Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 36D (General Relativity)

Set $f(r)=1-2M/r$ and use proper time along a timelike geodesic. In the equatorial plane, the radial Christoffel symbols needed for a circular orbit areThe radial geodesic equation with $\dot{r}=0$ gives $(fM/r^2){\dot{t}}^2-fr{\dot{\phi }}^2=0$. Outside the horizon this implies

For stability use the conserved energy $E=f\dot{t}$ and angular momentum $L=r^2\dot{\phi }$. The normalization of the timelike tangent in the Schwarzschild metric gives ${\dot{r}}^2+V(r)=E^2$, with the effective potential for timelike Schwarzschild geodesicsAt a circular orbit $V^{\prime }=0$, so $L^2=M{r}^2/(r-3M)$, requiring $r>3M$. Differentiate $V$ holding this conserved $L$ fixed and then substitute the circular value:The linear radial perturbation equation is $\delta \ddot{r}=-V^{\prime \prime }(r)\delta r/2$. Thus circular timelike orbits are stable for $r>6M$, unstable for $3M<r<6M$, with the marginal case at $6M$.

For the orbiting astronaut at radius $R$, use $d\phi /dt=\sqrt{M/R^3}$ to obtainThe static Earth twin has $d{\tau }_0/dt=\sqrt{1-2M/R_0}$. During equal coordinate-time intervals of the long circular-orbit phase, the astronaut ages less precisely whenThe outward and return transfers have finite additional proper times; the stated long-orbit comparison is the limit in which the circular phase dominates them. At equality the circular phase contributes no age difference, so transfer details can decide the finite-trip result.