This is the equatorial (spin-axis-perpendicular) plane of a rotating Kerr black hole — the same geometry the 3D simulator's disk and horizon sit in, viewed edge-on from above instead of in perspective. The outer horizon, photon sphere and ISCO all shrink toward the centre as spin a* increases (frame-dragging lets prograde orbits and photons survive closer in):
r₊ = M(1 + √(1−a*²)) outer horizon
r_ergo = 2M ergosphere, equatorial plane only
r_ph = 2M[1 + cos( (2/3)·acos(−a*) )] prograde photon sphere
r_isco = M(3 + Z₂ − √((3−Z₁)(3+Z₁+2Z₂))) prograde ISCO (Bardeen–Press–Teukolsky)
The ergosphere boundary sits at exactly r = 2M in this plane for any spin — a real feature of Kerr geometry, not an approximation — so as a* grows the gap between it and the shrinking horizon (where frame-dragging is strong enough that nothing can stay still, even light) visibly widens.
The bent light rays are a genuine numerical solution of the Schwarzschild photon-geodesic equation d²u/dφ² = 3Mu² − u (u = 1/r), integrated with RK4 for a fan of impact parameters. Rays with impact parameter below b_crit = 3√3 M spiral into the horizon; rays above it swing past and escape — exactly the boundary that produces a black hole's dark "shadow". This bending is computed for the non-spinning case for clarity; a full Kerr geodesic solve would make prograde and retrograde rays asymmetric, which this 2D panel does not attempt.
- M — black hole mass in solar masses; sets every length scale below via M(metres) = GM/c².
- a* — dimensionless spin 0 (Schwarzschild) to just under 1 (extremal Kerr).
- Radii are drawn to scale relative to each other (in units of M), not to a fixed pixels-per-km scale — the view stays framed the same way as you change the mass slider.