Canonical picture of a spinning black hole's spacetime structure: outer and inner horizons, ergosphere, and the prograde/retrograde ISCO radii — as functions of the spin parameter a. Cross-reference target for the other OCS tools that mention horizon or ISCO.
Equatorial looks down the spin axis (everything appears as circles); Polar looks edge-on and shows the ergosphere's characteristic equatorial bulge.
The Kerr metric (Roy Kerr 1963) describes the spacetime around any uncharged, rotating black hole. Its key structural surfaces — horizons and ergosphere — and the radii of marginally stable circular orbits depend on a single dimensionless spin parameter a = Jc/(GM²), ranging from 0 (no spin, Schwarzschild) to 1 (maximally rotating, naked singularity at the limit). Astrophysical black holes are bounded below 1 by Thorne's limit, a ≲ 0.998.
Outer horizon: r₊ = r_g(1 + √(1−a²)). The event horizon. Nothing inside can causally affect the outside. At a = 0 it sits at 2 r_g (Schwarzschild radius); at a = 1 it shrinks to r_g.
Inner horizon (Cauchy horizon): r₋ = r_g(1 − √(1−a²)). Inside this surface, the Kerr extension contains closed timelike curves and the singularity becomes timelike. For a < 1 it sits between 0 and r_g; at a = 1 it merges with r₊.
Ergosphere: the region between the outer horizon and the static limit r_erg(θ) = r_g(1 + √(1−a²cos²θ)). The static limit touches the horizon at the poles and bulges out to 2 r_g at the equator, producing the characteristic oblate shape visible in the polar-slice view. Inside the ergosphere, no observer can remain static — frame-dragging forces co-rotation with the hole.
Prograde ISCO: innermost stable circular orbit for matter orbiting in the same sense as the hole's rotation. Sits at 6 r_g for a = 0; shrinks to r_g for a = 1 (Bardeen, Press & Teukolsky 1972). This is the inner edge of any thin accretion disk.
Retrograde ISCO: same for counter-rotating orbits. Moves outward from 6 r_g (Schwarzschild) to 9 r_g (a → 1) — retrograde orbits get less stable as the hole spins faster.
Equatorial slice looks down the spin axis. All structures appear as circles. This is the natural view for thinking about accretion disks and orbital mechanics.
Polar slice looks edge-on, with the spin axis vertical. The ergosphere reveals its oblate (squashed-sphere) shape — bulging at the equator, flattening at the poles where it touches the horizon. The inner and outer horizons are still circles (axially symmetric), but the ergosphere is not.
The maximum fraction of a black hole's mass-energy that can be extracted via the Penrose process (test particle in, two particles out, one with negative energy) is η = 1 − √((1 + √(1−a²))/2). At maximal spin this is ~29% — three orders of magnitude above any chemical or nuclear conversion efficiency. This is the theoretical foundation of the BZ/Penrose Power Calculator.
Half a dozen OCS tools mention "ISCO", "horizon", or "ergosphere" — sometimes in Kerr, sometimes in Schwarzschild approximation, sometimes inconsistently. This tool provides the single canonical picture they can all reference. Tools 1 (BZ/Kardashev), 4 (Bekenstein–Landauer–Lloyd Panel 2), 6 (Time Dilation), and 11 (Tidal Disruption) all use Kerr ISCO or horizon radii computed identically to the values shown here.
Three independent methods constrain real BH spin parameters. Iron Kα reflection spectroscopy fits the relativistically broadened 6.4 keV line in X-rays from accretion-disk reflection; results: MCG-6-30-15 has a/M > 0.917 (Brenneman & Reynolds 2006), and a sample of ~25 AGN spans a/M = 0.7–0.99 (Reynolds 2021 review). Continuum-fitting models the disk thermal blackbody peak; works best for stellar-mass BHs in X-ray binaries (Cyg X-1 at a/M > 0.95). EHT polarimetry (Event Horizon Telescope 2021, ApJL 910:L12) constrained M87* spin orientation; spin magnitude not yet definitive but consistent with a/M ≳ 0.5. Most measured BHs cluster at high spin, consistent with prolonged accretion histories.
The LIGO/Virgo/KAGRA O4 catalog (closed Jan 2025) contains ~200 binary black hole mergers. The spin distribution of merger remnants peaks near a/M ≈ 0.7 (the natural value from orbital angular momentum at merger). Pre-merger component spins are individually less constrained but show a population mean χ_eff ≈ 0.06 with broad spread — consistent with either isolated binary evolution (small spins) or dynamical assembly in dense clusters (random orientations). The first BBH detection GW150914 had remnant spin a/M = 0.67 ± 0.05; the most extreme GW190521 had remnant a/M ≈ 0.7 at ~150 M☉.
Thorne 1974 (ApJ 191:507) showed that radiation back-reaction on accreting matter caps the spin a real astrophysical BH can reach at a/M ≈ 0.998, set by photon capture asymmetry in the accretion process. Above this, accreting matter spins the hole DOWN rather than up because of the increased capture of retrograde photons emitted by the disk. So the slider in this tool maxes at 0.998 not as a numerical limit but as the physically achievable ceiling. The Penrose extractable fraction at the Thorne limit is ~27% — almost the full theoretical maximum (29.3% at a=1). Whether any astrophysical hole has actually reached the Thorne limit is observationally constrained but not definitively resolved.