Effective potential for timelike Schwarzschild geodesics

h

\frac{1}{2} \overset{r}{˙}^{2} + V_{eff} (r) = \frac{E ^{2} - 1}{2}, V_{eff} = - \frac{m}{r} + \frac{h ^{2}}{2 r ^{2}} - \frac{m h ^{2}}{r ^{3}} .

A circular orbit has

h^{2} = m r^{2} / (r - 3 m)

and therefore requires

r > 3 m

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

∣ h ∣ < 4 M

. The limiting case

∣ h ∣ = 4 M

asymptotes to the unstable orbit at

r = 4 M

. A geodesic already inside the angular-momentum barrier can cross the event horizon with larger

∣ h ∣

, so the origin-at-infinity condition matters.

r > 3 M

\frac{d ϕ}{d t} = \frac{M}{r ^{3}} .

Its coordinate period is

2 π r^{3} / M

r > 3 M

, the orbiting clock measures

Δ τ = 2 π \frac{r ^{2} ( r - 3 M )}{M} .

The factor

r - 3 M

combines gravitational and orbital kinematic time dilation.

u^{''} + u = \frac{m}{h ^{2}} + 3 m u^{2}

1 - 3 m^{2} / h^{2}

to leading order. The perihelion advances by

6 π m^{2} / h^{2}

per orbit.

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