Black Hole

The physics

Every pixel traces a light ray backwards through the curved spacetime of a Schwarzschild black hole. No textures, no tricks: the lensing, the photon ring and the disk's asymmetry all emerge from the equations below.

Light bending

Null geodesics in the equatorial plane of the ray obey Binet's equation, with u = 1/r in units of the Schwarzschild radius r_s = 2GM/c²:

d²u/dφ² = 3M u² - u

It is integrated per pixel with a fourth-order Runge-Kutta scheme, starting from the exact impact parameter b = r sin ψ / √(1 - r_s/r). Rays that spiral below the photon sphere at 1.5 r_s are captured; the black disk you see is the shadow, with apparent radius √27/2 r_s.

Rotation

The spin slider turns this into a Kerr black hole. Rotation breaks spherical symmetry, so rays are instead integrated in horizon-penetrating Kerr-Schild coordinates via Hamilton's equations:

H = ½ g^μν p_μ p_ν, g^μν = η^μν - f k^μ k^ν, f = 2Mr³/(r⁴ + a²z²)

Frame dragging pulls prograde light closer to the hole: the shadow shifts sideways and grows D-shaped, and the photon orbits split (at a/M = 0.9: 1.42 r_s prograde vs 3.42 r_s retrograde). The disk's inner edge follows the Kerr ISCO, plunging from 3 r_s down to 0.62 r_s at a/M = 0.998, with the exact circular-orbit kinematics Ω = ±√M/(r^3/2 ± a√M).

The accretion disk

A geometrically thin, optically thick disk extends inward to the innermost stable circular orbit, 3 r_s for zero spin. Its temperature follows the Shakura-Sunyaev profile:

T(r) ∝ [ (1 - √(r_in/r)) / r³ ]^¼

Each patch orbits at the relativistic Kepler rate Ω = √(M/r³). The observed frequency shift combines gravitational redshift and Doppler effect:

g = √(1 - 3M/r) / [ √(1 - r_s/r₀) (1 - Ω L_z/E) ]

Observed brightness scales as g⁴, which is why the side of the disk rotating toward you glows brighter and bluer: relativistic beaming. Colors come from Planck's blackbody law evaluated at the Doppler-shifted temperature.

What you are seeing

• The thin bright circle hugging the shadow is the photon ring: light that orbited the hole before escaping.
• The arc above the shadow is the far side of the disk, lensed over the top.
• The arc below is the disk's underside, lensed from behind.
• Background stars smear into arcs near the shadow's edge.

Two extra render modes visualize the physics directly: the redshift map paints the disk by the frequency-shift factor g (blue = boosted toward you, red = redshifted), and the deflection map colors the sky by how far each light ray was bent on its way to your eye.

How big is it?

The geometry is scale-free: the same image describes any mass, with r_s = 2.95 km × (M/M_☉):

Controls

Drag to orbit, scroll or pinch to zoom. Arrow keys orbit, +/- zooms. The camera pose and spin live in the URL (?az=&pol=&dist=&spin=), so any viewpoint can be shared as a link.