Causal Structure

The causal structure is determined by propagation of light and is best understood in terms of \emph{conformal diagrams}. In this section we construct and analyze those for a number of important model cosmological solutions (which are assumed to be already known), following mostly the exposition of .

In terms of comoving distance $\tilde{\chi}$ and conformal time $\tilde{\eta}$ (in this section they are denoted by tildes) the two-dimensional radial part of the FLRW metric takes form \begin{equation} ds_{2}^{2}=a^{2}(\tilde{\eta})\big[d\tilde{\eta}^2 -d\tilde{\chi}^2\big]. \label{ds2Dconf} \end{equation} In the brackets here stands the line element of two-dimensional Minkowski flat spacetime. Coordinate transformations that preserve the \emph{conformal} form of the metric \begin{equation*} ds_2^2 =\Omega^{2}(\eta,\chi)\big[d\eta^2 -d\chi^2 \big], \end{equation*} are called conformal transformations, and the corresponding coordinates $(\eta,\chi)$ -- conformal coordinates.

Problem 1

Show that it is always possible to construct $\eta(\tilde{\eta},\tilde{\chi})$, $\chi(\tilde{\eta},\tilde{\chi})$, such that the conformal form of metric (\ref{ds2Dconf}) is preserved, but $\eta$ and $\chi$ are bounded and take values in some finite intervals. Is the choice of $(\eta,\chi)$ unique?

solution

Suppose $\tilde{\eta},\tilde{\chi}$ span infinite or semi-infinite values. Then we can always make the following sequence of coordinate transformations:

• Pass to null coordinates

\begin{equation} u=\tilde{\eta}-\tilde{\chi},\qquad v=\tilde{\eta}+\tilde{\chi}; \end{equation}

• Bring their range of values to a finite interval by some appropriate function, i.e.

\begin{equation} U=\arctan u,\qquad V=\arctan v. \end{equation}

• Go back to timelike and spacelike coordinates (this is not really necessary at this point and is done mostly for aesthetic reasons):

\begin{equation} T=V+U,\qquad R=V-U. \end{equation} Now the range of $(T,R)$ obviously covers some bounded region on the plane, while the radial part of the line element preserves its conformal form: \begin{equation} ds_2^2 \sim d\tilde{\eta}^2-d\tilde{\chi}^2 \sim du\, dv \sim dU\, dV \sim dT^2 -dR^2 . \end{equation}

As the choice of function $\arctan$ was rather arbitrary (though convenient), the choice of conformal coordinates is not unique.

In this section we will reserve notation $\eta$ and $\chi$ and name "conformal coordinates/variables" to such variables that can only take values in a bounded region on $\mathbb{R}^2$; $\tilde{\eta}$ and $\tilde{\chi}$ can span infinite or semi-infinite intervals. Spacetime diagram in terms of conformal variables $(\eta,\chi)$ is called conformal diagram. Null geodesics $\eta=\pm \chi + const$ are diagonal straight lines on conformal diagrams.