Scientists are intensifying efforts to test the boundaries of Einstein’s theory of general relativity, and recent observations of black hole shadows are providing a unique testing ground. A team led by Kourosh Nozari, Milad Hajebrahimi, and Sara Saghafi from the University of Mazandaran, collaborating with researchers from institutions including the National Observatory of Athens and the University of Science and Technology of China, has presented new solutions for rotating black holes that incorporate what’s known as ‘scalar hair’ – a property extending beyond the traditionally defined characteristics of mass and spin.
This research, published in arXiv on , is significant because it explores deviations from the ‘no-hair’ theorem, a cornerstone of black hole physics. The no-hair theorem states that all black holes are characterized by only three externally observable parameters: mass, electric charge, and angular momentum (spin). The introduction of scalar hair suggests the possibility of additional, independent properties influencing a black hole’s behavior and appearance.
For over a century, the Kerr metric has served as the standard description of rotating black holes, relying solely on mass and spin. However, alternative theories of gravity propose that black holes might possess additional characteristics, specifically a ‘scalar field’ which introduces this ‘scalar hair.’ This work focuses on ‘beyond Horndeski’ gravity, an extension of standard scalar-tensor theories, to investigate how this scalar hair affects the appearance of black hole shadows.
At the heart of this investigation is the Event Horizon Telescope (EHT) image of M87*, a supermassive black hole located 55 million light-years away. The EHT measured M87’s shadow to have an angular diameter of 42 ± 3 microarcseconds, with a circularity deviation of less than or equal to 0.1. These precise measurements provide a crucial benchmark for testing predictions made by modified gravity theories.
Researchers constructed rotating black hole solutions incorporating primary scalar hair within the framework of beyond Horndeski gravity, analyzing how this additional parameter alters the expected shadow characteristics. The presence of scalar hair induces measurable changes to the black hole shadow, specifically impacting its size and shape. Negative values for the scalar hair parameter enlarge the shadow and reduce its ellipticity, while positive values compress the shadow and increase its distortion.
By modeling M87* and applying the EHT’s observational constraints, the scientists mapped the viable range of parameters for both black hole spin and scalar hair. Current data do not rule out the existence of scalar hair, although the permissible values are restricted, particularly for positive scalar hair parameters. The predicted deviations caused by scalar hair are on the order of a few microarcseconds, a scale that is within reach of current and, especially, next-generation telescopes designed to image black holes.
The team employed a revised Newman-Janis algorithm (NJA) to derive the rotating black hole metric with scalar hair. The standard NJA, used to generate the rotating Kerr metric from a non-rotating Schwarzschild seed, can encounter complexities when converting metrics. To address this, a non-complexification process was applied, enabling the derivation of the rotating black hole metric with scalar hair. This allowed researchers to obtain a stationary, axially symmetric line element dependent on mass, spin, and the scalar hair parameters.
Detailed calculations of the Cauchy horizon and event horizon were performed, revealing that negative values of the scalar hair parameter enlarge the event horizon and diminish its oblateness, while positive values shrink the horizon and increase distortion. The study also examined the frame-dragging effect around these rotating black holes, quantified by the angular velocity of the ergosphere, providing another avenue for testing the predictions of beyond Horndeski gravity.
Analysis of photon motion around these black holes demonstrates that a positive scalar hair parameter shrinks the shadow and enhances its distortion, while a negative parameter enlarges the shadow and reduces its oblateness. At a coupling constant of λ = 1, the scalar hair parameter induces deviations of order 10-6 in the shadow radius, a value approaching the sensitivity of present instruments.
The significance of this work lies not merely in the existence of allowed parameter space for these ‘hairy’ black holes, but in the narrowing of that space as observational precision improves. Unlike previous studies, this work directly constrains the possible values of the scalar hair parameter using EHT data. As next-generation instruments come online, promising even greater resolution, these minute deviations are becoming increasingly accessible.
The research highlights a clear path forward for testing fundamental aspects of gravity using the shadows cast by these enigmatic objects. Since current observations haven’t excluded these alternative black hole models, future work should focus on refining both the theoretical predictions and the observational techniques. Beyond scalar hair, this framework could be extended to explore other modifications to general relativity, offering a powerful tool for understanding gravity in its most extreme environments.
