Black Hole Spacetime

Event Horizons, Ergospheres & Frame Dragging

Schwarzschild vs Kerr — Equatorial Slice

Event horizonErgosphere (Kerr only)ISCOTest particle orbits

Key Surfaces

Schwarzschild rₛ

Photon sphere

1.5 rₛ

Schw. ISCO

3 rₛ = 6M

Kerr r₊

M + √(M²−a²)

Static limit

2M (equator)

Extremal Kerr

a = M

What's Different

Schwarzschild describes a non-rotating, uncharged black hole — a single event horizon at rₛ, a photon sphere where light orbits, and a fixed ISCO at 6M.

Kerr adds rotation. The horizon shrinks and is wrapped by an ergosphere — a region where spacetime is dragged faster than any observer can resist, so no rest frame exists. Inside, you must corotate.

Kerr also reveals an inner Cauchy horizon r₋, prograde and retrograde ISCOs at very different radii (2.32M vs 8.7M for a/M=0.9), and the ability to extract energy from the spin.

Things to Notice

All real astrophysical black holes are Kerr — collapse preserves angular momentum

The ergosphere bulges at the equator and meets the horizon at the poles

Frame dragging extends well beyond the ergosphere, falling off as 1/r³

Prograde orbits can reach much closer than retrograde orbits

Penrose's process and the Blandford–Znajek mechanism tap rotational energy

Spin shapes the photon ring observed by the Event Horizon Telescope

Where We See It

M87* (EHT 2019)

First image

The first directly imaged supermassive black hole. The bright ring is the photon sphere, gravitationally lensed into a circle; the dark central shadow is ~2.6 × the event horizon, broadened by lensing. Mass: ~6.5 × 10⁹ M☉.

Sagittarius A* (EHT 2022)

4 × 10⁶ M☉

Our galaxy's central SMBH, imaged after years of effort. Variability complicates reconstruction, but the ring structure matches Kerr predictions for ~4 × 10⁶ M☉, with a spin parameter constrained to a/M ≳ 0.5 — moderately rapidly rotating.

GW150914 ringdown (2015)

First GW + Kerr test

The first detected gravitational wave. After two black holes merged, the remnant 'rang down' through Kerr quasi-normal modes. The frequencies recovered a single Kerr black hole of ~62 M☉ with a/M ≈ 0.67 — direct confirmation of Kerr geometry from a real merger.

Penrose process

Ergosphere physics

Inside the ergosphere, an object can carry negative energy as measured at infinity. Splitting an object so the negative-energy fragment falls in extracts rotational energy from the black hole. Believed to power the relativistic jets of active galactic nuclei.