Completing the systematic study of gravitational lensing for a camera near a fast-spinning Kerr black hole: caustics, critical curves, and multiple-image counts

The problem studied here was posed by James, von Tunzelmann, Franklin and Thorne (2015) in 'Gravitational Lensing by Spinning Black Holes in Astrophysics, and in the Movie Interstellar'. That paper develops the DNGR (Double Negative Gravitational Renderer) ray-bundle code used for the film, and explicitly leaves open a full, systematic study of gravitational lensing in the reversed observational geometry: light sources on the distant celestial sphere and a camera near a fast-spinning Kerr black hole. It maps primary-to-tertiary caustics and critical curves only at representative camera positions, and its counting argument for the observed multiple-image pattern rests on two unproved plausibility claims: that crossing into the astroidal primary caustic creates (rather than annihilates) an image pair, and that original order-n images lie between the order-n and order-(n+1) critical curves. No later work is known that systematically maps the near-camera caustic structure or settles these counting claims; the problem remains open.

Background

A Kerr black hole is the family of stationary, axisymmetric vacuum solutions of Einstein's general relativity describing a spinning black hole; it is characterized by its mass $M$ and angular momentum, with dimensionless spin parameter $a/M$ ranging from 0 (the non-spinning, spherically symmetric Schwarzschild solution) to 1 (extremal, maximally spinning). Gravitational lensing by such a hole bends light rays (null geodesics, the paths of light in curved spacetime) so that a viewer sees distorted, multiply imaged versions of background light sources. Almost all theoretical work before 2015 analyzed one observational geometry: a source near the black hole (for example a hot spot in an accretion disk) and an observer far away, effectively on the 'celestial sphere' at radius $r=\infty$. In that geometry, Rauch and Blandford (1994) and later Bozza (2008) mapped the optical caustics — the surfaces where the lensing map becomes singular and image magnification diverges — including their dependence on spin and inclination; Bozza, De Luca and Scarpetta (2006) extended the strong-deflection-limit analysis (an analytical approximation for light rays that loop many times around the hole on orbits close to the photon orbit, the family of unstable circular orbits of light at a characteristic radius) to generic observer positions, but only to second order in the spin and only for such highly bent 'relativistic' images; and Hasse and Perlick (2005/2006) proved by Morse theory — a topological method relating the number of image-forming light rays between source and observer to the critical points of the light-travel-time function on the light cone — that a light source anywhere in the domain of outer communication (the region between the event horizon and spatial infinity from which light signals can escape) of a Kerr-Newman black hole (the charged, spinning generalization of Kerr) produces infinitely many images for any observation event, a generic statement that does not classify where the images lie on the observer's sky.

The movie Interstellar required the opposite geometry: light sources on the celestial sphere (a star field built from the Tycho-2 catalogue, a survey of the brightest stars seen from Earth, plus an accretion disk) and a simulated camera moving near the hole at a substantial fraction of the speed of light. To render IMAX-quality, flicker-free images, the Double Negative team, with Kip Thorne, built DNGR (Double Negative Gravitational Renderer), which propagates not only individual rays but ray bundles — clusters of neighboring light rays whose cross-sectional evolution, computed via the geodesic-deviation equation (the equation governing how neighboring light rays separate), gives the solid-angle mapping and magnification of the sky. Several concepts from that work are needed below. The camera's 'local sky' is the sphere of viewing directions around the camera at an event; the camera's past light cone intersects the celestial sphere in a map from sky directions to source locations. Caustics of the past light cone (generically two-dimensional fold surfaces in Kerr — sheets across each of which a pair of images is created or destroyed — intersecting the celestial sphere in closed one-dimensional caustic curves rather than points) map to 'critical curves' on the camera's sky, generalizations of the Einstein ring of a non-spinning lens. Images are classified by their order $n$, the number of poloidal (polar-angle) turning points of the connecting null geodesic. When a source crosses a caustic, a pair of images is created or annihilated on the corresponding critical curve.

