Black hole photon ring beyond General Relativity: an integrable parametrization

TL;DR

The paper introduces the Kerr off shell (KOS) family, a symmetry‑driven parametrization of Kerr‑like spacetimes with two free functions that preserve Kerr’s Killing tower and keep the geodesic motion integrable. It derives a closed analytic expression for the parametric critical curve of the photon ring in terms of these functions, enabling a direct test of Kerr‑like geometries via the photon ring shape. Through four Kerr‑like examples (including Kerr‑MOG, Simpson‑Visser EOS, and two new radial/polar deformations), it shows that fitting the circlipse (phoval) to the critical curve can be highly degenerate, potentially matching Kerr and non‑Kerr spacetimes with different (M,a,α) without independent mass/spin information. The results underscore the need for independent measurements of mass and spin and for connecting geometric parameters to underlying gravity theories to meaningfully test GR with horizon‑scale imaging. The work provides a tractable analytic framework to explore beyond‑Kerr phenomenology and clarifies limitations in using photon‑ring shapes alone as a sharp GR test.

Abstract

In recent years, the shape of the photon ring in black holes images has been argued to provide a sharp test of the Kerr hypothesis for future black hole imaging missions. In this work, we confront this proposal to beyond Kerr geometries and investigate the degeneracy in the estimations of the black hole parameters using the circlipse shape proposed by Gralla and Lupsasca. To that end, we consider a model-independent parametrization of the deviations to the Kerr black hole geometry, dubbed Kerr off shell (KOS), which preserves the fundamental symmetry structure of Kerr known as the Killing tower. Besides exhibiting a Killing tensor and thus a Carter-like constant, all the representants of this family also possess a Killing-Yano tensor and are of Petrov type D. The allowed deviations to Kerr, selected by the symmetry, are encoded in two free functions which depend respectively on the radial and polar angle coordinates. Using the symmetries, we provide an analytic study of the radial and polar motion of photon trajectories generating the critical curve, to which the subrings composing the photon ring converge. This allows us to derive a ready-to-use closed formula for the parametric critical curve in term of the free functions parametrizing the deviations to Kerr. Using this result, we confront the circlipse fitting function to four examples of Kerr-like objects and we show that it admits a high degree of degeneracy. At a given inclination, the same circlipse can fit both a Kerr black hole of a given mass and spin $(M,a)$ or a modified rotating black hole with different mass and spin parameters $(M,a)$ and a new parameter $α$. Therefore, future tests of the Kerr hypothesis could be achieved only provided one can measure independently the mass and spin of the black hole to break this degeneracy.

Figures (5)

• Figure 1: Critical curve of the Kerr black hole for $M=1$, $y_\mathcal{O}=0$ and various values of $a$.
• Figure 2: Critical curve of the Kerr-MOG black hole for $M=1$, $a = 0.9$ and different values of $y_\mathcal{O}$.
• Figure 3: Critical curve of the Simpson-Visser black hole for $M=1$, $a = 0.9$ and different values of $y_\mathcal{O}$.
• Figure 4: Critical curve of the log-corrected black hole for $M=1$, $a = 0.9$ and different values of $y_\mathcal{O}$.
• Figure 5: Critical curve of the polar-corrected black hole for $M=1$, $a = 0.9$ and different values of $y_\mathcal{O}$. When $y_\mathcal{O} = 0$, the black hole is seen from its equatorial plane, and the polar corrections do not come into play: the critical curves for all values of $p$ are the same.