Black hole - 2D equatorial lab

Kerr geometry in the equatorial plane (G = c = M = 1)

A rotating black hole of spin a has its event horizon at r+ = 1 + √(1 − a²), a static limit (ergosphere edge) at r = 2 in the equatorial plane, a prograde and a retrograde circular photon orbit at rph = 2[1 + cos(⅔ arccos(−/+ a))], and the innermost stable circular orbit (ISCO) from the Bardeen-Press-Teukolsky formula. A disk that radiates from the ISCO outward releases the fraction η = 1 − EISCO of the rest mass: 5.7 % for a = 0, 42 % for a → 1 (prograde).

The rays are integrated numerically as null geodesics. In Mino time τ the equatorial photon motion with impact parameter b = L/E is (dr/dτ)² = R(r) = [r² + a² − ab]² − Δ(b − a)², dφ/dτ = −(a − b) + a(r² + a² − ab)/Δ with Δ = r² − 2r + a². Rays with |b| below the critical value are captured; for a = 0 that is 3√3 M. Far from the hole the deflection is the weak-field 4M/b.

Honest simplifications

Only the equatorial plane is shown, rays are not a rendered image of the sky, and the disk brightness uses a simple relativistic-beaming estimate (β from the circular-orbit speed in the Schwarzschild limit, intensity ~ g³) for illustration only. Disk thermodynamics, photon rings and the full 3D lensed image belong to the 3D page. Physical sizes use r_g = 1.4766 km per solar mass.

• Compare a = 0 and a = 0.99: the ISCO moves from 6M to about 1.5M and the efficiency jumps from 5.7 % to 26 %.
• Look at the rays: those on the side that co-rotates with the hole get closer before they are captured (smaller critical b).
• Choose the retrograde disk: the ISCO moves out to about 9M and the efficiency falls below 4 %.