Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 36D / a / Solution

This is the Schwarzschild-de Sitter spacetime. Its geodesic Lagrangian may be taken asTime-translation and rotational symmetry give three independent integrals of motion, which may be chosen asThe constant norm of the tangent is $g_{ab}{\dot{x}}^a{\dot{x}}^b=-\kappa$, where $\kappa =1$ for a timelike geodesic parametrized by proper time and $\kappa =0$ for a null geodesic. Substitution givesand hence