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A cosmic ray approaches a black hole almost tangentially. Instead of falling straight in, it lingers on an unstable orbit, like an acrobat balancing on a tightrope over an abyss. After a few loops, the light finally escapes toward a distant observer. Such rays produce multiple images of the same source, known as higher-order images.

Usually, the angle by which a black hole black hole deflects light grows smoothly, like the logarithm of how close the ray comes to the critical orbit. This law works well for non-spinning or slowly spinning objects. But when the black hole spins at the theoretical limit (called extreme), and the ray moves in the same direction as the spin, the critical orbit merges with the horizon. The usual description breaks down.

The authors showed that in this situation the deflection becomes much sharper. A power-law term adds to the logarithmic growth — the ray bends far more strongly. To correctly calculate the brightness and positions of higher-order images, extra terms had to be kept in the formulas. The scientists first worked out the case where light travels strictly in the equatorial plane of the spinning hole, then generalized the result to rays slightly tilted from the equator. This allowed them to compute what the caustics look like — places on the celestial sphere where images of the source become especially bright.

The new formulas were tested on specific solutions of Einstein's equations: Kerr, Kerr–Newman, and Kerr–Sen black holes. In all cases, the power-law contribution to light deflection was confirmed. Now, knowing the exact law, astronomers will be able to model better the shadows and rings around rapidly spinning black holes when telescope sensitivity allows us to see these fine details.