Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 34D (General Relativity)

Let $f(r)=1-r_s/r$ and parametrize the timelike geodesic by proper time $\tau$. Time-translation and rotational symmetry conserve $E=f\dot{t}$ and $h=r^2\dot{\varphi }$ in the equatorial plane. The normalized four-velocity obeysSubstitution and multiplication by $f$ give ${\dot{r}}^2+f(1+h^2/r^2)=E^2$, orThe dot denotes a proper-time derivative, and $E,h$ are energy and angular momentum per unit rest mass.

At a circular orbit, $V^{\prime }=0$. Differentiating the Schwarzschild effective potential at fixed $h$ givesA timelike circular orbit requires $r>3r_s/2$. Evaluating the second derivative after this substitution givesA stable nondegenerate minimum therefore requires $r>3r_s$. The boundary $r=3r_s$ is the marginal innermost stable circular orbit, not a strict stable minimum.

For Alice's circular geodesic, the conserved quantities or radial geodesic equation give $\Omega =d\varphi /dt=\sqrt{r_s/(2r^3)}$. ConsequentlyBob at rest far away has $d{\tau }_B/dt\simeq 1$. Thus the ratio of ageing rates isFor Alice's stable orbit $r>3r_s$, the denominator factor exceeds $1/2$, proving $k<\sqrt{2}$. Both gravitational time dilation and Alice's orbital motion enter this proper-time ratio.