Past exam of the mathematical physics course of the University of Oxford / 2007 / B / 8 / ii / Solution

Put $f=1-2m/r$, use proper time $s$, and denote its derivatives by dots. The timelike geodesic equations, away from the Schwarzschild coordinate singularity, arewith normalization $f{\dot{t}}^2-f^{-1}{\dot{r}}^2-r^2{\dot{\theta }}^2-r^2{\sin }^2\theta {\dot{\varphi }}^2=1$. Spherical symmetry permits an equatorial plane, $\theta =\pi /2$, and the cyclic coordinates give $\boxed{E=f\dot{t},J=r^2\dot{\varphi }}$. Normalization then gives ${\dot{r}}^2=E^2-f(1+J^2/r^2)$.

For $J\ne 0$ and $u=m/r$, $du/d\varphi =-m\dot{r}/J$. ThusAt $E=1$, $J=4m$, the right side factors as $2u(u-1/4{)}^2$. Set $q=2\sqrt{u}$, so $u=q^2/4$ and $(q^{\prime }{)}^2=(q^2-1{)}^2/8$. Integrating the regions $0<q<1$ and $q>1$ gives, up to angular shifts and time reversal,The constant $u=1/4$, or $r=4m$, is the corresponding circular timelike orbit. The hyperbolic-tangent branch lies at $r>4m$ and approaches that orbit from outside as $|\varphi |\to \infty$; the hyperbolic-cotangent branch lies at $0<r<4m$ and approaches it from inside. Near the limit $u-1/4$ decays exponentially as $e^{-|\varphi |/\sqrt{2}}$. Since $\dot{\varphi }$ tends to $J/(4m{)}^2=1/(4m)$, the limiting approach also takes infinite proper time. Neither nonconstant branch reaches and joins the circular orbit at a finite angular parameter.

Differentiating the orbit equation gives $u^{\prime \prime }=m^2/J^2-u+3u^2$. A perturbation about $u=1/4$ obeys $\delta u^{\prime \prime }=\delta u/2$, showing that the circular orbit is unstable. The two explicit curves are its separatrix branches, not ordinary small stable oscillations. For $E=1$ the energy is marginally bound, so the outer critical branch is the limiting orbit between scattering and Schwarzschild marginally bound capture from infinity. Time-reversed inner and outer branches describe the corresponding approach or departure from that unstable orbit. These are the Explicit separatrices of the marginally bound Schwarzschild orbit.