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What kind of global and causal structures does a Penrose diagram reveal?

How do I see (using a Penrose diagram) that two different spacetimes have a similar global and causal structure?

Also,

I have the following metric

$$ds^2 ~=~ Tdv^2 + 2dTdv,$$

defined for

$$(v,T)~\in~ S^1\times \mathbb{R},$$

e.g. $v$ is periodic.

This is the according Penrose diagram:

Is the Penrose diagram that I have drawn correct?

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1 Answer 1

A Penrose diagram of a metric $g_{ab}$ is used to represent the conformal structure of $g_{ab}$. Generally light rays move at $\frac{\pi}{4}$ from the upward vertical and the spacetime considered is spherical symmetric.

The metric, $\overline{g_{ab}}$, on the Penrose diagram satisfies: $\overline{g_{ab}}=\Omega^{2} g_{ab}$. This implies that timelike (null, spacelike) vectors remain timelike (null,spacelike). From this, one can see that all concepts given in terms of timelike curves or null curves as the sets $I^{+}, J^{+}$ will remain the same. So all, the causal structure given in term of those sets will be preserved.

The most common use for Penrose diagrams is to study the behavior at infinity of the different type of geodesics in maximally extended spacetimes. That is the reason the conformal boundaries $i^{o},i^{+},i^{-}, \cal{I}^{+}$ and $\cal{I}^{-}$ are such an important feature of the diagrams and they basically represent the 'boundary' at infinity of a certain class of geodesics.Two spacetimes have the same diagram if they are conformal this implies their behaviour at infinity is the same.

Having said that, there is fundamental information about the global structure of the spacetime and its causality that is not represented such as geodesic incompleteness (in the sense of distinguishing between singular point at finite distance and infinity) or isometries. The conformal structure preserves all information about angles, but loses information about lengths.

For the metric your asking for this paper might be useful.

Mistier C, Taub-NUT space as Counterexample to Almost Everything, Lectures in Math., Vol. 8, pp. 160-169.

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