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While reading the topic Penrose Diagram for Minkowski space-time I came across this sentence.

The essential idea for bringing in the points at infinity is to use coordinate transformations involving functions like $\tan^{-1}x$ which maps infinite interval into finite interval.

I don't understand why are we doing it. Why do we want to look at the causal structure at infinity? Space at infinity is of no significance because at infinity the curvature is flat. Am I right or wrong?

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  • $\begingroup$ If you are interested in the causal structure of spacetime as a whole, then it's really useful to be able to draw pictures of spacetime as a whole. $\endgroup$ – tfb Sep 8 '17 at 17:47
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We don't want to understand the causal structure of spacetime at infinity per se. What we want to do is we want to see what the causal structure of spacetime between the infinity and a considered object is. Now, an important feature in the causal structure of spacetime is that of a horizon. If a horizon is formed then the interior of the horizon can not influence the exterior. This implies that if a signal (say that of light) can not reach beyond a certain boundary when emitted from a certain region then we should conclude that there is a horizon in the game. Now, this requires us to check whether light signals emitted from a certain region can go indefinitely far away or not. A simple technique to do so would be to bring the infinities at a finite distance and directly see whether the light signals hit them or not when emitted from a certain region. This is why we care about bringing infinities to a finite distance even if we already know what the causal structure of infinities themselves is.

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