Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum: II. Black-Hole Images

According to the no-hair theorem, all astrophysical black holes are fully described by their masses and spins. This theorem can be tested observationally by measuring (at least) three different multipole moments of the spacetimes of black holes. In this paper, we analyze images of black holes within a framework that allows us to calculate observables in the electromagnetic spectrum as a function of the mass, spin, and, independently, the quadrupole moment of a black hole. We show that a deviation of the quadrupole moment from the expected Kerr value leads to images of black holes that are either prolate or oblate depending on the sign and magnitude of the deviation. In addition, there is a ring-like structure around the black-hole shadow with a diameter of about 10 black-hole masses that is substantially brighter than the image of the underlying accretion flow and that is independent of the astrophysical details of accretion flow models. We show that the shape of this ring depends directly on the mass, spin, and quadrupole moment of the black hole and can be used for an independent measurement of all three parameters. In particular, we demonstrate that this ring is highly circular for a Kerr black hole with a spin a<0.9M, independent of the observer’s inclination, but becomes elliptical and asymmetric if the no-hair theorem is violated. Near-future very-long baseline interferometric observations of Sgr A* will image this ring and may allow for an observational test of the no-hair theorem.

💡 Research Summary

The paper presents a concrete method for testing the no‑hair theorem using electromagnetic observations of black‑hole images. Building on the quasi‑Kerr spacetime introduced in “Paper I,” the authors add an independent quadrupole moment parameter ε to the usual Kerr metric, so that the total quadrupole is Q = −Ma² + εM². When ε = 0 the spacetime reduces to the exact Kerr solution; any non‑zero ε signals a violation of the no‑hair theorem.

Because the quasi‑Kerr metric is only a solution of Einstein’s equations up to quadrupole order, the authors restrict their analysis to moderate spins (a ≲ 0.4 M) and modest deviations (|ε| ≲ 0.5). Within this regime they perform full ray‑tracing simulations: photons are launched from a distant image plane, integrated backward through the spacetime using a fourth‑order Runge‑Kutta scheme with adaptive step size, and terminated at a thin equatorial emitting surface that represents an optically thin accretion flow. The initial photon momenta are taken perpendicular to the image plane, and a regular grid of impact parameters is used to map the observer’s sky to emission radii and azimuths.

Two robust observational signatures emerge from the simulations. First, the silhouette (or “shadow”) of the black hole changes shape when ε deviates from zero. Positive ε (a quadrupole larger than Kerr’s) makes the shadow prolate (elongated along the spin axis), while negative ε makes it oblate (flattened). This deformation is largely independent of the observer’s inclination and becomes noticeable for spins a < 0.9 M. Second, a bright “photon ring” appears at a projected radius of roughly 10 M. In the Kerr case this ring is essentially circular; its diameter depends almost exclusively on the black‑hole mass (variations with spin and ε are < 1 %). However, when ε is non‑zero the ring becomes elliptical and its centre shifts away from the image‑plane origin. The degree of ellipticity and offset scale with ε, providing a direct probe of the quadrupole moment.

The authors argue that the photon ring’s properties are remarkably insensitive to the details of the accretion flow because the ring is formed by photons that orbit the black hole many times, thereby accumulating a large optical path length in an otherwise optically thin plasma. Consequently, even highly variable or turbulent flows should still produce a detectable, bright ring.

From an observational standpoint, the paper focuses on the Galactic Center black hole Sgr A*. Its mass (~4 × 10⁶ M⊙) yields an angular shadow size of ~50 μas, well within the resolution of current and near‑future very‑long‑baseline interferometry (VLBI) arrays such as the Event Horizon Telescope. The authors discuss how multi‑frequency, time‑averaged VLBI imaging can suppress variability and enable precise measurements of the ring’s diameter, ellipticity, and centre offset. A diameter measurement would give the mass to a few percent, while detecting an ellipticity at the 0.1 % level would constrain ε to ≈ 0.1, thereby testing the no‑hair theorem at a new level of precision.

Limitations are acknowledged. The quasi‑Kerr metric loses validity for near‑extremal spins or large ε, and the simulations assume a geometrically thin, equatorial emission surface, neglecting three‑dimensional plasma dynamics, magnetic fields, and radiative transfer effects. Moreover, the claim of inclination‑independence of the ring shape rests on the idealized thin‑disk model; real accretion flows may introduce modest corrections.

Nevertheless, the study demonstrates that by measuring three independent multipole moments—mass (from the ring diameter), spin (from the shadow displacement), and quadrupole (from the ring’s ellipticity)—one can perform a purely electromagnetic test of the no‑hair theorem. This complements gravitational‑wave approaches and opens a promising avenue for probing strong‑field gravity with upcoming horizon‑scale imaging experiments.