← Atlas·Lensing

Gravitational Lensing

A massive body bends spacetime; light rays passing it are deflected by an angle 4GM/(c²b). From a distance, a point source behind the mass splits into multiple images — Einstein rings, arcs, twin stars. Drag the black hole; watch the background stars distort. At enough mass, the photon sphere becomes visible as a glowing ring around a dark central shadow — the EHT image of Sgr A*.

mass (Einstein radius) 0.15
stars 600
show photon sphere
why a star can become a ring
The deflection angle for a light ray passing a Schwarzschild mass M at impact parameter b is:

α = 4GM / (c² b)

Newton (treating photons as classical particles) gets exactly half of this. The other half comes from spatial curvature — a uniquely GR effect, confirmed by Eddington's 1919 eclipse observation, which made Einstein an international celebrity overnight.

For a point source directly behind the mass, by axial symmetry, light reaches the observer from every direction at the deflection-matched angle. The point becomes a ring — the Einstein ring. Slightly off-axis sources become arcs or multiple images. Lots of background galaxies + a foreground cluster = the famous "lensed arc" images from Hubble.

As mass grows, you also see the photon sphere at r = 3M. Light orbits in unstable circles here, so the lens picks up rays that have looped around the black hole once, twice, three times — each loop adding a sharper ring outside the shadow. The bright thick ring you see in the EHT images of M87 (2019) and Sgr A* (2022) is the first photon ring sum. The dark central disk is the black hole shadow — the projection of the photon sphere, with radius ~2.6 × R_s, slightly larger than the event horizon itself.