CQG+ Insight: Fractals and black hole shadows

In July 2016, Sam Dolan and I were invited to write an ‘Insight’ piece for CQG+, the companion website to Classical and Quantum Gravity. To read the article, please visit the CQG+ website or continue reading this blog post.

This has been a “miracle year” for relativity.

LIGO detected gravitational waves. The LISA Pathfinder mission demonstrated near-perfect freefall in space. And the era of gravitational-wave astronomy began in some style.

A century after black holes and gravitational waves were first predicted, we have learnt something truly mind-boggling: when two black holes collide, they shake the fabric of space-time with more power than is radiated by all the stars in the known universe put together!

The “chirps” from distant black hole collisions will travel for millions of years, at the speed of light, to reach our growing network of gravitational-wave detectors on Earth … and one day, out in space.

Next year, attention will turn to the Event Horizon Telescope (EHT): a global network of radio telescopes linked together to form an Earth-sized virtual telescope, using the technique of Very Long Baseline Interferometry. The EHT will take a picture of the region around the black hole at the centre of our galaxy, at exquisite resolution. Excitingly, this picture may deepen our understanding of the properties of supermassive black holes, and their curious features such as “shadows” and “light-rings”. We may even test the celebrated “no-hair conjecture” itself, as described by Tim Johannsen for CQG+.

So, what does a black hole actually look like? Imagine pointing a camera towards a black hole. The black hole will deflect and bend the rays of light from the distant stars. And it will cast a shadow too, by absorbing any starlight that strays too close. To simulate this effect, we could imagine tracing all the photons from the distant stars, to see whether they eventually reach the camera. But it is much more efficient to reverse this process, tracing the photons that pass through the camera’s lens backwards in time. Those photons which travel back to a black hole event horizon lie inside the black hole shadow — appearing as a dark region of our picture.

Now, suppose we could see a double black hole system — what would this look like? Thanks to advances in numerical relativity, the Simulating Extreme Spacetimes project have created fascinating videos of the mergers of pairs of black holes. Watch here to see how the black holes warp space and time to deflect the light from distant stars. In the centre of these simulations, we see something very striking: the black hole shadow has distinctive eyebrow-like features. And occasionally, around the primary eyebrow appears another, smaller cousin.

It appears that the shadow of a binary black hole may actually be a fractal!

In our new paper “Binary black hole shadows, chaotic scattering and the Cantor set”, recently published in Classical and Quantum Gravity, we set out to explain the origin of these curious fractal features. We studied a toy model: a pair of charged black holes in static equilibrium. We explored the link between black hole shadows and the chaotic scattering of light, using some methods from chaos theory.

Imagine yourself to be a photon, flirting with a pair of black holes. If you want to dance with them, you must take a sequence of decisions. For example, “go around the top black hole, now switch to the other black hole, passing in the opposite direction”, and so on. By encoding each decision as a single digit, we can encode any light ray as a sequence of digits. The longer the sequence, the longer the photon spends dancing with the black holes, before either escaping, or plunging in.


Top left: A binary black hole shadow viewed side-on. We see two main shadows in the centre, surrounded by a hierarchy of eyebrows, whose eyebrows have eyebrows upon them, ad infinitum.
Top right: Symbolic dynamics to describe photon orbits around two black holes.
Bottom: Using symbolic dynamics as a guide to construct one-dimensional slices through a binary black hole shadow.

The black hole shadow may be constructed through a step-by-step process. At each step, we take one more decision, and add one more fragment to the shadow. A rather similar process generates the Cantor set, which is a famous example of a one-dimensional fractal. As a result, we expect Cantor-like fractal structure to emerge naturally in double black hole shadows.

To confirm our ideas, we made some idealised images of double black hole shadows. An example is shown in the figure. Each black hole has a primary shadow, either ring-shaped or globular, and then an infinite hierarchy of eyebrow-like fragments.

Unexpectedly, we also found that some parts of our shadow are highly chaotic; and that photons can be trapped in stable orbits.

But to find out more about this, you will just have to read our paper…


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