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

TL;DR

This paper proposes a method to test the no-hair theorem by analyzing black hole images, focusing on the shape and brightness of a ring-like structure around the shadow to measure deviations in the quadrupole moment.

Contribution

It introduces a framework to relate black hole image features to mass, spin, and quadrupole moment, enabling observational tests of the no-hair theorem using electromagnetic spectrum data.

Findings

Ring shape depends on black hole parameters.

Deviations from Kerr quadrupole cause ring ellipticity.

Future observations can test the no-hair theorem.

Abstract

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.