# Kruskal–Szekeres coordinates for Kerr metric

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## Main Question or Discussion Point

I am having trouble understanding the Kerr metric. One of the things which helped me understand the Schwarzschild metric is the Kruskal–Szekeres coordinates. In particular, the fact that light cones were still at 45 degrees was very helpful, and it was helpful to see that the singularity was a spacelike surface.

Does a similar diagram exist for the Kerr metric?

## Answers and Replies

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Orodruin
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Yes, I have seen those, but I was hoping for something a little more quantitative and less "schematic".

PeterDonis
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Does a similar diagram exist for the Kerr metric?
Section 3.6 of this paper derives Kruskal-like coordinates for Kerr spacetime; section 3.7 presents Penrose diagrams:

https://arxiv.org/pdf/1503.02172.pdf

There is one key thing about Kerr spacetime that the above paper does not appear to mention: a single Kruskal or Penrose diagram, since it only has two coordinates, cannot completely describe the geometry up to symmetries, since Kerr spacetime is not spherically symmetric, it's only axially symmetric. So to fully describe the geometry, up to symmetries, you need multiple Penrose-type diagrams. The most commonly seen one is a diagram of the equatorial plane of Kerr spacetime (that appears to be the one in the above paper), but that by itself doesn't tell you everything.