General Relativity: An Introduction for Physicists · Problem 15.32

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · de-sitter-event-horizon-line-element

Problem

Consider the cosmological line element

ds^2 = c^2\,dt^2 - e^{2t/b}(dr^2 + r^2\,d\theta^2 + r^2\sin^2\theta\, d\phi^2).

Light signals from a galaxy at coordinate distance

r

are emitted at epoch

t_1

and received by an observer at epoch

t_0

. Show that

\frac{r}{bc} = e^{-t_1/b} - e^{-t_0/b}.

For a given

r

, show that there is a maximum epoch

t_1

and interpret this result physically. Show that a light ray emitted by the observer asymptotically approaches the coordinate

r = bc

but never reaches it.