Event Horizon

Drag a black hole across the starfield and the light behind it does not vanish. It bends. Stars smear into arcs, double, and at a precise radius close ranks into a perfect ring of light, the photon sphere, the last orbit light itself can take before falling in.

The concept

Mass curves spacetime, and light follows the straightest available path through that curvature. Near a black hole the curvature is so steep that paths wrap around: you see stars that are actually behind the hole, lensed into rings, and a dark disk, the shadow, slightly larger than the horizon itself, ringed by light that looped before escaping. The experiment ray-marches that bending in a shader.

The math

For a non-spinning (Schwarzschild) mass $M$, the event horizon sits at the radius

$r_s = \frac{2GM}{c^2}$

A light ray passing at impact parameter $b$ is deflected by an angle that, far out, is

$\Delta\phi \approx \frac{4GM}{c^2 b} = \frac{2 r_s}{b}$

and as $b$ shrinks toward $\tfrac{3\sqrt{3}}{2}\,r_s$ the deflection diverges: light can orbit. That critical impact parameter is the edge of the shadow, which is why the dark disk you see is larger than the horizon, about $2.6\,r_s$ across.

Why it stays strange

The bright ring is not decoration. It is light from the entire sky behind the hole, funneled into a circle, plus light that orbited one or more times before escaping, stacked into ever-thinner sub-rings. The 2019 image of M87's black hole is exactly this: not a photograph of a thing, but a photograph of where light cannot go.

Further reading

• Misner, Thorne, Wheeler, Gravitation.
• Event Horizon Telescope Collaboration (2019), the first black-hole image.

@book{mtw1973gravitation,
  author    = {Misner, Charles W. and Thorne, Kip S. and Wheeler, John A.},
  title     = {Gravitation},
  publisher = {W. H. Freeman},
  year      = {1973}
}