The DeSitter spacetime line element in global coordinates is given by

$$ds^2 = -dt^2 + \frac{1}{H^2}\cosh^2(Ht)\left(d\chi^2 + \sin^2(\chi)(d\theta^2 + \sin^2(\theta)d\phi^2 \right).$$

The ranges of the coordinates are given by $- \infty < t < \infty$, $0 \leq \chi, \theta, \leq \pi$ and $0 \leq \phi < 2\pi$ and $H$ is just a constant. The goal is to find the appropriate coordinate transformations and draw and properly label the Carter-Penrose diagram.

My first attempt was to follow the approach Carroll uses in his appendinx H, where he works out the carter penrose diagram for the Robertson-Walker universe. That means we define a new time parameter, call it conformal time $\eta$, such that $dt^2 = \frac{1}{H^2}\cosh^2(Ht)d\eta^2$. This, however, already yields a hoplessly different function of which I am not even able to determine the range. It seems, however, that this is the only 'recipe' I managed to read about to construct the diagrams. If this fails, what else is there to try?

Comes my second attemt, which basically involved googling what the diagram should look like and how one goes about finding it. This involves defining a new time coordinate, $\eta$, defined by the relation $$\cosh^2(Ht) = \frac{1}{\cos^2(H\eta)}.$$ Using this it is straightforward to show that $\frac{1}{\cos^2(H\eta)}d\eta^2 = dt^2$. Thereforw we can write our line element in the following way

$$ds^2 = \frac{1}{H^2}\frac{1}{\cos^2(d\tau)}\left(- d\tau^2 + \sin^2(\chi)(d\theta^2 + \sin^2(\theta)d\phi^2) \right),$$

where we have simply rescaled the time coordinate $\eta$ to absorb the factor of $H$, $H\eta = \tau$. By the definition of $\eta$, we see that $0 < \tau < \pi/2$. Omitting the spherical part of the metric, together with the conformal factor, the Carter-Penrose diagram will look like a square, with coordinate ranges $0 < \tau < \pi/2$ and $0 \leq \chi \leq \pi$. This seems fine to me.

However, the trouble starts when I need to label the diagram. When Carroll derives the CP-diagram for Minkowski space, there are various regions that names: $i^+$ (future TL infinity), $i^0$ (spacelike infinity), $\mathcal{J}^+$ (future null infinity). And similarly for the respective past infinities. I do not really understand how he 'finds' these regions? And how can I 'find' these regions for the DeSitter diagram just constructed? On all the diagrams that I googled it seems they only ever indicate the $i^+$ and $i^-$ regions. Where are the rest of them?


I found this discussed in a couple of places, and not treated in the same way:

Hawking and Ellis, pp. 132-133

Penrose, Cycles of Time, p. 114

H&E draw it as an infinite strip with a timelike idealized surface $\mathscr{I}$ adjoined at one edge. (The other edge is an axis of symmetry.) The timelike infinities $i^+$ and $i^-$ are disjoint points.

Penrose draws it as a finite cylinder with $\mathscr{I}^+$ and $\mathscr{I}^-$ at the top and bottom.

In each case, I think it's pretty straightforward to verify that the labels make sense. For instance, null geodesics begin and end on the surfaces labeled $\mathscr{I}$ or $\mathscr{I}^\pm$. There can't be an $i^+$ adjoined to the diagram itself, because there are points outside your cosmological horizon, so that no matter how long you live, they will never be in your past light cone.


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