Schwarzschild Orbits
General relativity bends spacetime around a black hole. Light and massive particles follow geodesics — curves that extremize proper time. Unlike Newton, orbits precess and light curves around the photon sphere at r=3M.
Schwarzschild metric: ds² = −(1−2M/r)dt² + (1−2M/r)⁻¹dr² + r²dΩ²
Event horizon: r = 2M (red circle). Photon sphere: r = 3M (dashed).
d²u/dφ² + u = 3Mu² (light) | d²u/dφ² + u = M/L² + 3Mu² (matter)
Event horizon: r = 2M (red circle). Photon sphere: r = 3M (dashed).
d²u/dφ² + u = 3Mu² (light) | d²u/dφ² + u = M/L² + 3Mu² (matter)