Isometric embedding of the optical geometry of a Schwarzschild black hole into $\mathbb{R}^3$
The optical geometry of a Schwarzschild spacetime is the Riemannian metric whose geodesics correspond to light rays. For the equatorial plane, this geometry is rotationally symmetric with metric ds2=(1−rs/r)21dr2+1−rs/rr2dθ2. This program computes a numerical isometric embedding of this 2D geometry into R3 using a spring-charge mesh simulation on a half-edge data structure.
The embedding asymptotes to flat Euclidean space far from the black hole and to a hyperbolic funnel of curvature −1/4 near the event horizon. The photon sphere at r=3rs/2 appears as the neck of the surface. A dat.gui panel controls the charge repulsion strength, which governs mesh quality.
Controls: Click and drag to orbit, scroll to zoom.
