Timelike orbits and innermost stable circular orbits

Use proper length parameter $s=c\tau$ along the massive body’s worldline. Normalize its energy as total Killing energy divided by rest energy, and its angular momentum as angular momentum divided by rest mass times c. The first quantity is dimensionless and the second has length units. The unit timelike normalization then combines those constants into a radial equation with an effective potential. Keeping this normalization explicit avoids importing a Newtonian energy convention whose additive constant or factor of two differs.

A circular orbit requires zero radial velocity and zero radial force, so its energy squared equals the potential and the radial derivative of the potential vanishes. That derivative must be taken while holding angular momentum fixed. Solving it gives an angular momentum that depends on the chosen circular radius. Substituting this radius-dependent result before differentiating would instead compare different orbits and produce an incorrect stability test. The positivity and finiteness of the resulting constants already restrict the circular family.

The circular-family formulas below require radius greater than three m for finite timelike energy and angular momentum. Their divergence as that radius is approached signals the photon-orbit boundary, not the stable-orbit boundary. Linearizing radial motion about a circular solution shows that positive second derivative of the potential gives a restoring displacement. The sign changes at six m: outer circular orbits are radially stable, inner timelike circular orbits are unstable, and the boundary orbit is marginal at this order.

Thus there are timelike circular geodesics between three and six m, even though there are no stable ones there. The ISCO marks the loss of a local radial minimum, not the disappearance of every circular solution. Stability here means small radial geodesic perturbations of the ideal test-body orbit; an accreting plasma, a finite-mass companion, or a spinning body introduces additional dynamics. The coordinate angular frequency and the locally measured speed also use different clocks and cannot be equated by simply multiplying by radius.

The angular frequency relative to asymptotic coordinate time has the familiar square-root dependence on mass and radius, but its local interpretation changes with clock normalization. A static observer at the orbit measures a tangential speed larger than coordinate radius times coordinate frequency by the inverse square root of f. At the ISCO that speed is half the speed of light, despite the orbit remaining well outside the horizon. Such a local measurement is defined only for static observers in the exterior.

The difference between rest energy and conserved circular-orbit energy measures ideal binding energy relative to infinity. Evaluated at the Schwarzschild ISCO, it is about 5.72 percent of rest energy. Calling this a universal luminosity efficiency would add unjustified assumptions. A thin-disk estimate needs a stress prescription, radiative transport, and conditions near its inner edge; radiation reaction also changes the test-body evolution. The oscillator lab captures restoring motion with positive stiffness only, so it can illustrate the stable side but cannot locate an ISCO or display the unstable geodesic branch.