An Introduction To Black Holes Information And The String Theory - Susskind, Lindesay

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Before beginning the study of the quantum theory of black holes, one must first become thoroughly familiar with the geometry of classical black holes in a variety of different coordinate systems. Each coordinate system that we will study has its own particular utility, and no one of them is in any sense the best or most correct description. For example, the Kruskal–Szekeres coordinate system is valuable for obtaining a global overview of the entire geometry. It can however be misleading when applied to observations made by distant observers who remain outside the horizon during the entire history of the black hole. For these purposes, Schwarzschild coordinates, or the related tortoise coordinates, which cover only the exterior of the horizon are in many ways more valuable. We begin with the simplest spherically symmetric static uncharged black holes described by Schwarzschild geometry.