● Black Hole Simulator
Spacetime Metric: This engine integrates the Boyer-Lindquist form of the Kerr-Newman metric, describing a rotating, charged black hole. The line element is:
where Σ = r²+a²cos²θ, Δ = r²-2Mr+a²+Q², a = J/M (spin parameter), Q = electric charge.
Ref: Kerr, R.P. (1963). Phys. Rev. Lett. 11, 237. Newman, E.T. et al. (1965). J. Math. Phys. 6, 918.
Geodesic Integration: Null geodesics (photon paths) are traced by numerically integrating the geodesic equations using the Hamilton-Jacobi formalism with Carter's constant of motion:
A 4th-order Runge-Kutta integrator (RK4) with adaptive step control propagates rays backward from the camera plane through the curved spacetime. Impact parameter b = L/E determines photon capture.
Ref: Carter, B. (1968). Phys. Rev. 174, 1559. Bardeen, J.M. (1973). Black Holes (Les Houches).
Event Horizon & ISCO: The outer event horizon of a Kerr BH is r+ = M + √(M²-a²). The innermost stable circular orbit (ISCO) determines the inner disk edge:
where Z₁ = 1+(1-a²/M²)^(1/3)[(1+a/M)^(1/3)+(1-a/M)^(1/3)], Z₂ = √(3a²/M²+Z₁²). Sign: − prograde, + retrograde.
Ref: Bardeen, J.M., Press, W.H., Teukolsky, S.A. (1972). ApJ 178, 347.
Accretion Disk: Disk emissivity follows the Novikov-Thorne thin-disk model. Flux radiated per unit area:
Colour temperature T(r) ∝ F(r)^(1/4) via the Stefan-Boltzmann law. Doppler beaming and gravitational redshift modify the observed flux by factors dependent on the photon four-momentum and local velocity field.
Ref: Novikov, I.D. & Thorne, K.S. (1973). Black Holes (Les Houches). Page, D.N. & Thorne, K.S. (1974). ApJ 191, 499.
Doppler & Gravitational Redshift: Observed frequency ratio:
Combined with special-relativistic Doppler beaming (Penrose-Terrell rotation), the approaching (blue) side of the disk appears brighter due to relativistic beaming factor δ = ν_obs/ν_em.
Ref: Luminet, J.-P. (1979). A&A 75, 228. Müller, A. & Camenzind, M. (2004). A&A 413, 861.
Hawking Radiation: Black holes emit thermal radiation with temperature:
where κ_surface is the surface gravity. For Kerr: T_H = (r+ − r-) / (4π(r+² + a²)) in natural units. The simulation renders this as a faint quantum glow near the horizon.
Ref: Hawking, S.W. (1975). Commun. Math. Phys. 43, 199.
Relativistic Jets (Blandford-Znajek): Magnetically-driven jets extract rotational energy from the ergosphere via the Blandford-Znajek mechanism. Jet power:
where Φ_BH is the magnetic flux threading the horizon, Ω_H = a/(2Mr+) is the angular velocity, and κ ≈ 0.044 is a numerical factor. Jets are visualized as bipolar collimated flows with synchrotron emissivity.
Ref: Blandford, R.D. & Znajek, R.L. (1977). MNRAS 179, 433.
EHT Photon Ring: The photon sphere at r_ph = 2M(1+cos(2/3·arccos(-a/M))) gives rise to the characteristic photon ring seen in Event Horizon Telescope images. Successive sub-images arise from photons completing n half-orbits.
Ref: Event Horizon Telescope Collaboration (2019). ApJL 875, L1. Johnson, M.D. et al. (2020). Science Advances 6, eaaz1310.
Rendering Method: Each pixel casts a ray in the camera's local orthonormal frame. The ray is converted to Boyer-Lindquist coordinates via the observer's tetrad. Integration halts if r < r+ (horizon capture) or r > r_max (escape). The accretion disk intersection test uses the equatorial plane z=0 with inner/outer boundaries at r_ISCO and r_out.
Accuracy Statement: This simulation implements physically motivated approximations of general relativistic ray-tracing. It is intended for educational and illustrative purposes. Full accuracy requires GPU-accelerated integration (e.g. GYOTO, GRMONTY codes). Numerical integration uses 80-step RK4 per ray with adaptive sub-stepping near the horizon (step factor × 0.1 for r < 3M).
Coordinate System: Boyer-Lindquist coordinates (r, θ, φ) with geometrised units G = c = 1. Mass M defines the length scale: 1M = GM_BH/c² ≈ 1.477 km × (M/M☉).
Limitations: (1) 2D screen-space rendering uses analytic approximation for gravitational lensing beyond 3M. (2) MHD turbulence in the corona is parameterised, not solved. (3) Hawking radiation is rendered symbolically (temperature computed exactly, visual intensity is illustrative). (4) Charge effects (Reissner-Nordström) affect horizon/ISCO calculations but are not yet fully incorporated into geodesics.
Author: Developed by Astrophyzix Digital Observatory. Physics verified against published GR literature. All equations implemented in JavaScript float64 arithmetic.
Version: 2.1 — Kerr-Newman metric, RK4 geodesics, Novikov-Thorne disk, BZ jets, Hawking radiation, EHT photon ring, multi-spectral rendering.
License: Astrophyzix License. Free for educational use on Astrophyzix.org - Not to be used on other platforms without permission. Cite: "Black Hole Simulation by Astrophyzix Digital Observatory , 2026." Physical formulae are from peer-reviewed sources cited above.
Primary references: MTW "Gravitation" (1973); Chandrasekhar "The Mathematical Theory of Black Holes" (1983); Luminet (1979); EHT Collaboration (2019).