Albert Sneppen’s 2021 paper, “Divergent reflections around the photon sphere of a black hole”, published in Scientific Reports, offers a novel analytical and physical perspective on the behavior of light near black holes. The study employs only basic mathematical tools to describe these phenomena: the core advancements are physical and philosophical.
At its heart, Sneppen’s research explores the trajectories of light—here known as null geodesics—in the curved space-time surrounding black holes. Specifically, it examines how photons behave near the photon sphere: the region where gravity is so intense that light can orbit the black hole. The study reveals that as photons approach this critical radius, their paths exhibit exponential divergence or convergence, depending on their initial conditions.
To model these dynamics, the paper introduces a second-order differential equation to model photon trajectories. The exponential terms derived in the equations correspond to tangible effects: the formation of multiple, increasingly narrow rings of light—known as photon rings—around a black hole’s shadow. Understanding the behavior of light near the photon sphere aids in interpreting the images of black holes and the surrounding accretion disk structures.
The study’s findings suggest that each additional orbit of light around a black hole results in an image that is a factor of exp(2pi) closer to the event horizon; the boundary to its inescapable gravity. This exponential scaling provides a framework for understanding the brightness and spacing of photon rings, which are features in high-resolution black hole spectrometry.
Extending the study to Kerr black holes, which possess angular momentum, Sneppen demonstrates that the exponential behavior of photon trajectories persists, albeit with modifications due to the black hole’s spin. This generalization underscores the physical nature of the findings, as it accounts for real-world complexities in astrophysical black holes, which often rotate, sometimes quite quickly. Analysis of the light rings around black holes could refine our understanding of cosmic distance-finding and enrich our understanding of the early universe’s physics.
Sneppen’s work, while powerful and elegant in its mathematical clarity, focuses on the idealized Kerr black hole model. However, for this theory to achieve comprehensive astrophysical relevance, it will need to be adapted to the more complex and less symmetric black holes we observe.
For example astrophysical black holes often exist in environments full of perturbations: they accrete matter, radiate energy, and interact with other celestial objects . These effects can break the symmetry that underpins the Kerr solution and therefore alter the behavior of photon trajectories.
Photon ring structures, lensing patterns, and time delays may all be measurably different in non-Kerr space-times. Consequently, Sneppen’s exponential divergence formula for light paths near the photon sphere will likely require modification or generalization to remain predictive and relevant.
Realistic black holes may also possess electric charge or exhibit deviations from the Kerr solution due to nearby matter, accretion disks, or alternative gravity theories . These “non-Kerr” black holes—such as the Reissner–Nordström or Kerr–Newman solutions—represent essential extensions of idealized models.
We’re most familiar with stellar-mass black holes that form from the gravitational collapse of massive stars at the end of their life cycles. These black holes have masses ranging from about three to twenty times that of the Sun and are the most commonly observed type, especially in X-ray binaries and through gravitational wave detections . Sneppen’s work may help to advance knowledge of the photon sphere of these relatively common entities.
It’s possible that Sneppen’s approach may work for the very slowly rotating supermassive black holes that reside at the centers of galaxies and have masses ranging from millions to billions of solar masses. Primordial black holes, though hypothetical, are another category that Sneppen’s work could eventually help unravel . These are proposed to have formed during the early moments of the universe due to density fluctuations in the primordial plasma. Their masses could range from subatomic scales to thousands of solar masses, and they are considered potential candidates for dark matter. They may also have light spheres that could be analyzed by careful adaptation of Sneppen’s thesis.
While relatively simple mathematics is involved, Sneppen’s approach for understanding the trajectories of photons surrounding black holes has the potential to aid in the interpretations of high-energy bursts from deep space that may be generated by black holes. His work may also be applied to microscopic black holes that are equivocally predicted by quantum theory. Modifications to Sneppen’s theory will be required to investigate these microscopic black holes, and also the supermassive black holes that inhabit the centres of galaxies.
Written by Jonathan Kenigson, a mathematician and statistician who writes for the advancement of the public understanding of scientific topics.
Edited by Emma Walsh, an undergraduate studying Neuroscience and an Online News Editor for EUSci.
References:
Cardoso, V., & Pani, P. (2019). Testing the nature of dark compact objects: a status report. Living Reviews in Relativity, 22(1), 4.
Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes. Oxford University Press.
Johannsen, T. (2016). Testing the No-Hair Theorem with Observations of Black Holes in the Electromagnetic Spectrum. Classical and Quantum Gravity, 33(12), 124001.
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.
Sneppen, A. (2021). Divergent reflections around the photon sphere of a black hole. Scientific Reports, 11, 13243. https://doi.org/10.1038/s41598-021-93595-w
Leave a Reply