General relativity · gravitational lensing

Black Hole Light Lab

Every pixel launches a ray from the observer and integrates it backward through a nonspinning Schwarzschild spacetime. A ray can fall through the horizon, cross the accretion disk, or escape toward a gravitationally distorted celestial sphere.

Spacetime modelSchwarzschild

Shadow diameter

d_{\mathrm{sh}}

$6\sqrt{3}\,r_g\approx10.39\,r_g$

Viewing inclination

i

$i=4^{\circ}$

Disk inner edge

r_{\mathrm{in}}

$r_{\mathrm{in}}=6.0\,r_g$

Approaching-side shift

g_{\max}

$g_{\max}=0.73$

Physics tutorial

Imaging a black hole: the horizon is not what we see

BackgroundA black hole emits no light of its own. A telescope records the directions and intensities of distant light and hot surrounding matter after curved spacetime redirects them. The central darkness is therefore not a photographed horizon surface; it is the black-hole shadow cast by captured light paths.

Why it mattersSeparating the shadow, photon sphere, and multiple disk images turns a spectacular picture into a measurement of spacetime geometry.

Start with the essentials

Focus question: Why can a geometrically thin, nearly edge-on accretion disk appear above, below, and close to the center of the black-hole shadow?
One-sentence intuition: Every screen pixel corresponds to a null geodesic arriving at the observer. Integrating it backward reveals whether the light came from the celestial sphere, crossed the disk, or fell through the horizon.

Core mathematical model

Null geodesic of light

k^\nu\nabla_\nu k^\mu=0

The photon four-wavevector is parallel transported along itself. Numerically integrating the spatial projection of this equation supplies the bent path for every pixel.

Schwarzschild black-hole shadow

d_{\mathrm{sh}}=6\sqrt{3}\,r_g\approx10.39\,r_g

Here $r_g=GM/c^2$ . The shadow is larger than the event horizon because critical rays are strongly bent and captured before reaching it.

Observed frequency-shift factor

g=\frac{\sqrt{1-2r_g/r}}{\gamma\left(1-v_{\mathrm{los}}/c\right)}

The numerator supplies gravitational redshift and the denominator supplies relativistic Doppler shift from orbital motion. The approaching side is usually brighter and bluer.

Innermost stable circular orbit

r_{\mathrm{ISCO}}=6\,r_g

Stable circular motion cannot extend inside this radius for a nonspinning Schwarzschild black hole, so the default disk begins here.

Common difficulties

The dark region is not a horizon photograph

Typical misconceptionThe rim of the central dark circle is the event horizon, so measuring it directly gives the horizon size.

Better mental modelThe dark circle is a shadow formed by critical light paths. For a Schwarzschild black hole its diameter is about ten point four gravitational radii, substantially larger than the horizon diameter.

The bright ring is not a fixed material loop

Typical misconceptionThe narrow light around the shadow is a physical tube of glowing matter at one fixed location.

Better mental modelNear-critical rays travel around the black hole longer and can sample the background or disk repeatedly. The narrow enhancement is primarily a structure of light paths, not a material ring.

Run the experiment

01
Begin with a grazing view
Choose Grazing cinema view and compare the direct disk image with the higher-order images lifted above and below the shadow.
What to observe: One planar disk seems to wrap around the black hole because light from its far side bends into the observer’s line of sight.
02
Raise the viewing inclination
Increase the inclination until the disk is seen more directly.
What to observe: The upper and lower arcs merge toward the direct image, while relativistic Doppler beaming makes the approaching side brighter than the receding side.
03
Move the disk inner edge
Move $r_{\mathrm{in}}$ outward from the innermost stable circular orbit.
What to observe: Central emission weakens and a wider cavity opens between the disk and critical curve, while the geometry-set shadow scale stays fixed.