Causal Structure

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The causal structure is determined by propagation of light and is best understood in terms of 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 [1].

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?

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.

Problem 2

Construct the conformal diagram for the closed Universe filled with

  • radiation;
  • dust;
  • mixture of dust and radiation.

Show the particle and event horizons for the observer at the origin $\chi=0$ (it will be assumed hereafter that the horizons are always constructed with respect to this chosen observer).

Problem 3

Closed dS. Construct the conformal diagram for the de Sitter space in the closed sections coordinates. Provide reasoning that this space is (null) geodesically complete, i.e. every (null) geodesic extends to infinite values of affine parameters at both ends.

Problem 4

Static dS. Rewrite the metric of de Sitter space (\ref{dSclosed}) in terms of "static coordinates" $T,R$: \begin{equation} \tanh (H_\Lambda T)=-\frac{\cos\eta}{\cos\chi},\qquad H_\Lambda R =\frac{\sin\chi}{\sin\eta}. \end{equation}

  • What part of the conformal diagram in terms of $(\eta,\chi)$ is covered by the static coordinate chart $(T,R)$?
  • Express the horizon's equations in terms of $T$ and $R$
  • Draw the surfaces of constant $T$ and $R$ on the conformal diagram.
  • Write out the coordinate transformation between $(\eta,\chi)$ and $(T,R)$ in the regions where $|\cos\eta|>|\cos\chi|$. Explain the meaning of $T$ and $R$.

Problem 4

Flat dS. The scale factor in flat de Sitter is $a(t)=H_\Lambda^{-1} e^{H_\Lambda t}$.

  • Find the range of values spanned by conformal time $\tilde{\eta}$ and comoving distance $\tilde{\chi}$ in the flat de Sitter space
  • Verify that coordinate transformation

\begin{equation} \tilde{\eta}=\frac{-\sin\eta}{\cos\chi-\cos\eta},\qquad \tilde{\chi}=\frac{\sin\chi}{\cos\chi-\cos\eta} \end{equation} bring the metric to the form of that of de Sitter in closed slicing (it is assumed that $\tilde{\eta}=0$ is chosen to correspond to infinite future).

  • Which part of the conformal diagram is covered by the coordinate chart $(\tilde{\eta},\tilde{\chi})$? Is the flat de Sitter space geodesically complete?
  • Where are the particle and event horizons in these coordinates?

Problem 5

Infinities. What parts of the spacetime's boundary on the conformal diagram of flat de Sitter space corresponds to

  • spacelike infinity $i^0$, where $\tilde{\chi}\to +\infty$;
  • past timelike infinity $i^-$, where $\tilde{\eta}\to -\infty$ and from which all timelike worldlines emanate
  • past lightlike infinity $J^-$, from which all null geodesics emanate?

Problem 6

Open dS. Consider the de Sitter space in open slicing, in which $a(t)=H_\Lambda \sinh (H_\Lambda t)$, so conformal time is \begin{equation} \tilde{\eta} =\int\limits_{+\infty}^{t} \frac{dt}{a(t)}, \end{equation} where again the lower limit is chosen so that the integral is bounded.

  • Find $\tilde{\eta}(t)$ and verify that coordinate transformation from $(\tilde{\eta},\tilde{\chi})$ to $\eta,\chi$, such that

\begin{equation} \tanh\tilde{\eta}=\frac{-\sin\eta}{\cos\cos\chi},\qquad \tanh\tilde{\chi}=\frac{\sin\chi}{\cos\eta} \end{equation} brings the metric to the form of de Sitter in closed slicing.

  • What are the ranges spanned by $(\tilde{\eta},\tilde{\chi})$ and $(\eta,\chi)$? Which part of the conformal diagram do they cover?

Problem 7

Minkowski 1. Rewrite the Minkowski metric in terms of coordinates $(\eta,\chi)$, which are related to $(t,r)$ by the relation \begin{equation} \tanh \tilde{\eta}=\frac{\sin\eta}{\cos\chi},\qquad \tanh \tilde{\chi}=\frac{\sin\chi}{\cos\eta} \end{equation} that mirrors the one between the open and closed coordinates of de Sitter. Construct the conformal diagram and determine different types of infinities. Are there new ones compared to the flat de Sitter space?

Problem 8

Minkowski 2. The choice of conformal coordinates is not unique. Construct the conformal diagram for Minkowski using the universal scheme: first pass to null coordinates, then bring their span to finite intervals with $\arctan$ (one of the possible choices), then pass again to timelike and spacelike coordinates.

Problem 9

Draw the conformal diagram for the Milne Universe and show which part of Minkowski space's diagram it covers.

Problem 10

Consider open or flat Universe filled with matter that satisfies strong energy condition $\varepsilon +3p>0$. What are the coordinate ranges spanned by the comoving coordinate $\tilde{\chi}$ and conformal time $\tilde{\eta}$? Compare with the Minkowski metric and construct the diagram. Identify the types of infinities and the initial Big Bang singularity.

Problem 11

Draw the conformal diagram for open and flat Universes with power-law scale factor $a(t)\sim t^{n}$, with $n>1$. This is the model for the power-law inflation. Check whether the strong energy condition is satisfied.

  1. V.F. Mukhanov. Physical foundations of cosmology (CUP, 2005) ISBN~0521563984