Black hole lensing — 2D light-ray lab

Switch to 3D
Controls
Live readouts
Numerical checks
What to try
  1. Drag on the canvas: the yellow ray swings from a gentle bend to a hairpin turn as b approaches 5.2 M.
  2. Press "Inside the shadow": every ray with b below 3√3 M falls in, however fast it travels.
  3. Compare the blue numerical curve with the orange weak-field curve: they agree far away and split near the hole.
  4. Change the mass: the picture is the same in units of M, only the kilometre readouts scale.
The science

Light follows null geodesics of the Schwarzschild metric. With u = 1/r and the azimuth φ as the independent variable the orbit equation is exact:

d²u/dφ² + u = 3 M u²    (G = c = 1, M = GM/c²) Weak field: α ≈ 4M/b + (15π/4)(M/b)²    Photon sphere: r = 3M    Shadow: bc = 3√3 M

The page integrates the equation with fourth-order Runge–Kutta from r = 3000 M (for the measured deflection) or r = 70 M (for the drawn rays). Four checks compare the deflection at b = 100 M and 200 M with the series, the capture threshold with 3√3 M, and the stability of the r = 3M circular orbit.

Simplifications: a non-rotating, uncharged hole; one plane of rays (spherical symmetry makes every plane equivalent); no accretion disk, redshift or polarisation; the 3D page’s warp is stylised, this page is the real geodesic.