Conformal compactification and Penrose diagram

Summary

A conformal compactification of a spacetime is a conformal embedding of it into a larger spacetime in which it has compact closure. In this way infinity becomes a finite boundary. Because the metric is only rescaled by a positive factor, light cones and hence the causal structure are preserved. A Penrose diagram is the two-dimensional picture of such a compactification, usually for a spherically symmetric spacetime, with light rays drawn at $45°$. It shows at a glance the global structure of a spacetime: its infinities, horizons and singularities.