Effective potential for timelike Schwarzschild geodesics

For equatorial timelike motion with specific angular momentum $h$,A circular orbit has $h^2=m{r}^2/(r-3m)$ and therefore requires $r>3m$.

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.

A timelike equatorial Schwarzschild geodesic with specific energy $E=1$ and angular momentum $J=4m$ obeys $(du/d\varphi {)}^2=2u(u-1/4{)}^2$ for $u=m/r$. Substituting $q=2\sqrt{u}$ reduces it to $(dq/d\varphi {)}^2=(q^2-1{)}^2/8$, integrating to the two displayed branches. One lies outside $r=4m$ and the other inside; each approaches the same unstable circular orbit exponentially in angle and only after infinite proper time. Linearization of $u^{\prime \prime }=1/16-u+3u^2$ about $u=1/4$ gives $\delta u^{\prime \prime }=\delta u/2$, confirming instability. These are separatrix approaches or departures, not stable oscillations around the circular orbit.

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.

A circular timelike geodesic of Schwarzschild radius $r>3M$ has coordinate angular velocityIts coordinate period is $2\pi \sqrt{r^3/M}$.

Over one circular orbit at radius $r>3M$, the orbiting clock measuresThe factor $r-3M$ combines gravitational and orbital kinematic time dilation.

The nearly circular orbit equationhas radial angular frequency $1-3m^2/h^2$ to leading order. The perihelion advances by $6\pi m^2/h^2$ per orbit.

For a small radial excursion about a stable circular timelike geodesic in Schwarzschild spacetime, the inverse radius oscillates with azimuthal frequency $\sqrt{1-6GM/(c^{2}R)}$. A satellite launched tangentially returns to its launch radius once per radial period. A later exact encounter with a point station at the launch azimuth requires the exact radial-to-azimuthal rotation ratio to be rational. An irrational ratio instead gives dense, arbitrarily close recurrence, which still threatens objects of finite size. The linearized rotation ratio is an approximation; exact orbital closure uses the actual orbit's radial advance.