Schwarzschild orbits and perihelion precession

For a spherically symmetric vacuum region with zero cosmological constant, the Schwarzschild geometry can be written using $f(r)=1-r_s/r$ and $r_s=2GM/c^2$. The coordinate r is the areal radius: spheres at fixed r have area $4\pi r^2$. It is not generally the proper radial distance measured along a static spatial slice. For an ordinary spherical star this vacuum metric applies only outside its surface, not throughout its matter-filled interior.

The familiar static chart applies to stationary observers only where $r>r_s$. A rotating source requires a different geometry, and a test body’s self-gravity is neglected here. Spherical symmetry lets us choose the orbit in the equatorial plane. Time-translation and rotation symmetries then supply two constants along an affine geodesic, making a four-dimensional motion problem considerably more manageable. Neither conserved constant should be confused with a locally measured coordinate velocity.

For timelike equatorial motion define specific energy $\mathcal E=fc^2dt/d\tau$ and specific angular momentum $\ell=r^2d\varphi/d\tau$. The normalization of four-velocity becomes a radial first integral. Far away, $\mathcal E$ tends to the special-relativistic energy per unit rest mass for a trajectory reaching that region. Near the source, a static observer’s measured particle energy also includes that observer’s gravitational clock normalization.

The effective radial potential includes a relativistic term absent from Newtonian mechanics. Setting $u=1/r$ gives an orbital equation with correction $3GMu^2/c^2$. For a weakly perturbed bound ellipse this correction shifts the radial oscillation frequency relative to the azimuthal angle, so successive closest approaches occur slightly beyond one full revolution. The orbital shape is therefore not one permanently fixed ellipse in an inertial plane.

To leading post-Newtonian order, the advance per radial orbit is $\Delta\varphi=6\pi GM/[a(1-e^2)c^2]$, where a and e refer to the leading Newtonian ellipse. The small parameter is $GM/[a(1-e^2)c^2]$, not just the eccentricity. A very eccentric orbit can visit strong gravity even if its semimajor axis is large. The formula excludes perturbations from other planets, source rotation, and source multipole structure.

Exact Schwarzschild geodesics also show that stable circular timelike orbits require $r>6GM/c^2$. The equality is marginal stability, while smaller circular timelike orbits that exist are unstable until the photon-orbit boundary at $3GM/c^2$. The event horizon is at $2GM/c^2$, a separate radius with a different meaning. Mixing these three scales hides the distinction between orbital stability, null trapping, and causal escape.