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

For the static-emitter redshift in a de Sitter patch, a stationary clock at radius $r_E$ satisfies $d{\tau }_E=\sqrt{{\alpha }_E}d{t}_E$. The geometry is stationary, so successive inward pulses emitted at the same radius have the same coordinate travel delay. Their coordinate arrival interval is therefore $\Delta t_O=\Delta t_E$. At the central receiver ${\alpha }_O=1$, givingFor $r_E=r_S/2$, this becomesThe pulses are redshifted at the centre. At fixed proper emission period, taking a stationary emitter toward the horizon gives $\Delta {\tau }_O\to \infty$, ${\nu }_O\to 0$ and $z\to \infty$. The inward coordinate travel delay also diverges. A stationary worldline at the horizon itself is not timelike, and the proper acceleration required to hold an emitter static diverges in this limit; the result concerns a sequence of idealized emitters with $r_E<r_S$, not a stationary physical lighthouse exactly on the horizon.