits the range of geodesics reaching an asymptotic observer. We solve the geodesic equations analytically and numerically in Eddington-Finkelstein-like coordinates, obtaining mutually consistent results that correct and extend previously available formulae. The resulting trajectories are used to construct simulated views for an observer in the negative-$r$ domain, revealing strong image distortion and inversion, with possible implications for analogous white-hole configurations.

Figures (14)

• Figure 1: The inner throat for different polar angles of an observer $\theta_o$ and rotational parameter $a/m=0.99$. The case $\theta_o=0 \, (\pi/2)$ corresponds to an observer located on the axis of symmetry (equatorial plane). For $\theta_o=\pi/2$, the inner throat vanishes.
• Figure 2: Plot of \ref{['eq:a_theta_pm_left_minus']} for $0\leq \theta_o<\pi/2$. The shaded area indicates combinations of $a/m$ and $\theta_o$ where two distinct geodesics with constant $\theta$ lie inside the inner throat. On the boundary curve, the left vertex of the inner throat exhibits a constant polar angle, corresponding to a geodesic with a constant polar angle captured in an orbit around the black hole. The vertical dotted line corresponds to $\theta_o=\pi/6$.
• Figure 3: The angles $\theta_\pm$ for the right vertex ($\alpha=\alpha_{\textrm{max}}$, $\beta=0$) of the inner throat for varying polar angle of the observer $\theta_o$. The rotation parameter is set to $a/m=0.99$.
• Figure 4: An example radial trajectory of a vortical null geodesic with impact parameters $\alpha=-0.15$ and $\beta=0.1$ within the inner throat for $\theta_o=\pi/4$. The black hole rotation parameter is set to $a/m=0.99$. Because we measure distances in terms of the black hole's mass $m$, we are plotting $r/m$.
• Figure 5: Two example $\theta$-trajectories calculated with \ref{['eq:theta_sol_uncorrected']} with $r_s=+\infty$, $r_o=-\infty$, $\theta_o=\pi/4$, and $a/m=0.99$. (a): The solution is smooth in the general case $\alpha\neq0$. The vertical line indicates one full period of $\theta$. (b): For $\alpha=0$, the trajectory is not smooth at $\theta_-=0$ (black, dashed). As discussed in the main text, we obtain a smooth solution by extending it to negative $\theta$ (blue, solid). The hemisphere defined by $\mathrm{sign}(\cos\theta)$ is unaffected by this. The vertical line denotes one half-period of the polar angle.