Schwarzschild marginally bound capture ($|h|<4M$)

A timelike geodesic with unit specific Killing energy can plunge from infinity into a Schwarzschild black hole only if its specific angular momentum satisfies $|h|<4M$. The limiting case $|h|=4M$ asymptotes to the unstable orbit at $r=4M$. A geodesic already inside the angular-momentum barrier can cross the event horizon with larger $|h|$, so the origin-at-infinity condition matters.

For an incoming timelike Schwarzschild geodesic with asymptotic Lorentz factor $k\ge 1$, the critical unstable circular orbit obeysAt $k=1$, $r_c=4\mu$ and $|h_c|=4\mu c$. For $k>1$, the critical radius lies between $3\mu$ and $4\mu$ and $|h_c|>4\mu c$. The marginally bound angular-momentum bound remains sufficient at larger energies but is no longer the exact threshold. Origin at infinity and an inward initial branch are essential; an orbit already inside a barrier is a different initial value problem.