Black Hole Lensing, Explained
After reading this you will know why a black hole bends the far side of its accretion disk up over the top of the shadow, where the √27 shadow diameter comes from, and how to read the bright ring and the lopsided disk in the simulator.
What the simulation shows
Point a camera near a Schwarzschild black hole and you do not see a black disk with a flat glowing ring around it. You see something stranger. The accretion disk is a thin, flat annulus that orbits in one plane, yet in the image its far side appears folded up and over the top of the dark shadow, and a second image of it curls under the bottom. A thin, very bright circle called the photon ring hugs the shadow's edge. One side of the disk glows brighter and bluer than the other.
Every one of these effects comes from a single fact: light does not travel in straight lines near a mass. The simulator fires one photon backward from each pixel of the camera and integrates its curved path until it either hits the disk, falls through the horizon, or escapes to the starfield. What you see is the true image a distant observer would record. No cheating, no painted-on halo.
The one hook: the far side comes over the top
Hold a coin flat and look at it edge-on. You see a line. Now imagine the space above and below the coin is curved so strongly that light leaving the coin's back edge can bend a full quarter turn and reach your eye from above. That is what happens here. The disk's back rim emits light upward and downward; the gravity of the hole redirects part of that light toward the camera, so the back of the disk appears lifted into an arch above the shadow. The famous image from the film Interstellar and the 1979 calculation by Jean-Pierre Luminet both show this arch. It is not artistic license. It is geodesics.
When this model applies and when it does not
The simulator uses the Schwarzschild metric: a black hole with mass but no spin and no charge. That is the right starting point for intuition and it is exact for a non-rotating hole. Use it to understand light bending, the shadow, the photon ring, and Doppler beaming.
Do not read it as a picture of a real astrophysical black hole in full. Real holes like M87* and Sagittarius A* spin, which drags spacetime around and makes the shadow slightly off-center and asymmetric (the Kerr metric). The disk here is an idealized thin, glowing annulus with a procedural turbulence pattern; real accretion flows are hot, magnetized, three-dimensional, and radiate across many wavelengths. The color map from white-hot inside to deep orange outside is illustrative, not a fit to a spectrum.
The toy model gets the geometry right and the physics of single photons right. It does not model the gas dynamics, magnetic fields, or emission spectrum of a real disk. Treat the image as a correct map of light paths through curved space, not as a photograph.
The formula behind the bending
Write u = 1/r, the reciprocal of a photon's distance from the center, and let \phi be the angle swept around the hole. The path of light in Schwarzschild geometry obeys a single ordinary differential equation.
Here r_s = 2GM/c^2 is the Schwarzschild radius, the radius of the event horizon. The left side, u'' + u = 0, is exactly the equation of a straight line in polar coordinates: with no mass, light goes straight. The whole effect of gravity sits in the term on the right, \tfrac{3}{2} r_s u^2. It is small when r is large (weak field) and grows fast as the photon approaches the hole.
That right-hand term is the general-relativistic correction. Newtonian gravity applied to light predicts only half of it, giving a deflection at the Sun's limb of about 0.87 arcseconds. The full equation above doubles it to 1.75 arcseconds, the value Eddington's 1919 eclipse expedition measured. The factor of two is the signature of curved spacetime.
Where the shadow and photon ring come from
Set the acceleration to zero, u'' = 0, and you get the condition for a circular photon orbit: u = \tfrac{3}{2} r_s u^2, which solves to r = \tfrac{3}{2} r_s. This radius is the photon sphere. A photon aimed just right can loop around the hole one or more times here before escaping toward the camera. Those looping photons pile up into the thin, bright photon ring you see touching the shadow.
The dark shadow is larger than the horizon because light that would have reached your eye gets swallowed. A distant observer sees a black disk whose apparent diameter is \sqrt{27}\, r_s \approx 5.196\, r_s. Work the number: \sqrt{27} = 3\sqrt{3} = 3 \times 1.7321 = 5.196. That is the size the Event Horizon Telescope measured for M87* in 2019, matching the prediction to within the error bars.
A worked example with the demo defaults
Load the simulator with its default settings and reason through the picture before it finishes rendering.
