Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 309 / 2 / ii

Set $f=1-2M(v)/r$, so $f_r=2M/r^2$ and $f_v=-2M^{\prime }/r$, where $M^{\prime }=dM/dv$. Dots in this solution mean derivatives with respect to proper time $\tau$. The geodesic Lagrangian becomesThe Euler-Lagrange equation for $r$ gives $\ddot{v}+(f_r/2){\dot{v}}^2-r({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)=0$. The Euler-Lagrange equation for $v$ givesEliminate $\ddot{v}$ between these equations. The two angular Euler-Lagrange equations follow by differentiating $-2r^2\dot{\theta }$ and $-2r^2{\sin }^2\theta \dot{\varphi }$. Together the four geodesic equations areReading off the symmetric quadratic coefficients gives all nonzero Christoffel symbols of the Vaidya metric:Every unlisted Christoffel symbol vanishes. The lower-index symmetry is that of the Levi-Civita connection. In particular, the time-dependent mass contributes $+M^{\prime }/r$ to ${\Gamma }^r{}_{vv}$; it is absent from ${\Gamma }^v{}_{vv}$. There are eleven independent nonzero entries, or fifteen when both lower-index orders are counted.