Drag the spin slider to spin up a 3D black hole and watch its photon-orbit and innermost-stable-orbit (ISCO) rings shrink for prograde paths and grow for retrograde paths, computed live from the exact Kerr formulas.
Every radius above is computed live from the exact Kerr black-hole solution (Bardeen, Press & Teukolsky 1972) at the spin you pick - not looked up from a table. At zero spin the numbers match the textbook Schwarzschild values (photon sphere 1.5 Rs, ISCO 3 Rs); as spin rises toward the astrophysical Thorne limit, the prograde rings pull in toward the horizon while the retrograde rings push outward.
Drag to orbit and scroll or pinch to zoom. For the artistic, non-spinning accretion-disk view see the Black Hole 3D Explorer; for the general spacetime-curvature grid see the Spacetime Curvature 3D Explorer.
Photon Sphere & ISCO 3D Explorer
This browser explorer spins up a 3D black hole and shows exactly where light and matter can still orbit it, computed live from the real Kerr equations as you drag the spin slider.
Every non-spinning (Schwarzschild) black hole has two fixed boundaries around its event horizon: the photon sphere at 1.5 Schwarzschild radii, where light can loop in an unstable circular path, and the innermost stable circular orbit (ISCO) at 3 Schwarzschild radii, inside which no stable orbit exists for ordinary matter. Spin the black hole and both boundaries split into a prograde and a retrograde version, following the exact closed-form Kerr solution published by Bardeen, Press and Teukolsky in 1972: as the dimensionless spin parameter a* rises toward the astrophysical Thorne limit of about 0.998, the prograde photon orbit and ISCO pull in toward the horizon (down to 1 and 1.237 gravitational radii), while the retrograde versions push outward (up to 4 and 9 gravitational radii). The event horizon itself shrinks with spin, and a surrounding ergosphere - the region where frame-dragging forces everything, even light, to co-rotate - always touches the horizon at the poles and reaches exactly 2 gravitational radii at the equator, for any spin.
- Drag the Kerr spin parameter slider (a* = 0 to 0.998) to watch every ring update live
- Jump straight to a*=0 (Schwarzschild) or the astrophysical Thorne spin ceiling (a*=0.998)
- Compare prograde vs retrograde photon orbits and ISCO rings side by side, each colour-coded
- Watch the purple frame-dragging arrows spin faster near the horizon as spin increases
- Read the exact computed radii in the facts panel, in both gravitational-radius (M) and Schwarzschild-radius (Rs) units
- Drag to orbit, scroll or pinch to zoom, fullscreen with the panel still readable
- Runs fully in the browser with the vendored three.js engine - no account, no upload
| Figure (a* = spin, M = GM/c^2, Rs = 2M) | Value |
|---|---|
| Photon sphere, Schwarzschild (a*=0) | 1.5 Rs = 3M (unstable circular photon orbit) |
| ISCO, Schwarzschild (a*=0) | 3 Rs = 6M (innermost stable orbit for matter) |
| Photon orbit at extremal spin | prograde -> 1M, retrograde -> 4M |
| ISCO at extremal spin | prograde -> 1M, retrograde -> 9M |
| ISCO at the Thorne astrophysical limit (a*~=0.998) | prograde ~= 1.237M (Kerr formula, computed live) |
| Ergosphere, equator (any spin) | 2M (fixed static-limit radius) |
Kip Thorne showed in 1974 that ongoing accretion-disk torque limits a real black hole's spin to about a* = 0.998 - true extremal spin (a*=1) is never physically reached, which is why the slider stops just short of it. The 2019 Event Horizon Telescope image of M87* and the 2022 image of Sagittarius A* both show a bright ring consistent with the general-relativity-predicted lensed photon orbit, the strongest present observational support for the geometry this scene visualizes. Frame-dragging - the effect pulling the arrows around the ergosphere - is the same Lense-Thirring effect that NASA's Gravity Probe B mission measured directly around Earth between 2004 and 2011, to about 19% precision; near a spinning black hole the same physics is vastly amplified by strong gravity, not separately measured.
For the artistic, non-spinning accretion-disk view of a black hole, open the Black Hole 3D Explorer; for a general teaching grid of spacetime curvature, open the Spacetime Curvature 3D Explorer.
Everything renders on your device with WebGL. The 3D engine loads once (about 0.7 MB) and is cached.
This is an idealized Kerr-vacuum-solution explorer, not a numerical-relativity or ray-traced simulation - the four radii shown are exact analytic GR results at the spin you pick, but the ergosphere is drawn as a simplified oblate-spheroid approximation of its true shape, and the orbiting markers are teaching cues, not a photon or particle trajectory integrator.
Frequently Asked Questions
What is the photon sphere?
The photon sphere is the radius at which light can orbit a black hole in an unstable circular path. For a non-spinning black hole it sits at 1.5 Schwarzschild radii (3 gravitational radii).
What is the ISCO?
The innermost stable circular orbit (ISCO) is the smallest radius at which ordinary matter can still orbit stably. For a non-spinning black hole it sits at 3 Schwarzschild radii (6 gravitational radii); inside it, orbits decay and matter spirals in.
Why do the rings split into prograde and retrograde versions?
Spin drags spacetime around with it. An orbit moving the same way as the spin (prograde) can stay stable closer in; an orbit moving against the spin (retrograde) is pushed to stay farther out. The exact Kerr formulas (Bardeen, Press and Teukolsky, 1972) compute both.
Can a black hole really spin all the way to a*=1?
No. Kip Thorne showed in 1974 that ongoing accretion-disk torque limits a real black hole's spin to about a*=0.998 - the slider stops just short of true extremal spin for that reason.
What is the ergosphere?
The ergosphere is the region around a spinning black hole where frame-dragging is so strong that nothing, not even light, can stay still relative to distant stars. It touches the event horizon at the poles and reaches exactly 2 gravitational radii at the equator, for any spin.
Is frame-dragging a real, measured effect?
Yes. NASA's Gravity Probe B mission measured this same effect (called the Lense-Thirring effect) directly around Earth between 2004 and 2011, to about 19% precision. Near a spinning black hole the same physics is vastly amplified by strong gravity - it has not been separately measured there.
Has anyone actually observed a photon sphere?
The 2019 Event Horizon Telescope image of M87* and the 2022 image of Sagittarius A* both show a bright ring consistent with the general-relativity-predicted lensed photon orbit - the strongest present observational support for this geometry.
How is this different from the Black Hole 3D Explorer?
The Black Hole 3D Explorer is an artistic, non-spinning accretion-disk-glow scene with no spin parameter. This page is a spin-parameter Kerr geodesic explorer - it computes the exact photon-sphere, ISCO, horizon, and ergosphere radii live from the real equations as you change the spin.