sabfodel / physics

project · active · Rust

gargantua-rs

Ray tracer for light around a Schwarzschild black hole. No physics libraries.

Rendered Schwarzschild black hole — a dark shadow ringed by a thin accretion disk, with the far side of the disk lensed into an arc above the hole.
Fig. 1 Render from v0.1. The arc above the shadow is the far side of the disk, bent over the top by the hole; the glow beneath it is light that looped underneath.

result Rays with impact parameter below b = (3√3/2) rₛ ≈ 2.598 rₛ are captured — the shadow size the theory predicts, checked in the test suite.

source ↗ sabfodel/gargantua-rs

I wanted to know how the image in Interstellar was computed, so I had a go at it myself.

The physics

Working in units G=c=1G = c = 1 with rs=1r_s = 1, the null geodesic equation of the Schwarzschild metric reduces to a single Newtonian-looking equation:

d2xdλ2=−32 h2 xr5,h=∣x×v∣\frac{d^2 \mathbf{x}}{d\lambda^2} = -\frac{3}{2}\,\frac{h^2\,\mathbf{x}}{r^5}, \qquad h = \lvert \mathbf{x} \times \mathbf{v} \rvert

where hh is the conserved angular momentum. The r−5r^{-5} term is pure spacetime curvature: it doubles Newton’s prediction for light bending and creates the photon sphere at r=1.5 rsr = 1.5\,r_s, where light can orbit the hole.

What it does

Every pixel is an initial condition. The renderer launches one photon per pixel backwards from the camera and integrates its path. A ray either falls through the horizon (black), hits the thin accretion disk, or escapes to the background sky.

  • Hand-written RK4 integrator, generic over any acceleration field
  • Thin disk and procedural starfield
  • Parallel pixel evaluation with Rayon
  • PNG output at any resolution

What I learned

Conserved quantities do the heavy lifting. Using the conserved angular momentum hh turns the geodesic equation into one compact ODE — faster and numerically better behaved than the full set of Christoffel terms.

Backwards is the right direction. Camera to source, not source to camera — you only compute rays that actually land on screen.

Design for the next metric. New spacetimes plug in by implementing one trait; the integrator and renderer never change. That is what will make the Kerr version possible.

Limitations

  • Fixed step size. Rays grazing the photon sphere need finer steps than the rest — adaptive stepping is the next version.
  • No Doppler beaming or gravitational redshift yet, so the disk is unrealistically symmetric.
  • Non-rotating hole only.

Next

  • v0.2 — adaptive step size, photon-sphere diagnostics
  • v0.3 — Doppler beaming and gravitational redshift on the disk
  • v0.4 — Kerr metric: frame dragging and an asymmetric shadow

References

  • A. Riazuelo, Seeing relativity — I, arXiv:1511.06025
  • O. James, E. von Tunzelmann, P. Franklin, K. S. Thorne, Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar, arXiv:1502.03808