May 21, 2024

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Formula for rings of light around the black hole – mathematical equations explain the distance between the photon orbits at the event horizon

Optical time capsule: just outside the event horizon lies a narrow zone in which light is captured and orbits the black hole several times. As a result, rings are created from images of the universe of different ages – like a visual time capsule. A researcher has now unraveled how far these photon spheres are from one another and what mathematics are behind them. His formulas make it possible to calculate the distance between the rings of light at different black holes.

When light hits a black hole, it is usually either swallowed directly or deflected and thrown back into space. But there is a third possibility that Albert Einstein predicted: just outside the event horizon there is a narrow zone in which photons can circle the black hole several times. With each orbit they move a little closer to the “last photon orbit” – the photon sphere from which the light is swallowed.

Photon ring
The light of a galaxy creates a thinner, more distorted image of its source with each orbit of the black hole.© Peter Laursen

What determines the distance between the photon orbits?

This creates a sphere of ever thinner, narrower photon rings that contain light and thus images of different ages. Each ring is an increasingly delayed and compressed snapshot of the universe, like a visual time capsule. “An observer sees in them the entire surface of the event horizon and the entire universe visible from there in infinitely repeating images,” explains Albert Sneppen from the Cosmic Dawn Center in Copenhagen.

But how are these light rings staggered? From Einstein’s field equations it can be deduced where the last photon orbit lies in classical, non-rotating black holes. The distance between the light rings spiraling towards this last stable orbit has already been calculated. Accordingly, each ring is separated from the next one by a factor of e – and thus about 500 times. What was missing so far, however, was a coherent and comprehensive mathematical equation to deduce why this is so.

Two rival exponential equations

Sneppen has now provided exactly this mathematical basis and its proof. “It is a fantastic beauty to finally understand why the images on the black hole repeat themselves in such an elegant way,” explains the astrophysicist. “The equations give us analytical insights into the previously developed solutions. In addition, this now offers new opportunities to check our ideas about gravity and black holes. “

Specifically, the young researcher discovered that the angle of deflection and all orbits of light in the vicinity of the last photon sphere can be described by two rival exponential functions. Depending on the position of the photon and its trajectory, the balance between these equations shifts and describes the deflection angle in relation to the last photon orbit – both inwards when falling to the event horizon and outwards when it is finally ejected.

Also applicable to rotating black holes

As previously calculated, these formulas show that the individual light rings around a non-rotating black hole are each 535 times away from their predecessor orbit. In addition, Sneppen’s equations can also be used to determine the photon orbit around rotating black holes and any other non-spherical symmetrical spacetime sinks, as he explains.

“It turns out that with a rapidly rotating black hole you no longer have to get closer by a factor of 500 to find the next orbit, but significantly less,” reports the astrophysicist. “The next image can then be only 50, five or even only twice closer to the last orbit around the black hole.”

Asymmetry offers opportunities for observation

And there is another difference: if you look at the “equator” of such a rotating black hole, the repeating photon rings appear asymmetrical. “The prograde copies of a light source will repeat themselves more quickly than the retrograde copies,” explains Sneppen. “This asymmetry has potentially far-reaching significance for the observation of these phenomena.” Because it means that the prograde rings of light around a rotating black hole will probably be easier to see with future telescopes.

In fact, the astronomers of the Event Horizon Telescope (EHT) have calculated that the outermost, youngest photon ring could already be detected if a radio telescope on the earth’s surface is coupled with a second in earth orbit. In order to see the next older photon ring, the second telescope would have to be on the moon. (Scientific Reports, 2021; doi: 10.1038 / s41598-021-93595-w)

Source: University of Copenhagen – Faculty of Science