Reproducing the default view
- The disk spans an inner edge at
3r_s and an outer edge at12r_s. The inner edge sits at the innermost stable circular orbit for a Schwarzschild hole. Measured in shadow radii, the shadow's edge is at \sqrt{27}/2 \approx 2.598\, r_s, so the disk's inner rim at3r_s sits just outside the shadow, and the photon ring at1.5r_s radius lies well inside it. - Orbital speed follows the Keplerian rule. At the inner edge, v = \sqrt{GM/r} with r = 3 r_s = 6GM/c^2 gives v = c/\sqrt{6} \approx 0.408\, c. Gas at the inner edge moves at roughly 41% of light speed.
- Doppler beaming scales observed brightness by about the Doppler factor cubed. For an approaching speed of 0.408\, c viewed near edge-on, the boost factor is roughly \left(\tfrac{1}{1 - 0.408}\right)^3 \approx 4.8 on the approaching wing and the receding wing is dimmed by a similar factor. So one side can outshine the other by more than a factor of ten.
- Increase the inclination toward edge-on and the arch of the far side rises higher over the shadow. Near face-on it flattens into a nearly circular disk with a hole punched in the middle.
The table below lists orbital speed and the approaching-side beaming boost at three radii, so you can check the lopsided brightness by eye.
| Radius (r_s) | Speed (v/c) | Boost, approaching side |
|---|---|---|
| 3 | 0.4082 | 4.802 |
| 6 | 0.2887 | 2.776 |
| 12 | 0.2041 | 1.989 |
Reading and interpreting the image
Once the run plays, decode what you see in this order.
- Shadow
- The central dark region, apparent diameter \sqrt{27}\, r_s. Its size barely changes with inclination because it is set by geometry, not viewing angle.
- Photon ring
- The thin bright circle right at the shadow edge. It is light that looped near 1.5\, r_s. It stays sharp and circular from every angle.
- Upper arch
- The far side of the disk, lensed up and over the top. Its brightness and height grow as you tilt toward edge-on.
- Lower band
- A second lensed image of the disk seen under the shadow, formed by light that passed below the hole.
- Bright wing
- The side of the disk rotating toward you, brightened and blue-shifted by Doppler beaming.
Common mistakes when reading the picture
Do not read the arch above the shadow as a separate ring of gas. There is only one flat disk. The arch is the disk's own far side, bent into view. If you mentally flatten the image back down, the arch and the lower band collapse into the single annulus that is actually there.
A second error is expecting the bright wing to switch sides when you change zoom. It does not. The bright wing is set by the direction of rotation relative to your line of sight, which zoom does not change. Only inclination and the sense of rotation move it.
A third error is thinking the shadow shrinks as you pull the camera back. In an idealized image the shadow's angular size shrinks with distance, but its size relative to the disk stays fixed at \sqrt{27}\, r_s. In the simulator the zoom control changes the field of view, so the shadow and disk scale together.
Related simulations on this site
If the curved-space geodesics here interest you, the Galaxy Collision Simulator shows gravity acting on matter rather than light, with tidal tails from many bodies. For sensitivity to initial conditions, the Double Pendulum splits nearby starts into a diverging swarm. And for the wave side of physics, the Wave Interference ripple tank builds fringes from overlapping sources.
Frequently asked questions
Why does the black hole look bigger than its event horizon?
Because gravity bends the paths of photons that pass close by, more of your view falls into the hole than the horizon alone would block. The dark shadow has apparent diameter \sqrt{27}\, r_s \approx 5.2\, r_s, while the horizon diameter is only 2\, r_s.
What is the bright ring touching the shadow?
The photon ring. It is light that orbited near the photon sphere at 1.5\, r_s before escaping to your camera. Photons can loop around several times there, so the ring is actually a stack of ever-thinner sub-rings.
Why is one side of the disk brighter?
Doppler beaming. The gas orbits at a large fraction of light speed, up to about 0.41\, c at the inner edge. The side moving toward you is boosted in brightness by roughly the Doppler factor cubed, near a factor of five at the inner edge, and blue-shifted. The receding side is dimmed and reddened.
Is this what a real black hole looks like?
The geometry and the single-photon physics are correct for a non-rotating hole. Real black holes spin, so their shadows are slightly asymmetric, and real disks are turbulent magnetized plasma rather than a smooth glowing annulus. The image is an accurate light-path map, not a photograph of any specific object.
Does the shadow size depend on the black hole's mass?
The size in units of r_s is fixed at \sqrt{27}. The physical size scales with mass, since r_s = 2GM/c^2. A more massive hole has a proportionally larger shadow but the same shape.
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