Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 36D (General Relativity)

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

For a nonradial null geodesic,It tends to $-\infty$ as $r\to 0^{+}$, rises to one maximum, and tends to $-L^2{H}^2$ as $r\to \infty$. A circular null geodesic requires $V^{\prime }(r)=0$, soSince $V^{\prime \prime }(3m)<0$, this photon sphere is unstable. Its energy satisfies $E^2=L^2(1/(27m^2)-H^2)$.

The function $f$ tends to $-\infty$ both as $r\to 0^{+}$ and as $r\to \infty$. Its only positive stationary point isand it is a maximum withTherefore $f$ has two positive simple roots exactly whenAt $r_c=1/(\sqrt{3}H)$,Since $f$ is positive precisely between its two positive roots, this proves

The simple zeros $r_{-}$ and $r_{+}$ are Killing horizons: respectively the black-hole event horizon and the cosmological horizon of the static region.