Spacetime Curvature Well
The rubber-sheet analogy: mass curves spacetime, and light follows the curved geometry
Photon sphere 1.5 Rs
Event horizon 1.0 Rs
Accretion disk
Illustrative Schwarzschild light tracks: finite boundaries, normalized distances and an experimental massive-particle branch.
The rubber-sheet analogy: mass curves spacetime, and light follows the curved geometry
Why is the critical b about 2.598 Rs while the photon sphere is at 1.5 Rs?
The impact parameter describes an incoming light ray’s conserved angular momentum relative to its energy; it is not the ray’s closest distance to the hole. In the nonrotating Schwarzschild model, the unstable circular light orbit is at 1.5 Rs. The critical incoming photon parameter is 3√3/2 Rs, about 2.598 Rs.
This page samples a finite exterior orbit equation, not a complete ray-traced black-hole image. The massive branch keeps an inherited initialization that has not been validated for a physical 0.3c particle. Disk colors, camera motion and playback duration are illustrations; no elapsed physical time, spin or error estimate is computed.
At 10 solar masses the page uses Rs ≈ 2.95 × 10 = 29.5 km, so the photon sphere is about 44.25 km and the nominal critical impact parameter about 76.64 km. Increase mass to 20: these kilometer scales double, while a track drawn in Rs units keeps the same shape. Compare b = 2.50 with b = 3.00 in photon mode and inspect the reported numerical boundary.
With Rs = 1 and geometric mass M = 0.5, the photon branch integrates u″ = −u + 3Mu² for u = 1/r using fourth-order Runge–Kutta angular steps of 0.005. It starts at r = 25 Rs and stops near r = 1.064 Rs inward, near 25 Rs outward, or after 22,000 steps. Classification now describes that termination, not an impossible horizon crossing.
The launch angle rotates the orbital plane coordinates in degrees; it does not vary the initial direction relative to the same position. Mass changes the kilometer label, not the normalized integration. CSV coordinates are in Rs and use the saved shot controls; changing a slider after a shot does not alter its export.
NASA distinguishes the event horizon, photon sphere and surrounding luminous matter. The drawn disk is decorative, and the blue circle marks a radius rather than an observed photon ring or black-hole shadow. NASA: black-hole anatomy
The University of Colorado Schwarzschild visualization describes the unstable light orbit at 1.5 Schwarzschild radii and distinguishes nonrotating geometry from rotating black holes. A trajectory near that radius is not evidence of a stable orbit. JILA: journey into a Schwarzschild black hole
Event horizon, photon sphere and luminous surrounding material; does not validate this solver.
Nonrotating geometry and the unstable circular light orbit at 1.5 Rs.