Schwarzschild Geodesics: In the Schwarzschild spacetime ds²=−(1−2M/r)dt²+(1−2M/r)⁻¹dr²+r²dΩ²,
geodesics are characterized by conserved energy E and angular momentum L. The effective potential is
V_eff(r) = (1−2M/r)(1 + L²/r²) for photons (ε=0) and V_eff = (1−2M/r)(1 + L²/r²) with −1→0
adjustment for massive particles. The photon sphere at r=3M is an unstable circular orbit
where photons orbit indefinitely — light with impact parameter b=L/E near 3√3 M spirals around.
The ISCO (innermost stable circular orbit) at r=6M is the last stable orbit for massive particles;
below this, matter inevitably plunges into the black hole. The event horizon at r=2M is the
point of no return where escape velocity equals c.