Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum: II. Black-Hole Images (1005.1931v2)

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.

• The paper demonstrates that deviations from the Kerr metric’s quadrupole moment can alter black hole shadow shapes.
• It employs quasi-Kerr spacetimes and numerical simulations to reveal how asymmetries in the photon ring indicate potential no-hair violations.
• The study highlights the role of high-resolution VLBI in empirically testing black hole metrics and refining general relativity.

An Analysis of Testing the No-Hair Theorem via Black Hole Imaging

The paper by Johannsen and Psaltis focuses on a particular astrophysical hypothesis known as the no-hair theorem. This theorem postulates that black holes are completely defined by only two quantities: their mass and spin. According to general relativity, any additional multipole moments of a black hole's external field are determined solely by these two parameters. The paper investigates the potential violations of this theorem by examining black hole images, employing a method that hinges on electromagnetic spectrum observations.

Summary of Method and Results

The authors introduce a framework involving quasi-Kerr spacetimes that allows an independent evaluation of black hole metrics, specifically focusing on the quadrupole moment. They posit that deviations of this parameter from the Kerr metric could manifest as alterations in the shapes of black hole images, producing observable characteristics in electromagnetic observations. The paper uses numerically generated images to highlight how such deviations can lead to prolate or oblate shadows, dependent on the sign and magnitude of the quadrupole deviation.

A critical feature of these images is the presence of a bright photon ring, predominantly influenced by the mass, spin, and quadrupole moment. This photon ring remains nearly circular for typical Kerr black-hole configurations with a spin $a\lesssim0.9M$, regardless of the observer’s viewpoint. Conversely, violations of the no-hair theorem would render this ring asymmetric and elliptical.

Implications and Potential Developments

The findings imply a promising approach to empirically testing the no-hair theorem. The sensitivity of the photon ring’s properties to minute variations in the black hole’s quadrupole moment suggests that future high-resolution imaging, particularly via very-long baseline interferometry (VLBI), could provide critical insights. These observations could either reinforce the prevailing understanding of general relativity or open avenues for exploring alternative gravitational theories.

Practically, this research sets the stage for using black hole imaging to derive the mass, spin, and potential deviations in quadrupole moments of astrophysical black holes. The paper highlights the future capability of VLBI to resolve such details, notably describing the setup required to image the supermassive black hole at the center of the Milky Way, Sgr A*. The implications extend beyond validating the structure of known black holes; they could potentially challenge or refine aspects of general relativity concerning strong gravitational fields near event horizons.

Future Prospects

Looking forward, the methodology outlined by Johannsen and Psaltis suggests a growing role for observational astrophysics in the direct testing of fundamental theoretical predictions. As observational technologies advance, enabling finer resolution and sensitivity, the empirical scrutiny of black hole metrics becomes increasingly achievable. This paper lays a conceptual and methodological groundwork for testing general relativity's predictions in extreme environments and may help identify deeper nuances in gravity's role across cosmic scales.

In summation, while the paper does not immediately transform the field, it offers a substantive approach to a longstanding theoretical proposal, providing actionable directions for future empirical inquiries into the nature of black holes and the verification of gravitational theories.