Schwarzschild Light Bending

A horizontal plane wave of photons enters from the left and meets a non-rotating black hole at the origin. Geometric units G = c = M = 1. Each photon obeys a null geodesic with conserved energy E and angular momentum L; its fate is fixed by the impact parameter b = L / E. Photons with |b| < 3√3 ≈ 5.196 cross the photon sphere at r = 3 and are swallowed (red); photons with |b| > 3√3 are deflected (blue), with photons just above critical looping the photon sphere multiple times before escaping. Drag the sliders to vary the photon count and the impact-parameter range.

Figure 1. Plane wave of null geodesics in the Schwarzschild equatorial plane (M = 1). Method: per-photon velocity-Verlet on the radial Hamiltonian H = p_r²/2 + L²(1−2/r)/(2r²) from shared/js/engine/symplectic.js; angular coordinate φ advances as L/r².

N 41

b_max 9.00

Show b_crit = 3√3 reference lines and critical photon

WHAT TO TRY

Raise the photon count N: the plane wave fills in and you see the sharp divide at the critical impact parameter b_crit = 3 sqrt(3) M. Rays inside it spiral in and are swallowed (red), rays outside whip around and escape (white).
Watch the rays that graze b_crit: they loop multiple times around the photon sphere at r = 3 M before escaping, the strong-lensing winding that makes a black hole shadow.
Compare the swallowed and deflected counts in the readout: they partition exactly at b_crit, which is the angular size of the shadow a distant observer would measure.