Geodesics in Kerr spacetime are the free-fall paths of particles and light around a spinning black hole. Unlike the Schwarzschild case, they are not confined to a plane: orbits precess and oscillate in latitude, and photons can orbit multiple times before escaping. This notebook integrates those trajectories and renders the gravitational lensing they produce.

Equations of Motion

The Kerr metric has three conserved quantities along every geodesic: energy , angular momentum , and the Carter constant . With all three in hand the equations separate, with and decoupling completely:

where

Mino showed that reparameterizing by Mino time — absorbing into the affine parameter — fully decouples the and equations, making each an independent 1D problem. Energy and angular momentum follow from Killing symmetries (time-translation and axial); for bound orbits. confines motion to the equatorial plane; allows polar oscillation. The normalization for massive particles, for photons.

Integration Method

At turning points — where or — the radial and polar velocities must reverse sign. Rather than tracking sign changes explicitly, we differentiate once more and integrate the second-order system:

Now and pass smoothly through zero with no root-finding required. The full state is integrated with an adaptive Cash–Karp RK4(5) scheme at relative tolerance .

Boyer-Lindquist coordinates become singular at the event horizon (), so integration stops there. The outer shaded surface is the ergosphere — the region inside which frame-dragging is so strong that no observer can remain stationary, and from which rotational energy can be extracted via the Penrose process.

Gravitational Lensing

The same geodesic equations render what a Kerr black hole with an accretion disk actually looks like. A WebGPU fragment shader traces one null geodesic per pixel backward from the camera. Rays crossing the equatorial plane within the disk are shaded with a temperature profile and Doppler-beamed by the orbiting gas. The approach follows James et al. (2015).

Ray tracing method

The ray tracer uses the same Carter equations, specialized to null rays (, so only and matter). Each pixel’s ray direction is converted from Cartesian to Boyer-Lindquist, the null condition fixes , and are read off. The same second-order Mino-time formulation handles all -bounces automatically; the 5D state is advanced with fixed-step RK4, with step size shrinking as near the horizon.

The brightness asymmetry comes from the relativistic Doppler factor , where is the prograde Keplerian frequency. On the approaching side and the disk blazes; on the receding side it fades. Observed intensity scales as from the Lorentz invariant (Cunningham 1975). The thin bright ring at the shadow edge is the photon ring — light that has orbited once or more before escaping.

References

• Carter (1968) — discovery of the Carter constant and separation of the Kerr geodesic equations via the Hamilton–Jacobi method
• Bardeen, Press & Teukolsky (1972) — locally non-rotating frames in Kerr spacetime; Keplerian orbital velocity and ISCO formula used in the disk model
• Cunningham (1975) — relativistic transfer function for Kerr accretion disks; source of the intensity scaling from the Lorentz invariant
• Mino (2003) — Mino-time reparameterization that fully decouples the radial and polar equations
• James et al. (2015) — null geodesic ray tracing for the Interstellar visual effects (DNGR); inspiration for the backwards per-pixel ray tracer here
• Cash & Karp (1990) — the embedded RK4(5) scheme used for adaptive step-size control in the geodesic integrator