How black holes bend light

Black holes bend light because mass curves spacetime, and light always takes the straightest path it can through curved spacetime, so near a black hole that path is a curve. The closer a ray passes, the more it bends, until inside a certain distance it cannot get away at all and falls in.

What is gravitational lensing?

Gravitational lensing is the bending of light by gravity. Any mass does it: a star, a galaxy, a cluster of galaxies, a black hole. When a massive object sits between us and something more distant, it acts like a lens. It can magnify the background source, smear it into arcs, or split it into several images.

The bending comes from general relativity, Einstein's theory of gravity. Mass and energy curve spacetime, and light follows the curve. There is a handy way to picture it: around a mass, space behaves like glass whose refractive index rises as you get closer. Light slows down and bends toward the denser glass, which means toward the mass.

How did the 1919 eclipse prove light bends?

During the total solar eclipse of May 29, 1919, the Moon blocked the Sun's glare, so stars right next to the Sun could be photographed. If gravity bends starlight, those stars should appear shifted slightly outward from where they sit at night. Expeditions to Brazil and the island of Principe, organised by the Royal Society and the Royal Astronomical Society and masterminded by Arthur Eddington, measured the shift. The results favoured Einstein's prediction over the Newtonian one (NASA, ESA).

The shift is tiny. For a ray grazing the Sun the bending is 2 r_s/b, where r_s is the Sun's Schwarzschild radius (about 2.95 km) and b is how close the ray passes (the Sun's radius, 695,700 km). That works out to about 1.75 arcseconds, less than a thousandth of a degree (Sun figures from the NASA Sun fact sheet).

What are Einstein rings and arcs?

When a distant galaxy sits almost exactly behind a nearer galaxy, its light bends around the lens on every side and arrives as a ring, called an Einstein ring. Hubble's Sloan Lens ACS (SLACS) survey searched about 150 galaxies for them; see NASA's gallery of Einstein rings.

Alignments are rarely perfect, so most lensed galaxies show up as bright curved arcs around a galaxy cluster, or as several copies of one object. Radio astronomers found the double quasar Q0957+561, and the Einstein Cross is a single quasar lensed into four images by a galaxy in front of it (NASA, Nature's boost: gravitational lenses). Astronomers use these lenses as natural telescopes that magnify galaxies too faint to see otherwise.

What is the photon sphere?

Close to a black hole, bending stops being a small correction. At 1.5 times the Schwarzschild radius (r = 1.5 r_s) gravity is strong enough to hold light in a circular orbit. That shell is the photon sphere. The orbit is unstable: nudge a photon inward and it falls in, nudge it outward and it escapes.

For light coming in from far away, what matters is its impact parameter b, the distance by which it would miss the center if it went straight. In general relativity any ray with b below (3√3 / 2) r_s, about 2.6 r_s, crosses the photon sphere and is captured. Rays a little above that threshold can loop part or all of the way around before escaping, which is why the image of a black hole shows a thin bright ring at its edge.

Why does light bend twice as much as Newton predicted?

You can get a bending angle from Newton's gravity alone, by treating light as a fast particle that falls a little as it passes the mass. That gives r_s/b. General relativity gives twice as much, 2 r_s/b, because gravity bends space as well as time: half the effect is the slowing of clocks near the mass (the part Newton also captures) and the other half is the stretching of space itself, which has no Newtonian counterpart. The 1919 eclipse was decisive because it could tell the two numbers apart.

Closer in, the next correction appears. To second order the deflection is

so a ray passing at 50 r_s bends by 0.0400 + 0.0012 = 0.0412 radians, about 2.4 degrees. The game's test suite checks its tracer against this formula.

How does Bent Light compute it?

Bent Light does not fake the curves. Its physics, in src/physics.ts, uses the fact that a static black hole bends light exactly like a lens with a refractive index. In isotropic coordinates (where ρ is the distance from the center) that index is

and a ray's unit direction t turns as dt/ds = ∇ln n − (∇ln n · t) t. The game steps each ray along with a midpoint (second-order Runge-Kutta) step that is 3.5 px long in open space and shrinks to 6% of the distance to the nearest black hole, down to 0.25 px, so strong bends near a hole stay accurate. With several black holes on screen their ln n add up, the standard weak-field superposition. For a single black hole this reproduces the 2 r_s/b deflection, the photon sphere and capture. The photon sphere at r = 1.5 r_s sits at ρ = 0.933 r_s in these coordinates; the black core in the demo is drawn at exactly that size.

Where does the game bend the rules?

The game takes two liberties so the puzzles stay playable, and the demo above shows the second one.

• Pull fades at the dashed circle. In the game, each black hole's pull fades smoothly to zero across the outer quarter of its dashed circle, so a well only bends light near itself. Real gravity reaches out forever. The demo turns this fade off.
• The accretion disk eats grazing light. The glowing disk swallows any ray that comes within 2 r_s of the center in isotropic coordinates, which is about r = 2.53 r_s, well outside the photon sphere. Traced with the game's code, a ray at b = 3.2 r_s is swallowed and one at 3.3 r_s escapes after bending about 74 degrees. So rays with b between about 2.6 and 3.2 r_s are eaten in Bent Light but would escape, sharply bent, in real general relativity. It keeps players from cheating a level by skimming a beam past the core. Tick Pure general relativity under the demo to remove the disk and watch those rays loop past instead.

Everything else, every curve you see in the game and the demo, is the real thing.