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

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.

For radial null geodesics in a de Sitter static patch, the conserved time quantity and the stated normalization at the origin give $\alpha \dot{t}=1$. The null condition then givesThe emitted outgoing ray selects the positive branch, soThe ingoing branch would have $\dot{r}=-c$ instead. Set the affine parameter to zero at emission. Then $r=c\lambda$, and integrating $dt/dr=1/[c(1-r^2/r_S^2)]$ givesThe coordinate time diverges as $r\to r_S$, although the outgoing affine parameter reaches $r_S/c$ there. This is a coordinate-horizon effect: a local stationary observer inside the patch measures radial distance $dr/\sqrt{\alpha }$ and proper time $\sqrt{\alpha }dt$, whose ratio along the ray is still $c$.

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.