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It seems that rotational symmetry is not necessary, since people draw Penrose diagrams for Kerr black holes. If you don't have rotational symmetry, how do you know what 2-surface is represented by a given point on the 2-dimensional diagram? Would this have to be defined on a case-by-case basis?

Conformal flatness doesn't seem to be necessary, since the Schwarzschild spacetime isn't conformally flat.

Asymptotic flatness isn't necessary, because we can make Penrose diagrams for FRW spacetimes.

IIRC Penrose distinguishes strict conformal diagrams from diagrams that aren't strict. (I may be misremembering the word.) How does this factor in?