Past exam of the general relativity and cosmology course of the University of Oxford / 2012 / 2 / a

Take the unhalved geodesic LagrangianThe cyclic coordinates $t,\varphi$ give $d(2c^2\alpha \dot{t})/d\lambda =0$ and $d(2r^2{\sin }^2\theta \dot{\varphi })/d\lambda =0$. The other two Euler-Lagrange equations areThese reproduce the supplied geodesic equations. Expanding the derivatives and comparing with ${\ddot{x}}^{\mu }+{\Gamma }^{\mu }{}_{\nu \rho }{\dot{x}}^{\nu }{\dot{x}}^{\rho }=0$, in coordinates $(t,r,\theta ,\varphi )$, gives all nonzero Christoffel symbols:Here ${\alpha }^{\prime }=-2r/r_S^2$. This is the static patch of de Sitter spacetime, with $0\le r<r_S$ for the stationary observers used below. Apparent divergences at the polar axis, origin or horizon belong to the chosen coordinate chart; they are not themselves evidence of a curvature singularity.