For a camera on a circular equatorial geodesic orbit at Boyer-Lindquist radius $r_c = 2.60M$ (Boyer-Lindquist coordinates being the standard coordinates for the Kerr spacetime, generalizing the Schwarzschild radial coordinate) around a hole with $a/M = 0.999$ — a camera kept outside the ergosphere, the region near a spinning hole where no observer can remain static — the paper found that the primary caustic is a very small astroid (a four-cusped curve whose sides are fold caustics), while higher-order caustics wrap around the celestial sphere many times, because light rays can be temporarily trapped in prograde (co-rotating with the hole), nearly circular orbits near the photon orbit before escaping. The practical consequence is a striking multiple-image pattern near the flat, left edge of the shadow: three images of the galactic plane between the primary and secondary critical curves, and eight between the secondary and tertiary. The authors' explanation of this pattern is an extension of a counting argument of Bozza, and they state it is 'highly plausible but not fully rigorous', listing exactly what is unproved. The wormhole companion paper (James et al. 2015, American Journal of Physics) covers the separate wormhole visualizations and does not address this black-hole lensing structure.

Problem Statement

Complete the systematic study of gravitational lensing for the observational geometry initiated by James, von Tunzelmann, Franklin and Thorne (2015): light sources on the celestial sphere at $r=\infty$ and a camera near a fast-spinning Kerr black hole, with the camera outside the ergosphere and in arbitrary motion; for this study the required motion states are the static, prograde circular-geodesic, and zero-angular-momentum (locally non-rotating, FIDO) cases, the last with angular velocity $\mathrm{d}\phi/\mathrm{d}t = \omega$ (the frame-dragging rate), each handled explicitly; other motion states lie outside the required scope. Concretely, for the Kerr spacetime with spin $0 \le a/M \le 0.999$ (the paper's representative case is $a/M=0.999$):

1. Map, systematically rather than at representative points, the caustic curves that the primary and higher-order caustics of the camera's past light cone trace on the celestial sphere, and the corresponding critical curves on the camera's local sky, as functions of the camera's Boyer-Lindquist radius, inclination, and motion state, and of the spin; include how the astroidal primary caustic and the multiply sky-wrapped higher-order caustics grow and deform as the camera moves closer to the hole. The map spans the spin range $0 \le a/M \le 0.999$ and camera radii from just outside the ergosphere out to $r_c = 10M$; each motion state is covered on its own valid positional sub-domain — the circular-geodesic camera only in the equatorial plane, the static and zero-angular-momentum (FIDO) cameras at all inclinations from equatorial to polar. The map resolves the astroidal primary cusp and each sky-wrapping of the higher-order caustics, with refinement of the sampling not changing the reported caustic topology.

2. Establish, in place of the paper's plausibility argument and by mathematical proof, the rules that govern multiple imaging in this geometry: (i) whether every crossing of a caustic branch by a source, from outside the astroid to inside, creates a pair of images (with annihilation occurring on exit), as claimed; and (ii) whether the original order-$n$ images (geodesics with $n$ poloidal turning points) wind up between the order-$n$ and order-$(n+1)$ critical curves on the camera's sky, as claimed.

3. Derive from (1) and (2) the resulting image-multiplicity rules, settling in particular the paper's prediction of three images of a galactic-plane (equatorial) source between the primary and secondary critical curves and eight between the secondary and tertiary critical curves near the flat shadow edge for $a/M=0.999$ and a camera on a circular equatorial geodesic orbit at $r_c=2.60M$ — proving the counts, correcting them, or exhibiting where and why they fail, including the role of light rays temporarily trapped in prograde near-circular orbits and of the multiple wrapping of higher-order caustics around the celestial sphere. All of (1), (2)(i), (2)(ii) and (3) are required and must each be established — no proper subset resolves the task — and this item requires both the general image-multiplicity rules and a settlement (proof, correction, or refutation) of the two quoted counts at the stated configuration.

References

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