The following is part of a lecture series on black holes. In this particular lecture, we will try to maximally extend the Schwarzschild geometry using Kruskal–Szekeres coordinates. Moreover, we will gain a more qualitative understanding of black holes through drawing Penrose-Carter diagrams. The outline of this lecture is as follows. First, we will give a brief summary of what we learned in the last lecture. Then, we will explore what goes wrong with Eddington-Finkelstein coordinates. Next, we will refresh our memory of some properties of conformal transformation, as they will be crucial in helping us develop an understanding of Penrose-Carter diagrams. Afterwards, we will draw the Penrose-Carter diagram for Minkowski spacetime. Thereafter, we will introduce Kruskal–Szekeres coordinates and use these coordinates to draw Kruskal–Szekeres diagrams which are maximally extended geometries for the Schwarzschild metric. And finally, we use conformal transformations to convert Kruskal–Szekeres diagrams to Penrose-Carter diagrams for Schwarzschild spacetime.
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