Black Hole Simulator: Light Bending, Orbits and Shadow

A free black hole simulator: watch light bend around a Schwarzschild black hole and find the shadow and photon sphere, follow orbits that precess, plunge or sit at the innermost stable orbit, see a ray-traced picture of a disk and lensed stars, and compute the size, tides and temperature of any black hole. Includes the optical analogue: a glass-like medium that bends light the same way.

Light arriving from the left passes a non-spinning black hole (Schwarzschild). Rays that pass close enough are bent more and more, and below a critical distance they fall in: that edge is the shadow. Move the pointer over the picture to choose a ray.

Chosen ray: distance from the axis (b)–in Schwarzschild radii, rs

Bending (Einstein)–

Newton's guess–light as a particle: half of Einstein's

Closest approach–

Chosen ray: b = rs Positive is above the hole, negative below. The shadow edge is at ±2.598 rs.

Number of rays:

Zoom: rs across

Photon sphere (1.5 rs) Shadow edge (2.6 rs) Optical analogue: same rays through a glass-like medium Try it in Light in a Medium

What this shows

A black hole is a region of space where gravity is so strong that nothing, not even light, can leave. This page uses the simplest one: Schwarzschild's solution of Einstein's general relativity, a hole that does not spin and has no charge. Its only property is its mass, and everything else follows from one number, the Schwarzschild radius r_s = 2GM/c². For the Sun that is 2.95 km; for the Earth, 8.9 mm. All distances on this page are measured in units of r_s.

light path \(\displaystyle \dfrac{d^2u}{d\theta^2} = -u + \tfrac32\, r_s\, u^2,\qquad u = \dfrac1r\)

shadow \(\displaystyle b_c = \dfrac{3\sqrt3}{2}\, r_s \approx 2.598\, r_s\)

weak bending \(\displaystyle \alpha \approx \dfrac{2\, r_s}{b} = \dfrac{4GM}{b\,c^2}\)

orbit \(\displaystyle \dfrac{d^2u}{d\varphi^2} = -u + \dfrac{r_s}{2L^2} + \tfrac32\, r_s\, u^2\)

Bending light

Far from the hole a ray is bent by a tiny angle, twice what Newton's gravity would give (this was the 1919 eclipse test of general relativity). Closer in, the bending grows fast. At 1.5 r_s there is a circle, the photon sphere, where light can orbit the hole on an unstable path. A ray that comes in with an impact parameter b smaller than about 2.6 r_s is captured; one just outside winds around the hole before escaping. The captured circle, seen from far away, is the shadow that the Event Horizon Telescope photographed for M87* and Sagittarius A*. Its size is 2.6 r_s, about 2.6 times bigger than the horizon itself, because the hole also swallows light that was never aimed at it.

The optical analogue

In suitable coordinates, light near a non-rotating black hole behaves exactly like light in a transparent medium whose refractive index grows towards the centre: n(ρ) = (1 + r_s/4ρ)³ / (1 − r_s/4ρ). Tick “Optical analogue” on the first tab: the dashed rays come from a glass-like medium, not from curved space, and they land on top of the real ones. (The test suite checks that they bend by the same angle.) The Light in a Medium simulator has a black-hole scenario that uses this index.

Orbits

For a planet the orbit is nearly Newton's ellipse, but the closest point creeps round a little on every lap: Mercury's creep (43 arc-seconds per century) was another early proof of general relativity. Near a black hole the effect is huge. Two radii matter: no circular orbit is stable inside 3 r_s (the ISCO, where the orbital speed is half the speed of light), and a particle with less angular momentum than 2 r_s·c always falls in. Orbits of a particle with a clock aboard show time dilation too: the clock slows as it nears the hole, and the animation shows both the clock on the particle and what a distant observer would count.

The picture

The picture is made by shooting one light ray backwards from each camera pixel and following it with the equation above until it hits the disk, falls in (black) or escapes to the starry sky. The glow of the disk is brighter on the side turning towards the camera (Doppler beaming) and dimmer, and redder, deep in the gravity well (gravitational redshift). Real accretion disks are thicker, hotter and turbulent, and real holes usually spin; this is the idealised thin-disk picture used in textbooks.

How accurate is it?

The geometry is exact for a Schwarzschild hole: the light and orbit equations are integrated numerically, and the test suite checks the shadow size (3√3/2 r_s), the weak-field bending, the periapsis advance, the ISCO and the optical analogue against the textbook values. What is simplified: no spin (Kerr), no charge, a razor-thin disk with made-up brightness and colours, no light from the disk's own gravity or from the plasma around it, and a static sky. It is a teaching tool, not a research code.

Things people ask

Is the shadow the horizon? No. The horizon is at 1 r_s. The shadow is the apparent size on the sky, 2.6 times bigger, because of the bending of nearby light.
What happens at the horizon? For someone falling in, nothing special is felt there (for a big hole). A distant observer sees their clock slow and the light from them redden and fade, and never sees them cross.
Why do small holes pull you apart? Tides: gravity is stronger at your feet than at your head. Near a stellar-mass hole the difference tears you up well outside the horizon; near a supermassive hole you could cross the horizon unharmed. The Numbers tab computes where a 2-metre body feels a lethal tide.
Do black holes evaporate? In theory, by Hawking radiation, but so slowly for any real hole (a solar-mass one is colder than the cosmic microwave background) that it does not matter. The Numbers tab shows the temperature and the (enormous) lifetime.

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