Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 36A / Solution

Vary the curve by $\delta x^c$, fixing endpoints. Integration by parts in the kinetic functional gives its Euler-Lagrange equationsMultiplying by $g^{cd}/2$ and symmetrizing the velocity product givesThere is no term proportional to $\dot{x}$ from parameter reparametrization, so this is the affine geodesic equation.

In units $c=1$, the static spherical vacuum metric is Schwarzschild with $f(r)=1-2GM/r$. For example, starting from $-e^{2\Phi }d{t}^2+e^{2\Lambda }d{r}^2+r^{2}d{\Omega }^2$, the vacuum radial equations give $[r(1-e^{-2\Lambda }){]}^{\prime }=0$ and ${\Phi }^{\prime }+{\Lambda }^{\prime }=0$. Asymptotic flatness and the Newtonian mass fix $e^{2\Phi }=e^{-2\Lambda }=1-2GM/r$, giving the displayed metric. Time and azimuthal coordinates are cyclic in its geodesic Lagrangian, soNormalize the velocity norm to $-k$, with $k=1$ for proper-time parametrized massive motion and $k=0$ for null motion. In the equatorial plane this gives ${\dot{r}}^2+f(k+h^2/r^2)=E^2$. For a nonradial orbit, $h\ne 0$ and $u=1/r$ imply $\dot{r}=-hdu/d\varphi$. ThereforeDifferentiating with $f=1-2GMu$ yieldsIt extends across isolated turning points by continuity. The original PDF's later $f=1-GMu$ is inconsistent with this claimed equation: that choice gives $u^{\prime \prime }+u=kGM/(2h^2)+(3GM/2)u^2$. The intended Schwarzschild factor is $1-2GMu$, which we retain.

For massive motion put $\ell =h^2/(GM)$. The Newtonian solution is $u_0=(1+e\cos \varphi )/\ell$. The resonant cosine forcing in $3GM{u}_0^2$ is $6GMe\cos \varphi /{\ell }^2$, producing the secular correction $3GMe\varphi \sin \varphi /{\ell }^2$. Resumming it into a shifted phase gives the precessing-ellipse approximationFor finite eccentricity there are also bounded corrections of this same small order: one particular correction is $3GM(1+e^2/2)/{\ell }^2-GM{e}^2\cos (2\varphi )/(2{\ell }^2)$, besides adjustable homogeneous terms. Thus the simple displayed ellipse captures the secular precession rather than the complete first-order shape correction. Consecutive periapses are separated by $2\pi /\alpha$, so the Schwarzschild perihelion precession per radial orbit is approximately $6\pi GM/\ell$, forward in the direction of motion.