Penrose diagrams for non-spherically symmetric spacetimes As far as I have seen, Penrose diagrams are composed for spacetimes where there is spherical symmetry. The angular degrees of freedom are suppressed so as to understand the causal properties of spacetimes. So I was just wondering: what happens if there is no spherical symmetry in a spacetime? How will one go about drawing its Penrose diagram? Especially when there are defects in the spacetime e.g. point defects?
 A: Not every spacetime admits a Penrose diagram. As you noticed, there might be issues when the spacetime is not "symmetric enough", as one is then not able to suppress coordinates. In general, Penrose diagrams will be useful when there is some way to suppress enough coordinates such that you can get some information from a two-dimensional manifold.
An interesting paper teaching how to construct these diagrams in quite some generality is Walker, M. (1970) Block Diagrams and the Extension of Timelike Two-Surfaces. Journal of Mathematical Physics 11, 2280–2286. From its introduction,

An exhaustive global analysis of a solution of Einstein's gravitational-field equations is usually a formidable task. In certain special cases however, there are intrinsically singled-out 2-dimensional submanifolds of the 4-dimensional space-times which can be simply analyzed. Examples are the symmetry axis of an axially symmetric static or stationary solution or the 2-surfaces containing the repeated principal null directions in a type [22] solution.

Hence, you do need some amount of symmetry to be able to make a Penrose diagram. In a spacetime like the Universe we live, for example, there is not enough symmetry to be able to depict the entire spacetime in a two-dimensional diagram. It does not need to be spherical symmetry, but there must be some way of restricting the metric to a physically interesting two-dimensional submanifold. The paper I mentioned does discuss a little bit of the Kerr spacetime, which is not spherically symmetric, in its last section. In short it restricts its attention to the symmetry axis, and mentions it had been analyzed before by Carter.
