Measuring the Shape of Kerr Black Holes at the Photon Orbit

The bright ring-like structures observed in the images of M87* and SgrA* captured by the Event Horizon Telescope strongly support the validity of general relativity. Lensed images of the emission region, often referred to as photon rings in this context, are a direct consequence of the unstable dynamics of null geodesics near the spherical photon orbit in the Kerr spacetime. The order of the lensed image can be characterized by the number of half-orbits the photons complete before reaching the observer, with higher-order photon rings produced by null geodesics that circle the black hole more times. However, low-order rings are significantly influenced by the astrophysical environment. Measuring the Lyapunov exponent requires probing the exponentially small differences between successive photon rings or between photon rings and the shadow. We investigate potential astrophysical sources of systematic error the estimation of Lyapunov exponent, including the location of the observed emission, and especially at low photon ring order. We show that it is nevertheless possible to measure this purely gravitational quantity to roughly 10% and 1% systematic uncertainty by resolving the n=2 and n=3 photon rings with the shadow size, respectively. Therefore, the forthcoming black hole imaging efforts to capture, even if indirectly, the n=2 photon ring can result in a measurement of the Lyapunov exponent that is not limited by astrophysical uncertainties.

💡 Research Summary

This paper investigates how the Lyapunov exponent—a measure of the instability of the photon sphere in a Kerr black hole—can be extracted from the radii of successive photon rings observed in horizon‑scale images. The authors begin by reviewing the theoretical foundation: in Kerr spacetime, null geodesics that linger near the spherical photon orbit are unstable, and a small radial perturbation δr₀ grows as δrₙ ≈ e^{γ n} δr₀ after n half‑orbits, where γ is the Lyapunov exponent. Consequently, the apparent radius of the n‑th photon ring follows Rₙ = R_∞ + ΔR e^{-nγ}, and the ratio of successive ring‑to‑shadow separations tends to e^{γ} as n → ∞.

In practice, however, two major sources of systematic error arise. First, the “geometric” ring radius (R_geo), defined by an ideal null geodesic that executes exactly n/2 half‑orbits, differs from the “observed” radius (R_obs) because real emission originates from an extended accretion flow. Photons emitted at a range of radii r_em contribute to each image order, broadening the observed ring’s radial extent, especially for low‑order rings (e.g., n = 1 spans 4.30–6.17 GM/c²). Second, the Lyapunov exponent is formally defined in the high‑order limit; using low‑order rings (n = 1, 2) introduces a bias because the exponential convergence is not yet fully realized.

To quantify these effects, the authors adopt a simplified model: a single, azimuthally symmetric emission ring at radius r_em in the equatorial plane, observed from the polar axis (appropriate for M87*). They compute R_geo and R_obs for a range of spins a and emission radii, using the exact Kerr expressions for the critical radius r_γ and the Lyapunov exponent γ (Eq. 7). Their numerical analysis shows that as n increases, R_obs converges rapidly toward R_geo, and the inferred exponent γ_obs approaches the theoretical γ. Specifically, measuring the n = 2 photon ring yields an estimate of γ with roughly 10 % systematic uncertainty, while the n = 3 ring improves this to about 1 %—even after accounting for the spread caused by the emission region.

The paper also discusses observational feasibility. A polar observer simplifies the geometry because the shadow radius R_∞ aligns with the image plane, reducing projection effects. For M87*, the next‑generation Event Horizon Telescope (ngEHT) is expected to achieve the angular resolution and dynamic range needed to isolate the n = 2 ring, either directly or via indirect signatures in the visibility domain. In contrast, Sgr A* suffers from interstellar scattering that blurs structures below ~0.6–1.2 µas, making detection of even the n = 1 ring challenging with current ground‑based facilities.

Finally, the authors argue that the Lyapunov exponent provides a purely gravitational observable, independent of the astrophysical details that dominate shadow‑size measurements. By demonstrating that systematic astrophysical uncertainties can be constrained to the 10 %–1 % level, they establish photon‑ring spectroscopy as a viable tool for testing the Kerr metric and, more broadly, general relativity in the strong‑field regime. This work thus paves the way for future high‑resolution black‑hole imaging campaigns to deliver not only images of shadows but also quantitative probes of spacetime stability.