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Flat space · no lensing

Schwarzschild · null geodesics

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Drag to orbit · wheel to zoom · Space = pause disk. Units: G = c = 1, lengths in M (r_s = 2M).

Light rays past a Schwarzschild hole

Time dilation & redshift

Black HoleSchwarzschild geodesic ray tracer · u = 1/r orbit equation integrated per pixel

Camera

Distance

Inclination

Black hole & disk

Mass / scale

Disk inner r

Disk tilt

Beaming

Bloom

Lensing comparison

Grid warp Photon ring glow Animate disk Auto-orbit

Performance

Resolution

Adaptive resolution

What to look for

Shadow: the black disc is not the horizon (r = 2M) but the capture cross-section, radius b_c = 3√3 M ≈ 2.6 r_s — about 2.6× larger.
Disk over and under the hole: rays from the back of the disk are bent over the top (and a second, thinner image under the bottom). Switch to Split to see flat space for comparison.
Doppler beaming: the disk rotates counter-clockwise (seen from +y), so the approaching side is brighter and bluer, the receding side dimmer and redder. Slide Beaming to 0 to remove it.
Gravitational redshift dims and reddens light from the inner edge, noticeably at small r.
Photon ring: a razor-thin bright ring at the shadow edge — light that looped near r = 3M before escaping. The Einstein ring appears in the starfield; enable Grid warp to see space-time mapped onto the sky.
Move the camera nearly face-on: the lensed stars form a distorted mirror image of the whole sky inside the Einstein ring.

Key formulas

r_s = 2GM/c² = 2Mhorizon; photon sphere r = 3M; ISCO r = 6M

d²u/dφ² = −u + 3Mu², u = 1/rnull geodesic (integrated in the shader by RK4)

(du/dφ)² = 1/b² − u²(1 − 2Mu)first integral; capture if b < b_c = 3√3 M

g = √(1−2M/r_e)/√(1−2M/r_o) · 1/[γ(1 − v·n)]total frequency ratio; v = √(M/(r−2M)) (Keplerian)

I_obs = g⁴ I_em, T_obs = g T_embeaming & colour shift

T ∝ r^−3/4[1 − √(r_in/r)]^1/4thin-disk (Novikov–Thorne-like) temperature profile