TRACKER-1
48.8566°N / 2.3522°E

Observing / Simulation

Relativistic black hole

A ray-tracing engine that bends every light ray around a Kerr black hole. Set the mass, spin, inclination and camera, and understand the shadow, the photon ring and the accretion disk.

Every pixel is a ray of light whose path is traced through curved spacetime. Drag inside the image to orbit the black hole, scroll or pinch to zoom. Then set the mass, the spin and the disk in the panel: the image recomposes live, as a real observer would see it.

The 4.3-million-solar-mass black hole at the centre of the Milky Way, imaged by the EHT in 2022 as a ring.

Sagittarius A*— fps
Observed black hole
4.3 × 10^6 M☉
Mass sets the size: the Schwarzschild radius grows in proportion.
0.900
A spinning black hole drags space along and moves the disk’s inner edge (ISCO) closer.
Camera
Azimuth 0°Elevation 50°Distance 20.0 r_s
Drag inside the image to orbit, scroll or pinch to zoom. Physical settings are in the right-hand panel.
16.0 r_s
Outer edge of the accretion disk, in Schwarzschild radii.
6,200 K
Temperature of the inner edge as shown. The cooler outer edge shifts towards orange.
1.00
1.00
Computed physical quantities
Event horizon9.12 million km0.718 r_s
Innermost stable orbit (ISCO)14.74 million km1.16 r_s
Period at inner edge9.8 min
Speed at inner edge95.0 % c
Maximum brightening× 29.1
Real disk temperature3.9 × 10^5 K
Hawking temperature1.43 × 10^-14 KUndetectable: this black hole needs 10⁶⁷ years to evaporate.
d²u/dφ² = −u + 3M u² · r₊ = M + √(M² − a²) · I_obs = I_emit · g⁴ · g = √(1 − r_s/r)

What you see

The central shadow is not the black hole: it is the absence of light coming from behind it. Rays grazing the horizon are swallowed, and those that should pass above reappear below. The thin bright rim bordering the shadow is the photon ring: light that looped around the black hole before reaching you.

The disk looks folded above and below the shadow: that is its far side, its light bent towards you. Its left half is far brighter than the right, because matter there races towards you at a sizeable fraction of light speed — the relativistic Doppler effect, exactly the asymmetry observed by the Event Horizon Telescope.

The hue ranges from orange to bluish white: the hotter, faster inner edge is shifted towards blue, the cooler outer edge towards red.

Why

Far from the black hole a light ray travels straight. Near it, you must integrate its path step by step: this is the relativistic Binet equation for a photon, u″ = −u + 3Mu², solved separately for every screen pixel. Each step turns the ray; some loop several times before escaping, others cross the horizon and vanish.

Observed intensity scales as g⁴, where g combines two effects: the Doppler effect (matter approaching looks brighter and bluer) and gravitational redshift (near the horizon time slows, light reddens and dims). The fourth power makes the asymmetry dramatic: one side of the disk outshines the other.

Rotation enters the horizon radius r₊ = M + √(M² − a²), the ISCO radius (Bardeen–Press–Teukolsky) and the Keplerian angular speed of matter. The higher the spin, the closer the disk’s inner edge and the faster matter orbits.

Sources: Kerr metric in Boyer–Lindquist coordinates; null geodesics for light transport; Bardeen–Press–Teukolsky (1972) ISCO radius; Novikov–Thorne (1973) thin-disk model; blackbody colour from Planck’s law. The disk is a thin plane with simplified radiative transfer, and geodesics are integrated in the Schwarzschild approximation: the shadow therefore stays nearly circular instead of Kerr’s “D”.