The science
Kerr black hole. In units G = c = 1 and lengths in rg = GM/c², a hole with dimensionless spin a* has horizon r₊ = 1 + √(1 − a*²) and rotates the space around it: the horizon angular velocity is ΩH = a*/(2 r₊) (in c/rg). The ergosphere (shown optionally) is rE(θ) = 1 + √(1 − a*² cos²θ).
ISCO. The innermost stable circular orbit (Bardeen, Press & Teukolsky 1972) shrinks from 6 rg (a* = 0) to 1 rg (a* → 1, co-rotating) and grows to 9 rg for a retrograde disk. A thin disk radiates ηr = 1 − √(1 − 2/(3 rISCO)) of the rest energy it accretes (5.7% to 32% here).
Blandford–Znajek (1977). Magnetic field lines threading the horizon are twisted by frame dragging at ΩF ≈ ΩH/2 and carry a Poynting flux. To lowest order P = (κ/4πc) Φ² ΩH² with κ ≈ 0.05 (0.044 for a parabolic field, 0.053 for a split monopole); this lab adds the higher-order spin correction f(ΩH) = 1 + 1.38 ω² − 9.2 ω⁴ of Tchekhovskoy, Narayan & McKinney (2010). That gives P ∝ a*² for slow spin and a steeper rise near a* → 1, because ΩH = a*/(2r₊) grows faster than a* as the horizon shrinks (f < 1 partly offsets this).
MAD. When the disk drags in as much magnetic flux as it can hold (a magnetically arrested disk; Narayan et al. 2003, Tchekhovskoy et al. 2011), the flux saturates at ΦMAD ≈ 50 (Ṁ rg² c)1/2 (Gaussian units). Then η = P/(Ṁc²) ≈ (κ φ²/4π) ω² f with φ = 50, which reaches 100% and more for a* ≳ 0.9: the jet carries more energy than the gas brings in, because the extra energy is spin energy of the hole. The slider fΦ is Φ/ΦMAD; weakly magnetised (SANE) disks have fΦ ≪ 1 and η ∝ fΦ².
Mass scaling. At fixed field strength B, Φ ∝ B rg² ∝ B M² and ΩH ∝ 1/M, so P ∝ B² M² a*²; at fixed Ṁ/ṀEdd and fΦ the power scales as M, i.e. P/LEdd is the same for 10 M☉ and 10⁹ M☉. Every characteristic time scales as rg/c = 4.9 µs × (M/M☉). This "mass-invariant" picture underlies the black-hole fundamental plane and the timing scaling relations (Merloni, Heinz & Di Matteo 2003; Falcke, Körding & Markoff 2004; McHardy et al. 2006). Recent (2026) reports that black holes from stellar to supermassive appear to launch jets under similar rules fit this picture; this lab illustrates the scaling, it does not reproduce those observations.
Beaming and superluminal motion. A jet with Lorentz factor Γ, β = √(1 − 1/Γ²), seen at angle θ has Doppler factor δ = 1/[Γ(1 − β cosθ)]. A continuous jet brightens as δ2+α, a discrete blob as δ3+α (α = 0.7 assumed); the counter-jet is boosted by δ(π − θ). The display uses a compressive tone map (power 0.25) so the faint jet is still visible; the numeric ratio in the calculator is the true value. Light-travel-time effects make a knot appear to move at βapp = β sinθ / (1 − β cosθ) (Rees 1966). The maximum, Γβ, occurs at cos θ = β. Check: GRS 1915+105 (Mirabel & Rodríguez 1994), β = 0.92, θ = 70° gives 1.26c, close to the reported 1.25c.
What is simplified. The 3D picture uses a Newtonian-like Doppler shift for disk gas, no gravitational lensing or ray tracing (the photon ring is a decorative circle of ≈ 5 rg), a flat disk and a fixed field geometry (paraboloidal field lines, twist at ΩH/2). Disk temperature follows a Novikov–Thorne-like profile. The BZ formula is a scaling fitted to GRMHD simulations (McKinney & Gammie 2004; Tchekhovskoy et al. 2011), not a first-principles result; the physical jet may also draw power from the disk wind (Blandford & Payne 1982) and the real jet base is far more complex. Below ≈ 1% of Eddington the flow is a hot, geometrically thick, radiatively inefficient flow (as in M87* and Sgr A*, Event Horizon Telescope 2019, 2021); the picture dims the thin disk and brightens the corona to hint at this. For a* = 0, PBZ = 0 and no BZ jet is drawn. Jet Lorentz factors are free parameters here, not predictions.
Sources. Blandford & Znajek 1977; Bardeen, Press & Teukolsky 1972; Rees 1966; Tchekhovskoy et al. 2010, 2011; Narayan, Igumenshchev & Abramowicz 2003; Mirabel & Rodríguez 1994; Event Horizon Telescope Collaboration 2019, 2021 (M87*); Reid et al. 2014 (GRS 1915+105 mass).